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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.

*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.

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:

http:/www.photodady.com/sub_440_432_426_420.m

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?

http://newsoffice.mit.edu/2009/bio-electronics-0603

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, 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.

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.

http://www.noiseaddicts.com/2008/07/the-sound-that-shouldnt-be/

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.

### NATURAL OCCURRENCE

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.

### HOW WE HEAR HARMONIES IN TRIADS

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)

Thanks, glad you like the articles :).

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.

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.

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 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.

http://www.photodady.com/subharmonics_rev2.pdf

Enjoy, and keep up the great work!

*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 :-)