Why do octaves sound equivalent?

Question

It is, I think, a perfectly clear observation that one note an octave above another note sounds as if it were the same in a certain sense; while they are by no means the same exact note, they are named with the same letter, and one can for example take any piece of music consisting of two parts and translate one part an octave up, leaving the other the same, and the piece will still work.

At first, I was satisfied with what I guessed was the reason: that the overtones of the second note are a subset of the overtones of the first, and so translation by them ought not change whether or not the music works.

But this argument also holds for the octave plus a fifth, since the ratio of the frequencies here is essentially 3 instead of 2. It no longer remains true that these notes are named the same (which in itself is just semantics), and more importantly it's false to say you can take any piece of music consisting of two parts and translate one part up an octave and a fifth, with the piece still working.

I was wondering then what the actual explanation is, or alternatively if someone could provide a convincing argument that the phenomenon is merely an illusion.

Answer 1

There are indications of an underlying neurological (and arguably evolutionary) basis for perceiving octaves as equivalent, see for example this discussion. This phenomenon is pretty fundamental in that it is also seen in monkeys and other mammals, but not (apparently) in some songbirds. There has been quite a bit of work on the neurological basis for octave equivalence, however I'm not aware of corresponding work that assesses the neurological basis involved in the perception of other intervals.

Answer 2

It is, I think, a perfectly clear observation that one note an octave above another note sounds as if it were the same in a certain sense.

It's certainly common for people to perceive things that way, but it's not universal. Here's a question from someone who complains that they don't hear things that way, for example!

shared harmonics alone can't be seen as a definitive reason to see an octave as special...

• ...for the reason that you pointed out (that for a timbre with all overtones, a higher note with a fundamental frequency that is the same as that of one of the lower note's higher (>2) harmonics will also have a subset of the lower note's harmonics)
• ...because for sounds that don't have the full set of overtones, the 'subset' thing doesn't work - a sound with only the first, third, and fifth harmonics won't share any component pitches with the same timbre sounded an octave up. This is the case for a closed-pipe instrument, like a clarinet, though it's worth pointing out that most instruments do have both odd and even harmonics present.

The idea also doesn't work for octaves for sounds with enharmonic partials, although cultures that use such sounds (e.g. Javanese) often use different scales - so this could be seen as an exception that proves the rule.

while they are by no means the same exact note, they are named with the same letter

We have to remember that the letters are a culturally-specific thing. The reason that notes an octave apart have the same letter is closely related to the fact that Western music culture assumes an octave-repeating scale. You don't have to have an octave repeating scale...

Octave equivalency is a part of most "advanced musical cultures", but is far from universal in "primitive" and early music. The languages in which the oldest extant written documents on tuning are written, Sumerian and Akkadian, have no known word for "octave".

...but the strength of the octave relationship means that an octave repeating scale tends to work well for more harmonically sophisticated music where groups of notes have to sound good together. For example, if we consider a base note C3, the fifth (G4) up from the octave (C4) also itself has an octave relationship with G3, which in turn has a strong relationship with C3. If you had a scale that repeated around a 3:1 ratio rather than 2:1, I don't think things would be as 'tight'.

Also, while your 'octave plus a fifth' relationship clearly doesn't have octave equivalence, the next harmonic would be a two-octave relationship - again, this isn't enough to say that the octave is qualitatively and distinctly special, but it does point to the strength of the octave compared to other ratios.

A natural occurrence of the octave as somewhat 'special' is with a flute-like instrument, where blowing harder gets you up an octave higher with each fingering - maybe this could be an influence on the adoption of octave repeating scales too.

one can for example take any piece of music consisting of two parts and translate one part an octave up, leaving the other the same, and the piece will still work.

Probably subjectively true in many cases, but as Kilian Foth pointed out in the comment, there are cases where it may not subjectively work as well, depending on the voicings, harmonic movements, and timbres involved.

In summary, I don't think you can say that an octave relationship displays an objective qualitative 'equivalence' that another simple ratio doesn't. It's more the fact that the octave relationship is stronger than other relationships that leads us to the idea of the octave-repeating scale, and to commonly-perceived subjective equivalence of the octave.

Answer 3

The frequency of a pitch is n. the frequency of a pitch an octave higher is 2n. So, yes the harmonics are going to be very similar, but the first harmonic of the original pitch IS the second pitch in frequency.

What you say about an octave and a half (but not exactly, that's a tritone) has caught out several singers in my past, where they pitch on a 4th or 5th, instead of the correct note, and they seem to be stuck in that new, but related, key. Odd.

EDIT: is it a coincidence that people sing in the octave most comfortable? As in, children will naturally sing one, sometimes two octaves above a tenor who is singing with them, not even giving it a thought. Likewise, the lower voices will drop the melody an octave automatically if singing along to something too high.

Answer 4

There are 2 sides to this question:

a) What is the same in the tones in an octaves, which isn't the same in other intervals? (physics)

b) Why are we able to percieve this? (psychology)

I'll try to answer the first part of the question: What really is the same are the overtones.

Suppose note 1 has a frequency of n then it's overtones are: 2n, 3n, 4n, 5n, 6n, 7n, 8n,... Note 2, an octave away, has frequency 2n and the overtones: 4n, 6n, 8n,... All the overtones of note 2 are present in note 1.

Now take note 3, a fifth away from note 1: It's frequency is 3n/2 and it's overtones are: 3n, 9n/2, 6n, 15n/2, 9n, 21n/2, 12n,... Only some of the overtones of note 1 are present in note 3. This is what makes it different.

But... there is also something in our brain that makes notes 1 and 2 more similar than note 3. Because when we hear pure tones without overtones (like generated by a computer) we still are able to registrate this 'sameness'. So while there is a physical reason for the sameness it doesn't have to be actually present in the sound. Evolutionary out brain has learned to associate a sameness to the octave that isn't there in the fifth. Why this is I can only guess..

Answer 5

Awesome question and sadly the significance of it seemed to be missed by a lot of people here. Saying its double the wavelength doesn't explain anything since light at double the wave length looks nothing alike. I've wondered this a lot. Its different to the question "why certain intervals sound better than others". A lot of people are trying for a false equivalence with these 2 questions. The later question clearly has a large content of subjectivity in it where-as 2 notes separated by an octave must be objectively similar in a deeper way since they offer no opportunity for harmonic clash. Adding an additional identical note but at another octave does nothing to change the major, minor, or key of a melody and yet every other note can. This is anecdotally proven by people singing along to a song in whatever octave they find most comfortable and nobody balking at their tone-deafness.

I have perfect pitch and even I find it hard to distinguish octaves occasionally when other perceptive elements come into play. For example, a baritone voice straining for a high note and a soprano singing this same note within their comfortable range may initially be perceived by me to be a different octave. This shows me that notes across octaves can be so similar that our brains are forced to look for other cues as to which octave is being voiced.

The reason octaves sound very similar must be because of how our ears/brains process sound. My guess is probably because our auditory cortex is very tiny compared to our visual cortex. As information gets more and more attenuated, our brain looks for ways to simplify. It will be looking for "sameness" across swaths of information that is actually very different. What better "sameness" to pick than something that is an exact multiple of another? Our brains can break out of an attempt to qualify it and just perceive it as "same but higher/lower". Consider how most people can't even tell any notes apart except when heard together within a short period of time. All these are clues on how limited our experience of music and sound is compared to our visual field.

Answer 6

For male voices (and perhaps sounds produced by other large beasts), the overtones or harmonics can be less attenuated than the fundamental pitch spectrum in certain environments and over certain distances. Human brains have been evolved to hear a male voice as the "same voice" even in those environments where the octave overtone and other harmonics propagate over distances far more strongly than the fundamental, or even when the fundamental doesn't carry at all. This tracking of a series of harmonics being recognized as the same "voice", with or without the fundamental pitch frequency being present in the spectrum, probably is part of the mechanism where humans might perceive some form of equivalence between a melody and the same melody an octave (or multiple octaves) up.

Answer 7

I've been meaning to make a test program, where one hears two sine waves (no harmonics to give away the octave) and attempts to tune them an octave apart. My vague memory of others having done this is that there is a tendency to make the octaves a bit wide. I'd like to replicate this and see if it is true or not.

If it is, there are two things that follow:

1. there's probably no neurological basis for octave receptors of any great precision (which I recall is generally accepted, but my study of the subject is from 1980's and my memory hazy);
2. when attempting to tune your instrument to others, it is best to try and listen to the fundamental tone rather than let yourself get caught up in the overtones which might nudge you sharp (a personal theory and experience).

EDIT (2021) I came across some very interesting research on the "stretched octave" phenomenon. Octave stretching phenomenon with complex tones of orchestral instruments I am wrapping up a pilot test of my own, am hoping to run it soon, pertaining to the interaction of timbre with pitch perception.

If there is a direct neurological basis for octaves (i.e., it's not just the result of learning), I guess it is possible that the biology could be off by a bit. The evolutionary genetics involved perhaps only needs to be fairly accurate, not precise.

Answer 8

I might be wrong, but I have thought of this myself and explained it to myself like this:

https://www.wolframalpha.com/input/?i=plot+%28sin+x+%2B+sin+2x%29%2F2,+sin+x

As you can see, a sum of a wave and its double is quite close to the original if you compensate for amplitude. So, singing along in a different octave will sound the same together if you compare it to the original.

But again, I might be wrong.

Answer 9

In addition to the geometrical explanations above, I'll just add this perhaps obvious perhaps accidental fact: women's voices and men's voices tend to be, on the average, about an octave apart. Women and men sing an octave apart in just about every musical culture of the world.

Answer 10

Really, the whole thing in overtones. When you hear 440hz you hear 220hz so. But usually you cannot to recognize 220hz because it is quieter.

Answer 11

Well, I am not sure but: If you have string like in a guitar or something. and you play it somewhere. The next octave will be for example a meter away and the third octave will be two meters away. That's more figurativ and somehow like that what Stian Yttervik said. And I am sure, you can see something like that if you open up the wing of a grand piano and watch the strings vibrating to the keys you play.