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I'm a graduate student in mathematics doing a bit of research in signal processing and Fourier analysis and I've come across a question that could probably be better answered by a physicist:

Is the phenomenon of octave equivalence (the psycho-acoustical sensation that pitch is subject to an equivalence relation whereby two frequencies are considered "the same" if they differ by a factor of a power of 2) a product of our biology alone or is it an artifact of something more basic in acoustics?

That is, could someone point me to a mathematical statement from which the sensation of octave equivalence could be "read off" in the matter that, say, the phenomenon of beats can be found in (I believe) the double angle formula? Or, does the sensation merely arise from the structure of our cochlea? Vaguely speaking, is the phenomenon somehow external or internal to our biology?

This is all very ambiguous, but I hope that something meaningful can be extracted from it. If it helps, I know that thus far octave equivalence has been found in rats and human infants.

Edit: Also, if anyone can point me to any pertinent literature, preferably in physics or applied mathematics, I'd be very grateful. Thank you.

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Problem with explanations below is that they assume octave similarity as EQUIVALENCE :) Octave notes are similar YES but it is NOT equivalence, we are able to distinguish them! So technically speaking there is no problem, we still hear them as different(slightly) sounds -- no problem. Question is why ALL sound sensations start to "repeat" when we double the frequency. In other words -- why sound "colors" start to repeat and are not continuous as normal colors. –  Asphir Dom Feb 26 '13 at 23:06
    
I meant equivalence in the mathematical sense, an equivalence relation. Of course frequencies differing by a power of 2 are not identical, but they are, so it seems, equivalent in this other sense. –  Alexander Sibelius Feb 27 '13 at 19:43

5 Answers 5

up vote 10 down vote accepted

Here is one reason: a note with a fundamental frequency of 100 Hz will have harmonics at 100 HZ, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, etc., while a fundamental of 200 Hz has harmonics of 200 Hz, 400 Hz, 600 Hz, etc. These are a subset of the harmonics of the 100 Hz note an octave below. The human auditory system detects the pitch of the fundamental largely by inferring it from the harmonics. You can test this by playing sine waves at, say, 400, 500, 600, 700 and 800 Hz simultaneously - you will hear it as a note of 100 Hz, even though the fundamental is not really there. Physically, the shape of the cochlea does something very like a Fourier transform on the incoming sound. If you "unroll" it, the cochlea is effectively a long tube of slowly varying width, and the width of each part of the tube determines its resonant frequency.

The upshot of this is that one reason for the auditory similarity between a note and its octave is due to the fact that they share so many harmonics. If you play a note at 100 Hz and then simultaneously start playing a note at 200 Hz, it makes some of the original note's harmonics louder but doesn't introduce any new ones.

Of course, the same would be true with if the new note had a fundamental of 300 Hz rather than 200. However, in this case, while the high note still shares all its harmonics with the low one, but the low one only shares a third of its harmonics with the high one. Perhaps this is why we perceive the octave as a more consonant interval.

This can also help to explain the consonance of other intervals. For instance, the harmonics of a note at 150 Hz (a perfect fifth above 100 Hz) are 150, 300, 450, 600, 750, 900 Hz, etc. You can see that it has a lot of harmonics in common with a 100 Hz note, and this is part of the reason we find these two notes consonant and, in some ways, quite similar-sounding to one another. But there are less harmonics in common between a 100 Hz fundamental and a 150 Hz one than there are between a 100 Hz and a 200 Hz fundamental, which perhaps is why the octave sounds so much more consonant than the fifth.

It's worth noting that automatic pitch-detection algorithms also find octaves (and fifths) "similar" to one another, in the sense that one of their most common errors is to mis-classify a note as one an octave above, effectively because they fail to notice the even harmonics.

Another interesting thing to note is that grand pianos are tuned to a scale that repeats at an interval slightly greater than an octave. This is because the heavy strings don't quite obey the ideal string law, and have harmonics that are slightly "stretched". So if you want the piano to sound in tune, you have to stretch the octave slightly as well.

However, despite this physical basis for the equivalence of the octave, there's a big cultural element in it as well. Not all musical traditions place the same emphasis on consonance as western music, and not all human cultures consider notes an octave apart as being equivalent to one another. Some composers have even experimented with scales in which non-octave intervals (such as an octave plus a perfect fifth, i.e. a factor of 3 rather than 2) play the same role that the octave usually does.

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do you have a referecne for your final statement/ –  Arnold Neumaier Nov 18 '12 at 12:35
    
@ArnoldNeumaier it's something I've read in a few different places - but I had a look for examples of non-octave scales just now and the only ones I could find were 20th century Western inventions, which makes me a bit less sure about it. I've edited the answer accordingly. –  Nathaniel Nov 18 '12 at 14:10
    
@Nathaniel, I gave a +1 for the very interesting details. But still, all you state could hold introducing a factor 3 instead of 2 in all the numbers. –  Eduardo Guerras Valera Nov 18 '12 at 14:12
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@Eduardo well, I guess with a factor of 2 you have that the higher note shares all its harmonics with the lower one, and the lower one shares half its harmonics with the higher one. With a factor of 3 the high note still shares all its harmonics with the low one, but the low one only shares a third of its harmonics with the high one. Perhaps this is why we perceive the octave as a more consonant interval. –  Nathaniel Nov 18 '12 at 14:25
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@Nathaniel Oh, trust me. Maple has convinced me. But, Helmholtz's name would probably carry more weight on paper than mine! Haha. –  Alexander Sibelius Nov 26 '12 at 5:14

The paradoxical perception of "sameness" between frequencies in a factor-of-two relationship is resolved by delineating the types of information involved:

  • primary sensation; the stimulus itself. Here we may note that pitch perception is maintained, and we clearly hear two different pitches, one higher, the other lower

  • meta-information; the information about the relationship between the tones, which registers as zero

We may also make the following observations (with respect to pure-tone frequencies):

  • harmonic consonance and dissonance are actually degrees of equivalence, or difference (depending on one's perspective - ie. octaves have zero difference, and tritones are very inequivalent).. because maximum consonance is equivalence (the 1:1 interval doesn't strictly count as an harmonic relationship, being an interval of zero)

  • this valuation we apply to tonal relationships is more fundamental than audition itself; that is, the 'stuff' of harmonic relationships is discrete from the stuff of auditory stimuli

  • the percept is likely a form of metamery, or channel convergence

  • it has analogs in other modalities, and in time as well as space

  • factor-of-two synchrony is the simplest form of frequency relationship, hence it may thus seem unsurprising that it forms the default backbone of tonality and rhythm

  • visuotemporal and visuospatial bandwidths are also octave-centric; the wagonwheel illusion recurs at all factors of two of the initial fundamental frequency where motion detection goes to zero, and our optical bandwidth falls neatly within one octave (it seems compelling to presume that were it larger, the resulting equivalencies would represent erroneous information)

  • the corticothalmic network has a one octave bandwidth (indeed, its myriad feedback and feedforward loops may present a further basis for affinity with the harmonic series, which is reproduced by recursive sub-division by two)

  • the peripheral nervous system has a two-octave bandwidth

  • all animals tested for the equivalence have produced positive results - however given these other points, it almost seems trite to mention

  • it seems likely that this equivalence principle is an important key to solving the binding problem, and that all factors of two of a given fundamental, within discriminable range, for all sensory modalities and for all species, register as "zero difference" in much the same way - so all animals hear octave equivalence, and no animals have optical bandwidths exceeding one octave, for example. All hitherto unestablished sensory bandwidths will be found to conform, be it chemical, electrical, magnetic or mechanical

  • it also seems likely to be intrinsic to multicellular information processing per se; corresponding to baseline connective and impulse rate entropies

In short then, it's neither exclusively physical, biological nor psychological... rather, it is informational, an inherent cognitive limitation, integral to multi-cellular frequency analysis-based processing

Sentient aliens (presuming they're multi-cellular) will be subject to the phenomenon, and will make music by sub-dividing octaves in time and space

This same informational "zero point" courses throughout our psyche - all words and phrases we know are bound to it and by it: all the information we process is written upon it

We'll know we've cracked hard AI when our creations are also subject to it..!

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Visit http://www.a3ccm-apmas-eakoh.be/index.htm

ISBN 978-90-816095-1-7

           Applying physics makes auditory sense

           A New Paradigm in Hearing

           Willem Chr. Heerens

           and

           J. Alexander de Ru

           ©2010 Heerens and De Ru

“The incoming sound signal is transformed into the sound energy signal inside the cochlea. It is this signal that evokes both the mechanical vibrations in the basilar membrane and the corresponding electrical stimuli in the organ of Corti, stimuli that are subsequently sent to the brain in a frequency selective manner.”

Mathematically, this signifies that the mammalian cochlea differentiates and squares the incoming sound pressure signal.

In terms of physics, it means that a sound energy signal is offered to the organ of Corti. Functioning as a Fourier analyzer, the organ of Corti subsequently converts these incoming signals into the sound energy frequency spectrum that is transferred to the auditory cortex in a frequency selective way.

Salient experimental results so far • For residual tone complexes – harmonic series where the first harmonic or fundamental is missing – the differentiating and squaring process in the cochlea reconstructs perfectly the corresponding but missing fundamental. • Contrary to the conclusion that an early neural mechanism is responsible for the mystery of the inferential pitch, strong evidence exists that the cause for this reconstruction of the virtual or fundamental pitch is hydrodynamic in origin.

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For the reasons for the divergence, look up the comma of pythagoras.

For an interesting perspective that reaches beyond, try reading Drew Hempel's Against Archytas: How the West Lost Alchemy or Paranormal Complimentary Opposite Harmonics - its a bit conspiratorial, but some interesting concepts there - basically that 'healing music' is decidedly non-western due to the removal of the harmonics because of the comma of pythagoras.

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I'm all but certain that no one has a satisfactory answer to this question. Our ability to make organized sense out of music is almost impossible to explain. The problem is that it is only in the last few several hundred years that we have had the technical capacity to create the extremely intricate music that exists today, and yet the evolutionary capacity to interpret and appreciate that music must obviously have been hardwired in over a period of not less than hundreds of thousands of years. The question is: what drove evolution to equip us with a skill that would only become useful long after that skill had been fully developed?

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The most convincing hypothesis I've heard regarding octave equivalence is that it is simply more efficient for an organism to condense the pitch range it encounters. That is, as opposed to processing each frequency as a distinct value, the organism interprets the pitch as an equivalence class, determined by the octave relationship. This has to do with using minimal energy or something like that. But, the question remains: why the octave? Why not the fifth, for instance? –  Alexander Sibelius Nov 18 '12 at 4:51
    
You say "the question remains... why not the fifth?" What's the question? Octaves sound similar, but not identical. So do fifths. Both facts seem to be explained by the harmonic relationship, along with non-linearity in the inner-ear response so that even pure external tones produce responses at harmonics within the cochlea. –  Foster Boondoggle Nov 19 '12 at 19:40
    
The question concerns the hierarchy that seems to be at play. For instance, we speak of octave equivalence but not "fifth equivalence". I suppose Nathaniel's answer gives a partial answer to why this is the case. –  Alexander Sibelius Nov 19 '12 at 20:19

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