Pure Tones For Healing - Harmonious Lissajous Knots - Pendulum Waves - Sloth Canon Music From Numbers
    By Robert Walker | May 4th 2014 02:44 PM | 18 comments | Print | E-mail | Track Comments

    What ties together Lissajous knots, the harmonic polyrhythms of Theremin's rhythmicon, and a way of adding sounds to Pendulum waves? And do they have healing properties? And what is the musical maths of sloth canon number sequences? 

    One of my first posts here was on the Music And Mathematics Of Fractal-Like Sloth Canon Number Sequences. But since then I've been posting mainly about Mars and space colonization. Today though, I just launched a kickstarter to help get my Windows programs running on the Mac, and thought it was a nice opportunity to talk again about some of these ideas on the interface between music and maths, especially focusing on the harmonic polyrhythms, which tie together Lissajous knots, Theremin's rhythmicon and the pendulum waves.


    This is an idea explored by Henry Cowell, in a famous book called  New Musical Resources, and he commissioned the Russian inventor Theremin - the same chap who made that instrument that you play just by waving your hands in the air around it - to make the rhythmicon, long before personal computers in the 1960s. 

    It lets you hold down any notes in the harmonic series and each note plays at a different tempo depending on its frequency. They interact to make complex and interesting rhythms, "polyrhythms" in fact - a kind of mash up of independent rhythms played simultaneously.

    Rhythmicon demonstration

    Bounce Metronome has its own version of Theremin's rhythmicon which you can try out for yourself. It doesn't emulate everything, doesn't have that rather wonderful drop in pitch as the machine slows down :).

    Many cultures have polyrhythms. You get them in Western music also - not so much as elsewhere - but 3:4 and 3:2 is common. Chopin makes a lot of use of polyrhythms, including sometimes playing a row of say 22 notes over 12, ( Chopin's Nocturne op. 9 no. 1 in B flat minor) though meant to be played with rubato, i.e. not in strict tempo.

    Some composers such as Ligetti made a lot of use of them. He was influenced by the Efe people - the "pygmies" and other polyrhythms in Subsaharan Africa.

    Efe People's Rhythms and Polyrhythms

    It's a playlist of an hour and a half of Pygmie music with lots nice rhythms including many polyrhythms - they are noted for them. The tuning of the music is also interesting for those interested in microtonal music.

    See Efe People and Efe People of the Ituri forest for more about the people who make this music 

    Polyrhythms are also used a lot in Indian music.


    You may well know the Lissajous patterns you get in 2D. Plot two frequencies against each other, one horizontally, and one vertically, and you get a pattern like this

    The idea is simple. Think of a point that vibrates five times from left to right for every time it vibrates three times back and forth horizontally. Then this is the path it would trace out on a sheet of paper.

    You can make the Lisssajous patterns using an oscilloscope. Originally they were made using mirrors attached to tuning forks.

    Here is a later version, using two mirrors attached to vibrating wires at right angles.

    Which makes the Lissajous pattern like this:

    To find out more with wonderful pictures of these ancient Lissajous mechanical devices, in this article Lissajous Figures by Thomas Greenslade

    For a while, Lissajous patterns were used for fine calibration of new tuning forks. Here are some tuning forks with mirrors attached from the late nineteenth century.

    Nowadays you can do it much more easily with lasers, like this

    Lissajous Curves with Tuning Forks

    And here is an example with the two tuning forks in frequency ratio of 8  9

    Lissajous Curve Ratio 8 to 9

    But what if you add a third wave? If you want to represent musical harmonies with Lissajous figures that's going to happen often.

    That's a question I had to tackle with my "Tune Smithy Lambdoma". 

    This is a program to show an arrangement of musical pitches used for music therapy, attributed to Pythagoras. What he actually taught is a matter of mystery, as no writings survive (for an overview of modern scholarship about what he actually taught, see Pythagoras (Stanford University) ). All we have are his sayings, and the accounts of later scholars.

    In the case of the Lambdoma, then the basis of this attribution is a commentary by the Greek Iamblichus regarding the Lesser Arithmetic of Nicomachus, which was studied by German scholars of the nineteenth and twentieth centuries, starting with the dedicated amateur nineteenth century scholar Albert von Thimus, who were first to introduce the Lambdoma matrix into modern times. For a detailed account of this possible connection of the Lambdoma with the teachings of Pythagoras, see "Musica Universalis: from the Lambdoma of Pythagoras to the tonality diamond of Harry Partch" by Scott Eggert.

    Again mathematically it's very simple. The numbers show the ratios of the frequency of that square to the 1/1, which you can set to any arbitrary pitch, say 440 Hz or middle C, or 256 Hz or whatever.

    Then the diagonal row running up from the bottom corner to middle right is a musical harmonic (or overtone) series 1/1, 2/1, 3/1, ... just multiples of the original frequency. 

    (If you are a mathematician, well it's related to, but not the same as the harmonic series of mathematicians which is the sum of all those terms).

    The other diagonal row is a subharmonic (or undertone) series 1/1, 1/2, 1/3, 1/4, ...

    It's easy to see, that all the other rows in both directions are the same sequence of pitches, just shifted up or down. For instance the second row diagonally up to right is the same as the first row but with all the numbers divided by 2, so shifted down an octave.

    So, the end result of this simple process of putting all the ratios of the numbers 1 to 8 in a lattice like this is an interlocking array of harmonic series going up to the right, and subharmonic series going up to the left.

    Note, this Pythagorean Lambdoma tuning shouldn't be confused with the Pythagorean twelve tone tuning of Medieval Music and it's generalizations, a different tuning system, based only on multiples of 3 and 2.


    It turns out that the purest major chord you can make uses the fourth, fifth and sixth harmonics 4/1, 5/1, 6/1. Similarly you find the purest minor chord as 1/4, 1/5, 1/6.

    So the arrangement of pitches here is a sort of criss-crossing pattern of major and minor chords - except, that it also extends beyond the harmonies of western music to encompass the seventh harmonic, here as the 7/1 and the 1/7 and its multiples 7/2, 7/3 etc and 2/7, 3/7 etc.

    The harmonies of Western European music are restricted to approximations of intervals made up of ratios of multiples of the primes 2, 3, and 5 - and in the middle ages was even more restricted, to just the primes 2 and 3 - but music of other cultures explores the higher harmonics. You can think of them like a kind of extension of our major and minor to higher harmonics in the harmonic series.

    Barbara Hero took this idea from Greek mathematics and added details to do with colours, then the Lissajous patterns, and then recently, made this connection with harmonic polyrhythms like the rhythmicon. I just wrote the software to help put her ideas into practice in this case.

    This pattern is closely related to Harry Partch's tonality diamond, but not exactly the same. 

    Harry Partch, remarkable mainly self educated genius, one time hobo (during the great depression), instrument maker, and pioneer of much of modern microtonal music theory.

    In the tonality diamond, you ignore octave repeats, reduce everything into the octave and arrange in order of increasing size for convenience of musical playing. 

    For more about the connections between the Lambdoma and the tonality diamond see Scott Eggert's article again.


    It was easy to draw Lissajous patterns for Barbara Hero's ideas, for two note chords, but for a three note chord you end up with something like this

    That's with two of the waves one way, and one the other way.

    You can also try the idea of putting them in three different directions, equally spaced and you get

    They are nice enough patterns for sure, but don't really show the connection between the three waves of the triad so clearly.

    So anyway the composer Charles Lucy of Lucy tuned lullabies suggested, why not put the third axis into 3D? I don't know why I didn't think of that. Did that and here is the result, for the same example, a major chord. This video clip also plays the notes of the chord

    Frequencies in ratios 1/1 : 3/2 : 5/4, Lissajous 3D, Polyrhythms with tempi in same relationship 

    It gives a far clearer visual representation of the relationships between the three frequencies.

    Later, I found out that this is an old, known idea, called "Lissajous knots" by mathematicians. I actually studied a bit of knot theory, in my own time for interest, but missed out on this.

    Knot theory is about the study of tangled closed loops in mathematics. It's a bit different from the everyday idea of a knot which relies on friction to join ropes to each other and to other things. A mathematical knot means a tangled loop of string with no ends to it., which can't be untangled except by cutting it.

    Mathematicians then say that two knots are "the same" if you can make one into the other by any amount of stretching and disentangling, or untying, but with no cutting permitted. It's a tricky subject, with surprising, unexpected connections with particle physics, specifically a three dimensional topological quantum field theory called the Chern–Simons theory through the Jones polynomial which is used nowadays to resolve many questions in knot theory.

    If you make the third frequency quite high, you get rather fine 3D representations of many common knots like this

    Here is the same knot in Lissajous 3D

    821 knot

    Maths details for anyone who wants to input Lissajous knots into the Lissajous 3D program: the Lissajous knots are described using five parameters  nx, ny, nz and  ϕx,  ϕy.

    To display in Lissajous 3D, you enter the  nx, ny, nz as across, up and in, in the shape window. You set the "across" phase shift to 90 + 180*ϕx / π . the "Up" phase shift to 90 + 180*ϕy/π and the "In" phase shift to -90 degrees

    In this case, it's 3, 4, 7 for across, up and in, and then phase shifts 90 + 180*0.1/PI across, 90 + 180*0.7/PI up, and 270 in. You can just enter those as formulae as Lissajous 3D recognizes formulae input to any of its text fields. 

    I'll add a window to make it easy to enter these figures for lissajous knots, no need to use these formulae, along with a droplist of examples in next version of Lissajous 3D.

    Many simple knots can't be made as Lissajous patterns, such as for instance the trefoil knot, you can prove that no Lissajous pattern in 3D can make this. In other words, no matter how you stretch and pull this shape around, so long as you are not permitted to cut it and rejoin the pieces, it will never make a Lissajous pattern.

    Trefoil knot, example of a mathematical knot that can't be made as a Lissajous pattern

    But there are infinitely many Lissajous knots you can make, of course and they are interesting because of their high degree of symmetry.


    The Lambdoma arrangement of pitches and the Lissajous patterns are used in music therapy by various music therapists in the States and in Taiwan. They have various ideas about why they work and how to use them. 

    As someone with a scientific background, then I have no idea what some of them mean by the things they say. For instance I simply can't attach any meaning to the idea that particular pitches resonate with atoms in your body, or that particular frequencies help with DNA repair of various parts of the body (I may not be describing their ideas accurately given I don't understand any of it but - that sort of thing). 

    The wavelength of a 256 Hz tone for instance is 331.2/256 or 1.29 meters (that's at 0°C) - what can that possibly do to a DNA molecule of width about 2.5 nanometers (nanometer is a billionth of a meter)?

    But what if they are onto something? If not presented in a way that's easy to fit within present day science. Perhaps it is more like a scientific metaphor or poetry at present?

    I'm prepared to keep an open mind, there are many things we don't understand. As Isaac Newton is reported to have said

    I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.
    Tung Ping Chau, Hong Kong
    Tung Ping Chau, Hong Kong 

     I think, in many ways, we are like that still, given the vastness of the universe, so little we really understand.

    We haven't yet sent anything as far as the nearest star, and our solar system itself is hardly explored at all close up. Everything else is inference from patterns of light, fast moving energetic particles, and radio waves and such like, detected from the distance by our telescopes. Also limited in time, only been doing science for a few centuries, and written records go back only thousands of years, a millionth of the age of the universe. Limited in energy, we can use only a tiny fraction of the solar energy that falls on the Earth, never mind the amount of energy that is emitted by the sun.

    Also, we haven't done any scientific experiments that last for as long as two centuries

    The Oxford electrical bell, which has been ringing continuously since 1840, possibly for 15 years longer, (with occasional interruptions when humidity is high), needs no winding up, often cited as the longest running scientific experiment.

    If we need to do an experiment on a longer timescale than that, for instance, to leave an ocean full of organics for a hundred million years to study how evolution gets to the primitive cells from amino acids, we have no way of doing it. We just have to hope that nature has done the experiment for us. We are also limited in the range of extremely short time scales and high precision in space, such as the Plank scale, as this requires vast amounts of energy.

    And our universe is so young also, compared for instance with the trillions of years lifetimes of an orange or red dwarf star, with the largest stars shining brightly still and many young stars newly forming all the time. How can we imagine we really have a clear understanding of everything that we can study in our science? It's no surprise that along with all our knowledge, we also have countless mysteries we don't really understand.

    Indeed, a rather whmisical but fun thought. If we meet ETs, what if they don't have scientists, like us, and in their place have artists, poets, musicians, and therapists who think in similar ways. perhaps talk about harmonies resonating with atoms in their bodies and so forth? And perhaps build spaceships out of music or poetry? And warp the universe with thought and ideas rather than technology? After all in some ways, according to some ways of looking at things, thoughts are more fundamental than our technology philosophically speaking.

    You get a few ideas like that explored in science fiction sometimes. I don't know how likely it is, that we might meet ETs like that, but if it does happen, we might find each other mutually pretty incomprehensible to start with.

    I wouldn't want to say at all that we ditch science of course. When it is at its best, it is just rigorous thinking, and making sure you don't fool yourself with wishes and delusions. If it was possible to navigate the galaxy with spaceships made of poetry and music, you could look at how that works as a scientist also. It's not really at all limited to conventional ideas of what science is about.

    Einstein's quote, similar to the Newton and also often quoted brings this out, the value of science and rigorous thinking in all its forms.

    "One thing I have learned in a long life: that all our science, measured against reality, is primitive and childlike -- and yet it is the most precious thing we have."

    I wonder sometimes though if we might need new methods of rigorous thinking that, for some reason, we simply haven't invented yet. Science does seem rather limited when it comes to holistic things, such as music therapy indeed. You can't really do a double blind experiment for instance, it's not like pills where the control is a pill that looks the same but hasn't got the active ingredient. 

    Anyway, whether their explanations do have some meaning I can't grasp, I can well understand that the harmonies of the Lambdoma matrix and the Lissajous patterns could have healing properties - just because of the healing effect of listening to harmonious sounds and watching gentle smooth curves as you play on the keyboard.

    You can give it a go yourself with the Tune Smithy Lambdoma and see if you think it does have a healing influence. 

    Also you could have a listen to this video as a kind of intro to the many ideas around about musical pitches with healing properties (nothing to do with Barbara Hero) demonstrating some pure ratio intervals played.

    Do you feel healing effects of these musical intervals when they are played? Many of those commenting on the video do. 

    If you do think that things like playing musical tones, for instance, may have profound healing properties on the body - then as scientists, or scientific by inclination, is there any way we could look into it? 

    Can we hope to find out in any rigorous way if these musical tones in harmonic relations do help to heal the body (and mind also especially mental trauma)? And if they do, can we maybe make new scientific discoveries that can be of value to the music therapists that work with them? Or is it just not amenable to that sort of study?

    What are your thoughts on all this - do say in the comments below.


    I wonder if you've seen these?

    Pendulum Waves

    It's a simple enough idea. Just set up a whole bunch of pendulums, each a different length. They will all have different periods and so gradually go in and out of sync. But the visual effect is rather lovely.

    Anyway so a while back we had a discussion on facebook, some of my musiican composer friends, about, could you make a version of these with sounds?

    The answer is, yes, actually they are closely related to the rhythmicon. The natural thing is to assign a frequency relating to the tempo of the pendulum just as for the original rhythmicon.

    I tried that, you can listen to it here if interested. But then, the composer  Elaine Walker (no relation, it's a common name especially here in the UK) suggested I do it as a harmonic series starting from the first,instead of the 52 harmonic, after I posted the first clips to facebook, and of course, that's much more harmonious.

    The results were these "sonified pendulum waves"

    The Sound of a Pendulum Wave - As Bounces with Harmonic Series Pitches

    Another way of showing these "sonified pendulum waves"

    Spiral version of Sonified Pendulum Wave - rhythms start 52, harmonics 1 to 16 - Bounce Metronome 


    The other fun thing you can do with the software is to make "sloth canon" musical number sequences. 

    Here is an endless tune for unaccompanied violin, shows how the melody line goes on and on with no exact repetition. It's based on a strict sloth canon though don't expect to spot that easily.

    You hear the tuning first (a scale that switches direction and repeats at the harmonic fourth 4/3 instead of the usual octave), then the short seed phrase, and then the tune that results from putting that phrase through the magic of the Tune Smithy sloth canon process.

    Endless movement for unaccompanied violin - fractal tune made with Tune Smithy 

    And this one, where I've used Lissajous 3D for the animation

    Breathing space - with Navigating the pacific by stars wind and waves

    And here I put the sloth canon through a sequence of chords in Tune Smithy, a very famous chord progression called La Folia.

    Tune Smithy tune: la folia chords with Lissajous 3D

    The important thing to realise here, is that there is no composition technique going on at all. Tune Smithy, which creates these sequences, has no "knowledge" of composition technique built into it at all.

    It just blindly starts with a musical phrase and piles copies of it on top of each other in a similar fashion to the Koch snowflake to make this music.

    You don't need to be a musician or composer or have any musical training to try it out, just need to know what you like to hear. It's just like the artistic Mandelbrot Sets and such liek which again don't need any artistic training or capabilities, except to know what you like to see.

    Indeed, the musical and visual fractals seem to go rather well together. Here we see them together, the 3D fractal here is a fractal made with Mandelbulb by Torsten Stier

    Torsten Stier's video again, this time with the Golden Ratio Cello tune.

    Mandelbulb 3D fractal by Torsten Stier with Golden Ratio 'cello tune from Tune Smithy 

    As usual, the entire tune is generated fractal fashion from the short musical phrase you hear at the beginning.
    It's from the Fractal Forums where enthusiasts are exploring new ideas of 3D fractals such as this one, continually coming up with new and amazing 3D fractals.

    I won't go into the mathematics of the Tune Smithy sloth canons, and how they relate to the work of the Danish Composer Per Norgard, just now, suffice it to say that you can get these sloth canons from some really simple number sequences, such as the number of 1s in the binary expansion of a number - that generates a self similar sloth canon number sequence which you can turn into satisfying music to listen to. For more about it see  Music And Mathematics Of Fractal-Like Sloth Canon Number Sequences


    Here is the Kickstarter project which inspired this post, project video:

    And the project itself here:

    Bounce Metronome, Tune Smithy, Lissajous 3D... on Intel Mac! 

    If you want to support it to help get it underway it's much appreciated, you don't have to be a Mac user - and do get an unlock key for all my software as a reward which works on Windows and Linux as well. Also, if you know anyone who is likely to be interested in this, especially Mac musicians, do share it with them. Perhaps we can help make this happen :).

    I've got plenty more can talk about related to the project, especially would be great to talk a bit about some of the rhythmic ideas such as the fibonacci rhythms, euclidean rhythms, and the Ho Ho Chi rhythms, along with fibonacci tonescapes and other things mainly on the interface of music and mathematics. I've been meaning to do an article about them for some time; this may be a good time to do it.


    You can get Tune Smithy at, Bounce Metronome (for the virtual Theremin and harmonic polyrhythms) from (look for Theremin in main window droplist after you install), and the FTS Lambdoma from the Tune Smithy download page for the Lambdoma.

    They come with 30 day free trials which you can choose to start at any time, also you can renew them as often as you like by asking for a new test drive. So, if you just want to try out my software for fun, or to follow up what I say in this article, please feel you have plenty of time to explore these ideas.


    Google (v-c)/c=2.48e-5 to see SLAC's E158 data match CERN's "loose cable" and expose the harmonic comma responsible for gravity in 2D.


    Okay, turns up search results about the "faster than light" neutrino results - which were eventually found to be erroneous with the original team retracting their results see Faster-than-light neutrino anomaly. I don't know anything about a harmonic comma though. You do get "commas" in music theory.

    E.g. if you go up by four fifths C G D A E using pure 3/2 intervals, ratio between third and second harmonics - and then drop down by a pure 5/4, ratio between the fifth and fourth overtone, and then drop down by as many octaves as necessary to get to the same octave, you find the final note is slightly sharp compared with the C you started by, with the ratio of frequencies 81/80. Musicians call that the syntonic comma.

    Another comma, the Pythagorean comma tells you that if you go up by 12 pure fifths and then reduce to an octave, the result is slightly sharp compred with the original note you started from.

    The two of those together are the reason why you can't have completely pure major or minor chords (in the sense of matching the major and minor chords in the harmonic and subharmonic series) in the twelve equal system, or in any tuning system with all the steps equally spaced. That's why for instance the pure tones in that music of the spheres video can't be played exactly in the twelve equal tuning system.

    That's interesting and relevant to this article. But I don't know of any connection to gravity in 2D or any other number of dimensions.

    How can we imagine we really have a clear understanding of everything that we can study in our science? It's no surprise that along with all our knowledge, we also have countless mysteries we don't really understand.

    How refreshingly brilliant! Too bad this attitude is not more pervasive.

    I first came across Lissajou curves through our school Scientific Society, so this brings back memories.  But I’m not really into “mathematical music”, especially not Schoenberg et al.

    However, I did look out some stuff from the Efe people of the Ituri forest, and most excellent it is.

    The life of Léon Theremin himself is fascinating reading.

    Robert H. Olley / Quondam Physics Department / University of Reading / England
    The other diagonal row is a subharmonic (or undertone) series 1/1, 1/2, 1/3, 1/4, ...

    Funny thing, a very common misconception is that the subharmonic series consists of integer multiple denominators or an inversion of the harmonic series. (for that matter the harmonic series is also a trivial case when a waveform interacts with itself or its reflection) It is actually the result of sum and difference frequencies. Frequency division is a misnomer and is in practice arrived at by the detection, or rectification, of the combination of two separate frequencies less than one octave apart.

    For instance take frequency F and superpose it with another frequency, say a perfect fifth, F*3/2, then either eliminate the negative going wave (half wave rectification) or multiply the negative portion of the wave by -1 (full wave rectification) and then do an FFT. You will see the resultant subharmonic at nearly the same magnitude as the original signal. Now pass the waveform through a low pass filter to eliminate all but the subharmonic (in this case it is F/2) and you now have frequency division. Use a bandpass filter and you can pick any of the sums, differences, or sums and differences of any other sum or difference. Hence the harmonic series interacting with itself ends up being: F, F+F, F+F+F, or F, 2*F, 3*F, and so on.

    This process is actually called modulation and demodulation in radio circles. As is obvious the subharmonic series (as well as the complete harmonic series) is NOT discrete, in fact it is quite obvious that the complete series is actually quite continuous when the waveforms are unbounded as with radio waves (otherwise a simple radio would not function) as opposed to them being constrained by fixed nodal points as in most musical instruments. i.e. a reed, a resonator, or any stringed instrument.

    The ONLY way to extract subharmonics from the actual instrument is by the combination of two or more waveforms less than an octave apart. In practice the subharmonics that we normally perceive are most likely demodulated in the ear drum as Helmholtz originally suspected long, long ago. I also suspect that it is actually the subharmonics that are mainly responsible for dissonance, the perfect fifth has the fewest and most symmetrical harmonic frequency domain pattern and is generally accepted to be the least dissonant.

    Thanks for some very interesting articles.

    I have an GNU Octave (MatLab) script that can be used to demonstrate, actually this script uses the natural atmospheric resonances of 8, 14, and 20 Hz. I was trying to figure out why the 432 Hz. A4 tuning has become so popular in underground circles for alternate tunings. I hope this link works:

    Okay, in this article by the subharmonic series, or undertone series I'm using the word as often used in music theory as a mainly theoretical thing. It is simply defined as the intervals you get by inverting the harmonic series. See Undertone Series (on wikipedia) - I'm using it in that sense.
    So for instance if you invert the intervals of a just intonation major chord 1/1 5/4 3/2 (which you also get as 4th, 5th and 6th harmonics of the harmonic series), then you get 1/1 4/5 2/3, or from the bottom, multiplying through by 3/2 (which raises the pitch of all the notes by the same amount) 3/2, 6/5, 1/1, a minor chord. 

    Result is that if you look at all the possible dyads that you find in a minor chord you get all the same component intervals as a major chord, i.e. 6/5, 5/4, and 3/2, but in a different order.

    In the same way, take any selection of intervals from the harmonic series, and invert them all, and the result has all the same dyadic harmonies as the original, but sounds different because it has different triads and chords of more than three notes. 

    As for physical subharmonic series, I don't know about that. You get a minor chord in the overtones of a modern church bell sound but that's because the bells are carefully crafted and tuned to make that happen.

    Yes can get hear difference tones, if you play pairs of notes, e.g. less than an octave apart, and those - they are part of the same harmonic series e.g. if you play a 1/1 and 5/4, you hear a difference tone two octaves below the 1/1, as if they were the 4th and 5th harmonics of that tone. 

    Is that what you are talking about? 
    Ya, I know, cold shoulder, I expect it ;-) just for reference here is an MIT article showing how a model of the inner ear is being used in order to create a highly optimized broadband radio signal demodulator.

    Funny that academia still takes over a century to investigate classical physicists' work, but I take great joy in seeing that they (modern theorists) always end up reverting to those very sound classical theories and experimental results when they need a new breakthrough. :-)

    By the way, it was Nikola Tesla who first asserted that the earth's atmosphere has natural resonances, unfortunately he was blinded by his obsession with numbers that were divisible by three, he postulated the frequencies of 9, 12 and 18 Hz. he was so very damn close you still have to give him accolades for this unwarranted original concept which only had applications for his own bizarre, but very sound, unorthodox theories.

    Sorry, just replied to your comment. Didn't mean to ignore you, just got a lot on right now with the kickstarter and various other things.
    Sorry, I don't understand really what you are saying, as you'll see from my new reply to your comment. Do say a bit more if I haven't answered. That's a really interesting link on a kind of radio receiver based on principles of the inner ear cochlea. Thanks.
    My point is, perhaps a bit long winded, that when theory does not correspond to practice then the theory should not be falsely perpetuated. There really is no real world justification for the inversion of the harmonic series, it only contributes to misunderstandings in practical (sub)harmonics.

    I may stand corrected, it appears as if the body of certain violins may actually be capable of performing demodulation and thus produce subharmonics directly. It is a widely accepted misconception that these subharmonics are not natural, when in reality there has been plausible explanations for them since the mid 19th century. These explanations have simply been ignored, for the most part, by modern theorists who seem to love to poo-poo classical theorists.

    Again thanks for the brilliant articles.


    Okay, as I understand it, it's not a theory, just a mathematical definition useful in music theory for describing certain things, at least as I'm using it here.

    You need to call it something, and that then makes it easy for instance to describe the way the Lambdoma matrix is constructed as using the sub harmonic series one way, the harmonic series the other way and the result comes from multiplying the two together.

    If played as pairs you get the same harnonies as for the harmonic series, in a different order. And -  is no question - the subharmonic series has the same two note chords, the same dyads, as the harmonic series.

     The main thing is, what about triads - when you play as triads then the triads you get in the subharmonic series, are minor sounding, and not so bright and clear as the major chords.

    So, as you are doing, we can debate about how and where they can occur in nature. Whether they do or not, then for sure the overtone series is much more common.


    Okay yes, here she is playing subharmonics on violin

    Website about it here: subharmonics

    Here is another related video about using a tuning fork and a sheet of paper to create subharmonics

    I've just tried this on my big weighted tuning fork, can't seem to do it yet, got the 1/2 and 1/4 but not yet the 1/3.


    I understand that the whole area of why and how we come to hear things as harmonious is very complex and intricate. But one interesting theory is Paul Erlich's Harmonic Entropy - which only models some aspects of how we hear harmony.

    So - this picture shows his theoretical calculations, where the highest peaks show the most harmonious triads.

    Lower interval in the triad plotted horizontally, upper interval plotted diagonally upwards to right.

    There the major chord is that very prominent 4:5:6, highest peak in the picture. 

    The minor chord is the 10:12:15 - in symmetrical position - if you reflect the image in its diagonal, then that's where the minor chord is. Notice that it's shown as a promontory on the slope of the 6:7:9 chord - that's the septimal minor 1/1 7/6, 3/2 - so according to this way of analyzing the septimal minor is far more harmonious than the usual "just intonation" inversion of the major chord.  With many shades of minor you can explore  all harmonious in different ways.

    Though it's not meant as a complete model of perception of harmony, it does fit quite well with the way many perceive harmony. Which, as you say, doesn't have a prominent subharmonic series component at all.

    More here

    Psychological basis of dissonance (Wikipedia)

    And Paul Erlich (Wikipedia)

    Thanks, glad you like the articles :).

    Thanks, a lot more info to digest.

    It appears as if you are working in the Pythagorean scale using perfect ratios of 3/2's. I'm not sure, I'd have to look into it a little deeper, but I don't think that by simply scaling by perfect integer ratios in the equal tempered diatonic scale, with a note spacing factor of 2^(1/12) the most widely used western scale, that you will arrive at the proper frequencies. For instance the perfect fifth ain't so perfect, since its frequencies are separated by exactly 2^(7/12) rather than by exactly 3/2. Pythagorus was a freaking genius!

    I may be splitting hairs here but harmonics, as I understand them, are naturally derived directly from a vibrating object(s), it sounds as if you are using arbitrary, albeit very interesting and quite useful, means of translating a signal into a different portion of the audible spectrum, but is not really a natural phenomenon.

    The other long standing, and very controversial, phenomenon in music are phantom notes or notes that are perceived to be heard between two or more notes played more than an octave apart. These are also very closely related to subharmonics, but as I demonstrate in my paper the phantom note is actually out of tune, off by several cents. I'd have to look into it deeper but I think that they may actually be spot-on in the pythagorean scale.

    Yes, you are right. The pure fifth 3/2 is 702 cents (to nearest cent) so a fiftieth of a semitone sharp compared to twelve equal. The pure major third, ratio of the fifth to the fourth harmonics, is 5/4, is 386 cents, so 14 cents flat, about a seventh of a semi-tone flat.
    So, as you say, the difference tones for an equal tempered fifth or major third are going to be a few cents out of tune in twelve equal. But if you play a pure major third or fifth, then it will be in tune.
    Photodaddy, are you on facebook? Several of my facebook friends could probably answer better than I can, as they research into this sort of thing. If you like to link with me there, then I can introduce you to them and the groups they belong to.
    I'm on facebook as Robert Walker - post public posts usually - if you are there as well just ask to be friends on facebook and I'll accept and can take it from there.

    One of them, Paul Erlich, suggested I say this to you, for some reason he is finding this page doesn't load properly on his computer right now, anyway suggests I say this to you:
    "One of the combinational tones or heterodyne products that is important with loud sounds indeed falls between two input frequencies over an octave apart -- the first-order difference tone -- and it falls below the lower of the input frequencies if they are less than an octave apart. Other combinational tones are sometimes louder than this one. These phenomena are quite well-understood. With Pythagorean harmonies the pitchs formed by the combinational tones don't fit the chord very well, while with a harmonic series harmony they fit perfectly. However there is a completely different phenomenon, the phantom pitch or virtual fundamental, which doesn't necessarily agree with any of the combinational tones in pitch."
    But - obviously will be easier to continue this conversation there if you are on facebook given that for some reason he can't post here.

    BTW the way the tuning of the Lambdoma described in this article shouldn't be confused with the "Pythagorean tuning system" made of stacked 3/2s, which is a medieval tuning system attributed by medieval theorists to Pythagoras. In that tuning system, then the "pythagorean major third" 81/64 is 408 cents of course a long way away from the 386 cents of 5/4.
    I just finished comparing the frequencies of the two scales and found that in the equal tempered scale has no intervals that produced difference frequencies (subharmonics or phantom notes) that were in-tune. The pythagorean scale has three intervals that do produce difference frequencies that are in-tune, the major second, the perfect fourth and the perfect fifth.

    I then generated these tones in Audacity to listen to these three intervals in each scale and found that qualitatively these tones in the Pythagorean scale sounded sterile, neither consonant or dissonant, like listening to a pure octave interval. While the equal tempered scale had some sort of dynamic qualities to it. This actually bolsters by own theory that it is the detuned subharmonic content in musical intervals and chords that contributes heavily to the quality of dissonance.

    Thanks again!


    Yes, that's right. Only low numbered pure ratios can produce those difference tones. Things like 81/64 are interpreted instead as approximations to 5/4. I think you missed one out though, the Pythagorean system's 9/8 is a pure ratio also, produces a difference tone four octaves below. If you play a high pitched dyad it's clear to the ear though a bit faint.

    Yes some people find pure tone ratios too restful and harmonious to listen to. As you say, it's not going anywhere, no dissonance. In Western music we almost never hear chords that sound like that except for the octave - and sometimes in choral singing. In medieval music based on the pythagorean system then they have the 3/2 and 4/3 as a point of rest, no use of 5/4 so that is an example of a Western tuning system that works that way.

    Anyway I think, depends on the composer, tradition, what you are used to, what you want to achieve, you can use a tuning system with some pure ratios in it, one with none at all except for octave, have even the octave detuned also as in Gamelan music

    Another thing some composers try is to use pure ratios but go way up the harmonic series. For instance David Beardsley in his pieces such as "as beautiful as a crescent of a new moon on a cloudless spring evening" which uses a very large 128 note "harmonic fragment" tuning. Or the composer Jacky Ligon who explored ratios of extremely high numbered primes way up into the thousands, where you can't expect to hear that it belongs to a harmonic series. At that point you are into "rational intonation" like the Pythagorean.

    When I listen to a chord carefully on a steady sustained timbre, no change in the quality of sound - then the main thing I hear myself is the pattern of beats. I hear many different beats at different frequencies and different timings for each one, creating a polyrhythmic texture. I think that might have something to do with the perceived dissonance as well. That doesn't go so far as an attempt at a theory of dissonance, I haven't ventured into the field myself :). It's just a personal observation, something that doesn't seem to be mentioned much in these discussions, the polyrhythmic interaction of different pitched beats especially when you are close to a just intonation harmony.

    There are several components to it anyway. Paul Erlich in his theory makes it clear he is modelling only one aspect of the experience of dissonance.

    In case you are interested i wrote a paper a few years ago that extensively explores harmonics and subharmonics. The paper also includes a very simple electronic circuit that can extract subharmonics from any audio signal.

    Enjoy, and keep up the great work!

    Thanks, it looks like an interesting paper, I'll enjoy reading it.
    Just thought that I would provide a link to the FIRST place that I have found that freely admits that wave propagations in the real world are not discrete, in other words continuous integration is necessary: (this separates unbounded waves such as light waves, radio waves, and acoustical waves from bounded nodal point waves emanating directly from any instrument, with a few rare notable exceptions previously discussed)

    Individual plane waves have infinite length and infinite duration. They do not exist in isolation except in our imagination. Moreover, a waveform constructed from a discrete sum (as in the previous two sections) must eventually repeat over and over (i.e. it is periodic). To create a waveform that does not repeat (e.g. a single laser pulse or, technically speaking, any waveform that exists in the physical world since no light source repeats forever) we must replace the discrete sum (7.1) with an integral that combines a continuum of plane waves. Such a waveform at a point r can be expressed as...

    Quote from BYU online "Physics of Light and Optics" 2013 edition section 7.3 (pp. 174)

    Again, I take great joy when modern theorists must revert back to classical theories in order to resolve real world problems :-)

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