Here it is in full flow on the second note, which was the note favoured in his rendition…

We now switch to the table of partials and find something rather queer happening. Gone are the musical intervals of the perfect fifth (ratio 1.50) and octave (ratio 2.00) between the three main notes; in their place are the rather interesting ratios of 1.17 and 1.62 – and all this despite a sensible integer progression for the partials of note 1.

This requires a fresh packet of bourbons!

We see that the three sets of partials line up nicely, and there is a juicy surprise in store for us when we determine nearest modern musical notes…

By Jove, they all come out with A=432Hz tuning as the nearest standard! I’ve gone the whole hog and designated this as an A=432Hz instrument, putting the appropriate design target frequencies down in my tangy orange box as before.

The minimal frequency differences are quite telling. In terms of musical intervals this instrument is capable of sounding a Pythagorean minor third (ideal ratio = 1.19) and a Pythagorean minor sixth (ideal ratio = 1.58) above an A#4.

Quick Take

My one line quick take on this is that we are listening to an A=432Hz herald trumpet in A#(4), capable of sounding a Pythagorean minor 3rd and a Pythagorean minor 6th, with complex acoustic properties pertaining to that of the conical tube.

Cogitation & Cocoa

We seem to have two very different beasts here – a silver device sounding the authoritative but somewhat simplistic stamp of A=440Hz Pharaonic power; and a complex, coppery – possibly bronze - device producing a fabulous clutch of A=432Hz tones. Chalk and cheese!

Male and female?

Sun and Moon?

Empowering and calming the masses?

Power-up and power-down?

If we are going to blow our own trumpet perhaps we need more than one.

Big Fat Problem

As regards linking the trumpets to outbreak of war and unrest there is one big, fat problem…

Sadly, outbreak of something aggressive and horrid like war is rather a common event in the affairs of modern humans, which makes coincidences with sounding of the trumpets more than likely, especially if they’ve been handled behind the scenes many times. I guess that all of this magical powers business is a darn good line to take to foster public interest (and raise dosh for the museum).

Conspiracy types will probably assume the global elite, some 3,000 years after their construction, are still sounding the very same trumpets in secret when they fancy a decent spot of subjugation of the masses and a bit of lucrative death and destruction. My take on this is to ask why don’t these globalist evildoers make a few more trumpets and make life easier for evildoing? And the very keen among them ought to be sounding such trumpets every week, not every now and then. I guess you can’t get the staff these days…

Kettle On!

A delicious benefactress very kindly sent me a YouTube link to a remarkable BBC broadcast originally made in 1939 in which these two trumpets were sounded for the first time in 3,000 years. Here’s what Wiki has to say…

Tutankhamun’s trumpets are a pair of trumpets found in the burial chamber of Tutankhamun, king of Egypt. The trumpets, one of sterling silver and one of bronze are considered to be the oldest operational trumpets in the world, and the only known surviving examples from ancient Egypt. The trumpets were found in 1922 by Howard Carter. Both of them were sounded for the first time in over 3,000 years to a live audience of an estimated 150 million listeners through a BBC broadcast aired on April 16, 1939. The trumpets were played by a Bandsman, James Tappern of Prince Albert’s Own 11th Royal Hussars regiment. The recording was recently featured, and can be heard on the BBC Radio program series “Ghost Music.”

Claims of Magical Powers

There have been some claims made by Zahi Hawass, former Minister of State for Antiquities Affair, and Egyptologist Hala Hassan, curator of the Tutankhamun collection at the Egyptian Museum, that the two trumpets contain “magical powers” and have the apparent ability to summon war. The evening they were first played in 1939, the power cut out at the Cairo museum five minutes before the scheduled air, and the BBC were forced to record the sounding of the trumpets by candle light. Five months after the radio broadcast, Britain entered World War II and the war in Europe began. The trumpets were again said to have been played before the 1967 Six-Day war, before the 1990 Persian Gulf war, and most recently, the bronze trumpet was played one week before the Egyptian Revolution of 2011 by a Cairo museum staff member to a Japanese delegation. This same bronze trumpet was subsequently stolen from the Cairo museum during the Egyptian looting and riots of 2011, yet was mysteriously returned to the museum some weeks later.

…rather interesting, n’est pas? Here’s a photo of James Tappern taken during the broadcast, which gives some idea of their size:

Jiggery-Pokery

A little bit of jiggery-pokery enabled me to capture the YT video as an MP3 sound file and run this through my studio DAW with its armoury of analytical tools. It’s always a good idea to know what you’re dealing with when throwing spectral analysis about, so here’s what we’re dealing with…

Long, thin, (natural) herald (a.k.a. fanfare) trumpets like these are capable of playing 6 to 8 usable notes, these being derived from the upper partials of the natural harmonic series. The skill in obtaining these rests entirely with the ability of the player to produce the right air pressure through correct lipping – this is no mean feat and requires years of strengthening lip, jaw, throat and face muscle not to mention all that support from the thorax.

The easiest notes to play are the fundamental tone, a tone a perfect fifth above the fundamental, and the octave of the fundamental… these are what I shall be looking for.

Silver Trumpet

Here is the spectral plot for the silver trumpet’s lowest note (aka fundamental). Along the bottom axis is frequency, ranging from 100Hz to 3,800Hz. The height of each peak is a measure of the sound energy present at each frequency, with the highest peak topping out at -21dB on this particular recording:

The orderly nature of the harmonics is readily apparent. One curious feature is the double-peaked nature of some of these. As I shuttled through the recording, the two slightly different peaks would come and go. My engineer’s ear winced at the significant wow & flutter present in the recording, and this is what we are seeing here. The instability in tape machine transport mechanisms of the day is giving rise to two slightly different peaks as the tape slows and speeds up. A bit of fiddling on my magic DAW soon sorted this!

Now for our first table, revealing measurements of the three notes sounded by James Tappern and their harmonics (aka partials):

I am sure that partials extend well beyond number 7 with the real thing blasting at us, but we are talking about 1939 sound recording technology here! Nevertheless, this is plenty sufficient to see what is going on.

According to the laws of acoustics we should see an integer progression in the ratio of partials produced by the first and lowest note and, indeed, this is what turns up. The first partial of the first note is the fundamental – what the trumpet has been tuned to. We can see it going a bit flat at partials 6 and 7 (ratios should be precisely 6.0 and 7.0 respectively).

The second note should ideally sound a perfect fifth above the first, the idealised musical ratio here being 3:2 (a.k.a. 1.50) in terms of frequency. We see that it falls a tad short of the mark at 1.46 (730Hz/501Hz).

The third note should sound at the octave, but this is even flatter with a ratio of 1.91 instead of the perfect 2.0. Such flatness could be poor old James Tappern struggling but I suspect it is also partly due to the design. The trumpet is a narrow conical tube opening into a rather wide conical horn. These are not ideal shapes for accurate tone reproduction, and modern brass instruments follow an exponential taper. It does the job pretty darn well though!

We can plot this data on a pretty graph and run a linear regression to get the very best estimates of the fundamental frequencies for each note:

With three accurate estimates for the three fundamentals in our eager hands we can now look to see how they compare with our modern notions of musical scale:

Voilà! This is primarily an A=440Hz trumpet in ‘B’, yielding a 2.25Hz difference between note 1 frequency as measured (497.25Hz) and the theoretical ideal (495.00Hz) if we opt for Pythagorean temperament.

Note 2 (the perfect fifth) - which is a tad flat - thus turns up closer to the lower A=432Hz tuning. Whether this is by deliberate design or an acoustical artefact of a conical tube - or James Tappern’s quivering lip - we cannot be sure.

With note 3 (the octave) swinging back to A=440Hz tuning, my guess is that this was meant to be an A=440Hz instrument but is hampered in accurate tonal reproduction because of the simple conical design. No doubt the player of 3,000 years ago was sufficiently skilled to lip those notes well-up and good. With this in mind I’ve tucked the ideal design frequencies down in the tangy orange corner of the table above.

Quick Take

My one line quick take on all this is that we are listening to an A=440Hz herald trumpet in B(4), that sounds both the perfect fifth and the octave with reasonable precision, whilst exhibiting the classic acoustic properties of an open tube.

One cheeky notion I had when seeing these figures for the first time is to forget our modern notions of a musical scale and think of this as a trumpet designed to play at 500Hz, 750Hz and 1,000Hz. There is no time to ponder on deep matters yet though, for we must take a look at the copper (bronze?) trumpet…

Kettle On!

Dominant Frequency

As it stands the BBC/CSETI recording offers 18 very high frequency bursts, 18 high frequency bursts, 18 mid frequency bursts and 12 low frequency bursts, making 66 bursts in total. Frequency analysis was undertaken for all 66 bursts to determine the dominant tone frequency, with the mean value of the fundamental frequency calculated for each of the four burst groups. The table below summarises an initial key result that raised my left eyebrow…

…we are looking at an octave expansion of the frequency 333Hz! The column on the left reveals what was measured and the column on the right is the mathematical doubling of the value 333. Not being a New-Ager I’m not familiar with the intricacies of 333 as an esoteric code so I decided to ask Google AI, and here’s what it said:

The number 333 is widely interpreted in numerology and spirituality as an angel number signifying growth, creativity, and divine support. It is believed to be a message from the universe or ascended masters encouraging you to align your mind, body, and spirit and trust your intuition during times of change.

Key meanings include:

• Spiritual Alignment: It signals that you are supported by higher powers and should focus on self-love, personal growth, and balancing your inner truth with external actions.

• Creativity and Expression: The sequence is a call to tap into your creative potential, communicate your ideas boldly, and pursue projects that bring you joy.

• Career and Finance: Seeing 333 suggests you are on the right professional path, encouraging you to aim high, trust your talents, and embrace opportunities for financial abundance through strategic risk-taking.

• Relationships: In love, it promotes harmony and joy, urging you to inject playfulness into partnerships or approach dating with an open, creative mindset.

• Twin Flame: For those connected to a twin flame, it indicates harmony and deep understanding, serving as a reminder to also focus on individual personal growth.

Ultimately, 333 is viewed as a positive omen of encouragement and optimism, reminding individuals that they are never alone and that their current struggles are leading toward a more fulfilling future.

Crikey. I need a cake after that!

The Cuckoo

The lowest and slowest tone group (sequence #4) helps us to understand what is going on warble-wise and anybody hearing this is bound to have the cuckoo spring to mind. The cuckoo sings an interval of a minor third and this is what frequency analysis confirms, with a mean of 394.7Hz dropping to a mean value of 333Hz in a repeated pattern of around 20 ‘calls’. The interesting thing here is that 394.7Hz is the musical note ‘G’ in a scaled tuned to A=444Hz, with 333Hz being the musical note ‘E’ in the same scale.

I took temporal measurements of the ‘Cuc’ and ‘Koo’ elements and entered these into a spreadsheet to determine what hidden mathematical treats may be lurking. I wasn’t disappointed and shall now resent a dense table of numbers that define the detailed structure of the call as measured (left hand columns) and as fancifully hypothesised (right hand columns):

As we may clearly see all measurements I could muster came close to a mathematical ideal, either in terms of duration or frequency (1/duration), with many significant values rearing their head for students of numerology.

Measuring the frequency of the ‘cuc’ and ‘koo’ components proved tricky owing to the indistinct audio of the ‘cuc’, this being but a fraction tone burst at just 0.1790 seconds long in the slowest sequence (#4). The trained musical ears of my musician wife proved to be the business here, identifying the interval as a minor 3^rd. The fine structure of the start of the initial very high frequency call burst (sequence #1) can be seen in the following oscilloscope trace:

Here we see a series of insect-like blips that, when slowed down eight times, sound very much like a cuckoo. Each blip is spaced at 0.806 seconds giving this blip burst a warble frequency of 1.24Hz. The first complete phrase is somewhat complex in structure and the following oscilloscope trace shows this:

Visually speaking this reminds me very much of the sound structure of a living bird though, as we shall see later, this sound was also transmitted on radio frequencies - if that’s not food for thought I do not know what is!

A Hidden Message?

With 333Hz hypothesised as the building block for the BBC/CSETI tone sequence, and my delicate mind disturbed from learning of the mysterious death of Paul Vigay, I sat wondering how and why the BBC, Advanced Space Systems, Thanatos Industries and UFOSI had all come to differing conclusions on the fundamental frequency of the White Crow crop circle sound sample. Were these organisations trying to hide something or could there be something about the sample that was throwing analysis?

I considered the prosaic possibility that Paul Vigay’s WAV file itself was throwing Adobe Audition out in some technical manner i.e. software wasn’t handling the file encoding properly. This is when I discovered something that may be of profound significance (or not); it appears that Paul had encoded the bit rate of the WAV file to a completely non-standard value…. 333kbps.

I can tell you now that seeing this gave me Goosebumps from head to toe, and I had to reach for the custard creams. Was Paul trying to tell us something important before he ended up floating face down?

United Crop Circle Harmonic Scale

The last finding stunned me sufficient to warrant an extra pot of tea. Here I was, working steadily toward the hypothesis that the BBC/CSETI tones had been artificially processed in the studio to generate the remarkable relationship with 333Hz, but now it would seem that all balls are still very much up in the air and not landing any time soon.

One thing is certain is that these crop circle tones have not been produced by a simple mechanical or electronic oscillator or anything analogue like a flute, or anything capable of producing a complex waveform like a violin. They very much resemble bird song in their richness, nuance and form, yet they are not produced by any warbling bird we know. Neither can a warbling bird reach 200dB or emit radio frequencies.

I decided to see if I could reconcile the differences between the tone frequencies as recorded by the White Crow team and the BBC crew, so I drew up a shortlist of harmonic peaks that I had managed to identify other than the dominant tone. Following a hunch I decided to derive a harmonic series for the fundamental of 111Hz and colour code this with the tone harmonics I had managed to identify and measure within both sound samples. What transpired clean blew my socks off!

Below is a united crop circle harmonic scale that has been colour coded light blue for all frequencies detected within the BBC/CSETI tone sequence and purple for the White Crow tone sequence. This one scale alone encapsulates all that has been recorded both in terms of dominant tone and discernible harmonic content. We now see that the White Crow sample sits at the 33^rd harmonic of the scale with the BBC/CSETI sample sitting at several harmonics, and up to the 23^rd partial. We may conclude that both tone sequences arise from one and the same unified harmonic family.

In acoustics it is traditional to notate the first partial as H0, in which case the White Crow frequency sits at H33 – a very Masonic number and being the vertebrae count in the human spine. Fans of Tesla coils and vortex-based mathematics may wish to note the incidence of 3, 6 and 9 as the digital roots for each frequency (these reductions are also known as Pythagorean roots).

Not exactly a keen follower of New Age trends and notions, I didn’t realise at first that this scale was well known in such circles, being referred to as the Angelic scale, and particularly good for communing and summoning angels. Blimey!

Me King Lears

Just after crunching the grand unified scale I went into the studio here to double-check my findings using a different (professional) audio editing suite that is more than capable of handling non-standard WAV encoding. This confirmed the results obtained using cheapo Adobe Audition, but the darn curious thing is that my left ear instantly starting ringing with a pure tone after playing the scale back using a sequence of pure sine waves. It felt as though a ‘silent tone’ had passed through my head and was physically sitting in/on my right ear, which felt decidely odd.

No doubt there’s more to explore. In the meantime…

Kettle On!

I have no idea who or what makes crop circles; we know some are human-made, and we know some are publicity stunts, but there are some that totally baffle. One thing I do know is that crop circle designs generally would make wonderful cake decorations (or biscuit patterns for that matter).

Here’s how Google AI sums up the situation this year:

The 2026 crop circle season in the UK has been active since early April, with a notable uptick in sightings reported across Wiltshire, Somerset, and Oxfordshire. The season began with oilseed rape circles in Ilchester and Waden Hill in April, transitioning to barley and wheat as the summer progressed.

Key 2026 Sightings and Events:

Early Season (April–May): The first circle appeared on April 4, 2026, in Ilchester, Somerset. Subsequent early sightings included a dumbbell design at Waden Hill, Wiltshire (April 29) and a large pentagonal design at Jack’s Plantation, Wiltshire (May 8).

Mid-Season Highlights: A mysterious eye-shaped crop circle was discovered near Salisbury, Wiltshire on June 22, 2026, with astrophotographer Tom Malone noting it was his fifth sighting of the year. Other notable mid-summer designs included an elaborate 10-fold design at Hackpen Hill (June 26), which was confirmed to be manmade for TV, and a double spiral in Switzerland (June 23).

Late Season (July): The season peaked in July with multiple complex designs, including a 13-point star and crescent moon at Roundway (July 18) and a long pictogram of symbols at Fox Hill, Wiltshire (July 21).

Season Overview:

The 2026 season saw a mix of mysterious formations and confirmed manmade circles. Farmers and organizations like the National Farmers Union have highlighted the potential for criminal damage when circles are created without permission. The season typically concludes with harvest in mid-to-late August, after which the fields are cleared for the next year. Resources like Temporary Temples and The Crop Circle Connector provide ongoing documentation and photography of these phenomena.

Here’s the latest pattern, dated 24^th July:

Not the best cake design, I admit, but I’m now thinking of a series of savoury platters. It would also be a great cover for an album – check this story out.

And now for that sound sample…

In The Beginning

Colin Andrews appears to be the man at the centre of all things crop sound-wise in the UK so I searched his website for information - click here for a decidedly fruity link. What delighted me no end is that the handwritten letter that is featured is from a bod called Gunner Sanberg, and was penned at my old stomping ground of Sussex University. Yaroo!

Herewith some pertinent extracts nicked from Colin’s site concerning the inadvertent recording by the BBC of what was to become a crop circle initiatory tone sequence used by Dr Steven Greer and those engaged with the Center for the Study of Extraterrestrial Intelligence (CSETI). For those new to this game CSETI is a full-on CE-5 crew (Close encounters of the 5th kind); we’re talking human-initiated contact with ET life-forms. Yikes!

SIDE NOTE:

NOTE: When the BBC sound was being recorded as it happened in the field, fault lights came on the TV camera and the camera was destroyed. When Pat and I appeared in the BBC studios ‘live’ in Birmingham as the recorded sound sequence and film was shown to the public for the first time, the studio lost power as this sequence came on air. Generators kicked in but a temporary blackout on the program was witnessed throughout the country.

The photograph below was taken by Pat during the filming. We have no idea what the object is in the sky. During the writing of this article, the sound posted here on my site was being played by Pat Delgado in his home in England and the computer locked up mysteriously, which it has not done before.

After I transferred the original BBC and White Crow recording onto my computer for the top of this page, I played it back and my own computer also crashed and re-booted itself with a serious e)ror message. This it has never done in its 6 year history. I am conveying only the facts to you here. If this string of experiences are co-incidences then so be it.

Colin Andrews - 7.0 PM Feb 11, 2009.

Operation White Crow

The Operation White Crow experience was by all accounts a spooky event for those concerned, with the sound appearing as though a tactile, discreet form just before the appearance of a circle in a nearby field. If you want to zip down a rabbit hole, then try this link; meanwhile, herewith a weeny extract I nicked from a website that no longer exists (cropcircleresearch.com):

One of the best documented examples of sound in relation to crop circles was reported by Colin Andrews and consisted of a number of short bursts of oscillating tones (or ‘trilling’ noises) recorded on audio tape. This was recorded during what was known as Operation White Crow, which took place in Hampshire during the summer of 1989. Operation White Crow took place at a location known as the “Devil’s Punchbowl” at Cheesefoot Head, near Winchester in Hampshire, in the summer of 1989. It was organised by crop circle research Colin Andrews and a number of colleagues. It was the first of a number of dedicated crop circle surveillance operations and consisted of a number of vehicles, work stations and cameras, monitoring a popular circles prone location.

Some more fascinating snippets from Colin’s archives…

4. Crop Circle Sound

4.1. History
Colin Andrews, the founder of Circles Phenomenon Research (CPR), first heard a trilling sound in a field near the village of Kimpton, Hampshire, England in June 1987. Walking alone through the cereal crops, Andrews heard an electro-static, or cracking, cricket-like sound from a nearby ring. Within seconds the sound became so loud. 200 decibels, as to frighten him. The sound was not heard again until 1989. Then, at his night surveillance project, ‘Operation White Crow”, two more incidents with very similar sounds occurred.

Near the project completion, more than six researchers heard the sound. The sound suggested behavior In that it moved in linear and non-random movements. It stayed for several minutes and then headed east. At first light a new circle was reported on a hill in the direction in which the sound had traveled. At dawn, George Winfield, a man from the surveillance group, and Andrews heard the sound again as they drove to the circle. After stopping the car, they taped the sound.

The sound occurred for the third time two weeks later. The BBC was conducting an interview with Andrews in Beckhampton. The sound was received as a radio signal on the microphone aerial. At the same time as the reception of the sound, the fault lights on the BBCs computer-controlled camera flashed, and all the circuits went out simultaneously. Following the recording of the sound the camera had to be rebuilt (13:2,3].

Analysis By The BBC

Below is the analysis carried out on the sound that was recorded during a BBC television interview with Colin and Pat in a large crop circle at Beckhampton, Wiltshire. Pat felt a static field around him as noise bars broke across the TV camera. Within seconds the camera was destroyed. This will be subject to a future article but here for comparison is the result of analysis. NOTE how very close the frequency is to the Cheesefoot Head sound. BBC: 5.2 Khz / Cheesefoot Head: 5.4 Khz.

Analysis By Others

4.2. Previous Research
Two previous research teams, Advanced Space Systems and Thanatos Industries, also performed sound analysis on the White Crow Sound. An auto power spectrum of this sound at normal speed revealed one distinct peak frequency. Advanced Space Systems determined this peak at 5.6 kHz, while Thanatos Industries reported 7.2kHz.

4.3. Sound Analysis
The tape recording was a crop circle sound at normal speed. It also contained the sound incrementally slowed at 50%. This reduction in speed was to show the unnatural characteristic of the sound. In Figure 4.0, a time series analysis of the White Crow sound is plotted. The frequency of the crop circle sound is approximately 4200Hz.

UFOSI has acknowledged that a bird song may be the source of the White Crow sound. In turn, we researched several commercial audio CDs looking for similar trilling.

Right, that’s yer lot - it’s now my turn for a fiddle…

Analysis By Moi

Having obtained a copy of the original White Crow WAV file from investigator Paul Vigay's cropcircleresearch.com website before it disappeared, I ran the sound sample through Adobe Audition software and came up with the following spectral plot:

This now presents us with a deep mystery! According to Colin Andrews, Advanced Space Systems determined the dominant frequency of the White Crow audio sample to be 5.6kHz, while Thanatos Industries reported 7.2kHz, with Peter Chow from the BBC plonking down the middle at 6.6kHz and UFOSI offering 4.2kHz. This is utterly baffling since the above frequency analysis I made myself using the original sample clearly shows a peak around 3.7kHz. How is it possible that experts in three competent organisations could have missed the mark so badly?

I can see how Peter Chow of the BBC may have reached his 6.6kHz conclusion since a harmonic of 6,692Hz can readily be seen in the power spectrum in the above plot; but this is the first harmonic of the series and not the fundamental. Neither the 5.6kHz claimed by Advanced Space Systems nor the 7.2kHz claimed by Thanatos Industries come anywhere close to anything within the actual original sound spectrum, and quite how they can differ by as much as 1.6kHz beggars belief!

This brings us on to the analysis undertaken by UFOSI who offer 4.2kHz, claiming it is merely bird song and neatly providing evidence in form of the blurry graph labelled Figure 4.0: Auto power Spectrum of White Crow Sound. However, the fundamental harmonic within the original recording is most decidedly not 4.2kHz but 3.7kHz. Shift the White Crow material 0.5kHz to the left where it genuinely should sit, with a peak at 3.7kHz, and you expose the issue: no species of bird found in the region sings in this manner. This critical correction to UFOSI’s dismissive report is exactly what Paul Vigay discovered days just before he was found floating face down in the sea. Here’s a sobering note pinned to Colin Andrew’s page:

CSETI Tone Sequence

Overview

Not having access to the CSETI CE-5 contact app for the purpose of this analysis, I relied upon the initiatory crop circle tone sequence as presented in this mirror on YouTube…

When displayed as a spectral plot it looks like this…

In the first tone phrase we see three very high frequency bursts, followed by three at a lower frequency, followed by three at an even lower frequency, followed by another three and a final three. This descending phrase is presented six times but it is not identical in the number of lowest tone bursts.

We know that the original tone burst was short-lived and high frequency, so what we have here is that original burst that has been cloned and progressively reduced in frequency using studio techniques. Each of the four tone bursts is exactly one octave below the other and exactly half the duration – time and frequency compression being the easiest way of processing high frequency audio simply by halving tape speed playback.

In the next instalment I’ll dig deeper into the numbers and likely bore everyone silly.

Kettle On!

We left-off with me saying:

...the [infinite] exponential slope of the mathematical relationship connecting all genuine Tibetan bowls is encoding the ratio of the perfect fifth.

What I was about to type is that this means the bowls are tuned to be a perfect fifth apart - all of ‘em! This fact has been right under our noses all along, for if we look back at table 9 we can see the frequency ratios of TB7 – TB4 are:

849.43/572.45 = 1.48

572.45/380.38 = 1.50

380.38/257.36 = 1.48

Tying my wild and exciting notions together in one mathematical statement, we end up with this equation for (hypothesised) perfection in sound healing:

I call this TBM-1 (Tibetan Bowl Model 1)

… is this what they were trying to achieve with all that careful blending and hammering of precious metals?

I decided to run the equation through the spreading spreadsheet to see what values it would produce, and it didn’t disappoint:

The modelled frequencies for the first four chakras emerge as perfect integer values. On top of this ALL modelled chakra bowl frequencies emerge as VBM1 matrix harmonics. Tasty.

In addition, we should note that 364.5 is the number of days in a solar year according to Vedic and Ancient Egyptian tradition, and is a value found in the works of Philolaus and others2. This is the stuff to get excited about!

We have unearthed the mysterious cycle of fifths in ancient Tibetan singing bowl frequencies, make no mistake.

And that’s it for this mini series, folks; time to put the kettle on…

Kettle On!

…unless you are a paying subscriber, in which case I have lined-up up a little treat.

A Treat For Paying Subscribers

Read more

A similarly zoomed section for TB7 reveals the same story of ghosted harmonics:

With this knowledge I was able to revisit the spectral trace and correctly rebuild tables 4 and 5, and produce corrected versions of figures 3d and 3g. Here they all are…

Just in case anyone is wondering what the ghost harmonic series for TB4 and TB7 looks like, here is a summary table. This reveals a non-musical smattering of additional bowl resonances. These could easily be a function of the sounding stick smacking into imperfections rather than the acoustic dynamics of the ringing bowl itself. Careful inspection of the spectrographs for all other bowls reveals that ghost harmonics exist for these, the difference being they were sufficiently pronounced on TB4 and TB7 to throw my eyeballing. Extended critical listening on studio monitors revealed buzzing sounds created by the stick vibrating against the bowl, along with strange echoing tones present in the reverberation signature.

Here endeth the mystery of the bent bowls!

The wonderfully linear harmonics found with all bowls now enables us to replace point estimates of fundamental frequencies with more accurate regression modelled estimates.

The Secret of the Unbent Bowls

Here is figure 1 and table 1 again with point estimates for primary frequency replaced by the more accurate unbiased regression modelled estimates…

This little bit of modelling has further emphasised the difference between A=432Hz and A=440Hz tunings; aside from TB3 it is now quite clear all bowls are much closer to A=432Hz tuning than A=440Hz, with three being tuned to within 1Hz!

All seven bowls exhibit the same mathematical relationship with each other, but of the slightly modified form:

The outlier in terms of tuning as well as graphical representation is TB3 and I can’t help but wonder if this is a rogue bowl – a bowl with dubious A=440Hz provenance.

I thus repeated the regression model with TB3 removed…

The mathematical relationship is now of the form:

The zeroth partial – what I have called the root of all things – now stands at 48.223Hz. Now, this is rather interesting… especially when considering my findings regarding 24Hz/48Hz and the Bolivian stone panpipe. My mind wandered a little as I pondered this equation, throwing up titbits such as 0.4114 being nearly the value of √2 - 1 (0.4142). I got very excited indeed when I realised this relationship…

…for 1.5 is the precise frequency ratio of the perfect fifth, no less!

To cut a long story short what this means is, the [infinite] exponential slope of the mathematical relationship connecting all genuine Tibetan bowls is encoding the ratio of the perfect fifth.

Kettle On!

Here is a time slice of TB1 at its most harmonically rich. To the far left is traffic rumble we can ignore. The fundamental frequency of 79.12Hz already measured is the peak over on the left labelled ‘primary tone’. There are two energetic peaks to the right as equally as loud as the fundamental at around -18dB that I’ve labelled ‘strong harmonics’. Together, these three strong peaks define the harmonic backbone of the bowl. Sometimes these are the only harmonics that audibly sound out but at other times a whole family of weaker micro-harmonics joins them in zinging fashion.

The coming and going of weaker harmonics does not make analysis easy! The three strongest harmonics were noted first, and then audio files were scrubbed (shuttled back and forth) to see which of the weaker harmonics were persistent enough to worth noting. The idea behind this fiddling was to reduce artefacts caused by the leather stick squealing, bowl scrapes and so on. After several hours of concentrated effort the most truthful harmonic series for each bowl was nailed up to the 12^th partial.

Now for the acid test – do these 7 sets of partials represent a linear harmonic function (Pythagorean) as found for all manner of strings, pipes and plates or do they represent something spooky, spherical and otherwise non-linear like Helmholtz resonators and Schumann Resonances?

I decided to stick with TB1 and throw all the necessary numbers into one diagram; these being the measurements from table 4 alongside theoretical values derived from some number-crunching in a spreadsheet.

AHA!

Tibetan bowls, despite being goodly round entities of ancient strangeness actually ring, shudder and ping in perfectly Pythagorean fashion just like any old string or pipe.

But let’s just check out the family of partials for all bowls in case there is some strangeness therein…

Actually, before that I’ll just mention a numerical quasi-coincidence that caught my eye. Let’s start with the simplified equation for the expansion of a series of harmonics to predict Schumann Resonance:

Empirical observation reveals that for typical atmospheric conditions on Earth, the value of f(0) approximates to 5.54Hz; a figure which corresponds to the zero order spherical harmonic. This is also called the cavity constant. As it so ‘appens, as I fiddled around with TB1, the ringer-dinger bowl for the root of our chakra-tastic energetic beingness, I discovered it had to have a cavity constant of 55.95Hz or thereabouts in order to ring the first partial at 79.12Hz. It was a bit of an eerie moment realising these cavity constants were almost exactly a factor of 10 different. I’m not quite sure what this means (cue theremin music…).

Now to check those partials. Given the enormous errors I could make in eyeballing the proper sequence of partials from complex spectral output, anything approaching a straight line was considered fabulously fortuitous. This being the case in most cases, I decided to ram a least-squares regression model through each series to obtain unbiased estimates of the fundamental frequency; these being more accurate than the point estimates of table 1. TB1 offers up a nice example of this in figure 3a, where we see that the regression coefficient representing the slope (and thus first partial) is 76.489Hz as opposed to 79.12Hz.

Now to examine TB1’s six comrades…

All bowls were well-behaved in that they produced a linear series of harmonics in accordance with acoustic theory apart from TB4 (heart) and TB7 (crown). TB4 was reputed to be the oldest bowl, dating back some 400 – 500 years, and was well-wacky in terms of harmonic spectrum. So wacky in fact, that I’ll need to devote a separate section to it.

Before I do that, let’s just take a quick look at the interval ratios and see how they stack-up against acoustic theory…

According to acoustic theory for linear oscillators, the ratio of the frequency of each subsequent partial to the first partial is a simple integer increment. i.e. if the first partial sounds at 100Hz then the second partial will sound at 200Hz, third at 300Hz and so on, thus ratios will be 200/100 = 2; 300/100 = 3 and so on. We can see this progression of ratios in table 5 for all bowls apart from TB4 and TB7.

TB4 starts well enough, with a ratio of 2.0 but then harmonics seem to come lumped into three disparate harmonic sections – as though the bowl has three components. TB7 doesn’t start at all well with a ratio of 1.5 (the perfect fifth) but one thing that caught my eye is that there seems to be two interwoven series sounding together, which I’ve coloured.

The possibility of TB4 and TB7 blending two or more harmonic series is most intriguing but I shall need to revisit the spectral analysis just in case I’ve been misled by very weak harmonics.

Kettle On!

Nice, but a tad bland even with a fancy purple backdrop. But… the graphical form of this reveals something totally and utterly stunning…

All seven bowls - regardless of date of manufacture, location of origin, maker and constituent metal components - form a striking mathematical relationship with each other sufficient to fit an exponential model of the form:

This is breath-taking and utterly profound in its implications! That there is an underlying mathematical design element for the primary frequency of Tibetan bowls that crosses all aspects of their origin is a bit of a corker. These are clearly frequencies that matter; not only that but we now have a mathematical description guiding the creation of the ‘perfect’ fundamental frequency for use with each chakra in accordance with the Bon-Pot tradition.

Before we move on, let’s just take a quick look at the implications of that exponential model, starting with a table comparing model values with those measured.

That’s not too shabby. The largest difference is seen with bowl TB3 (solar plexus chakra), with the actual bowl sounding lower than that expected from a pure mathematical consideration – I’ll come back to this later. A rather geeky consideration thrown up by equation 1 is the notion of the ‘zero chakra’, which has been estimated by my modelling to be 44.4Hz. I have no idea what a zero chakra might mean, if anything, but we might argue that something vibrating around 44Hz is fundamental in some esoteric way, this essentially being the zeroth root of all things.

Let’s roll our sleeves up now and test the hypothesis that these bowls are tuned to A=432Hz…

Are these A=432Hz bowls?

This is not a straightforward question to answer. The single value of note ‘A’ at 432Hz means nothing unless we consider the scale structure upon which the rest of the notes/harmonics are built. Our modern scales, when transferred to a polyphonic sounding device such as a piano keyboard, become Equal Temperament in nature; that is, the scale is fractionally adjusted, being based on equally-spaced values derived from calculation of the twelfth root of two (1.059463094); there being twelve semitone divisions of the octave, which represents a doubling in frequency. If you have a trusty hand-held calculator to hand jab in 1.059463094, then raise this to the power of 12… with luck you’ll arrive at a value of 2.00 or as near as dammit.

There are other tunings we can consider such as Just Intonation and Pythagorean which produce slightly different note spacing to Equal Temperament, but since Jamie Buturff was using a Korg CA-30 tuner operating with Equal Temperament tuning, we shall start by following in his footsteps with the Korg CA-1 (an incarnation of the CA-30).

My plan was very cunning yet very simple – to stick a Seiko SAT500 tuner set at A=440Hz next to a Korg CA-1 set to A=432Hz and see which of these hit their ‘bang in tune’ green light most of the time when I played the tracks of the Tibetan Chakrra Meditations CD back.

After a thrilling fight going seven rounds of bong! and ting!, it was the CA-1 set to A=432Hz came out tops, revealing A=432Hz tunings for bowls TB2, TB5, TB6 and TB7. The SAT500 set to A=440Hz came up trumps for TB1, TB3 and TB4.

In summary, we could say that Jamie Buturff was a little over-enthusiastic in his hypothesis that these Tibetan bowls are A=432Hz tuned: it ain’t necessarily so! We can now become geeky and set out a table depicting this battle numerically, with winners for each round depicted in green:

If we take absolute values of differences and sum them we find that A=432Hz tuning returns a total harmonic discrepancy of 20.29Hz, whereas A=440Hz tuning returns a total harmonic discrepancy of 46.87Hz. Thus, we observe closer tunning with A=432Hz then we do with A=440Hz though, TBH, both of these modern frequency standards may be red herrings when it comes to ancient harmonic requirements.

We can see that A=440Hz equal temperament tuning is particularly wayward for the highest three sounding bowls i.e. those frequencies affecting throat, brow and crown chakra functioning. Curiously, the fundamental of the supposed oldest and most venerated bowl – TB4 – is pretty much precisely tuned to A=440Hz. Fascinating indeed, and a poke in the eye for Buturff.

These are all points worth thinking about with many biscuits, but to answer this question thoroughly we need to analyse all significant partials and not just the fundamental frequency. This leads us to a consideration of spherical vs. linear harmonics.

Kettle On!

Wiki points out that modern Western occultism views chakras as energy-centres that actually exist within the body, yet the concept started out as a meditation aid in the form of psychospiritual energy-centres in the subtle body. Quantum physicists wouldn’t have a problem with this, since I gather that we largely don’t exist (and neither does the world around us). It turns out that I am a conglomerate of probabilistic fuzz typing away at a conglomerate probabilistic fuzz on a big ball of conglomerate probabilistic fuzz. To be nit-picky precise about this I ‘exist’, but only in the form of probability distributions that surround a big bunch of excruciatingly small nuclei (so they say). In essence, I am not a series of spining atomic balls and voids, but a metaphysical concept that fills the whole. And not just the whole of ‘me’, for that conceptualisation extends to an infinite number of positions across the entire universe, but with rapidly decreasing likelihood of finding a bit of me everywhere. The same goes for that biscuit next to me; at a very small probability we are not separate objects in time or space, and if I reach over and nibble it, then that probability changes dramatically. In many ways whirling psychospiritual energy-centres make more sense, and offer less doctorate-level woo-woo.

Wiki further points out that chakras (literally translated as ‘wheels’) have become associated with colours, and with correspondences expanding to a wide range of alchemical and astrological systems, together with links to gemstones, homeopathy, the Tarot, and the Kabbalah. In this series we are going to find this correspondence extends to metals, to sound, and to resonant frequency – something I had not appreciated until now.

Close Encounters of the Bowel Kind

My first close encounter with a Tibetan bowl was about a million years ago. We wanted to get my brother-in-law something a bit different for his 40^th birthday, and chanced upon a beautifully hammered brass bowl in a small shop that no longer exists. It took some time for us to get used to rubbing a wooden stick around a metal bowl at a constant rate, but once the knack was achieved the ensuing ‘glow’ of sound was immediate. A sure winner! My brother-in-law was delighted with his unusual gift, but I don’t think the bowl does anything these days except collect dust on their coffee table.

In 2012 – the great year of the end of world according to an understanding of the Mayan long count calendar - I experienced a close encounter of the bowel kind with another such bowl. My niece, in training for a therapy diploma of some sort, had got me stretched out on a table in the hallway while she sounded a bowl and moved this over my person. The well-padded green leatherette table was rather comfy, and with eyes closed the whole experience was deeply relaxing. As the bowl reached my left ear, the glowing sound suddenly became overwhelmingly powerful. At this point I felt (or at least I sincerely thought I felt) something small and very tangible detach from the left side of my body and streak away like a shot. There was a loud BANG from the kitchen and we both jumped, then started laughing. We cautiously went to investigate but found nothing. We have no idea to this day what that was all about. It may well have been coincidence but it certainly left an impression – and I’m glad my rather relaxed bowels held up! My niece would later come to relate a number of curious experiences with clients and Tibetan bowls that were sometimes hard to explain away.

Tibetan bowls were to leave my life for several years, until their return with my work on spherical harmonics, certain of which is now published. The more I thought about Tibetan bowls and acoustics the more I realised we might be missing something profound. I understood that they were originally prayer devices, and I began wondering if they were esoterically and exoterically linked to the Schumann Resonances of the Earth, and to human brain function. In essence, I saw our skulls as being sacred bowls held within the sacred bowl of the ionosphere and wondered wildly on the matter. Do patterns of neural firing in our CNS mimic the resonances of a Tibetan bowl when in deep meditative states? Could the acoustic resonances of bowls entrain our brains in this manner? All very curious.

I decided to take the plunge and purchase my own very bowl – one tuned to the rather esoteric notion of A=432Hz no less! Armed with Mrs Dee’s chromatic tuner and a table of Pythagorean-derived frequencies covering all key musical intervals, I set out through the mists and rain of the South West for the largest collection of bowls I could find. Yes indeed, this was to be found in Glastonbury. Pinned on a frame on a stout post in the middle of the shop was a chart clearly telling folk what bowl was needed and why. I apparently needed seven – one for each chakra – each being tuned to a note on our modern western diatonic musical scale (C, D, E, F, G, A, B), apparently. I felt moved to harrumph loudly but refrained on account of Mrs Dee giving me a look.

I sounded a few of the better-looking hand-hammered bowls and noted they were all settling down into modern A=440Hz ISO tuning. This didn’t feel like cricket to me, so I asked the owner if they stocked anything in non-ISO tuning. She pointedly thrust a finger at the scribbled chart that told all, and without saying a word! In turn I pointed down to my home-brewed Pythagorean A=432Hz frequency table and explained I needed something that resonated harmonically with these numbers. This was a social, and, possibly, cultural faux pas on my part because she stared me down into a whimpering cotton-headed ninny muggins before telling me categorically that GOD HAD TOLD HER THE RIGHT FREQUENCIES! Her nostrils flared during delivery, and I decided not to thrust and parry with the Oracle herself in her Temple, even though I was itching to point out that the God she had spoken with was strangely keen on the same ISO standards as was the Asian sweat-shop manufacturer she was buying from.

Disgruntled at the poor quality of bowls flooding the New Age big business market, the matter slipped to the back of my mind, until I chanced upon an interesting chap called Jamie Buturff, who triggered my interest once more in a flurry of fascinating A=432Hz tuning commentary. In one of his YouTube clips he mentioned a CD called ‘Tibetan Chakra Meditations’ by Christa Michell and Ben Scott (ORP 58432), that allegedly revealed that ancient bowls were A=432Hz tuned. It should go without saying that this CD simply had to be most desirable pressie from Santa that year, pipping the fabulous Korg CA-1 tuner - a fine little device capable of A=432Hz tuning (most guitar tuners cover the range A=435Hz – A=445Hz) - into second place on the priority pressie list that sat by the chimney. Though released back in 1999 you can still buy decent second-hand copies of this unique CD from sellers on Discogs, and tracks are also available for download from Amazon Music. The venerable bowls that were lovingly and expertly sounded on this CD were owned by Ben Scott, and the sleeve notes had this to say about him and his bowls:

Ben Scott has lived through nine decades. An office boy at 17, the RAF at 20, senior command during the war years. In the 1960s Scott started on the ‘Seeker’s Path’, studying paranormal and psychic Eastern religions and practices made him a yoga and meditation teacher. After ending his business life, he took off around the country seeking kindred spirits in ashrams, monasteries and communities. “I see everyone I meet as my teacher”, Scott says. “Many of the bowls used on Tibetan Chakra Meditations date back to the sixteenth century. The ‘Heart Chakra’ bowl is estimated at four to five hundred years old”.

“The singing bowls were originally used in Tibet by shamans of the Bon-Pot religion which preceded the Buddhist religion. Tibetan singing bowls were used for the invocation of good spirits for the harvest, for curing disease, for banishing evil spirits and negative energies. They were made with seven metals corresponding with the seven planets and seven chakras: gold, silver, copper, mercury, iron, tin and lead. The very large bowls were struck and kept by the monks in their own quarters for magical and devotional use. In 1949, the Chinese invaded Tibet and large numbers of the lamas fled, bringing bowls with them. Many bowls have come to the West through these monks”.

“The bowls are very idiosyncratic and, on the recording, there is great variation of harmonics of both bass and high frequencies. Sometimes there are curious buzzy sounds at nodal points and strange ‘creaky’ sounds but these are all characteristics of the live played bowls. The bowls were played for a continuous five minutes for each chakra. This was then overlaid with two more stereo tracks of the same bowl, played live.”

I figured Ben had sadly passed from this life several years before I chanced upon his 1999 recording back in 2015, which somehow spurred me into spending time with his recordings, both sonically and statistically. I suggest at this point that we grab a cuppa and then I’ll reveal what I found…

Spectral Analysis

I’ve covered this concept before in several reports but here is a refresher in one short paragraph…

Spectral analysis is taking a snippet of sound and working out what precise frequencies are present (measured in cycles per second aka Hertz) and how strongly these are represented in terms of sound level. There are two ways of presenting the information graphically, depending on whether we want to see how the sound spectrum varies over time, or whether we just want an accurate plot of all harmonics that are present. It is one thing to determine what note a particular bowl may be tuned to, but it is quite another thing altogether to determine what family of harmonic frequencies it produces.

Each bowl will be referenced by its identification number, viz:

• TB1 - Root chakra

• TB2 - Sacral chakra

• TB3 - Solar plexus chakra

• TB4 - Heart chakra

• TB5 - Throat chakra

• TB6 - Third eye/brow chakra

• TB7 - Crown chakra

There now follows a series of high-resolution screenshots of the spectrograph of each bowl along with commentary. The horizontal axis represents time, whilst the vertical axis represents frequency (Hz). The sound energy present at each moment is represented by a rather large number of coloured pixels. Black means silence. Very low sound energies – seriously quiet stuff - are painted with purples, which pass up through the colour spectrum into reds, oranges, yellows, and eventually whites as the sound crescendos. Steady tones look like long, thin horizontal bars. Upward streaks mark places where the stick catches on the bowl to produce the ‘buzzy’ and ‘creaky’ sounds. Faint blotches down in the 40 - 60Hz region are ‘moans’ – and quite possibly external noise like traffic rumble. The tingsha (ritual hand bell) sounded at the end of each recording appears as harmonic dashes in the upper right-hand portion – I’ve marked these out for TB1.

Fundamental Frequency

The first numerical data I wish to rub the wet towels over are the estimated fundamental (primary) frequencies for each bowl. I say estimated because it should be noted this can never be a single, exact value. Despite what fancy digital tuners and iPhones may display to umpteen decimal places, all such algorithms rely on Fast Fourier Transform (FFT) techniques and these require a ‘gridding’ parameter with which to slice and analyse the signal. Under magnified levels of scrutiny, all FFT frequency peaks are thus plateaus embracing a range of values for which a single point estimate is usually derived (a best guesstimate). Here is an example of a high-resolution FFT sweep zoomed in on the fundamental frequency of TB1…

DING!

Drat - I’ve run up against the email limit, so my first table of juicy numbers will have to wait until the second instalment…

Kettle On!

A: YES!

We’ve already mustered evidence of tendencies for bottles to cluster around a 24Hz cellular matrix, and have unearthed an epochal step change in frequency that occurred between 1465BCE and 930BCE, which provided somewhat fascinating evidence of an average 1440Hz tuning for the older bottles – a figure that is utterly 24Hz compatible and splendidly Biblical! We’ve also seen a progression in cultural median frequency that also seems to adhere to my 24Hz matrix hypothesis. In contrast, we have discovered that Pythagorean scale tunings of A=432Hz, A=440Hz and A=444Hz don’t apply.

Given the importance of cultural differences, I sat for a while with a table of cultural mean values and a large mug of tea. After I while I decided to round the numbers to the nearest integer and have a fiddle with the spreadsheet. What intrigued me immensely was the fact that every single rounded frequency value was exactly divisible by 2. Curious!

If there is indeed a deep design structure we might wonder if its purpose was to generate beat frequencies. I realised that analysis of the entire cross-cultural sample of 69 bottles would yield pretty nonsensical results, so opted to focus on the 31 bottles of the Chimú culture. The smallest beat frequencies are what we’re interested in if we are talking about brain entrainment, and in order to calculate these all I had to do was sort the bottle frequencies in ascending order then derive a differenced series. Here’s what came out of the oven…

We’re certainly down in brain entrainment territory with a minimum beat frequency of just 5Hz, but the maximum of 114Hz is well beyond anything we acknowledge today. The average beat frequency comes in at 49.7Hz but since the arithmetical mean can be distorted by extreme values we can opt to consider the geometric mean or median. The median is my favourite summary statistic in wild data sets, and this indicates a ‘centralised’ beat frequency value of 38.6Hz. Here are the data in graphical form…

The first frequency band covering 0 – 12Hz has been chosen to coincide with the brain wave entrainment range common to many psychoacoustic experiments. Despite this, only two bottles were capable of generating such a low frequency. We observe a preponderance of beat frequencies in the 12Hz – 24Hz band (6 bottles), with 36Hz – 48Hz (5 bottles), and 60Hz – 72Hz (5 bottles) also being popular. This is not psychoacoustics as we know it!

Q: Were the bottles meant to be used as musical instruments?

A: NO!!!!

We have already unearthed evidence of a musical octave and perfect fifth within the beat frequencies of the bottles when blown en masse. Let’s now continue in the same vein, but looking for musical intervals between bottle frequencies themselves.

To keep computation manageable, I decided to stick with just the sample of 31 Chimú bottles, since these alone would provide no less than 465 musical ratios when taken a pair at a time in all possible combinations. This means we’re looking at a pure culture – which is desirable. It more importantly means we’re looking at one of the most recent of ancient Peruvian cultures in a time period where musical form and rudimentary theory was certainly extent throughout the rest of the world (700 CE – 1470 CE).

Octaves, perfect fourths and fifths are characteristic of all musical forms throughout the world regardless of originating culture. This is because nature itself vibrates this way: these intervals are a mathematical property of the cosmos. If these bottles were meant to be musical instruments, we should expect to see a high proportion of these intervals emerging when they are blown in pairs; followed by intervals that promote joy (major musical intervals) or melancholy (minor musical intervals).

All possible musical intervals were calculated using a 31 x 31 matrix. These were then examined for correspondence to Pythagorean 12-tone interval ratio values – a mathematical construct that was known 1,000 years before the first Chimú bottle had been cast. Precise ratio matches were noted (see column headed ‘Exact’) along with two permissible levels of error for comparison (see columns headed ‘Within 0.01’ and ‘Within 0.02’).

A total of 71 exact matches were found, which represents just 15.3% of the available data (71/465). When the criterion was relaxed to musical ratios lying within ±0.02 of a Pythagorean value this rose to 70.5% (328/465). That being said, table 20 clearly shows a preponderance of very small ratios, the most popular being the minor second at 1.05. The data are presented in graphical form in figure 9:

The relative absence of musical intervals between the perfect fifth and octave is striking and this alone tells us that the bottles were never meant to be used in a musical context. The significant clustering of ratios down below the musical ratio of 1.20 is also indicative of vessels that were never meant to please in a musical fashion.

To my mind this analysis is sufficiently clear enough for us to realise the Chimú were not chasing musical joy or melancholy through these bottles. They were probably not designed with musical purpose in mind but something else entirely.

Q: Can we identify the cultural origin of the three unknown bottles?

A: Theoretically, yes, but we need more data

There is a rather powerful and fancy statistical technique called discriminant analysis that enables prediction of group membership from a set of independent variables. However, as regards Peruvian bottles, we only have two such variables, these being frequency and sound pressure level. Frequency was found to be strongly associated to culture but sound pressure was not. In terms of discriminant analysis, we therefore only have one useful variable – frequency – and this alone doesn’t have sufficient discriminatory power to identify the origins of the three missing vessels. But we can mimic fancy discriminant analysis by using a simple boxplot and a pink ‘ruler’…

The box representing the frequency range of the three unknown bottles on the far right of this plot can be seen to overlap with the lower end of the frequency range for the Chimú and Inca bottles. On this basis alone we can say the unknown bottles are most likely to have originated from one, or both, of these two cultures.

Q: How does the range of the Peruvian bottles compare with the Bolivian Stone panpipes?

A: There is considerable overlap with the oldest four Peruvian cultures

Figure 11 paves the way for a rather simple and effective way to answer this question in a single plot. That is to say, the nine resonators of the Bolivian stone panpipe were treated as though they were nine Peruvian bottles. Simples! In-keeping with the use of raw measurements throughout this study I decided to ditch my fancy M-24 Bolivian pipe model values and favour use of the actual measurements by Putney & Benoit that were adjusted for damage.

Isn’t that a corker of a plot? The nine resonators of the Bolivian panpipe sit neatly within the ranges for Vicus, Moche, Gallinazo bottle cultures. Aside from the wayward Salinar bottle, these three represent the oldest Peruvian cultures, with cultural mid-points spanning 1465BCE – 2165BCE. The correspondence in frequency between the Bolivian panpipe and the oldest Peruvian whistling bottles is stunning and may even give us a date fix for the mysterious Bolivian black stone culture.

Cognac & Chocolatey Treat

A gazillion pages of statistical output is an awful lot to swallow, so let me offer a chocolately treat and cognac at the end of our meal. We’ve seen that these bottles are nothing much to do with musical scales, and can be used to induce beat frequencies over a faily wide range that is not normally associated with psychoacoustics. We’ve noted an epochal change in frequency that divides the more recent Chimú, Chancay and Inca cultures from those of the more distant past, when lower fundamental frequencies were the order of the day. We’ve gone so far as to tease out a model indicating an annual frequency rise of 0.91Hz, thereby supporting the intuition of ceramicist Don Wright, who feels the bottle frequencies need to evolve with humanity. The sonic overlap with the ancient and enigmatic Bolivian panpipes has been thrilling, as is yet more evidence supporting a 24Hz sound matrix.

Daniel Statnekov tells us from experience that seven seems to be the right number of bottles to sound together for maximum effect. More than this and people seem to generate what he describes as “chaotic states”. I note with interest that he has spotted seven small niches in the wall behind the main altar at Machu Piccu, each precisely of the right size to store a whistling bottle. In his book Animated Earth he mentions that many vessels produced up to seven partials during acoustic tests, which would naturally interact with those of other bottles. His supervising professor noted that the resulting sound was similar to that produced by specially-trained Tibetan lamas, who are able to chant in a unique way so as to make overtones audible, i.e. they effectively sing their own chord. Words like AUM/OM are rich in overtones and are probably chosen for this very purpose.

Statnekov’s account of his encounters with a remarkable woman called ‘Yeti’ are also somewhat fascinating. She tells him of the legend of the acoustical key and the hidden importance of the site:

‘The legend foretells that one day a certain sound will unlock an ancient door that leads to an altogether different reality… a different dimension’

‘The ruins aren’t really important in themselves, except that they physically draw us to a location where the old masters discerned an energy fluctuation that affects consciousness’

Of the use of the bottles, she remarks:

‘They were probably designed to manifest an increasing potential when played in certain places – megalithic ruins for example’

The point she repeatedly made to Statnekov was that the vessels need to match up with a special place – like a key and a door - and she felt Peru was no longer the door. In relation to this, Statnekov writes:

‘Anthropologists in the remote villages of the Andes have uncovered a myth that gave direction to my fantasy. One upon a time a mythical being called Inca-Ri founded the city of Cuzco at the spot where a gold rod embedded itself in the earth. He was killed by a white chieftain and the dead man’s head was buried near Lima. Since then the head has slowly been growing and one day it will return, complete with its body.’

Well, there’s funky for you! The best course of action now is to invest in a set of bottles and see what happens, meanwhile I’ll pack my tea urn and head for Tibet.

If you’ve enjoyed this mini-series you may want to read two rare items by Daniel K. Statnekov that I saved to HDD way back; the first being his 1974 transcript for the Journal of Transpersonal Psychology, and the second being an unfinished essay dated 1981.

Kettle On!

A: NO, neither are they A=440Hz or A=444Hz tuned

Though an understandable question to ask, it’s not easy getting a sensible answer. A musical scale doesn’t just require a reference pitch but a starting point for this and some method of temperament. For one thing, setting the tuning reference note at the major sixth of the scale gives slightly different results than if we set it to the unison (the ‘transposition chestnut’), but we shall leave that chestnut unpeeled for now. As regards temperament, the oldest surviving method is Pythagorean which, weighing in at around 500 – 550 BCE, is at least contemporaneous with the Inca culture – but of course Pythagoras could easily have been given his scale by sages in Egypt where he allegedly trained, i.e. we have no idea how old they may genuinely be. In any case, they are the very best ancient scale I can readily muster for study of the Peruvian bottles.

To make things juicy I decided to run with the three scale tunings that figure prominently in my work and that of others delving into all things psychoacoustic, these being A=432Hz, A=440Hz and A=444Hz. A complex spreadsheet matrix was devised which enabled me to discern which scale offered the closest note value for each Peruvian bottle. In this way each bottle was assigned ‘closest scale’ along with the exact frequency deviation from this. Altogether, this became a rather juicy bit of analysis in more ways than one!

First up, let’s take a look at which scale tuning came out tops…

A=432Hz as it happens, with 31/69 (44.9%) of bottles being assigned this tuning as possessing the nearest note to the actual bottle frequency. A=440Hz came out the least popular scale, second to the most curious A=444Hz tuning.

We can now look and see if there is an association between scale tuning and epoch – i.e. did things change scale-wise over time?

The answer is nothing changed.

50.0% of bottles from the pre-epoch era were tuned to A=432Hz, whereas 43.8% of bottles from the post-epoch era were so tuned, this difference not reaching statistical significance (p=0.838). We can repeat this sort of analysis and see what the cultural preferences were (if any) …

So, then… we’ve got Chimu, Gallinazo, Huari, Recuay and Vicus all favouring A=432Hz, but we should not read too much significance into this because sample sizes are rather low. Perhaps more importantly is the issue of whether A=432Hz bottles were more accurately tuned to this scale or not, as the case may be. To answer this all we have to do is conjure another of those lovely boxplots:

We can see that the spread of frequency differences for the A=440 Hz bottles is less than that of the others, whereas the spread for A=432 Hz is the greatest. Such variation in variance calls for another of those ANOVAs…

...Ooohhh, it comes close to being statistically significant at the 95% level of confidence! With p=0.088 there is an 8.8% chance the variation we see in the data could have arisen by chance. If we stick rigorously to convention, we have to reject our hypothesis that there is a difference in accuracy of scale tuning, but there is also a vague hint that A=440Hz bottles may well have been intentionally tuned that way. Another way of looking at this is to flag all bottles that have been tuned to within 10Hz of the nearest scale value and run this through a crosstabulation…

We can see that 60.0% of A=440Hz bottles are tuned to within 10Hz, compared with 48.4% for A=432Hz bottles and 30.4% for A=444Hz bottles. However, these differences did not reach statistical significance at the 95% level of confidence (p=0.176).

Looking at the overall picture we could say that only 31/69 (44.9%) of bottles were intentionally tuned to something we would now recognise as a Pythagorean scale, regardless of reference pitch. These being in the minority we must conclude the bottles were not intentionally tuned to any such standard scale.

I ran a final crosstabulation looking at tuning accuracy and epoch but this returned nothing, with p=0.571, this lowly value meaning they were just as inaccurate in the pre-epoch era of great antiquity as the more recent post-epoch era. A final, factored histogram should wrap things up for now…

We can see the bunching of A=440Hz bottles to the left and the large spread of A=432Hz bottles compared to this. I don’t know about anybody else but this spread of tunings across three reference scales is again telling me more or less one thing: these bottles do not appear to be tuned to any form of Pythagorean musical scale.

Q: Are the bottles following a 24Hz matrix of frequency values?

A: YES! There is evidence of clustering around frequencies divisible by 24

This was quite a technically challenging question to answer. I decided to approach this by finding the nearest frequency that was an exact multiple of 24Hz, then calculating the frequency difference between this and each bottle frequency. This would give me a difference distribution that I could examine for statistical significance. If bottles were intentionally designed to lie close to a 24Hz matrix value then, instead of a random spread of frequency differences, we’d see evidence of aggregation i.e. more than expected numbers of bottles lying close to frequencies that are multiples of 24Hz.

Before I got stuck-in I knocked-up a simulation table to better understand the mechanics of how a whole load of bottle frequencies could vary in relation to an underlying 24Hz matrix. One way to think of this is that the bottle frequencies (when rounded to nearest integer values) are permitted to increase in steps of just one unit, whereas the 24Hz matrix frequencies must increase in steps of 24 units.

As I suspected, there is an indefinite repeating pattern behind this, which I present in tabular form below. In this table integer frequencies are used to make presentation simpler – but we need to remember that the single integer value of 2400 represents every possible bottle frequency value from 2399.500 to 2400.499 – the table is thus a schematic covering the essence of all the wartiness of whistling reality!

We see that the maximum frequency difference attained can only ever be 12Hz, at which point the matrix unit must flip to the next value, so that the difference progresses back down to zero for the next numerical cycle.

In probability speak we can see that a nominal frequency difference of, say, 10Hz or less is represented in no less than 21 of the 24 rows. We shouldn’t therefore be at all surprised to find many of the bottles sounding within 10Hz of their nearest 24Hz matrix value – an important point! In fact, it turns out that 63 of 69 bottles (91.3%) do this very thing, demonstrating how easy it is to fall foul of a tasty but false conclusion.

If we get really picky we can count the bottles which are tuned precisely to a 24Hz matrix value. Table 13 tells us the odds for this happening are 1 in 24 (0Hz difference), thus with 69 bottles we would expect to see 2.875 bottles this well-tuned (1/24 x 69). As it happens there are 5 bottles that fall within this super-picky criterion, which seems a bit more than expected by chance. Bunging these odds and observations into a Binomial probability calculator we get a p-value of p=0.1607 – a tantalising figure which hovers reasonably close to statistical significance.

Let’s now progressively relax our super-picky criteria and see how we fare…

This table offers up a rather pleasant surprise! We can see that as we progressively broaden our criteria for frequency difference, there is a corresponding rise in the number of bottles expected by chance. However, observed numbers of bottles is consistently higher in each case. This discrepancy between observed and expected numbers of bottles reaches the 95% confidence level of statistical significance in 6 instances – which are shaded pale yellow.

At a frequency difference of 6Hz things really hot up! We observed no less than 50 bottles tuned to within 6Hz of nearest 24Hz matrix values when only 37.375 bottles would be expected by chance. This discrepancy is highly statistically significant at p=0.0014. This suggests there is a tendency for bottle frequencies to cluster around a 24Hz frequency matrix, with no less than 50/69 (72.5%) being tuned to within 6Hz of a 24Hz matrix value. We cannot say whether this was intentional but it (apparently) remains an evidential fact.

With an overall mean frequency of 2378Hz, a discrepancy of 6Hz in tuning represents an overall error rate of just 0.25% - this would be an extraordinary feat of achievement for clay work, which suggests the whistling mechanisms may have been carefully selected for mounting in the bottle.

Whatever the reason, and however it was achieved, some sort of non-random selection process causing clustering around 24Hz matrix values is clearly evident.

Quite how they knew which whistle to select is anyone’s guess – though Don Wright was using a keenly developed self-induced state of synesthesia. It would be darn interesting to see how his modern bottles compare.

With a 24Hz frequency matrix now in play, I decided to see if this bore any relation to other factors such as cultural preference, epoch and sound pressure level. After a few crosstabulations and stats tests it was clear it wasn’t related to anything in particular – it just was.

Darn! I’ve hit the email limit again!!! Time to grab the biscuits…

Kettle On!

A: YES – enormously so!

We can dial-up a one-way Analysis of Variance (ANOVA) to crunch the stats in one swift move. The corresponding statistical output is below, which reveals a p-value of p<0.001 (the ‘.000’ in the column marked ‘Sig.’)

We can also eyeball cultural frequency differences using a boxplot. In these handy little jobbies a thick black band represents the median, the shaded box represents the interquartile range (IQR), the ‘whiskers’ represent a nominal spread of values that is 1.5x the IQR, and numbered circles represent individual case outliers that lie beyond 1.5x the IQR:

I love these simple boxplots because they pretty much say everything you need to know in one neat graphic. For example, we can see that Gallinazo, Huari, Mocha and Vicus cultures made the lowest frequency bottles, and with similarly overlapping ranges. Chancy and Recuay cultures also bear likeness to one another in terms of the spread of frequencies, as do Chimú and Inca.

Q: Are there cultural differences in sound pressure level?

A: Not really. There is no evidence of this apart from two ‘outlier’ cultures

Here we go again, straight in with the ANOVA output…

There is not even a sniff of a statistically significant difference here, and especially so with a p-value pf 0.379. Here with the corresponding boxplot that tells the same story:

Q: Does fundamental frequency change w.r.t. geographical location?

A: Unanswerable, but it looks like it does not.

A tricky one to answer because culture and geographical location are largely bound-up with each other and thus confounding. In stats-speak these are correlated variables, which means that the cultural differences we’ve seen will invariably be reflected in location (and vice versa). That being said, let’s take a look at the hard data by way of a mini frequency table and have a look-see:

Aha! Now then… only one area – the North coast – bears a (mildly) sufficient number of different cultures to make this worthwhile, so I’ll pop out a boxplot for us to eyeball:

The North coast bears the widest box and longest whiskers of all, which suggests that cultural differences outweigh regional ones. In fact, we see as much variation in this region as all other regions put together! This fact alone would suggest there is no geographical specialness when it comes to frequency – only culturally-driven factors - but, TBH, the data are a wee bit thin on the ground.

Q: Did the fundamental note of the bottles change over time?

A: YES, but so did culture and bottle design!

Garrett & Statnekov have put it this way…

It is apparent that the earlier cultures tended to produce double-noted, low frequency, “enclosed-type” whistles while the later cultures generally produced single-noted, high-frequency, “exposed type” whistles.

In terms of rigorous statistical analysis this question faces exactly the same confounding situation as the last. If we do find a chronological difference it will also be a cultural difference (and vice versa). Nevertheless, Garrett & Statnekov do clearly state that subsequent cultures progressively altered the frequency and design of their bottles, and this will be most interesting to probe.

To try and keep this clear and simple I’ve determined the chronological mid-point of each culture and converted this from Garrett & Statnekov’s AD/BC convention to number of years Before the Current Era (BCE). Big numbers thus mean darn old!

This boxplot is darn interesting for starters…

As Garrett & Statnekov conclude, we see a clear and definite tendency toward lower frequencies for older cultures; aside, that is, for a sample of just one bottle from the Salinar culture, which is an outlier in many respects. Looking at this my eyeballs tell me there is not a trend so much as a step change between 1465 BCE and 930 BCE. This means we can concoct a simple binary variable to denote this step and use it as a factor in analysis. I shall call it… epoch! When we split the sample in this manner, we find a total of 48 bottles for the post-epoch period (the most recent cultural period) and 18 bottles for the pre-epoch period (the most ancient cultures). We can now use this rather useful binary variable to explore the data further - a crude but handy factor like this can lift the lid on things which may be hidden.

In terms of sound pressure level, there was no statistically significant difference between pre and post epoch periods (p=0.222), as we may expect. Frequency is a different beast, and below is a summary of the mean epochal differences, these being highly statistically significant (p<0.001):

That there pre-epoch mean of 1440.62Hz is most interesting.

Not only is 1440Hz a Biblical-flavoured number but it is also precisely divisible into units of 24Hz, 48Hz, 72Hz, 48Hz and 96Hz – numbers we have come across before in relation to the design of the Bolivian Stone panpipe. In fact, 1440Hz is precisely the frequency of (modelled) resonator #4 of the M-24 Bolivian panpipe!

The post-epoch mean of 2673.51Hz, however, looks very nothing much, and we find the nearest 24Hz matrix value coming in at 2664Hz. This is the equivalent to a piece of cold toast with margarine on a rainy day.

There Is Another…

Another way of dealing with scatter in the frequency data is to calculate cultural mean or median values and have a look at how these measures of central tendency progress over time. I’ve opted for cultural medians to reduce the impact of outliers, and I also decided to exclude the single Salinar bottle because of its wayward sonic behaviour. As result, out popped a rather splendidly munch-worthy figure…

Each point on this represents the median frequency for a specific culture, there being eight cultures in total (Salinar being excluded). These are plotted against something called the ‘cultural mid-point’, which is simply the chronological half-way mark for each culture. We get a nice progression clearly showing a gradual rise in preferred frequency over time. A linear regression (black line) drops in on top of this with ease, and with decent statistical significance (p=0.01).

This is telling us that median cultural frequency, on average, has risen by 0.911Hz per year - or 91.1Hz every century. The intercept of 3170.2Hz indicates what the median frequency would be in the current era. We have no idea what brought about this gradual change, but what I find intriguing is that 3170.2Hz is just 2.2Hz away from 3168Hz… a 24Hz matrix value point! Should we be whistling at 3168Hz these days to obtain the same psychoacoustic effect?

I say this because ceramicist Don Wright (apprentice to Daniel Statnekov) had this to say on his website…

The bottles recently experienced a rapid evolution in their effect and appearance. While writing an article for New Frontier Magazine, Don suddenly began to have profound insights into how the bottles should be altered in their tuning and construction. He realized that his previous method of precision constructing and tuning the bottles had frozen them into the evolutionary level of one of the last cultures (Chimú) to make them. The new insights enabled him to allow the bottles to once again freely evolve to be in harmony with the present.

His new approach to producing the bottles involves tuning them while in a self-induced synesthesia perceiving state, intuitive irregular incision of resonator apertures, and the use of copal and frankincense, sacred resins of Peru and Arabia. This evolution has resulted in the bottles producing a deeply profound subjective as well as objective experience as opposed to a predominately objective experience. The time of the onset of the expanded/altered state has changed from around 30 minutes to only 2 or 3 minutes. The experience is much more entheogenic in nature. It now appears to be accumulative in that a person becomes more sensitive and responsive to the effect. The experience now seems to also have residual effects of sensitizing a person’s perception of colors, sounds (music), and the emotional states of others.

Just in case Don Wright’s work piques interest, here’s what Brave Search AI has to say about him and his whistling bottles…

Don Wright is a renowned ceramicist and creator of contemporary Peruvian Whistling Vessels, also known as ‘huaco silbadores’. He is recognized for crafting vessels tuned to produce binaural beats — a psycho-acoustic effect that can influence brainwave activity, promote relaxation, and induce altered states of consciousness. His work is deeply rooted in ancient Andean traditions, where these vessels were used in shamanic rituals for spiritual journeying and healing.

Wright’s vessels are typically made in sets of four or more, as they are designed to be played together to create harmonic, resonant effects. He uses sacred clay sourced from specific sites and often incorporates personal items, such as hair, into the vessels to create a spiritual connection between the maker and the instrument. His creations are inspired by ancient Moche, Chimu, and Inca designs, and he has been instrumental in reviving the true purpose of these vessels—once mistaken as mere utilitarian water containers.

His work is featured in various publications and interviews, including the ‘Songbird Ocarina Podcast’, where he discusses the spiritual and healing power of the instruments. You can find his original pieces through his website, entheosound.com, or via his Facebook page, Peruvian Whistling Vessels.

NOTE: According to his Facebook page Don has not been at all well, with his last post there dated August 2021. His website has now been decommissioned by the ISP, but back in 2014 I saved this leaflet of his to HDD.

OK, so I think that’s enough stats for the time being (and I’ve just reached the email limit)… I’ll be back with more slides and stats in part 3, so stay tuned…

Kettle On!

Peruvian whistling bottles first came to my attention back in the Autumn of 2013, when I started statistical modelling of the rather enigmatic Bolivian black stone panpipe that was featured in appendix I of Putney & Benoit’s strange tome: Sanskrit. Here’s what Putney & Benoit have to say about these curious artefacts…

The rediscovery of the physiological effects of this ancient class of instruments occurred through the research of S Garrett and D Statnekov. The researchers studied the frequency variation in 73 whistling ceramics from 9 distinct Andean cultures, spanning some 2,000 years. The tuning of instruments was determined to be Culture-specific, as all of the instruments of anyone culture were tuned together into precise frequency ranges.

A general trend observed was a steady increase in the chosen frequency over time, the earlier Vicus, Moche and Huari near 1320 Hz, the Recuay near 2000 Hz and the later Chancay, Chimu and Inca bottles near 2670 Hz. The fact that all cultures of the region employed this instrument, each with its own specific frequency range, attests in itself to the significance of the group frequency identity.

Further experiments in playing several instruments from one culture showed an increased effectiveness, often inducing extraordinary personal experiences and group reflections. The prime number of instruments in any group is said to be 7, sacred to Andean belief. The achievement of unified group consciousness may also be reflected in deeper ways in the societies who shared in their use. The collective result of this kind of societal tuning is that one brainwave pattern is simultaneously shared by all. Ancient neural synchrony technologies are indeed the prime key to enhancing human consciousness, opening another dimension of awareness and communication.

Daniel K. Statnekov, back in 1987, was the man responsible for their re-labelling as an ancient psychoacoustic device. Up until then museum curators had simply labelled them as novelty pouring bottles, the development of which was said to brought about by a “happy accident”1. Daniel wrote an account of his experiences with these bottles in a book – Animated Earth – I and think it worth pasting his very first experience down below, so you can see just how remarkable it was:

‘Alone in the living room of my stately home, safely surrounded by my possessions, I took a deep breath and blew into the ancient vessel. For nearly a minute the sustained, piercing note of the whistle enveloped my head. Again and again I repeated the process, immersing myself in the all-pervasive sound. I felt as if I were trying to drown out the thoughts of my failure… that somehow the sound would relieve me of having to face the emptiness of my life with its trappings of a success that wasn’t mine. By the seventh or eighth breath something happened. Suddenly, in the middle of a breath, my mind let go with a ‘snap,’ and I perceived myself as a light; that is, I had become a moving luminescence apart from what had been my physical body. As I travelled through a blue-black darkness, illuminating—with a diffuse glow—the path through which I moved, I sensed the space around me to be the universe.

I knew that I was moving very fast—perhaps at the speed of light—yet my sense of motion was not much different from my memory of driving down a country road in an old friend’s Model T Ford. Surprisingly, I was not afraid. It was a pleasant experience, exhilarating. Secure in my new form, I felt supported by the pulse of energy. Every now and then I passed another body of light; some were brighter than others, I thought of them either as planets or more distant stars. This new dimension seemed almost familiar and it did not occur to me to question whether or not I could return to my physical body.

I don’t know in what direction or for how long I travelled in this manner, but suddenly I perceived an inky, black cloud looming in my path. There was no way to avoid it, and although the appearance of this apparition was no less strange than the rest of my experience, I was terror-stricken. As my body of light touched the shape, I knew in my deepest core that unless I escaped at once, I would be absorbed into the cold blackness forever. This realization brought forth a burst of energy, and with one last desperate effort, I recoiled from the dreadful form like a ‘spring’ of light. I crashed back into my body.

I arrived in my living room to find myself sitting upright in a blue, Martha Washington chair, still holding the artefact in my hands. My heart was throbbing and perspiration had soaked through my shirt. In a single moment, as if a casual glance, I saw the enormity of my hypocrisy and the shallowness of my life’

Now THAT is what I call remarkable! He was never to experience anything quite like this again, though the sound continued to profoundly affect him and others. Here’s his account of the first time two bottles were played together:

‘An other-worldly sound engulfed the room, as if a cyclone or whirlwind had arrived in the middle of Manhattan. Beginning as a low rumble, the sound rose rapidly in pitch to become a full-blown maelstrom that sounded like a swarm of bees…but it was a swarm of bees buzzing around the inside of my head. We played several breaths together in this fashion, and with each breath the strange sound reappeared, flooding my brain with its distinct sensation. I could actually feel the sound travelling from one side of my head to the other little a gentle massage of the brain. I felt as if waves of energy were washing my mind, sweeping away thoughts. I knew that what I was hearing was articulate and sentient beyond my projection into it’

Well that did it… I simply had to obtain a copy of his sole research paper from the Journal of the Acoustical Society of America2, and scour this for frequency data!

Methodology

Scientific reports in academic journals need to be brief and to the point. I know this all too well having written a few of my own, and having assisted in the preparation of several others over the years. It came as no surprise, therefore, to discover that Garrett & Statnekov had only provided one table of hardcore numerical data – and even this was a summary of average values. Herewith that one and only data table:

Fortunately, they had provided one priceless figure – figure 7 – which incorporated all of the raw data that I needed. To turn the symbols of this plot into accurate frequency and sound pressure level estimates I resorted to cunning use of a professional vector graphics package, as I had done for the Bolivian Stone Panpipes. To this end, a high-resolution image of figure 7 was pasted into an A4 layout file, locked into place, and framed to provide reference coordinates. It was a simple matter of then zooming in to each point, centralizing a red aiming circle, and reading-off the cursor position to three decimal places. These were then scaled to provide estimates of the original raw data.

Here is my trusty aiming system in action - below is how the aiming circle (a vector object that can be placed with precision) looked under high magnification…

In the space of four hours I was able to accurately pick-off every single one of the 69 bottles represented in Garrett & Statnekov’s figure 7, then calculate pretty precise values for both frequency and sound pressure level.

NOTE: The following abbreviations for originating culture may be found in statistical output, figures and tables:

Verification of Dataset

The first question in my mind was how accurate was my methodology in attempting to reproduce the original raw data? I had certainly bagged all 69 bottles without fuss, but decided the acid test would be to attempt to reproduce Garrett & Statnekov’s table 1 using my estimates of their data.

Here is my own table 1, which compares their original summary statistics with those obtained through my graphical estimation. As can be readily seen, the estimates I have derived are quite robust indeed, with a difference of just 2Hz for mean frequency and precisely the same mean sound pressure level – quite an achievement – and a glowing testimony for this cunning method of nicking other people’s data!

Some minor differences may be expected through the process of measurement error and these will find their way into discrepancies in standard deviation. I thus ran a paired samples T-Test on standard deviation values to ensure all was well, and indeed it was. Herewith the stats output:

‘Pair 2’ corresponds to standard deviation values for original and estimated fundamental frequencies, with ‘pair 4’ comparing standard deviation values for original and estimated sound pressure levels. The key column is the one on the far right which provides the all-important ‘p-values’, and which indicates that both pairs of values are not statistically different in the slightest.

Thus, we can now put the kettle on, open a nice tin of biscuits and enjoy another analytical ride. Here are some questions in the back of my mind for starters…

• Are the cultural differences noted by Garrett & Statnekov statistically significant?

• Are there cultural differences in sound pressure levels?

• Does the fundamental frequency change with respect to geographical location?

• Did the fundamental frequency of the bottles change over time?

• Are the bottles A=432Hz tuned?

• Are the bottles following a 24Hz matrix of frequency values?

• Can any deep design structure be discerned?

• Were the bottles meant to be used as musical instruments?

• Can we identify the cultural origin of the three unknown bottles?

• How does the frequency range of the Peruvian bottles compare with the Bolivian Stone panpipes?

Though not the same thing, the following demonstration of whistling water vessels by Old Peru Replicas is sure to delight - you can even treat yourself to one from their Etsy shop!

Kettle On!

Well, the taut string of a venerable monochord, lyre, guitar or lute is a visual example of perfect straightness. Strings are straight – we know that for sure. We also know that if we divide one of these strings into two with our finger (or capo, or moveable bridge) our plucking then produces the musical interval of the octave. In this instance the length is exactly halved and the frequency of vibration is exactly doubled – hence the musical ratio of 1:2 denoting the octave. The same goes for the all-important perfect fifth, which is created when we place our finger/capo/moveable bridge etc exactly two-thirds of the way along the string – hence the musical ratio 2:3.

Because the monochord/lyre/guitar/lute is a linear harmonic system we can go on indefinitely halving each half of the string, thus moving up through the octaves with ease and precision. Equally, we can go on indefinitely two-thirding the string to steadily cycle higher and higher through the perfect fifths. String lengths and frequencies they will generate can be predicted with ease using nothing more than a basic calculator. Flutes, tin whistles, recorders and pipes (organ, pan, Scottish or exotic) are all examples of linear harmonic systems acting just like their stringed chums. These tubular lovelies possess a moving air column which is excited into vibration by our very breath to produce glorious sound.

Note I carefully use the word circle here and not spiral. Dividing strings and shortening pipes generates a circle of ever-higher octaves and perfect fifths within a linear system. The ratio of 1:2 forever remains 1:2, progressing in exact glory, as does the 2:3 – a pipe or string reduced to 1/4 of its length sounds a note two octaves up. A pipe or string reduced to 1/8 of its length sounds a note three octaves up. Simple, clean and lovely!

Milk bottles and orbs

Instead of blowing across a panpipe or twanging a lute let’s put these down for the moment and pick up a metal sphere with a small hole in it. When we blow across this strange little chap to sound a milk-bottle-esque note-of-sorts all the familiar harmonic linearity of pipes and strings no longer applies; it resonates and produces a set of musical tones in a totally different way. Gone are octaves with a ratio that is forever 1:2, and fifths with a constant ratio of 2:3.

Take a basketball, throw it hard to the floor then quickly put it to your ear. If you are quick enough, you’ll hear it ‘ring’ for a split-second after the impact. It doesn’t ring with one simple frequency either; a whole harmonic series of tones are produced that are quite unlike those of pipes or stringed instruments. Experiments reveal the ringing of objects like basketballs conform to the strange realm of spherical harmonics1 .

It isn’t just plastic basketballs and metal orbs either; all concentrations of energy field, fluid and gas whether they be electrons, stars or plasma conform to the same laws of spherical harmonics. TV transmitters, sonar and radar all follow the same harmonic laws2.

Perhaps the most famous set of spherical harmonics are the Schumann Resonances. This is an electromagnetic ‘ringing’ of the ionosphere surrounding the earth, which is caused by continual discharge of lightning and other ionic interactions of solar and galactic origin. If you want to know what this would sound like within the audible sound spectrum rub your finger round a large, high-quality wine glass. The characteristic ‘angelic bell’ sound common to all of these resonators is the product of spherical harmonics at play. I recommend folk seek out and listen to the sonified Schumann Resonance online – it is strangely moving3.

Some have noted our brain wave frequencies come in bands that closely resemble Schumann Resonances and propose that they are linked in profound ways. This could indeed be the case, but it could also be that our brain simply follows the same divine laws of spherical resonance: perhaps it is the divine law itself that unites us.

Getting mathematical

Helmholtz is the guy who got heavily into the field of non-linear harmonics in the 1850’s after fiddling with metal spheres with little holes (called Helmholtz resonators). In order to fathom what was happening in theoretical terms he had to invent a slice of complex maths. Together, with mind-bending related mathematical constructs such as spherical Bessel functions, Legendre polynomials and wave equations, this pioneering work went on to inform the newly emerging discipline of wave-particle physics.

One way of looking at the issue is to say the harmonic mechanics of resonating strings and pipes can be approximated by a two-dimensional conceptual consideration. Linear maths and good old pencil and paper will suffice in explaining how they work. But when it comes to explaining resonating blobs of energy/matter we need to move to spherical harmonics, and a three-dimensional conceptualisation. Here’s a sweet little 2D diagram of some of the possible modes of vibration of a string:

Here is an even nicer picture of some of the possible modes of vibration of a 3D blob of energy/matter:

The top blob represents the musical interval of the unison at 1:1, whereas the next row of three blobs represents three possible derivations of the musical interval of the octave at 1:2. We no longer have one obvious and easily calculated octave but three with at least two different values for maximum permitted wavelength. If this looks awkward, try calculating the permutations for the perfect fifth at 2:3! It is important to realise that this is not just abstract theory – empirical studies of wave-like structures confirm this is more or less how things actually shake-up and fizz about when wobbling about in nature.

Schumann leads the way

The Schumann Resonances follow the law of spherical harmonics, so I started out with these hoping I would get a lead to something I could understand – spherical harmonics is graduate-level theoretical physics after all! Not long after I was looking at this interesting online tit-bit4…

Not only are we are about 75% water, we are also literally swimming in the Schumann Resonances. These electromagnetic frequencies, mostly infrasonic and unheard, occur in the cavity between the earth’s crust and the ionosphere about 35 miles above the earth. They are activated by lightening storms and the sun. The basic frequency - 7.8Hz - creates a standing wave around the earth (travelling around the earth 7.8 times/second). The figure fluctuates by plus or minus 0.5Hz daily, as the ionosphere fluctuates. In addition to this base frequency of 7.8Hz, there are quasi-harmonics at (rounded-off) 14, 19, 25, 30, 36, and 42Hz, corresponding to the following tones: B/C, A, D#, G/G#, B, D, and E/F. So far no F#s. But a curious thing happens when these frequency numbers are octavised up. Instead of the usual doubling to create the octave above (i.e the octave above 440 = 880) - the method used in linear harmonics and ‘standard’ physics - when working with the harmonics of a sphere the rules are different. Instead of reaching an exact doubling of the original number, a slight collapse takes place and the upper number is less than an octave. When you ‘octavise’ the Schumann frequencies into our hearing range, six of the seven tones are F#s, A#s and C#s. The seventh is a D#. As they ascend they become even more like the ‘pyramid chords: F/F#, A/A#, C/C#. I haven’t a clue what this means, beyond imagining a theoretical tuning’ taking place on our physical plane. (See First, refs.)

I was surprised to learn that a composer – David First5 – had attempted to calculate spherical harmonics for Schumann Resonances after discussions with a world authority on the subject (Dave Sentman, the Professor Emeritus at the Geophysical Institute, University of Alaska). The internet is a wonderful thing… within a few minutes I had a printed copy of First’s paper on my desk. The title was incredible intriguing:

The Music of the Sphere: An Investigation into Asymptotic Harmonics, Brainwave Entrainment and the Earth as a Giant Bell

Within an hour I had crunched numbers in a spreadsheet; all sorts of little surprises were in store. Let’s start with the simplified equation for the expansion of a series of harmonics to predict Schumann Resonance:

Where ‘c’ is the speed of light and ‘a’ is the planetary radius. Empirical observation reveals that for typical atmospheric conditions on Earth, this value approximates to 5.5366Hz; a figure which corresponds to the zero-order spherical harmonic.

So, how well do values derived from this mathematical equation match observations? Very well indeed – take a look at table 1.

The first surprise popped out when I looked at the ratio of the first two harmonic frequencies - 7.83Hz and 13.56Hz – which gave the value 1.7318, this being the square root of 3. It transpires that every single frequency ratio (with respect to the fundamental harmonic) is an exact root of the integer progression: 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78... For example, 7.83Hz and 19.18Hz give the ratio 2.449, which is the square root of 6.

Something rather lovely happens at the eighth harmonic; the ratio of 7.83Hz and 46.98Hz is exactly 6.00 – and the square root of 36. This ratio is the spherical harmonic equivalent of the perfect fifth. In a linear harmonic series, the perfect fifth appears at the third, sixth and twelfth harmonic ‘slots’ but in spherical space it pops up at number eight as clean and clear as an Irish whistle. A cursory side-study revealed our familiar musical ratios of the linear harmonic series are embedded within the spherical series along with many other exotic ratios. The sphere seemed to be acting like the mother repository of all musical harmonics!

Another curious finding is that perfect integer ratios like 7.83Hz and 46.98Hz pop up throughout the harmonic series. There is a deep level pattern to this that I’m currently researching. Ratios divisible by 6 (the spherical equivalent of the perfect fifth) crop up with regularity in groups of three. These triple spots of ‘pure resonance’ may well be important in the configuration of stable matter. A delightful light-bulb moment happened when I decided to study what happens to the octave and fifth musical ratios in spherical harmonic space.

I had read the statement…

‘…when working with the harmonics of a sphere the rules are different. Instead of reaching an exact doubling of the original number, a slight collapse takes place and the upper number is less than an octave…’

…but what did Susan Alexjander exactly mean by this?

By fiddling with a large spreadsheet, I was able to track octave and perfect fifth ratio progressions as we know them. I find the results to be staggeringly profound, but there again I am quite a nerdbod, so let’s take our time over this juicy bone…

Spherical Musical Ratios

The octave is represented within the linear harmonic series as a set of doubled binary partials - 1:2:4:8:16:32:64 and so on. Our Pythagorean friend, the perfect fifth, is represented by the expansion of partials 3:6:12:24 and so forth. Let’s take a look at a table I baked aeons ago:

Thus, the ratio of partial 3 to partial 2 (3/2=1.50) gives us the interval of the perfect fifth; so does the ratio of partial 6 to partial 4 (6/4=1.50) and 12 to 8 (12/8=1.50). In each case, all the way up to infinity and back, the multiplication factor sits squarely at x1.5 for the interval of the perfect fifth and x2.0 for the octave.

Let’s now track this progression in spherical harmonic space, starting with the octave…

The octave series kicks off with the musical ratio 1:2 returning a value of 1.732 for the ratio in frequencies – this is the square root of 3! The next octave of 2:4 returns a ratio value of 1.8257. As we progress, we see the ratio of the octave asymptotically approaching its linear harmonic value of 2.000. After around the 1,000^th partial (representing the ratio 1024:512), spherical harmonic space becomes approximately linear.

Here is the progression for the perfect fifth:

This time the series kicks off with the interval 2:3 returning the square root of two in terms of frequency ratios! Again, after about 1,000 partials (representing the ratio 1536:1024), spherical harmonic space becomes approximately linear.

There is a profound depth to this I am only just beginning to understand. Firstly, it means that observation of high-order harmonics of any spherical resonating system will result in apparent linearity. Our ‘classical’ laws of music will apply and be seen to be truthful. This could also be why Sir Isaac Newton’s classical laws of bodily motion apply at the level of the macro but not the micro world of atomic wave-jellies.

A string stretched on a monochord produces linear, easily divisible harmonics. String dynamics in pluckable-sized space-time is at the macro level compared to the resonance of its components down at the atomic level. I would propose that a monochord the size of a decently-sized molecule – like DNA - no longer works linearly. Fingering the 12^th fret of DNA wouldn’t sound the octave!

Another way of viewing this is to imagine blowing-up a balloon. At first, we’d see and feel the curvy, rubbery roundness. When it got to 1,000 miles across after a lot of puffing, we’d not be able to discern its curvature as easily. The earth looks flat n’est pas? When we listen to a monochord, we are doing the equivalent of looking at the 1,000-mile balloon. We are listening to the macro level of the string and nicely carved wooden surround made of a zillion atoms-worth of resonant interaction.

Here is the deepest question we can perhaps ask: how does a vibrating string know where its centre is? How do individual atoms work together to determine where all the vibratory nodes are? In what manner must a string sense its length in order to produce an appropriate sound?

Sir Isaac Newton asked similar of light many years ago. He wanted to know why mirrors work, why glass partially reflected light and why different thicknesses of glass produced pretty diffraction patterns (equivalents of string nodes). Despite a couple hundred years deep searching from the finest brains, and the development of Quantum Electro Dynamics – arguably the finest piece of theoretical physics to date - we still don’t know the answer. What I do know is that I need a decent cuppa and a jammy dodger after all that thinking out loud…

Kettle On!

We’ve come a long way on this charabanc journey – six articles in total – and despite increased understanding and more careful data manipulation, we seem to have arrived more or less back where we started. But… the REALLY BIG NEWS from this final excursion is confirmation that the pipe was built on a 24Hz unit of cellular design. Evidence for this was provided in tables 3 and 4, but just to thrust the point home, here is a colourful chart showing all resonator primary frequency differences that are integer multiples of 24Hz…

Given this is the most accurate model of the pipe we can muster we may ask how its tuning compares with our modern notions of musical scale. To answer this, I have developed a cunning spreadsheet with lots of numbers. To keep things simple, I decided to calculate precise frequencies for both A=432Hz and A=440Hz reference pitches in equal temperament and Pythagorean tunings, thus giving me four sets of scale frequencies in total. This handy matrix enabled me to ascertain the closest note from each of these four musical scales as a match for the best estimates of resonator frequency. Herewith three tables revealing how close this panpipe comes to our modern musical notions:

As we can see it’s a pretty mixed bag with just over half of the resonators falling closer to A=440Hz tuning. Equal temperament seems preferred at 6 out of 9 resonators, with a slurry of notes. Forcing nearest note matches indubitably introduces frequency differences between sounded frequency and musical note frequency. Analysis of these differences is rather revealing…

The blue dotted line represents a zero-frequency difference and thus a perfect match between pipe tones and these two reference pitches. We see that the mean difference for the five resonators falling closest to A=440Hz tuning is well above the line – in addition its whiskers are well clear of touching the line. In statistical terms this means the pitch of these resonators is significantly different from A=440Hz at the 95% level of confidence. In contrast, the mean difference for the for resonators falling closest to A=432Hz tuning falls below the line, yet with whiskers touching. We thus conclude that there is no statistically significant difference. What we can do now is use those mean differences to adjust mean frequencies to determine what sort of tuning these pipes actually resemble. When we do this, we get A=430Hz or A=444Hz. Despite this suggestion I suspect this pipe is not tuned to any sort of a scale other than a pure mathematical progression. This neatly brings me to the concept of the M-24 panpipe.

The M-24 Panpipe

The hypothesis that this ancient Bolivian panpipe was designed around a 24Hz unit of design is quite a corker. So much so in fact, that I’m going to double-check this notion before moving on to reveal the theoretical pipe once more. I shall hitherto refer to my hypothesised cellular model as the ‘M-24 panpipe’. I can’t help but remind everyone of the remarkable coincidence in the numeric sum of the position value of the specially-identified triple-vented resonators…

3 + 5 + 7 + 9 = 24

First-up, we should note our investigative job is made much easier this time round because the secondary and tertiary tones are merely octave and twelfth intervals above the primaries. If there is any truth to M-24 the evidence must necessarily sit with the primary tones alone. We’ve also noted the wayward data of damaged resonator #8 and its possible impact on resonators #7 and #8. We’ve also noted that, despite belonging to the triple-vented family of resonators, tubes #7 and #9 share very similar sonic characteristics to the double-vented family. This means we can throw all but resonators #3 and #5 into the assessment pool to increase statistical power. Here is figure 5 again but with all the raw, unadjusted data we can muster…

Despite a wayward primary frequency value for damaged resonator #8 and slightly wobbly data from its connected neighbours #7 and #8, we still get a rather lovely linear series we simply cannot ignore. The regression equation for the best-fit line (depicted in blue) is:

Primary frequency = 1626.516 – 48.923 x resonator position (…Eq.2)

If we use equation 2 to calculate unbiased estimates of resonator frequencies we get the following:

Differences between measured and modelled frequencies for all resonators apart from damaged resonator #8 lie within the ±28Hz measurement error associated with transcribing data from the original sonogram (fig. 1). Interval differences are within a whisker of 48Hz and 96Hz values for a perfectly-formed M-24 pipe.

Without further ado I shall now reveal my very best guess at what a M-24 panpipe was meant to sound like. To kick this off, I’ve assumed that the primary frequency of resonator #9 – measured to be 1202Hz – was actually designed to be divisible by 24, and thus I’m pinning its theoretical value at 1200Hz once more. Theoretical primary frequencies were then derived using the following M-24 friendly equation (compare to Eq. 2):

Primary frequency = 1632 – 48 x resonator position (…Eq.3)

Secondary frequencies were derived using the following M-24 friendly equation:

Primary frequency = 3192 – 96 x resonator position (…Eq.4)

Primary and secondary values for resonators #3 and #5 could not be derived from regression equations and so the nearest multiple of 24 was substituted. Nearest multiple substitution was also used for all three tertiary frequencies. Herewith a summary of how much the raw data were swung toward a M-24 panpipe using these methods:

We know damaged resonator #8 produced wayward raw data, with resonator #9 producing a totally bizarre value for its secondary tone. We also suspect some issues with resonator #7 which was partly damaged. Aside from these expected differences there is only one sizeable difference and that is in the production of the secondary tone for resonator #4, which is 25Hz lower in the 24Hz cellular model. That being said, we must remember the transcription error rate for the sonogram is ±28 Hz. All in all, I’m rather pleased with how close the raw data sits with the M-24 panpipe model. Now we have rebuilt the pipe once again, let’s have a look at what secrets it holds…

As expected, beat frequencies now drop out in multiples of 24Hz but there are a couple of little gems in this data that are rather natty. The first to catch my eye was that beat frequencies for partial 3 (interval of the twelfth) against the tertiary tones – 96Hz, 72Hz and 48Hz – represent 4, 3 and 2 units of 24Hz respectively – a nice first touch! The cells shaded yellow align perfectly with A=432Hz tuning in that 1296Hz is E6 (representing the interval of a major 3^rd) and 1536Hz is G6 (representing the interval of the perfect 5^th). The green cell is just 3Hz in excess of F6, this being a near-perfect fourth. That being said, this is not really an A=432Hz pipe by any means. We may ask what sort of musical intervals are playable on the M-24 pipe. The answer is not many…

If we start at the lowest sounding resonator (#9), then blow others in succession, we arrive at the musical ratio sequence of figure 17. Leaping from resonator #9 to resonator #6 gives an approximate whole tone; leaping from resonator #9 to resonator #4 we get a minor 3^rd but that’s about it. The following matrix takes this to the extreme by analysing all playable combinations and comparing them to Pythagorean twelve tone tuning…

Orange cells indicate exact Pythagorean tunings, these being the Pythagorean major third (1:1.27), Pythagorean minor third (1:1.19) and unison (1:1.0). Pale yellow cells indicate approximate tunings for these plus the Pythagorean whole-tone (1:1.13) and minor second (1:1.05). Other than these the pipe is matrix of non-descript ratios. Overall, this would suggest the pipe was not designed as a musical instrument – certainly not one in terms of Pythagorean number harmonics in any case. For the overarching tonal structure, we now have two interlocking minor sixths and a pair of symmetrical perfect fifths encoded in the frequency matrix:

If I were to make a bold summary statement at the end of our journey it would be that the M-24 was never designed as a musical device. Instead, we have a rather dry mathematical instrument designed to kick out a modest range of micro-intervals whilst inducing fluttering beat frequencies of 24Hz, 48Hz, 72Hz and 96Hz. Goodness knows what for!

Kettle On!

Assessing the Triple-vented Resonators

According to my current pipe-plumbing hypothesis, the triple-vented resonators will be: #3, #5, #7 and #9 – these are also marked out with additional circles on the pipe itself, which suggests they should be considered together as a family. Let’s now take a look at the raw, unadjusted data from just these, noting that the secondary tone for resonator #9 has been argued to be much higher than it possibly should be, and noting that this resonator uniquely may have had a vent to the outside which has since blocked…

This is a most curious geometric pattern that I haven’t really appreciated before. No doubt there is a design structure underpinning it, but as it stands it is impossible to apply simple statistical modelling in order to understand any deep design issues that may be lurking. At first this would seem to present us with a brick wall but I couldn’t help noting something rather interesting that cropped up when modelling the data for the double-vented resonators, this being the simple fact that unadjusted primary frequency measurements for triple-vented resonators #7 and #9 came darn close to double-vented modelled values. Here’s a table explaining this more clearly:

We can see that the theoretical model value for resonator #7 – at 1288Hz - is only 20Hz less than that actually measured (1308Hz). The frequency difference for resonator #9 is a smidgeon out at just 10Hz. These differences are well within the ±28Hz estimates for measurement error made in report 5 and this encouraged me to assume resonators #7 and #9 were designed to mesh-in sonically with the double-vented resonator series. Here’s a figure explaining what I mean in graphical terms:

Every point on this chart is raw, unadjusted data aside from the corrected value for damaged resonator #8. The evidence for assuming resonators #7 and #9 fall within the same design parameters for the double-vent tubes is thus most compelling indeed!

At this stage in the proceedings the pipe is looking rather neat and tidy, with 7 of the resonators accounted for in terms of modelled primary and secondary tones. Herewith a splendid figure revealing the elegant but hidden acoustic design that has been unearthed so far:

We now need to consider how resonators #3 and #5 fit into this schema - very well indeed as it happens! Here is a chart revealing the modelled data for all resonators apart from #3 and #5, which present themselves in their raw, unadjusted state:

Again, this is very compelling evidence of an underlying design structure for primary and secondary tones, since the raw data values for resonators #3 and #5 align almost perfectly with modelled data for all other resonators. This also means the steady approach I am now taking in assessing this pipe is more likely to end with a truthful perspective.

With all eighteen primary and secondary tones assessed and understood in terms of a statistical model that is revealing the deeper design structure, we can now turn our attention to the three remaining tertiary tones of resonators #3, #5 and #7.

Assessing the Tertiary Tones

Although we only have three data points for this, the good news is that resonators #3 and #5 are totally undamaged and resonator #7 – though partly damaged – seems to be performing in line with expectations. Let’s start straight in with a figure revealing just how tidy the relationship between primary and tertiary tone is (Tertiary frequency = 2.928 x primary frequency (p<0.001) before we wade in with more statistical modelling:

I shall now conjure a little table to summarise just how nicely modelled and raw data fit together…

Differences of 16Hz and 19Hz are well within the accepted boundary for measurement error (±28Hz).

We now have a handle on everything aside from an assessment of the primary tones of resonators #3 and #5. As may be remembered from figure 8 these are tricky blighters to nail. Without any solid hypothesis on which to base further investigative modelling we shall have to leave them at their original measured values for now and see what the rebuilt pipe has in store.

Were Beat Frequencies Intentionally Designed?

To answer this question for beat frequencies against the octave we can either use the raw, unadjusted data and omit wayward resonators #8 and #9 or use the dataset with corrected frequencies for these resonators or use the fully modelled dataset. We can also be rather lavish and look at all approaches!

The ratio of secondary to primary tone frequencies must be precisely 2:1 to sound the perfect octave. In graphical terms this would be equivalent to the coefficient of the slope of secondary frequency plotted against primary frequency. We can then use least squares regression to obtain a best point estimate of this slope along with estimates of the 95% confidence interval.

Here’s the cunning bit… if the designers were attempting to create perfect, non-beating octaves then the 95% confidence interval for the slope would embrace a value of 2.000 i.e. we could be 95% certain that a perfect octave was attempted. On the other hand, if the 95% confidence interval for the slop did not embrace a value of 2.000 then we could say with 95% certainty that they never intended to design the pipe with perfect octaves in mind in the first instance.

The delicious thing about applying statistical theory at this point is that we can quantify and account for errors in manufacture undertaken aeons ago in order to see what they were up to.

In each case we can see the coefficient for the slope never embraces a value of 2.000, which means the pipe was never designed to create perfect octaves… those fluttering beat frequencies witnessed were most certainly intentional!

The very small standard errors involved mean there is little random scatter in the data, which means we have a very, very high probability of not getting this assumption wrong. Here’s the plot for the first result listed on the table above using the raw data as measured but excluding resonators #8 and #9:

Now that we’ve exhausted all modelling possibilities (at least for the moment), let’s go and rebuild the pipe, removing errors caused by damage, to see what sort of beat frequencies they were intent on creating…

Kettle On!

A Helping Of Acoustic Theory

Acoustic theory tells us that closed pipes produce odd harmonics (n=1, 3, 5, 7…) and are thus incapable of producing the octave when overblown. However, when overblown they will produce what is sometimes termed the interval of the twelfth – this being the perfect fifth above the octave.

For a while, I scratched my head as to how octaves can be produced from closed pipes, when it dawned on me that this was probably the precise function of those side holes mentioned by Putney & Benoit1. Here are their relevant words:

These strangely designed stone panpipes have resonators formed by interconnecting holes unlike the typical Andean open-ended panpipes used today. In fact, all five of these black stone panpipes have been fashioned with small holes connecting adjacent resonators at the bottom of their 2” drilled depth, allowing air blown into one pipe to exit through each of the two adjacent pipes.

At this point my rusty brain dimly remembered something called the Dolmetsch Recorder Modification – this is the small thumb hole on the back of every school recorder in the Kingdom that, when opened, enables the instrument to play in its second octave and thus children to terrify us with squeaky renditions!

Here’s what Wiki has to say…

Opening a small “speaker hole” at the Phi point, or shared “wave/node” position will cancel the fundamental frequency and force the tube to resonate at a 12th above the fundamental. This technique is used in a Recorder by pinching open the dorsal thumb hole. Moving this small hole upwards, closer to the voicing will make it an “Echo Hole” (Dolmetsch Recorder Modification) that will give a precise half note above the fundamental when opened. Note: Slight size or diameter adjustment is needed to zero in on the precise half note frequency.

Let’s go back to those 2” tubes of the panpipe for the moment and take a look at what frequencies they should theoretically produce. A closed tube will have resonances at:

Where:

f = Frequency (Hz)

L = Length of tube (m)

d = Diameter of tube (m)

v = Speed of sound in air (m s^-1)

n = Harmonic integer value (n=1, 3, 5, 7…)

If we assume the following:

L = 5.08 x 10^-2 m (2”)

d = see table 2

v = 343.2 m s^-1

n = 1 (for primary frequency)

Then these frequencies emerge for different tube diameters:

The cells shaded lemon represent the range of frequencies embraced by the pipe’s design - but there is a problem here! The lowest measured frequency for an undamaged resonator (#6) was 1332Hz – this would require a tube 34mm in diameter. This is clearly ludicrous and points to two rather important but hitherto ignored possibilities:

1. The tubes cannot have all been drilled to 2” as claimed by Putney & Benoit.

2. Adjacent tubes are working together to create primary frequencies and not just permit harmonics.

Taking the first assumption, we may ask what length of tube would be needed to produce a primary tone of 1332Hz on, say, a comfy-to-blow 12mm drilled aperture. It turns out we only need to adjust the length from 2” to 2.35” to accommodate this frequency i.e. make it around 9mm deeper. This doesn’t seem to be an awful lot of difference, and I thus wonder if Putney & Benoit were a little too rough and ready in their approximation of 2” for all tubes.

My brain does not wish to consider the second assumption at this point, since matters are already complicated enough!

What damage, where?

Let’s now pull up an over-size photo and have a very close look at where the damage appears to lie…

Surprisingly, resonator #9 looked totally unscathed despite its wayward sonic characteristics. Most damage appeared to be associated with resonator #8, with some damage appearing at resonator #7 – a paradoxical finding given the reasonable sonic performance of both of these. What makes guessing extra tricky is that we have no idea of what is happening down in those small interconnecting holes - but I did wonder if I’d overlooked something else in the acoustic design – something that also made resonator #9 stand out from the crowd. Here’s another revealing snippet from Putney & Benoit’s report…

The row of geometric patterning just below the resonators visually describes their linked airflows, yet also emphasizes the special harmonic output of 3rd , 5th and 7th tones.

Sounds straightforward enough, but when it comes to deciphering the precise symbology the task isn’t that easy! I’ve had a go and have come up with this…

…here we see that side holes join adjacent resonators together as Putney & Benoit seem to describe but my motif-inspired guess is that the odd numbered resonators are also linked to each other via a deeper set of vent holes. Close inspection of the motif under resonator #9 also suggests this may have been vented to the outside. Blockage of this may well be responsible for the wayward behaviour of resonator #9 despite the apparent lack of damage (at least that’s my theory for the moment).

Resonators #3, #5, #7 and #9 each thus receive a pair of deep vent holes which would seem to correspond to the two dots in the circle under each of these. This approach also means that resonators #2, #4, #6 and #8 will only possess two side hole vents – which again seems to find correspondence in the two-dot motif under these.

If we blow across resonator #1 we’ll get both the primary and octave sounding in this tube because of the vent hole. Air escapes sideways into resonator #2 and downwards into the base of resonator #3, one or both of which I then assume is capable of sounding a tone. This basal air feed is reminiscent of an organ flue pipe but without the mouth and lip sounding devices.

Sadly, my knowledge of instrument design and acoustics isn’t sufficiently advanced to determine whether all this is a sensible stab at guessing the inner workings of the panpipe. However, it has given me a valuable clue that resonator #9 may have a blocked vent to the outside affecting performance; it would also suggest that resonators #7, #8 and #9 might not be performing as intended, despite sonic appearances – which is my main reason for attempting to understand the hidden connectivity in the first instance. Suitably forewarned I shall now analyse the pipe for a third time!

Third Time Lucky

Regardless of the validity of my idea of pipe inner airflow (as outlined in the previous section), one thing is most clear: tones and their harmonics are created by air flows in multiple resonators. In this respect, the frequencies depicted in the sonogram of figure 1 are the result of interacting resonators. Thus, the lowest frequency of resonator #1 (what I have hitherto termed the primary frequency) and the highest frequency of resonator #1 (what I have termed the secondary frequency) are not necessarily sounds that arise within the same tube, nor are they necessarily produced by resonator #1 even! They are simply the sounds recorded from the entire pipe matrix when resonator #1 is blown. This is quite a head-mangling revelation.

In terms of analytical process, it means my two previous jolly outings – which have essentially assumed sound production within a single resonator – may have taken us up a leafy country lane that looks pretty but is not necessarily the destination as displayed on the header board. I shall now refund everyone’s ticket money and start the motor on our charabanc running once again with the single most important figure I can think of – the harmonic relationship between all low and middle tone frequencies for all resonators (figure 4). For convenience I shall continue to call the low, mid and high frequencies ‘primary’, ‘secondary’ and ‘tertiary’ but please remember these are not solely within-tube harmonics as we know them Jim – they are multi-tube harmonics.

It is abundantly clear - in this most vitally vital of figures - that seven red blobs line up in a near perfect alignment. Our assumption must be that these resonators are working together as the designer intended. Two red blobs stand out from the crowd over on the far left and these happen to coincide with resonators #8 (which has sustained the most visible damage) and resonator #9 (which I now suspect to have a blocked outlet vent). The output generated by blowing resonator #9 would also be dependent on correct functioning of resonator #8 and vice versa.

It is perhaps most surprising at first glance to see the performance of resonator #7 unaffected by damage to resonator #8. We must, however, note that resonator #7 is also dependent on the correct functioning of resonator #6, which is undamaged; the correct functioning of resonator #5, which is undamaged and the correct functioning of resonator #9, for which the main outflow pipe is also undamaged - at least according to my notion of how these tubes are interconnected. In fact, I’m going to take well-behaving resonator #7 as evidence supporting my deep plumbing ideas!

Resonator #8 comes a cropper primarily because its overall length is compromised (please refer to equation 1). A marginally shorter tube will result in a higher primary frequency and thus any octave produced by side hole positioning will be in error. This is precisely what we see in figure 4 – drop resonator #8 down a little in terms of secondary harmonics and it will fall neatly into line.

Resonator #9 remains a curiosity. The tube does not show any visible sign of damage but it is adjacent – and thus connected - to compromised resonator #8. It may well also possess a unique plumbing system, possibly being the only resonator to possess a side hole vent to the outside – a feature which may have enabled fingering by the player. As already discussed, the Laws of acoustic resonance tell us that closed tubes can produce the interval of the twelfth when overblown or the octave with a suitably-placed vent hole. The problem – it would seem - with this resonator is measured data do not comply with these laws, the primary tone being 1202Hz and the secondary tone being 2679Hz instead of an expected 2404Hz for the octave or 3606Hz for the twelfth.

At this point we need to consider multi-tube output: the primary and secondary tone outputs from blowing resonator #9 is the result of more than one tube! Thus, if resonator #8 is compromised, then resonator #9 will be feeding air into this via the side hole vent to sound a compromised octave – this is probably what we are seeing with the measured frequency of 2679Hz instead of 2404Hz. This would tend to suggest that the primary frequency output of resonator #9, at 1202Hz, is probably more or less what the maker intended – an assumption supported by it’s apparently undamaged tube.

At this point we’ll stop for sandwiches and have a quick recap:

1. Resonator #8 is visibly damaged and is probably sounding erroneous primary and secondary tones.

2. Resonator #9 is visibly intact and is probably sounding a correct primary tone but incorrect secondary tone owing to its side hole venting to resonator #8 (and possibly blocked third vent to the outside).

3. Resonator #7, despite being connected to damaged resonator #8, is probably sounding both primary and secondary tones more or less as the maker intended because of its connections to other undamaged resonators.

Now that we’ve got a better idea of which data to trust and why, we can start re-building the pipe again.

Assessing the Double-vented Resonators

According to my current pipe-plumbing hypothesis, the double-vented resonators will be: #1, #2, #4, #6 and #8. Let’s take a look at the raw, unadjusted data from just these, noting that both tones for resonator #8 may well be compromised…

The linear sequence with respect to position along the panpipe is quite striking and is undoubtedly revealing an obvious design element. A best-fit least squares regression line is presented in blue, this being derived from data for resonators #1 - #6 only. The first point of curiosity is that the measured primary frequency for resonator #8 is lower than expected. I shall attribute this contrary finding to hidden problems with the plumbing! When we place these data values in a table something else rather surprising happens…

We can see that the model is a darn good fit to the data arising from the first four resonators, being only 6Hz out at worst. We can also see that resonator #8 is sounding 46Hz lower than expected. The rather more interesting finding, however, is that the modelled frequency intervals are precise multiplies of 24Hz. This is good evidence to suggest my crazy notion of a 24Hz cellular design grid for the whole pipe may hold after all. Despite this excitement we shall proceed with caution! Let’s now take a look at the secondary frequencies and stick a regression line through these…

Resonator #8 is now behaving a touch better, with a secondary tone a little higher than expected; table 3 reveals this to be 33Hz higher than expected. An erroneously high secondary tone for this resonator is totally in-keeping with earlier assumptions. The column labelled ‘interval’ yields more evidence for a 24Hz unit of design, given that 96Hz and 192Hz are the nearest multiples of 24. I must say I find this mounting evidence for a 24Hz design quite compelling!

To round this section off I shall graph the modelled frequencies for both primary and secondary tones for all six double-vented resonators in a figure similar to figure 4 - herewith the pipe smartened up a little…

Kettle On!

We’ve already had a go at fixing damaged resonator #8 and sorting out its wayward neighbour resonator #9, but there may well be others slightly out of kilter, especially resonator #7 which is connected to #8. Looking back at figures 5 & 6 it would seem that resonator #7 is indeed sitting just a little high compared to the rest; assuming, of course, that it is part of the same harmonic family. If we assume that it is we can fix this in a jiffy - and here is the result…

…as square as square can be.

This leaves just two resonators to consider: #3 and #5. So far, everything has lined-up nicely with just a small adjustment to the observed sonogram frequency. We could stretch this assumption further and assume the same linearity should apply to resonators #3 and #5. This being so, we can churn-out a simple multivariate model that looks like this…

This model reveals that the primary frequency for resonators #3 and #5 sits 118Hz lower than the rest with a 95% confidence interval of [104Hz; 132Hz] and p-value of p<0.001 for this regression coefficient. We’ve now all the information needed to revamp the entire set of 21 frequencies and hopefully reduce measurement error. Here they all are, along with revised beat estimates:

This is quite curious! There seems to be a set of beat frequencies down in the 50Hz range and another at roughly twice this frequency but nothing at all to suggest alpha-wave entrainment in the 4Hz – 10Hz region. I couldn’t help but think I was over-looking something quite profound, yet all I could do at this stage was scratch my head.

Pipe Dreams

At just after 4am after a long day of yet more pipe analysis I woke with a strong impression of the word ‘SOMETHING’ being written in capital letters on a descending diagonal sequence that resembled organic cells. I jotted this down in my journal and promptly forgot about it since it didn’t make any sense. At 3:12am on the following day I was again awoken from a pipe dream. This time figure 8 from part 2 came into my head and rearranged itself within my mind’s eye. There and then I knew why the interval of the perfect fifth was special, why the pipe had nine holes, why the pattern of frequencies had been so chosen and how these relate to our base10 counting system. Only then did I realise that the word ‘something’ contained nine letters; I thus wondered what information those dream-intuited ‘cells’ were meant to convey.

Let us consider the musical ratio of the perfect fifth - 2:3 – for a moment…

Back in figure 8 we saw this ratio depicted four times in a symmetrical mirror image arrangement that overlapped. I called this fractal since the frequencies of the six resonators represented this ratio flipped-over and embedded within itself. In my dreamy mind’s eye I saw these pieces flying about until I understood how they fitted together. There is only one possible mathematical solution for mirror image fractal 2:3 symmetry and this requires a base 10 grid. Here’s a diagram explaining this a great deal better than a zillion words…

Here we see a row of red blobs symbolically representing the spread of frequencies observed in figure 8. Beneath these are the four ratios of 2:3 sitting together in the only possible spatial solution. Below this is the simplest numeric grid we can muster to permit this exact arrangement in terms of integers. Just as soon as this realisation struck home another came along, reminding me that the secondary tones of the pipe were twice the frequency of the first, whereas the tertiary tones were thrice the frequency – the ratio 2:3 again!

Figure 7 had kicked this all off, so I decided to reproduce this plot with the modelled frequencies…

We see a splendid progression with a slope coefficient pretty darn close to 2.0, indicating the ratio of the perfect musical octave. There does indeed seem to be a pattern with resonators #3, #5 and #7, with spacing between #8 and #5, #5 and #7 and #6 and #3 suggesting a regular sequence. Furthermore, this spacing sequence seemed to be half that exhibited by resonators #9 and #8, and their mirror counterpart #2 and #1. A small table should clinch it…

We are beginning to see multiples of 24Hz or as close as dammit!

Are 24Hz cycles an ancient psychoacoustic equivalent of Thom’s megalithic yard1?

I decided to put this hypothesis to the test by calculating all frequency differences in descending order just to see how close they came to multiples of 24Hz. The answer is… seriously close!

This is quite incredible.

It would appear the whole primary tone series from an estimated 1198Hz to 1576Hz has been encoded in intervals of 24Hz, there being 16 such units of 24Hz, making a tonal span of 384Hz – at least in theory! At 378Hz our estimation of the total primary tone span is only 6Hz out, which represents a measurement error of just 1.6% This also means the secondary tone series, at double the primary frequency, is encoded in 48Hz units, with the tertiary tone series at triple the primary frequency encoding 72Hz intervals. This is simply stunning and means we can now complete a hypothesised tonality schema that underlies the basic design of the pipe…

I say basic because this doesn’t take account of the progressive and careful de-tuning of both secondary and tertiary tones to produce the desired range of (as yet unfathomable) beats – the pipe is an unbelievable masterpiece of acoustic engineering! But here, in succinct graphical form, surely is the meaning of the cell symbology that initially came within my intuitive dream.

Curious Lump

A most curious lump of evidence for my 24Hz ‘acoustic cell’ hypothesis came when I took another look at the photo of the pipe and realised Putney & Benoit had not been accurate enough in their assessment of the artwork. They claim the importance of the three triple-tone resonators is keenly brought to our attention by three distinct circles in the engraving, which is re-drawn by them as below…

…but take a good close look at the photo of the pipe in part 1 and you’ll see a fourth circle directly under resonator #9.

Let’s try adding the numerical position values of all four circled resonators…

3 + 5 + 7 + 9 = 24

Coincidence?

My thoughts now turned to reconciliation of figures 13 and 15, since we have a 16 segment grid revealing the 24Hz design unit and a 9 segment grid underpinning interlocking 2:3 ratios. The simplest grid that would simultaneously accommodate both of these design elements is thus 9 x 16 = 144. Now there’s a Biblical number and a half!

In Revelation 14, verse 1 we read:

1. Then I looked, and there before me was the Lamb, standing on Mount Zion, and with him 144,000 who had his name and his Father’s name written on their foreheads.

3. And they sang a new song before the throne and before the four living creatures and the elders. No one could learn the song except the 144,000 who had been redeemed from the earth.

Ernest G McClain, the late Professor Emeritus of music at Brooklyn College, New York has a great deal to say about the musicality of the number 144. I would urge keen readers to grapple with his thought-provoking essays on decoding hidden musical scale references within the Bible, Rg Vega, Qu’ran and the Egyptian Book of the Dead. A good starting point are these webpages dedicated to his life and works, but this modest paper alone is capable of blowing your mind in just 12 pages.

A Theoretical Pipe

By now it should be clear that this ancient artefact was made by somebody with incredible technological skill, together with knowledge of mathematics and psychoacoustics. This is not the tool of a hairy-arsed human trying to fathom bronze smelting or flint-knapping. We can pretty firmly state that the pipe’s primary frequency range has been encoded in discrete 24Hz cells; and looking at resonator #9 again I would also suggest its observed primary frequency of 1198Hz was probably meant to be exactly divisible by 24 and thus possibly was meant to be 1200Hz through intended design. It is thus an easy task to reconstruct the entire set of idealised primary frequencies used by a forgotten civilisation.

Secondary frequencies are not so easy to jot down afresh. These cannot be exact doubles of the primary otherwise those fabulous beats would not occur. To understand the possible intended design element behind these I re-ran the regression first presented in figure 3, but with the modelled pipe data from table 4. This provided a coefficient of 1.967 and thus a means to calculate the entire set in a jiffy. Repeating for the tertiary frequencies produced our old friendly coefficient of 1.495. Using these, plus our understanding of the cellular nature of the pipe, we can reconstruct a theoretical version. Here it is…

Either this is all an incredibly large bottle of purest distilled nonsense with ice and lemon or something mind-boggling about this ancient artefact is being unveiled before our very eyes…

Kettle On!

I promised an X-Y plot and here it is:

…the primary frequencies of resonators #3 and #5 sure stand out as being much lower than their neighbours but their triple tone brother - Resonator #7 – seems to hover around the rest of the series. Damaged resonator #8 seems to sit a little low in the series when compared to #1, #2, #4 and #6, whilst resonator #9 looks reasonably placed in terms of primary frequency.

Here is the same graph for all nine secondary frequencies…

Again we see resonators #3 and #5 standing out with much lower frequencies, with resonator #7 lining-up with the rest. This time damaged resonator #8 seems to sit reasonably well with the rest but resonator #9 has gone wildly high – and especially so given it is the largest of the hollowed tubes! My first thought was to attempt to ‘mend’ resonators #8 and #9.

Time for some assumptions…

• Resonators #1, #2, #4, #6, #8 and #9 form a tone series that is progressive, linear and descending.

• The triple-tone resonators #3, #5 and #7 form a separate tone series that could be complex.

• All resonators have been crafted to reproduce the octave with a high degree of accuracy.

• The triple-tone resonators have been crafted to reproduce an additional perfect fifth above the octave.

• Damage to resonators #8 and #9 has affected tone production.

…with these in mind I used simple statistical modelling techniques to reconstruct what an undamaged panpipe may possibly have sounded-like. Here are the results for the ‘mended’ double-tone series of resonators (#1, #2, #4, #6, #8 and #9)…

Primary frequency symmetry

A hidden symmetric pattern now seems to emerge. Compare the graphical spacing between resonators #1 and #2 and #9 and #8; then between #2 and #4 and #6 and #8 in the above figure. This is quite exciting!

Analysis of the proportions of this spacing yielded even more interesting results:

Four ratios that may be observed within the frequency spacing pattern, namely 0.676, 0.661, 0.638 and 0.649 lie remarkably close to the idealised ratio of 0.667 for the musical interval of the perfect fifth; the average ratio for these four being 0.656. On top of this, the primary tonal range of the instrument from 1202Hz to 1572Hz represents the ratio of 0.764, which is intriguingly close to the idealised ratio of 0.750 for the musical interval of the perfect fourth. It is perhaps worth noting here that the centrally placed artwork on the pipe depicts three heads, each with two bejewelled eyes – a reminder of the 3:2 ratio of the perfect fifth no less!

Given primary frequencies for resonators #8 and #9 had to be modelled because of damage, one cannot ignore the possibility that these primary frequency clusters were possibly designed around symmetries for the perfect fifth, perhaps enclosing a tonal range of the perfect fourth. This also suggests modelling to mend the pipe isn’t perhaps far off what was originally intended.

If this is indeed the case, then we are also looking at a fractal musical device: symmetrical nested perfect fifths that interlock and sit within the capsule of a perfect fourth.

Time for a colourful diagram to explain this graphically...

The three primary tones of the three triple-tone resonators #3, #4 and #7 did not produce such an interesting result aside from coming close to representing the intervals of the Pythagorean diatonic semitone (theoretical musical ratio = 0.950; observed ratio = 0.951) and Just Intonation semitone (theoretical musical ratio = 0.960; observed ratio = 0.961).

Musical Pitch

Imagine blowing each resonator starting with #1 and working your way down to #9. This creates a sequence of eight musical intervals. The musical ratios for these are plotted below. A ratio large than 1.000 denotes a lowering of pitch and a ratio less than 1.000 denotes a raising of pitch. I’ve nominally set the ratio at resonator #1 to unity to get the sequence going. In figure 9 we can see the pitch ratio bounces up and down all over the place!

I’ve placed two dotted lines at the musical ratios of 1.125 and 0.949 which represent the Pythagorean 9:8 whole-tone and Pythagorean 243:256 whole-tone respectively. It is interesting to note that the entire primary frequency series bobs about between these two ancient and venerable tones.

Despite these connections with musical ratios I would tentatively suggest at this stage that this panpipe is not an early attempt at a musical scale.

Beat frequencies

According to Putney, Benoit and others beat frequencies are what these psychoacoustic pipes are all about. Conventional academic research suggests our auditory system calculates simple difference frequencies, such that tones of 400Hz and 410Hz - when played simultaneously - produce a perceived beat frequency of 10Hz. This is all very well for low frequency tones like 400Hz but what about those up at 2,000Hz and beyond1? How can a primary frequency of 1,572Hz ‘beat’ with a secondary frequency of 3,093Hz, as is the case for resonator #1?

The answer lies in the production of partials within each resonator. All resonating structures, whether strings, plates or air columns vibrate to produce a set of harmonics, the sum total of which produces the characteristic timbre of the instrument. Thus, the vibrating air column producing the primary frequency of 1,572Hz for resonator #1 also produces an audible harmonic2 at twice the frequency (3,144Hz). This second partial thus enables us to perceive flutter-like beat frequencies against the secondary resonator frequency of 3,093Hz; the difference between these being the beat frequency of 51Hz.

Each resonator will also produce a third partial, which produces the exact interval of the perfect fifth. In resonators #3, #5 and #7 this audible harmonic can beat against the tertiary tones to produce a second set of flutter-like beat frequencies.

Let’s have a look at a table of beat frequencies shall we?

What a big beat surprise!

Of twelve possible beat frequencies not a single one comes anywhere close to the psychoacoustic frequency required for brain-wave entrainment at the alpha level (~13Hz).

Bearing in mind we had to model frequencies for resonators #8 and #9, we witness a remarkable consistency in beat frequencies for the primary octave against the secondary tone – especially so for resonators #1 to #7, whose mean beat frequency is 50.9Hz, with a standard error of just ±2.0Hz. The mean beat frequency for all nine resonators is 48.0Hz with a standard error of ±2.5Hz, the 95% confidence interval being [42.3Hz; 53.6Hz]. Note also that secondary tones are lower than the primary octave in every single case.

An X-Y plot of beat frequencies against the primary frequency reveals another point of interest…

Here we see double-tone resonators depicted in red and triple-tone resonators in blue. We can see that beat frequencies for the triple-tone resonators tend to be higher than those for double-tone resonators, though this difference is not strictly statistically significant (ANOVA3; p=0.075).

Whichever way we look at beat frequencies, there’s no way the pipe seems to be primarily designed for producing beats down in the sub-10Hz region. Random errors in measurement could certainly halve estimates for beats but we’d still be looking at a pipe producing beats in the 20Hz – 25Hz region. Statistical modelling would seem to preclude even this possibility, however, since it has revealed systematic components within the data which suggest intentional design.

Putney and Benoit, who have experienced the psychoacoustic properties of the pipe firsthand state:

This class of instruments produces two slightly offset tones that interact to produce a third tone, like the flutter of wings, perceptible only to the listener.

At what frequency do bird wings flutter?

In an analysis of 32 morphologically diverse bird species representing 18 families, Pennycuick (1990) observed mean wing beat frequencies of between 2.24Hz and 9.19Hz4. These certainly lie within the range for alpha-wave entrainment but there is one very special bird depicted no less than three times on the pipe – the sacred hummingbird - whose wings beat at between 12Hz and 80Hz depending on the species. The arithmetic mean for this range of beat frequencies is 46Hz - wouldn’t a pipe producing beats of 50Hz or thereabouts offer a better approximation to the wing dynamics of this sacred being?

Taken together, these findings suggest intentional design for beats but not necessarily for alpha wave entrainment and other ultra-low frequency psychoacoustic effects as suggested by Putney, Benoit and others.

To be continued…

Kettle On!

Each curandero/curandera has their own personal and private set of icaros – maybe as many as 40 to 60 distinct melodies – that come to them over the years through non-normal experiences; and especially during Ayahuasca and other psychedelic ceremonies. The majority of icaros are whistled. In addition to whistling and singing, the curandero/curandera will use a chakapa leaf rattle, which produces a unique rustling, swishing sound. Chakapa sessions always finish with a sharp whistle-like ‘shush’ sound to send energies off to the tree canopy. A different form of recycling, perhaps, and one that is older than the wheelie bin.

SHUSH!

Whistling and loud ‘shushing’ sounds like these have intrigued me ever since I stumbled on Peter Kingsley’s commentary on Parmenides1 epic poem On Nature back in 2016…

There’s one simple detail in Parmenides’ account of his journey to the underworld that’s so easy to miss. During the whole of his journey there’s no mention at all of any noise — apart from one single sound. That’s the sound the chariot makes as the daughters of the Sun draw him along: ‘the sound of a pipe’. . .

After Parmenides mentions the sound of the pipe he uses the same word again to explain how the huge doors spin open, rotating in hollow tubes or ‘pipes’. This use of the word is extraordinary. It’s the only time in the whole Greek language that it’s ever applied to doors or parts of doors, and scholars have pointed out that Parmenides must have chosen it for a particular reason: not simply to describe what the doors look like but also to give a sense of the sound they make. On his journey everything that moves has to do with the sound or the appearance of pipes.

The word for ‘pipe’ that Parmenides keeps using is syrinx. It had a very particular spread of meanings. Syrinx was the name either for a musical instrument or for the part of an instrument that makes a piping, whistling sound — the sound called syrigmos. But there’s one aspect of these words that you have to bear in mind: for Greeks this sound of piping and whistling was also the sound of the hissing made by snakes. It would be so simple to dismiss as totally insignificant the fact that this piping, whistling, hissing noise is the only sound Parmenides associates with his journey to another world — except for one small matter. Ancient Greek accounts of incubation repeatedly mention certain signs that mark the point of entry into another world: into another state of awareness that’s neither waking nor sleep. One of the signs is that you become aware of a rapid spinning movement.

Another is that you hear the powerful vibration produced by a piping, whistling, hissing sound. In India exactly the same signs are described as the prelude to entering samādhi, the state beyond sleep and waking. And they’re directly related to the process known as the awakening of kundalinī — of the ‘serpent power’ that’s the basic energy in all creation but that’s almost completely asleep in human beings. When it starts waking up it makes a hissing sound. The parallels between standard Indian accounts of the process and Parmenides’ account of his journey are obvious enough; specialists in Indian traditions have written about them and discussed them. But what hasn’t been noticed is that the particular sound mentioned by Parmenides also happens to be the sound made by a hissing snake. . .

The sound of a syrinx was a call for silence. This is something that makes sense even on a very obvious level when you consider how hissing or whistling at people is still a way of silencing them. To ancient mystics and magicians the journey into a greater reality was a journey made through silence, in silence and into silence. The noise of a syrinx is the ultimate password. It’s the sound of silence.

Prior to my introduction to Peter Kingsley’s work I’d been coming across the importance of the high frequency-generating SH* syllable in certain words that shared the same Proto-Indo-European (PIE) language root – words like Shah, Czar, Tsar, Caesar, Kaiser, Shaman, Sire, Sir, and Sahib. Even the very Biblical and very lovely name Sarah means princess! To this list we perhaps should add Séa! (pronounced shar) - the ancient Celtic battle cry of empowerment.

Why do we let rip with an attention-grabbing ‘Shhhhh!!!’ to alert everyone to required silence? Why does it have the dramatic effect that it does? Why are we imitating the sound of the syrinx in calling for silence? And why are the important and powerful bods among us, in cultures around the globe, linked with the PIE root of SH*/ZH*?

Whistling Vessels

We shall now hang a left and consider what Alexander Putney & Suzanne Benoit have to say in appendix I (psychoacoustics & Earth Resonance) of their curious and rather glossy 2013 publication entitled Sanskrit…

The Inca, Chimu, Lembayaque and virtually all other Andean cultures crafted a multitude of ceramic whistling vessels whose simple design belies a penetrating sophistication. The vessels were blown into through a tube, and half-filled with water to regulate air pressure into a second chamber with whistles poised at exit vents, producing paired tones that engage human consciousness in astounding ways. These instruments produce a sensation of internal synchronization often described as an inner wind or resonance, having no reality independent of the perceiver. The enigmatic effect the vessels produce has been called a ‘phantom tone’ created by the hearing apparatus itself, having an ethereal existence within consciousness, as observed by German experimenter H V Dove in 1839. The tendency of the human mind to entrainment was first described as electroencephalographic (EEG) frequency-following response. The whistling vessels are described as ‘psychoacoustic’ for the resulting entrainment of human brainwave patterns into an alpha frequency range of 13-8 cycles per second (Hz) and below.

The whistling vessels are described as ‘psychoacoustic’ for the resulting entrainment of human brainwave patterns into an alpha frequency range of 13-8 cycles per second (Hz) and below. This class of instruments produces two slightly offset tones that interact to produce a third tone, like the flutter of wings, perceptible only to the listener. This elusive third tone is referred to as a difference tone, or beat frequency, as it occurs when the two offset frequencies phase in and out of synchrony. The effect is most powerfully experienced with each ear receiving separate distinct tones; a binaural beat is perceived that originates in the brainstem’s superior olivary nucleus, the site of contralateral integration of auditory input. Psychoacoustic whistles create an alpha wave environment that immediately brings the frequency of the human mind into its most propitious state, while the whistler themselves receives the strongest binaural experience. Although many double-tone combinations do produce beat frequencies in the human brain, precision tuning can be achieved using harmonic frequencies, so the audible tones are multiples of the desired beat frequency (e.g. 7 Hz beat = 77 Hz and 84 Hz audible). Induced low-level beta states have been demonstrated to improve human memory.

The rediscovery of the physiological effects of this ancient class of instruments occurred through the research of S Garrett and D Statnekov. The researchers studied the frequency variation in 73 whistling ceramics from 9 distinct Andean cultures, spanning some 2,000 years. The tuning of instruments was determined to be Culture-specific, as all of the instruments of anyone culture were tuned together into precise frequency ranges. A general trend observed was a steady increase in the chosen frequency over time, the earlier Vicus, Moche and Huari near 1320 Hz, the Recuay near 2000 Hz and the later Chancay, Chimu and Inca bottles near 2670 Hz. The fact that all cultures of the region employed this instrument, each with its own specific frequency range, attests in itself to the significance of the group frequency identity. Further experiments in playing several instruments from one culture showed an increased effectiveness, often inducing extraordinary personal experiences and group reflections. The prime number of instruments in any group is said to be 7, sacred to Andean belief. The achievement of unified group consciousness may also be reflected in deeper ways in the societies who shared in their use. The collective result of this kind of societal tuning is that one brainwave pattern is simultaneously shared by all. Ancient neural synchrony technologies are indeed the prime key to enhancing human consciousness, opening another dimension of awareness and communication.

Often used in conjunction with psychotropic plants, breathing and fasting methods, the central role of consciousness synchronization in the spiritual life of ancient Andean cultures is made remarkably clear by the staggering numbers of instruments produced and entombed with the mummified practitioners. This profound ancient psychoacoustic technology remains largely forgotten, as deep a secret as the one emanating from within every living cell. The experience of inner resonance induced by the difference tones is felt by many to be profoundly transformative in nature, in that it increases the coherence of neural function by synchronizing the left and right hemispheres of the brain. The full potential of this unified state of modified neural functioning is not known, but has been shown to allow increased visual and creative ability. Andean textile and masonry patterns present diverse variations of nested semicircles that are quite clearly seen as symbols for the hemispheric synchronization of the brain. The acoustic wave environment has also been shown to effect neurochemical balances, just as techniques using wave induction derived by Patterson and Capel, reported in 1980.

Recent findings on the Bolivian altiplano have included a more ancient form of psychoacoustic instrument, rendered in stone. A group of several intricately designed and engraved black stone relics was collected from an area of brushy highland, including a smoking pipe, two ceremonial cups, five psychoacoustic panpipes and five inlaid masks of which one is also a whistle. The set of unique artefacts appears to be carved from a finely grained type of black stone that eludes scientific identification. The stone masks are thin, yet still quite heavy, the eyes inlaid with turquoise irises. Elaborate psychoacoustic symbols lace the black stones’ polished surfaces, including zigzag, diamond, and square waveforms. Two of the masks also have three dots of the same inlaid turquoise extending below each eye, perhaps symbolic of the penetrating gaze of a supernatural being, distinguishing them from the other masks of the group. This pair of masks seems to represent oculate and human beings in opposition to each other. One appears to be an oculate being with characteristic triangular head, bearing its teeth as snake-like patterns exude from its eyes. By sharp contrast, the other being appears very human-like with a long face and mouth open in chant, and a whistle at the crown. Symbolic psychoacoustic animals grace the whistling mask, including the hummingbird and the cat, as well as a pair of spirals engraved near the ears of the mask’s wearer. The head of a llama forms a comfortable handle at the chin.

These five black stone panpipes are perhaps the most sophisticated psychoacoustic instruments yet discovered, and despite having sustained some minor damage they can still be played to profound effect. These strangely designed stone panpipes have resonators formed by interconnecting holes unlike the typical Andean open-ended panpipes used today. In fact, all five of these black stone panpipes have been fashioned with small holes connecting adjacent resonators at the bottom of their 2” drilled depth, allowing air blown into one pipe to exit through each of the two adjacent pipes. Interestingly, the panpipes’ tonal progressions are not ascending but staggered such that each tone is adjacent to two higher tones or two lower tones that receive airflow from the central tone being played. The curious result of the tunings of the connected air passages is the simultaneous production of three harmonic tones that entrain consciousness.

Wonky Boat

I don’t know about you but this sort of left-field wacko thing floats my wonky boat; though I always take a platter of freshly sliced skepticism and a hip flask with me just in case. I’ll be writing up my own modest research into those Peruvian whistling bottles in a future article but right now I want to focus on that utterly fascinating 9-tone oculate panpipe.

Here it is…

Ain’t that wonderful?

I simply had to have some decent raw data; and I obtained this in a most cunning manner… Putney and Benoit had provided a wonderfully clear sonogram of this instrument that I pasted into a vector graphics package in order to add precisely positioned overlays that were used to provide object coordinates that were transformed into kHz at the press of a button.

Here’s the sonogram in question…

…and now for some results.

Results & Analysis

The frequencies displayed in the sonogram above are given in table 1 below. Measurement error associated with the determination of a single frequency using graphical means was estimated to be ±28Hz. This, of course, excludes any measurement error inherent in the production of the sonogram itself.

Two resonators were said to be damaged, which may thus provide inaccurate frequency readings for the sonogram. From the photograph we can clearly see one of these is resonator #8, which would suggest resonators #7 or #9 may also have suffered impairment – something to bear in mind for later.

I decided to run an initial check for damage using a simple X-Y plot of secondary against primary tones, which would reveal any underlying discrepancies in the dynamics of tone production. The plot is a corker of clarity! Here it is…

A most remarkable linear relationship between primary and secondary tones is evident for the first 7 resonators; something the maker must have intended. We see the damaged resonator #8 sitting a little awkwardly in the left hand corner, along with wayward resonator #9 – which could suggest this was also damaged.

Leaving resonators #8 and #9 out of the picture and running a cheeky little linear regression, we get the following trend line and thus a neat statistical model linking primary and secondary tone production…

…a superb fit if I may say so myself – and with a Pearson correlation coefficient of 0.9995 (p<0.001) – we can’t really argue! But what does it mean?

OCTAVES - plain and simple: the secondary frequencies for resonators #1 to #7 are double the frequency of the primary. This fact is denoted by the coefficient 2.0002, with a standard error of 0.020 (p<0.001). The manufacturing tolerances of this pipe are astounding; here we see 7 resonators, all with slightly differing primary frequencies, but all crafted carefully to generate 7 octaves, all pitched slightly differently. But what about that -51.221Hz figure? This is a regression constant that can be ignored, since its associated probability value is a mere p=0.133 – which means it is not statistically significant at the universally accepted level of 95% confidence.

I repeated the cheeky analysis to establish the relationship between the three tertiary and secondary frequencies. Here it is…

First of all we can ignore that regression constant of 91.084Hz again, since its p-value is a pathetic p=0.856. This is so pathetic that we can safely run the regression again without it and tidy matters up rather more neatly. In graphical terms this is the equivalent of forcing the trend line through the origin of both axes. In common sense terms it means we are assuming zero tertiary frequency when the secondary frequency is zero – this makes terribly good practical sense! When we force the null point we get the rather simple linear equation:

y = 1.495 x

This regression coefficient has a p-value of p<0.001 and a 95% confidence interval [1.478; 1.512]. What does this mean?

It means the three tertiary frequencies have been consistently crafted to be almost precisely one and a half times higher in pitch than their corresponding secondary frequencies. This ratio isn’t just any musical ratio; it is the ratio of the perfect fifth

These pipes have been very carefully crafted to produce a series of 9 octaves and 3 perfect fifths, all at slightly different starting pitches. The question is… why?

Mending the pipe

Those differently pitched octaves intrigued me. I wondered if they encoded something other than an apparently random set of disparate pitches. Nothing about this beautiful object seemed random so far, so I had another peek at the data for the 7 resonators we presumed were undamaged.

The minimum and maximum primary frequencies for the 7 resonators were 1,257Hz and 1,572Hz respectively, giving a range of 315Hz, which is variation of about 20% in overall pitch across all resonators.

The ratio 1257:1572, at 0.7996 is most intriguing, since this is pretty much the ratio value for the musical interval of the major third (4:5 = 0.80). A slight tweak (that is well within the boundary of measurement error) provides the exact 4:5 musical ratio when primary frequencies are set at 1256Hz:1570Hz. This tantalising suggestion of intended musical design made it all the rather more annoying that the pipe had been damaged! I wondered if I could ‘repair it’ using statistical modelling and a few basic assumptions.

We know resonator #8 is damaged and have seen that resonator #9 stands out from the crowd for some reason, so I took a good hard look at all nine primary frequencies again. In the next article in this series I reveal the X-Y plot of these, expanded in scale to reveal differences that otherwise might be overlooked. Until then…

Kettle On!