# What is Octave Equivalence?

This is an updated copy of a question I asked on Physics Stack Exchange not too long ago. Since I work primarily in mathematics, I thought it would be a good idea to ask it here as well (especially after being inspired by the popularity of this question).

As part of some work I've been doing in signal processing and Fourier analysis , I recently ran into a couple of questions regarding "octave equivalence," the psycho-acoustical sensation that frequencies differing from each other by a power of 2 are somehow "the same." I'd like to incorporate this into a construction I'm doing in my work, although I'd prefer it to not be ad hoc or even mostly biological; that is, I'd like some kind of mathematical statement from which I could "read off" the phenomenon of octave equivalence in the same way that one can "read off" the phenomenon of beats from (if I recall correctly), the double angle formula.

There doesn't seem to be too much literature on this subject, but here is what I've found so far. Apparently, the cochlea in our ear does something like a windowed Fourier transform. After extracting the frequency content from an auditory signal, the brain infers the fundamental from the partials available. This is why when actually hearing frequencies of, say, 400, 500, ..., 900 Hz, we may perceive a fundamental of 100 Hz. So, according to this logic, the phenomenon of octave equivalence arises because the frequencies $f$ and $2f$ share so many partials within the given window, in fact, more than with any other frequency $nf$ for $n>2$.

Is this sufficient? I'm not sure it is. For example, I was playing around on Maple with pure sine waves, and it seems as if one's ability to perceive pitch affinities, as they're called, is significantly dulled when complex signals are not used. I'm not sure why this is the case.

Also, to anticipate people's statements, I do know that these matters have a cultural component as well. However, from what I've read, octave equivalence has been detected in rats and human infants, making it seem at least somewhat universal. What is done with this phenomenon, of course, is completely subject to the particularities of one's place int the world.

If someone could point me to anything in the mathematical physics literature on this topic, I'd be very grateful.

Thanks.

ISBN 978-90-816095-1-7 Applying physics makes auditory sense

Willem Chr. Heerens

and

J. Alexander 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.

• "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." It has been known for over 70 years that this phenomena is not occuring exclusively within the cochlea. Listening tests have been performed in which two sine tones are played to the individual ears of a listener, and yet the listener is still able to identify an underlying fundamental frequency. Hence, there is definitely some "mixing" distortion taking place in the brain. – Ryan Dec 15 '12 at 1:27

I think a good start would be Music: A mathematical offering by Dave Benson, particularly Chapter 4.

My recollection definitely agrees with yours that our perception of consonant octaves (and fifths etc.) is a phenomenon that occurs with complex sounds (pitches with overtone series included) rather than with pure sine waves.

I remember reading somewhere how investigating the specific waveforms of an oboe and a horn can show mathematically why a major third sounds good with the horn above the oboe, but a perfect fourth sounds good with the oboe above the horn. (Please consider every single specific detail in that sentence extremely suspect.)

Octave-equivalence is a learned phenomena. There is no mathematical reasoning for why we should hear a frequency ratio of 2:1 as more "equivalent" than 3:1 or 5:1, for instance.

Check out the Bohlen-Pierce Scale for an example of a scale which uses the tritave (frequency ratio of 3:1) as an equivalence interval rather than the octave. Many individuals (e.g. Elaine Walker, proponent of the BP scale) claim that they can hear tritaves as equivalent, after years of personal experience with the scale.

As for literature on the subject, I would recommend you check out Tuning, Timbre, Spectrum, Scale by William Sethares. If you cannot get a hold of this book, Sethares and Milne have many other related studies concerning the psychoacoustic effect of timbre on intervals.

• From the Wikipedia article: "The BP scale's use of odd integer ratios is appropriate for timbres containing only odd harmonics." The article on the psuedo-octave also points out that octave sensation is partly based on timbre. If the overtones are stretched or narrowed, then the tuning will have to be stretched or narrowed to create the same interval interpretation. This leads me to question your first sentence and even assert that there is physical, non-psychological basis for octave equivalence in a world where instruments with non-stretched even harmonics dominate. – Todd Wilcox Jan 9 '13 at 18:05
• @ToddWilcox: Then why do listeners identify octave equivalence using timbres without the 2nd partial present? Consider gamelan or slendro, for example. If there truly was a physical basis for octave equivalence, then it would be implied that balinese musicians would use an interval much wider than the octave for equivalence purposes. – Ryan Jan 10 '13 at 17:52
• I think my comment was worded poorly since it might seem like I'm saying octave equivalence is 100% physical, when in fact I believe it is partly physical and party psychological. If your answer was meant to say the same thing then we are in agreement, I read it as you saying it is 100% psychological. Otherwise, my comment was to say that tritave equivalence would be more natural for instruments without even harmonics, I see no reason why octaves would not sound equivalent on such instruments. But tritaves would be challenging on instruments with even harmonics. – Todd Wilcox Jan 10 '13 at 22:57
• Also, I personally find it difficult to accurately hear intervals on instruments with low or missing even harmonics in noisy environments (i.e., when other non-harmonic sounds are present). Some time spent messing with a drawbar organ while a drummer is warming up nearby has driven home for me the importance of timbre in interval recognition. It would also be quite a coincidence for separate cultures with wildly different tuning systems to equate octaves if the octave had no physical basis. – Todd Wilcox Jan 10 '13 at 23:02

Valuation of meta-information fails at all factors of two of a given fundamental within discriminable range, for all modalities and for all species. It's an inherent limit in multicellular information processing.

The paradox of octave equivalence is resolved by noting that the thing that sounds "the same", and the thing that sounds "different" are two different entities - the stimuli themselves, vs the information about their relationship. The stimuli remain different (higher and lower pitched), and the "sameness" is a more fundamental currency, not unique to audition...

Without a baseline of sameness we'd possess no empirical value of difference, or informational weight, and factors of two of a fundamental component are the simplest possible frequency relationships - all others are, by definition, more complex, resolving to ever-larger temporal integration windows.

IMHO the octave equivalence paradox is a big clue as to the nature of informational entropy in cognition, and a key piece of the puzzle in solving the binding problem...