Why is a doubling of frequency called an octave?

Question

I understand that when two frequencies have a ratio of 2^n:1 for any natural number n, they are said to be n octaves apart. Moreover, they sound "essentially the same" to our ear. However, why do we split an octave into eight notes in the first place? Consider two frequencies of 220 Hz and 440 Hz. How do musical systems select frequencies between these limits in a suitable manner?

On a related note, I sort of understand why the chromatic scale divides the range between 220 Hz and 440 Hz into 12 equal parts. This was nicely explained in this video and has to do with the fact that the twelfth roots of two are very close to rational numbers p/q for small p and q.

I do not understand where the octave (specifically, the number 8) fits into this picture. I am not a musician so if possibe, please define musical terms in terms of physics e.g. frequencies and harmonics.

Note: The answers here did not answer my question. The question is why 8 notes similar to how there exists an answer for why the chromatic scale has 12 intervals per octave.

Although it may have to do with perception, there is still a mathematical, physics, and biological consideration to this question. — Bingohank, Apr 17 '20 at 21:47
I really don't see how the linked question isn't a duplicate. It goes over where the octave comes in the same terms as below. — Dom, Apr 17 '20 at 22:53
@Dom the answer here has a lot of details that none of the answers there (I read through all of them when Googling this) have. Specifically, the why divide into 7 parts is answered here with reference to frequency ratios which I don't see there. — user1936752, Apr 17 '20 at 22:58
@user1936752 why are scales 7 notes is a separate question than than where the term octave came from which we've answered before here: https://music.stackexchange.com/questions/32971/why-does-the-scale-have-seven-or-five-notes-why-not-six and why there are 12 notes in an octave https://music.stackexchange.com/questions/24/why-are-there-twelve-notes-in-an-octave. We have all of these already answered on the site. — Dom, Apr 17 '20 at 23:02
@Dom those links (and links therein) are indeed useful. Unfortunately, I did not find them when I asked this question but thanks for pointing them out — user1936752, Apr 17 '20 at 23:25
Recent research in neuroscience suggests that the octave is NOT the same to our ears and that this is cultural, imposed on us as children. There are cultures where this is not the case and people do judge notes that are 2*f0 as being the same or having any quality that is similar. — , Apr 18 '20 at 00:24
@ggcg: we always new that other cultures are not so smart like us! ;) (joke!) Now I understand why my female teacher colleagues said I was asking too much when I proposed them to sing an octave lower! - But how about the octave transposition of brassinstruments with similar fingerings? — Albrecht Hügli, Apr 19 '20 at 06:22
I understand the need to understand the rationale behind all the naming conventions very well. However, you should keep in mind that much of the terminology was made up over multiple centuries, starting in the Middle Ages. So explanations involving modern physics, although useful, do not describe how the terms were conceived. This means that some of the question you ask won’t be answered satisfactorily, for example: the number 8 shows up simply because it was the eighth interval a guy who didn’t know the number zero could form using the scale system passed down to him – no physics there. — 11684, Apr 19 '20 at 11:16
On the passed down scale system: https://music.stackexchange.com/a/23953/3113 — 11684, Apr 19 '20 at 11:20
@11684. Great point. There has been an attempt by Helmholtz to provide physics based explanations of western music theory and they are somewhat successful. But the fact is that it evolved over 100s of years, is cultural by nature and Helmholtz's premise was ethnocentric. — , Apr 19 '20 at 12:29
@AlbrechtHügli brass (and woodwind) instruments can also be (and are) constructed with identical fingerings at relative pitches other than one octave. The saxophone family is an example. — phoog, Apr 19 '20 at 18:45
What I mean is: in China, India, Egypt, Greek, Israel and Rome they had woodwind and brass instruments but no Saxophones and to say that they didn‘t identify the octave as the same note is probably more speculative than my theory of 7 degrees inspired by the 7 weekdays and by the 7 planets. — Albrecht Hügli, Apr 19 '20 at 20:08
It's just the stupidity of one-based numbering. A pitch that rises by two diatonic notes is a "third" interval. We end up with an ascent of every 7 notes called octave (8), and then need a "rule of 9" to invert intervals. — Kaz, Apr 20 '20 at 15:25
@phoog Maybe you were thinking of clarinet? Because sax fingering is identical for octaves. (The two middle octaves, anyway.) — RedSonja, Apr 21 '20 at 05:07
@RedSonja No, I am thinking of saxophones. The assertion I was responding to concerned "octave transposition of brass instruments with similar fingerings," which I took to mean that a B-flat tuba has similar fingerings to those of a B-flat trumpet. Similarly, an E-flat saxophone has similar fingerings to a B-flat saxophone: transpose up or down by a fourth or fifth, and keep the same fingerings by switching the instrument. For that matter, a C trumpet has similar fingerings to a B-flat trumpet, so I am baffled by the comment. Perhaps I misconstrued it. — phoog, Apr 25 '20 at 02:14
No, I misunderstood you, quite right. Or you can read it both ways. I'd like to add that I play bari sax is E-flat, and I can play trombone music in the bass clef just by mentally removing 3 flats. Which adds a bit of interest. — RedSonja, Apr 27 '20 at 04:51

Athanasius · Accepted Answer · 2020-04-18T01:52:43.813

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First, let's be clear that the standard (major) musical scale divides the octave into seven parts, not eight. The word "octave" comes from eight, because a unison (two notes sounding at the same frequency) is considered to be a "prime" or kind of a "one" in the system, rather than zero. Thus, the first interval created between a note and the next note above it is called a "second," even though it's only one interval.

This is just a historical convention for naming musical intervals, which sort of originated by counting both endpoints in an interval. For example, a musical "fifth" (frequency ratio 3:2) is the interval created by two notes that are four steps apart. So, C-D-E-F-G creates a fifth between C and G. The interval is five notes, but there are only four "steps" (of various sizes).

Anyhow, you need to get past that quirky numbering system first. So, I believe your question then becomes: why do we divide the octave into seven parts?

There are many longer answers here on this topic. But the gist is that like the octave (frequency ratio 2:1), small whole-number ratios of frequencies are often heard as "consonant." So the ratio 3:2 between frequencies sounds good (and, as noted, creates an interval called a perfect fifth), as well as the ratio 4:3 (the so-called perfect fourth).

The ancient Pythagoreans recognized the importance of these intervals (2:1 octave, 3:2 perfect fifth, and 4:3 perfect fourth). They also recognized that the difference between the size of the 3:2 and 4:3 ratios was useful, an interval with a frequency ratio of 9:8, which eventually became known as a "whole step."

So far, if you combine these ratios within one octave, you can build a perfect fifth and fourth up from the bottom note, as well as a perfect fifth and fourth down from the top note. The meet in the middle around a 9:8 "whole step." In musical notes, this would for example outline the notes E-A-B-E within an octave. (As I noted in comments, this overall can be thought of as a 12:9:8:6 ratio between four notes, which was how the Pythagoreans thought of it.)

Again, all of this dates to ancient Greece and is built on fundamental consonant intervals/ratios. The question then is how to fill out the rest of the notes within that octave. And the Greeks had a lot of answers to that, with a lot of different tuning systems.

But one possibility that they settled on was called a "diatonic" system, which literally means "through whole tones" (i.e., those 9:8 ratios I mentioned). They took that interval that originated as a difference between the 3:2 fifth and the 4:3 fourth and started tuning 9:8 "whole steps" to create a scale, starting at the top.

In musical terms, this was like going E-D-C down the scale. But they had already built B and A after that. So then once they got to A, they built more whole tones going down A-G-F. Then you had a complete scale going down an octave: E-D-C-B-A-G-F-E. Most of the intervals were those 9:8 "whole tones." But a couple (C-B and F-E) were smaller intervals with really odd mathematical ratios. In effect, the Greeks didn't care about those ratios so much: they just cared about tuning the other notes, and those left over small bits were kind of like the "errors" that were left over in tuning.

What you then have is seven intervals within an octave creating the diatonic scale, which survives to the present day.

As to how this relates to the 12-note chromatic scale -- well, some Greeks (particularly a guy named Aristoxenus) realized that those little left over bits in the scale were about the size of intervals that could almost divide the octave into 12 equal bits. They were a little bit off, though. Similarly, those little bits were roughly one half the size of the 9:8 whole tone interval.

Over the centuries once harmony developed more, there were various reasons that the 12-note chromatic equal division seemed to be better than the one constructed with the simple 3:2, 4:3, and 9:8 ratios. That's a much more complex story (the story of musical temperament).

But hopefully this explains the gist of why a 7-interval division of the "octave" developed historically.

edited Apr 18 '20 at 01:52

answered Apr 17 '20 at 22:00

Athanasius

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Thank you for the answer. I have no musical education so pardon my basic questions as I try to get through your answer. What exactly did you mean by "If you combine these ratios within one octave, you can build a perfect fifth and fourth up from the bottom note, as well as a perfect fifth and fourth down from the top note"? I've got my two frequencies on either end of the octave, say 1 and 2. Which numbers in between are you selecting that satisfy the perfect fifth and perfect fourth ratios? – user1936752 Apr 17 '20 at 22:21
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No problem -- I tried to avoid a lot of music terminology but it was hard to avoid it completely. So, think of your octave frequencies as 6 to 12. (it's a bit easier with the numbers than 1:2). In that case, you can create a 6:9 perfect fifth (2:3), a 6:8 perfect fourth (3:4). From the top (12), you have 8:12 (a 2:3 fifth) and 9:12 (a 3:4 fourth). So, you have overall the frequency ratios 6:8:9:12, with the 8:9 whole step in the center. Does that make sense? – Athanasius Apr 17 '20 at 22:30
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Wonderful! Thank you - it'll take me a while to fully understand your answer but it's a goldmine of information :) – user1936752 Apr 17 '20 at 22:32
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Re: "seven parts, not eight". If you don't divide the octave at all, how many parts there are: zero or one? – piiperi Reinstate Monica Apr 18 '20 at 09:50
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@piiperiReinstateMonica the name octave comes from medieval mathematical theory in which there is no zero. This is the same reason for the French word for "fortnight" being *quinze jours* (fifteen days). The other issue is that of counting *points* (pitches) vs counting *spans* (intervals): the number of the former is greater by one. So in a system with no divisions of the octave, an octave comprises *two* pitches, a single pitch class, and one interval, analogous to the diatonic octave's eight pitches, seven pitch classes, and seven intervals. – phoog Apr 18 '20 at 16:33
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@piiperiReinstateMonica: Umm, one? I'm not sure what the point of your comment is. The diatonic scale divides the octave into seven parts, no? (I.e., 5 whole steps and 2 half steps.) – Athanasius Apr 18 '20 at 19:44
@Athanasius regarding your last comment, why do you say 5 whole steps and 2 half steps? I assume this part of your answer is related "Then you had a complete scale going down an octave: E-D-C-B-A-G-F-E. Most of the intervals were those 9:8 "whole tones." But a couple (C-B and F-E) were smaller intervals with really odd mathematical ratios". Are the C-B and F-E the half steps you meant? However, those ratios aren't really "half" the whole step of 9:8, are they? – user1936752 Apr 18 '20 at 22:17
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@user1936752: I was using modern musical terminology to explain my answer to another commenter. Modern music terminology generally assumes the 12-fold equal division you discussed in the question, where the C-B and F-E literally are half the size of the other steps. But no, in the ancient Greek tunings, those smaller intervals are not exactly half the larger "whole tones." Nevertheless, they were *about* half the size and became known as "semitones" or "half steps." At first, this terminology was approximate; in modern musical scales, it's often exact. – Athanasius Apr 19 '20 at 02:22
@phoog: *eight days a week!* – Albrecht Hügli Apr 19 '20 at 06:24
@Athanasius The comment was stupid stuff, I wrote one sentence and then something came up and apparently I had pressed Enter and didn't realize I had actually written the comment. The answer to the rhetorical question might have been, if you don't divide the octave, there are ... two points, the start and end point. If you see an interval as a frequency jump. ;) But one part. – piiperi Reinstate Monica Apr 19 '20 at 11:41
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Another point of view to the 7 vs 8 discussion: If we consider an octave divided into 7 parts, then 10 octaves is divided into 70 parts, but if we consider an octave divided into 8 parts, then 10 octaves is divided into 71 parts. One of these seems to make more sense. – JiK Apr 19 '20 at 14:05
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@JiK: Yes. I think the confusion (to the extent there actually is any here) is about what a "part" is. I'm referring to a "part" as an *interval*, which is what makes most sense when talking about "dividing" the octave. But the name of the octave comes from the number of *notes* contained within an octave span. As every interval has to have a start point and an end point, the number of *notes* will always be one more than the number of *intervals*. – Athanasius Apr 19 '20 at 15:01
A similar issue applies to floor numbers in buildings but there are two common conventions. In the US, the ground is usually called 1 and the first above it is 2. You climb 7 flights of stairs not 8 to get the 8th floor. Climb the same amount again and you get to the 15th and not 16th floor (actually you may get to 16 because 13 may be skipped). Over in Europe, floor 1 is the first above ground . Climbing 7 flights of stairs gets you to the 7th floor and another 7 flights gets you to the 14th (13 is not skipped so often). Musical intervals are like US floor numbering (except for 13). – badjohn Apr 20 '20 at 09:38

score 12 · Answer 2 · answered Apr 18 '20 at 02:47

The name "octave" comes from eight notes. (Like using "ocho días" as well as semana to mean week in Spanish.) Sometimes endpoints are counted. The octave is the eighth note (there are seven different diatonic notes.) Language need not be mathematically consistent. Still, the term "unison" seems more descriptive than "zero" or "nihil" for an interval of two equal notes. Linguistically, the term "unity" or "one" or "unison" have been used for centuries (and in different languages) to represent sameness.

Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/106946/discussion-on-answer-by-ttw-why-is-a-doubling-of-frequency-called-an-octave). — Dom, Apr 20 '20 at 13:18

score 7 · Answer 3 · answered Apr 18 '20 at 10:03

If we knew only the pentatonic scale the octave would be probably called the sixth. As the ancient Greeks had a scale of two tetrachords: (tetra = 4, chords = strings and as 2x4=8 it was later been called octava.

https://en.wikipedia.org/wiki/Musical_system_of_ancient_Greece

Actually there where 7 steps and the eighth degree was identical with the first. The system was constructed that the degrees found place between the lengths of a string and it's half length (which was identified as the same sound - but higher: this could have been 12, or even 32 because the Greek also knew 1/4 tones.)

Why 7 and not 5? or 9? or 12? or 32? or any other combination of tones and semitones? why is a tone a tone and a semitone is what it is? you can continue the questioning and researching.

I always thinks there are two reasons:

a) the 12 tones developed by the 7 fifths by Pythagoras. b) the 7 "planets"

In classical antiquity, the seven classical planets were (are) the seven moving astronomical objects in the sky visible to the naked eye: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. The word planet comes from two related Greek words, πλάνης planēs (whence πλάνητες ἀστέρες planētes asteres "wandering stars, planets") and πλανήτης planētēs, both with the original meaning of "wanderer", expressing the fact that these objects move across the celestial sphere relative to the fixed stars.1[2] Greek astronomers such as Geminus[3] and Ptolemy[4] often divided the seven planets into the Sun, the Moon, and the five planets.

https://en.wikipedia.org/wiki/Classical_planet

But what was first? the seven planets or the seven degrees? or the seven weekdays -> Week= Hebdomas (griech. εβδομάς): Sun-day, Moon-day, Mars-day, Mercur-day, Jupiter-day, Venus-day Satur-day, the other names are derived from German Gods

What was first: the 12 months or the 12 semitones between the repetition of a similar sound?

And the question should be: why has the division of a string length (or tube, or iron hammer (aerophones and idiophones) and its half length respectively half weight organized in seven intervals like it is - so that the repetition of the identical sound-impression is exactly an octave (an interval of eight degrees built by the seven other intervals, which have been derived from the overtone series and the rations of the string division.)

I think the answer to this question will be one of the seven world wonders ... hem, I mean the eighth.

"what was first? the seven planets or the seven degrees?": I have never heard this before. I can't find any evidence to suggest that the numerical similarity is anything other than coincidence. — phoog, Apr 18 '20 at 16:45
Now you have heard it, and if you had heard it before you wouldn‘t have heard it before then. ;) — Albrecht Hügli, Apr 18 '20 at 19:48
"Actually there where 7 steps and the eighth degree was identical with the first." I'm not sure this is true in any meaningful sense. While the Greeks recognized the specialness of the octave and its ratio, they didn't employ octave equivalence in the way Western music has since about the 11th century. The eighth degree had a special relationship with the first degree, but to the Greeks, it wasn't "identical" in any meaningful fashion. — Athanasius, Apr 18 '20 at 19:50
You mean they didn‘t have a sensorium to identify the similarity of the 1/2 string/tube length sound? — Albrecht Hügli, Apr 18 '20 at 19:57
@Athanasius is there any culture in which adult men singing with women or children do not sing an octave below them? — phoog, Apr 19 '20 at 23:38
"Now you have heard it, and if you had heard it before you wouldn‘t have heard it before then": but I've read a good deal of early music theory, and I suppose that if this idea had any basis in reality I would have encountered it before last week. — phoog, Apr 19 '20 at 23:41
@phoog isn't it quite common in most cultures for people with different voice ranges singing together to harmonize in all kinds of well-sounding intervals other than an octave? — Peteris, Apr 20 '20 at 00:01
@phoog: recognizing the consonance of an interval (and/or using said consonance in a practical fashion) is different from equating two notes as being "identical," which is the world Albrecht used. As Peteris notes, many cultures also tend to harmonize at various intervals. I don't necessarily know of a culture that actively *shuns* the octave, but choosing it as one possible harmonization interval is different from claiming two obviously distinct pitches are "identical." (My point was mostly about word choice -- as I said, the Greeks understood the special place of the octave in scales.) — Athanasius, Apr 20 '20 at 02:46

score 4 · Answer 4 · answered Apr 17 '20 at 21:53

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People found that using frequency ratios based on small integers (3/2, 4/3, 5/3 ...) produced pleasing results. They put these notes together and came up with a seven note scale that sounded good. The eighth note they called the 'octave'.

answered Apr 17 '20 at 21:53

PiedPiper

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Thanks for your answer. However, this doesn't quite explain why this led to the choice of the seven note scale. Could one, say, drop one of the notes or add another suitable frequency that satisfies the small integer ratio rule? This process would then divide the "octave" into an arbitrary number of notes. – user1936752 Apr 17 '20 at 21:56
Yes. You can take integral fractions and they will be multiples, ie. harmonics (5/2 f = 5*f/2). By the way, refering to the question,you can tell the difference between frequency octaves so I don't know what is meant by "essentially the same". – Bingohank Apr 17 '20 at 22:02
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@user1936752 Seven gives a result that works well. That's why the system has lasted so long. Adding or subtracting notes doesn't give a 'balanced' scale. You might want to try experimenting with different scales. – PiedPiper Apr 17 '20 at 22:03
You can not quite choose arbitrary frequencies, you can choose infinitely many divisions but they must be harmonics. The primary harmonics have greatest amplitude therefore are more apparent. – Bingohank Apr 17 '20 at 22:08
I rather suspect that the scales existed before the theoretical explanation of why particular notes were used. – phoog Apr 18 '20 at 16:35
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This is wrong. When ethnomusicologists study the scales used in various cultures, they do not find that this holds true. It doesn't even hold true for the major and minor scales. On the other hand, the perceptual similarity of notes differing by an octave *is* found to be a cross-cultural universal. – Apr 18 '20 at 19:50
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Do you have sources for your claim? As far as I know, it's very easy to show the mathematical, objective reason why there are 12 chromatic notes in an octave by using only 3/2 (=fifth) and 2/1 (=octave) ratios. Choosing 7 diatonic notes for a scale out of those 12 chromatic notes was purely subjective, though, and cannot be explained by math alone. – Eric Duminil Apr 18 '20 at 19:59
@BenCrowell This question (and my answer) is about the seven note scale used in western culture (and some others). – PiedPiper Apr 19 '20 at 06:52
@EricDuminil I don't claim there is a mathematical objective reason. The choice of seven notes was entirely pragmatic, but it's no coincidence that the notes are based on integer ratios. – PiedPiper Apr 19 '20 at 07:05
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@EricDuminil: If one repeatedly goes up by a 3:2 ratio, dropping back an octave as needed to say within a single octave, one will end up placing seven pitches before one would have to place a pitch between two pitches that are only a step apart. – supercat Apr 20 '20 at 00:36
@supercat ooooh. Excellent. Thanks. This is the first plausible mathematical explanation I've read for 7 notes. All the others I've seen had hidden historical or cultural assumptions, like starting with already named notes. – Eric Duminil Apr 20 '20 at 00:44
@supercat note that there's still an arbitrary decision : half steps are allowed, but not when consecutive. Or is there a mathematical justification for this choice? – Eric Duminil Apr 20 '20 at 06:52
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@supercat While that's an excellent explanation I suspect that the seven note scale had been decided on before people realized this. – PiedPiper Apr 20 '20 at 09:12
@PiedPiper: If one builds scales using 3:2 and 2:1 intervals, and wants to have a scale where consecutive notes have two kinds of intervals, that will occur when scales have two, three, five, seven, or twelve notes. If one starts building a scale at F, the eighth different note would be F#, which is further from F than to G. – supercat Apr 20 '20 at 16:44

score 3 · Answer 5 · edited Jun 17 '20 at 08:26

Lol, I can't use mathjax over here, as I would have if answering on Physics SE. But the answers here may have more musical background.

Whole number ratios work well because the wave forms combine to make repeating patterns, which the ear can process easily, meaning they sound good.

The chromatic scale is used because particular powers of 2^1/12 are very close to the ideal whole number ratios. We have

C: 1

C♯: 2^1/12 = 1.059

D: 2^2/12 = 1.122, close to ⁹⁄₈

D♯: 2^3/12 = 1.189, closish to ⁶⁄₅

E: 2^4/12 = 1.259, closish to ⁵⁄₄

F: 2^5/12 = 1.335, very close to ⁴⁄₃ (perfect fourth)

F♯: 2^6/12 = √2 = 1.414,

G: 2^7/12 = 1.498, very close to ³⁄₂ (perfect fifth).

For the purpose here, very close means that the combined frequencies (as in a chord) produce a slowly varying wave pattern, rather than a chaotic pattern heard as a dischord. No scale produces perfect combinations for all notes, but the chromatic scale maintains the same relationships whichever note one takes as the first "doh", which is important to music, but also imperfections to any scale are important because they help to convey mood.

The "white notes" C–D–E–F–G–A–B from the chromatic scale form the diatonic scale (seven notes + 1 for the octave), which can also be obtained from a sequence of perfect fifths

F—C—G—D—A—E—B

The real reason for adopting the chromatic scale has to due with instruments with fixed tuning, like the piano. Many other scales have been used, with different numbers of notes. Wikipedia lists:

Chromatic, or dodecatonic (12 notes per octave)

Octatonic (8 notes per octave): used in jazz and modern classical music

Heptatonic (7 notes per octave): the most common modern Western scale

Hexatonic (6 notes per octave): common in Western folk music

Pentatonic (5 notes per octave): the anhemitonic form (lacking semitones) is common in folk music, especially in Asian music; also known as the "black note" scale

Tetratonic (4 notes), tritonic (3 notes), and ditonic (2 notes): generally limited to prehistoric ("primitive") music

Monotonic (1 note): limited use in liturgy, and for effect in modern art music

I would say "tolerably close for some purposes" rather than "very close." In particular, the cube root of 2 is not particularly close to 1.25 and the fourth root of 2 is not particularly close to 1.2. — phoog, Apr 18 '20 at 16:37
True enough, and I though I had written "closish" rather than close, but I must have lost that somewhere. In music one is not just interested in the base note, but also in the relationship of notes to each other, which makes it a bit more complex. — Charles Francis, Apr 18 '20 at 16:55
-1 because this answer doesn't answer the question, because it is about 12-equal temperament, which is not relevant. — Rosie F, Apr 18 '20 at 17:32
@RosieF, good point. I had meant to cover the relationship (now added) but it looks like I ran out of steam. — Charles Francis, Apr 18 '20 at 17:59
You have it backwards. You start with an already constructed diatonic scale, start on 'F' (why?) and stop on 'B' (why?). You could also apply the circle of fifths to the whole chromatic scale, and "prove" that the diatonic scale should have 12 notes too. — Eric Duminil, Apr 18 '20 at 20:05

Why is a doubling of frequency called an octave?

5 Answers5

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