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    Presumably the latter, but I could be wrong. – Sophie Alpert Apr 26 '11 at 19:44
  • In addition to some good answers here - this book provides a fairly good explanation amazon.com/dp/0962949671/?tag=stackoverfl08-20 – Bella Jun 24 '11 at 19:51
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11 Answers 11


This requires an excursion into musical history.





There are equal temperaments other than the 12 tones per octave that will sound fine, but they don't generally have a integer number of notes per octave. Wendy Carlos experimented a lot with this, and made such scales as the Gamma scale with a slightly mind-boggling 34.29 notes per octave.

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    Actually this was known before Pythagoras. He was just the first whose followers wrote it down. Also, modern theory shows that small integer ratios are only applicable to harmonic sounds. Inharmonic sounds or sounds with only odd harmonics produce different scales. – endolith Jul 23 '12 at 18:10
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    Here's an image of just vs ET side by side flic.kr/p/7rNope – endolith Dec 22 '16 at 0:45
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    "But just tuning has a problem: you can basically only play the scale that the instrument is built for, because the intervals between the notes are different": actually, if you're playing music with harmonies of the sort that emerged during the European Renaissance, you can't even use just intonation if you stick to a single key, unless you avoid certain chords in that key. This answer skips the important and long-lasting period of unequal temperaments, which lasted from the beginning of the 16th century into the 19th, before the revival in the 20th. – phoog Oct 8 '19 at 3:38

This question on math.se is quite similar to what you're asking and the answers give a lot of detail:

Mathematical difference between white and black notes in a piano?


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  • I think this quote does not apply or explain the question. There is no coincidence here. It is by construction. – user50691 Dec 18 '19 at 20:21
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    @ggcg That the n-tone equal-tempered scale consists of frequency ratios of 2^(j/n) for integer values of j is by construction. That 2^(7/12) and 2^(5/12) are good approximations to 3/2 and 4/3, and that there are no similarly good approximations of these ratios in 11- or 13-tone equal temperament is a fact. And not a coincidence--it relates to the continued fraction of the base-2 logarithm of 3. That 2^(4/12) is a decent approximation to 5/4 is, however, a coincidence as far as I can see. Special properties of the number 12 are what make 12-tone equal temperament work reasonably well. – Will Orrick Dec 20 '19 at 9:30

Two points that may have not been completely answered.

  • Why is C major the reference scale for natural tones ?





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    In fact, only ut through la come directly from the hymn, which ranges only from C to A, but that was fine since the system that used these syllables comprised overlapping six-note scales called hexachords; these syllables were used alongside the letter names of the seven-note scale that seems to have preceded them. Ut was applied to F, C, or G. Si was added later when the hexachord system broke down and the syllables were applied to the seven-note scale. The major scale did not really exist at that time, however, since there were only four authentic modes and their plagal counterparts. – phoog Oct 8 '19 at 3:59


So why twelve notes?


This page (from which I quoted) provides greater detail: http://thinkzone.wlonk.com/Music/12Tone.htm

  • I don't think the twelve-tone scale was introduced as an equal-tempered scale. However, I imagine twelve fifths (of some size) would make a fairly "uniform" scale. – awe lotta Dec 30 '19 at 3:28










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    The frequencies don't need to be multiples of each other; they need to share a small common mutiple. See my answer here. – user28 Dec 9 '13 at 5:28
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    @nonpop is right. If we divide the octave into n equal intervals, it is not important for n to have many factors. 16et has no usable approximation to a perfect fifth. 30et has no intervals better than those of 15et, whose best fifth is 18 cents wide (12et's is 2 cents narrow). On the other hand, some equal temperaments with excellent intervals have prime n, for example 19et, 31et and 53et. – Rosie F Nov 13 '16 at 12:53
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    Yeah, I agree with @nonpop. There is something incorrect about this answer. None of the 12TET intervals "line up", the just tuning provide perfect alignment but causes other issues. The 12TET is a compromise. I've known people with perfect pitch who claim that ALL of the 12TET intervals sound dissonant. – user50691 Dec 18 '19 at 20:25

The reason is THE BRAIN. THe brain likes frequencies that are simple proportions. It thinks they go together. You should really ask, first, why are there octaves?

Well the octave represents a doubling / halving of hertz (cycles per second).

So, midi middle C is 256 hz, and if you know your computer numbers, you'll realise that the next octave C's are at 512, 1024, 2048, etc and the lower octaves are at 128, 64, and (pimp your ride) 32.

Earthquakes, by the way, show up at around 11 hertz.

Every society starts with the octave. 'Cos 1/2. Got it?

(I propose that the 2nd Viennese school abandon the octave by the way, and also tuning the instruments. Niether make any sense for them. The current state of things with octaves and tuning and suchlike is pure hypocrisy. Let it go, boys! Also scores. And playing in public. Nobody comes anyway.)

Hh HHm...

How to divide the octave?

If we start it on C and divide it into 3 (which is a nice brain-friendly proportion) we will get a lovely 3 note scale:

C, E, G#, C

How about dividing it in to four:

C, Eb, F#, A, C

"THat's nice", says the brain, "but it's too SYMMETRICAL. Both of these scales just seem to go on forever and ever, I can't tell what's what. I know! Why don't you mix and match the proportions so they are slightly more uneven? Then I can figure out the bass note.".

And thus was born the "Proto Major Thingy":

C, E, G, C

and the "Proto Minor Thingy":

C, Eb, G, C

"Hang on a bit", says the brain, "you missed a note, didn't you?".


"Between G and C, I'm pretty sure you had something between G and C".

C, E, G, A, C ?

"Thas NICE! Rock and Rollish. Go on then, what about the other one?"

C, Eb, G, Bb, C ?

"Hey, what's with the Bb? We never heard that before. What kind of proportion is that?"

"It's 10/12ths".

"You mean 5/6ths. Alright. Play it again".

C, Eb, G, Bb, C

"Kay, that's bluesy. All right! But its 70,000 years ago and there's loads of poor bastards buggerin' round the the scenery getting crunched and munched by sabre tooth tigers and suchlike. Lotta funerals. Mucho sadness. Like Trump nowadays, you should know! Need variety. "


"Show me."

C, D, E, G, A, C
C, D, E, G, Bb, C
C, Eb, F, G, Bb, C
C, Eb, F, G, A, C

"Whats the F proportion?"


"Great! I like it. 5 notes. Let's give it fancy Greek name. Tart it up a bit. Penta...?"


"That's wonderful".

"I was kidding. You know, too literal..."

"Nevermind. It's awesome. we'll go with Pentatonic. More! We NEED MORE! Now there's chieftains, mud huts, jewelry"

"I need some rules".

"kay. Er.. keep the Minor third or the Major third and the Fifth where it is, and just move the others about... I know, like this: move the seventh up, the sixth down, the fourth up, and the second down!"

C, D, E, G, A, C
C, D, E, G, Ab, C
C, D, E, G, Bb, C
C, D, E, G, B, C
C, Eb, F, G, Bb, C
C, Eb, F#, G, Bb, C
C, Eb, F, G, A, C
C, Eb, F#, G, A, C
C, Db, E, G, A, C
C, Db, E, G, Ab, C
C, Db, E, G, Bb, C
C, Db, E, G, B, C

"Hey, then if we superimpose them all we''ll get 12 subdivisions of the octave! Brilliant!"

C, Db, D, Eb, E, F, F#, G, Ab, A, Bb, B, C

"That's why I'm called the BRAIN, son. Oh, and you're welcome."

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    I appreciate the humor (right up my alley) but it may be a bit overdone for this site. What do you mean by "divide the C into 3?" – General Nuisance Dec 29 '16 at 5:38
  • @GeneralNuisance Likely means split the octave into three equal parts. – user45266 Apr 5 '19 at 4:14
  • Actually, in equal temperament, middle C is 261.63 Hz. – phoog Oct 8 '19 at 4:22
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    I do not think the premise is sound. – user50691 Dec 18 '19 at 20:14

For western music the greeks were the first to figure out the math that occurs naturally in the harmonics overtones generated by horns and other wind instruments. The Greeks applied the same mathematical ratios (golden ratio) to strings. Pythagoras invented the pythagorean tuning of (3:2) perfect fifths and Octaves (2:1) to match naturally occurring harmonic overtones. Later the greeks invented 7 modal scales based on pythagorean tuning. Seven Modes with eight notes in a scale. These scales were Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian. We still use Ionian (Major) and Aeolian(Minor). The flaw with natural harmonics is that the octaves between each mode were slightly off from each other. Aristoxenus in the 4th century BC invented the 12 tones between octaves in an attempt to use the same ratio between each note. Later Keys were invented to use these 12 tones as a home base for each scale. The problem was that by nature these keys are slightly off from each other. To solve this J.S. Bach in the early 1700's promoted the use of the Tempered Scale. He equalized the natural occurring gap between each of the twelve semitones. Brass instruments in the baroque period had a bag of different sized crooks to adjust for each key that they performed in. String instruments also had to retune for each key change. By using the tempered scale a performer could switch between all of the different keys without re-tuning.

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    Okay, good history, but why did Aristoxenus decide on 12 rather than 13 or 11? – user45266 Apr 5 '19 at 0:08
  • Aristoxenus wanted to use the same ratio of 3/2 math.uwaterloo.ca/~mrubinst/tuning/12.html explains the math behind it. – Stan Lyman Apr 5 '19 at 0:27
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    You should explain that in your answer, then. – user45266 Apr 5 '19 at 4:24
  • This answer has many incorrect statements. The golden ratio does not generally appear in harmony. Greek modes did not include Ionian or Aeolian (and Greek modes are not the same as those we learn today by those names; the Greek names were applied to four of those modes in the middle ages, while Aeolian, Ionian, and Locrian were developed later). There are 7 distinct pitches in a scale, not 8. Temperament was invented long before Bach, and the temperament favored by Bach was not equal. Brass crooks have nothing to do with temperament, and strings did not need to retune for each key change. – phoog Oct 8 '19 at 4:21

A simple picture is sometimes better than a huge explanation, so i'd also encourage to check the graphs in this link, you can mouseover the 10edo to the 19edo for example to see the differences between various divisions: http://www.tonalsoft.com/enc/e/edo-11-odd-limit-error.aspx ( just look at strongest consonances : 3 - 1/3** , 5 - 1/5 and 3/5 - 5/3 , the rest of the graph is really not important in comparison.)

Basically what it clearly shows, is that the 12-notes division is the only one to make the ratios 3/2 and 4/3 (the most importants*** after the octave) almost pure. And the thirds/sixths (ratios with the number "5", the next most importants***), are not so bad neither. No other division by a fair number of notes, 10 to 19, can even slightly approach this. This is mathematically remarquable and the reason why we use 12 notes and not 13, 11, or etc.

** ("1/3" just mean a 4/3 ratio with 2 octaves shifts, it's just the way they originally present the numbers.)

*** (What i mean is that if your brain wants to easily recognise and remember music, you rather need a big bunch of fifths, fourth and thirds to be more or less in tune, in your musical architecture, even melodic, otherwise it's mostly dissonant sounds, leading to noise, and hard to remember for your brain...)





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  • Distinct means where a change from one note to another is clearly identified. The more the divisions in a scale, the less distinct the notes become. Dissonant intervals maybe are easily identified as they are jarring, but in terms of how brain like harmony, the 7 intervals are musical and naturally melodic. Try singing a dissonant tune, and a melodic tune and you will know which one feels easier. pentatonic is a subset and has more distinct intervals that all the 7 notes of the scale. If you decided to add more stops in a scale like 20 for example, it will naturally become one long yawn – srinivas Oct 10 '19 at 7:56

Based on your wording of the question I would say that it is by design. It is not a coincidence that 12 half steps fit into an octave rather than 11 or 13. Though the details may change if one assumed just tuning I will explain assuming equal tempered tuning. First you should know that there is a continuum of frequencies and therefore pitches between any two notes. We have converged on a particular choice of pitch combinations for the western diatonic scale through centuries of experimentation. The notes in a scale reflect what is pleasing to the ear(s) for a particular culture. Over time Westerners standardized the half step by splitting the octave into 12 steps using the relation

f_octave = 2*f_tonic

they imposed the constraint that the ratio of two consecutive half steps be the same no matter where you start,

f_1/2 = r*f_tonic (this would be a minor second)

since we are forcing the number of 1/2 steps from tonic to octave to be 12 we get the relationship

r^12 = 2 or r = 2^(1/12)

IMO a few posts here are putting the cart before the horse. You cannot demonstrate that the octave has only 12 semitones using the above definition of a semitone. Rather you ask what does the ratio have to be in order to ensure there are 12 in an octave.

To that end there are all sorts of alternate chromaticisms that attempt to place N equal steps in an octave. These result in the tuning equation,

r = 2^(1/N)

There is a 24 TET containing 24 equal quarter steps in an octave. And you absolutely could build a scale with

r = 2^(1/13)

or some other root of 2. Of course these would NOT be 1/2 steps in the traditional sense of the term. Now the issue of how we got there is a longer story. Before 12TET tuning the Just major scale with 8 notes (including octave) have more than 5 accidentals. You can google this and find Wiki articles on the topic but there were, I believe, just scales with as many as 17 independent notes in the octave. Though all consecutive notes are probably slightly different ratio. Hence not really a 1/2 step. What you call a 1/2 step depends on how you learned the term.

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