# Why do octaves sound equivalent?

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.

• It's not true that octave-shifting a voice in a piece of music is always possible. It can turn a rule-abiding, harmonious counterpoint into a non-conforming and strange-sounding one. – Kilian Foth May 25 '16 at 7:10
• Have a look at the waveforms / frequency spectra. – OrangeDog May 25 '16 at 12:16
• I'm not convinced that they do sound the same. – Carl Witthoft May 25 '16 at 13:23
• Related: Why do harmonics sounds good? – Édouard May 25 '16 at 14:41
• @CarlWitthoft: Octave equivalence isn't the statement that notes separated by an octave sound the same. If I play 1000 Hz followed immediately by 2000 Hz, all humans can tell that there was a big change in pitch. Octave equivalence says that these two pitches are perceptually similar in a variety of ways. Depending on context, they can be easily confused with one another. Across all musical cultures, there is a strong tendency for them to be considered as having equivalent musical functions. Untrained singers singing in unison will actually often sing in octaves, without knowing or caring. – Ben Crowell May 25 '16 at 17:40

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.

• There has been quite a bit of work on the perception of other intervals. The results are qualitatively different than the results for octaves. Perception of non-8ves is different in different cultures and different between musicians and untrained people. Untrained people basically judge non-octave intervals to be consonant or dissonant based on whether they have harmonics that clash (frequency difference of >~1% and <~10%). E.g., they will judge sine waves at pitches of C and F# to be consonant. – Ben Crowell May 25 '16 at 17:32
• @BenCrowell References? esp. on "Perception of non-8ves is different in different cultures and different between musicians and untrained people" – Dave May 25 '16 at 17:37
• You may find these helpful. Kameoka, A. & Kuriyagawa, M. (1969a). Consonance theory, part I: Consonance of dyads. Journal of the Acoustical Society of America, Vol. 45, No. 6, pp. 1451-1459. Kameoka, A. & Kuriyagawa, M. (1969b). Consonance theory, part II: Consonance of complex tones and its computation method. Journal of the Acoustical Society of America, Vol. 45, No. 6, pp. 1460-1469. Cross-culturally, e.g., look at the difference between the treatment of intervals in European common practice, Gregorian chant, and gamelan music. – Ben Crowell May 25 '16 at 23:10

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.

• "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." Not for all wind instruments: clarinets for example overblow at a 12th... – Tim H May 25 '16 at 11:23
• @TimH I guess I'm not thinking of the clarinet as 'flute-like' - as I mentioned, the clarinet is a closed-pipe instrument, which will lead to the different overblowing behaviour as well as the particular harmonic series. – topo Reinstate Monica May 25 '16 at 13:01
• Clarinets have open pipes. Unlike the base of an organ flue pipe, however, the reed end of a clarinet acts as a non-inverting pressure wave reflector/amplifier. When a pressure wave reaches the far end, it is reflected, inverted, so each round trip will invert the wave once (at the open end). In a closed organ flue pipe, each round trip inverts the wave once, but at the base. – supercat May 25 '16 at 15:55
• There is a lot of evidence that octave equivalence is universal rather than culturally specific, that it occurs in both musically trained and musically untrained individuals, and that it has a neurological basis. The situation for other intervals, such as fifths, is qualitatively different. This answer sounds plausible, but it's simply wrong based on the scientific data. Dave's answer is correct. – Ben Crowell May 25 '16 at 17:26
• @Supercat whether you choose to call them "open" or "closed", both clarinets and saxophones have pipes with the same type of single-beating reed at one end. The reason one overblows at the octave and the other at the twelfth is because the bores are conical and cylindrical, not because of the reed. Also open and closed organ flue pipes only overblow at the octave and 12th in elementary physics textbooks. If you actually measure the frequencies of the harmonics of real organ pipes, they can be very different from the simple theory, as organ builders have known empirically for centuries. – user19146 May 25 '16 at 19:33

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.

• This is an important point in this question; that the frequency is doubled for each octave. – awe May 25 '16 at 10:49
• Octaves, 4ths and 5ths are the most consonant intervals: their waves harmonize best together physically, and also to our ears. Apparently our sound perception system follows the laws of nature, but I have never seen an explanation why exactly. – reinierpost May 26 '16 at 19:57

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..

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.

• There is no wavelength of visible light (390nm - 700nm) that is still visible when that wavelength is doubled (780nm - 1400nm), so we have no idea how light looks different when its wavelength is doubled. It's not a valid analogy. Your point about the role of psychoacoustics is valid, and just as much as we can't ignore the psycho- part (which you have reminded us), we also can't ignore the -acoustics part (which is part of the content of other answers). – Todd Wilcox May 26 '16 at 16:37
• @ToddWilcox Did you doctor the alleged range of the visible light spectrum to try prove your assertion that we can't visibly compare doubled light frequency? The lower end of the visible light spectrum (380) is in no way similar to the high end of red 760. Every source I googled had a wider visible spectrum than the one you presented. – Mike S May 26 '16 at 17:28
• I just pulled it off Wikipedia, which yes, isn't often the best source, but I assumed there wouldn't be too much variance. On the other hand, this page lists five other sources, and only one of the five lists a wavelength range where double the shortest is less than the longest (380 - 780 nm). Either way, the way the rods and cones in the iris "decode" light wavelengths is so different from how the basilar membrane in the ear "decodes" audio wavelengths, I still assert the analogy is not useful. – Todd Wilcox May 26 '16 at 18:01
• It does look like the CIE Standard Observer is assumed to have a range of 380 - 770nm as shown in the CIE 1931 Color Space. This color model, due to the use of the tristimulus model, does highlight the great difference between how the three types of cones work and how the basilar membrane works. This is also shown by there being no hearing equivalent to the line of purples. – Todd Wilcox May 26 '16 at 18:16
• Perfect 4ths and 5ths do nothing to make something major or minor. The difference in timbre between a high male voice and a low female voice may be what is muddying the water for you when trying to establish octaves sung - unison or not. – Tim May 30 '16 at 17:45

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 these environments where the octave and harmonics becomes much stronger than the fundamental, or where 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 frequency 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.

• SInce the 3rd harmonic is an octave plus a fifth, this might be why some learners accidentally sing or pitch a fifth up. – hotpaw2 May 25 '16 at 17:25
• This is somthing I mentioned in my answer. – Tim May 25 '16 at 22:22

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.

• But... the same would be true (even more so) for "non octaves" like: sin(x) + sin(3x/2). – Tim H May 26 '16 at 11:54
• I'll try to explain better then. To me the difference is striking between your example and mine. – Stian Yttervik May 26 '16 at 12:03
• Well, clearly not skilled enough in wolframalpha to visualize it online, and since I might be both wrong (will contemplate that) or not able to visualize it better I will leave it at that. Sorry, and thanks for the food-for-thought. – Stian Yttervik May 26 '16 at 12:17
• No, no, don't be sorry. Maybe I'm wrong... – Tim H May 26 '16 at 12:24

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.

• This hardly addresses the actual question. – Tim May 30 '16 at 17:47
• I disagree. The fact that men and women all over the world sing together an octave apart, and consider themselves to be singing "the same notes", seems to me to be likely part of the reason. – Scott Wallace Jun 1 '16 at 18:33
• The fact that they sing in octaves seems like it might result of octave equivalence rather than a cause. To clarify, were these people's voices measured independently, or is it measured while they are singing together? – awe lotta Dec 5 '19 at 17:21
• @awelotta - Both. While it varies worldwide, speaking voices of men and women are roughly an octave apart in adults. Singing voices also, whether solo or together. Obviously, the octave equivalence effect reinforces this, but it's also in part a biological accident. – Scott Wallace Dec 7 '19 at 14:07

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).

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.

• Igor, welcome to the site! Can you elaborate on your answer? Usually if your answer this short it means there is more that you could say on this but you're not saying it. We would rather have the whole story in each answer. What do you mean by "when you hear 440 Hz you hear 220 Hz so"? Do you have any sources you can site that help explain your answer? – Todd Wilcox May 27 '16 at 15:04

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.