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I know that one scale consists of 12 half-tones. But my question is still: Why? Why not 13 or 11?

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Do you mean "given the interval we call the 'half step,' why do 12 of them make an octave" or "given the interval we call the 'octave,' why do we split it into 12 half steps"? –  rshallit Apr 26 '11 at 19:41
    
Presumably the latter, but I could be wrong. –  Ben 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 –  DRL Jun 24 '11 at 19:51
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Another in-depth answer can be found here. A nice demonstration of other tunings is here. –  Michael Litvin Dec 7 '13 at 21:30

6 Answers 6

up vote 45 down vote accepted

This requires an excursion into musical history.

Originally, instruments were made to simply play notes that sounded "right" together. Why some notes sounded right and others wrong wasn't of great concern for most of humanity's history, until Pythagoras, (yes, the guy with the theorem) noticed that it had to do with intervals, and made a music theory based on perfect fifths. This theory had its problems, however, and was improved upon by later people, eventually ending up on what is called a "just intonation"

Basically, notes sound harmonious if the frequency of the notes is close to a simple interval, like 3/2 or 5/4. These theories were important because it meant it was possible for different instrument makers to make instruments that could play scales together, thereby making orchestras.

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. If you play a tune on the wrong scale, it will sounds out of tune. This means that if you want to sing along with the instrument, you have to find a singer whose range fits the song in the scale the instrument is built for. You can't transpose the song to fit the singer. Also, musicians were exploring the limits of what you could do with just intoned instruments.

So out of this came then the equal temperament. It splits the scale into equal intervals, meaning you can transpose a tune into other keys, and also means you can do dramatic chord changes and other interesting things. You can indeed split the octave into 11 or 13 notes if you should wish to do so, but to most people it will sound out of tune. But when you split it into 12 notes, you get close enough to the seven notes of just intonation for it to be bearable, except to some unlucky few supposedly burdened with overactive perfect pitch. The five tones that are in between the basic seven are, as expected, called "half-tones".

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

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Would you happen to know of any software that would allow me to play around in different temperaments, preferably taking input through a MIDI keyboard? –  Babu Apr 28 '11 at 23:39
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@Babu: with a little editing, this should be asked as a separate question. Playing with different temperaments and microtone music is of general interest for many musicians. –  ogerard Apr 30 '11 at 8:42
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there was a lot of practical and theoretical exploration going on for centuries but equal temperament came specifically out of the standardization of keyboard instruments (especially church organs), the question of fretted instruments and the renewal of a mathematical approach of tonality (see Mersenne treatise for instance) –  ogerard Apr 30 '11 at 8:47
    
@Babu: Yes there is. Do I know of any off the top of my head? No. Would it be an good question? Yes. –  Lennart Regebro Apr 30 '11 at 9:18
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That's the whole point. Small integer rations = harmonic sound. I don't see what is modern with that. :-) And how do you know people knew it before Pythagoras if they didn't write it down? –  Lennart Regebro Jul 23 '12 at 20:29

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?

What's going on here is a massively convenient mathematical coincidence: several of the powers of 2^(1/12) happen to be good approximations to ratios of small integers, and there are enough of these to play Western music.

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Yep, that's basically the explanation for the ratios I mentioned :) –  Matthew Read Apr 26 '11 at 19:57
    
Indeed, but the linked answer gives way more detail. Your answer is good, though. –  Ben Alpert Apr 26 '11 at 19:58
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Yeah, I went "Shoot!" after you posted this because I had seen that (great) Math answer before, and forgotten about it. –  Matthew Read Apr 26 '11 at 20:01
    
I think more fundamentally, (3/2)^12 (129.75) is close to a power of two (128). Thus, the fifths on a 12-note equal-tempered scale have a ratio of 1.498:1 (ideal would be 1.5:1), which is closer to perfect than for any other reasonable number of notes. –  supercat May 19 '13 at 17:47
    
I've read discussions of 19-TET (19-tone equal temperament) in which a diatonic scale would have five "large" intervals of 3/19 octave and two "small" intervals of 2/19 octave. Such a scale would be amenable to normal music notation if one regards e.g. C# and Db as being 1/3 step apart. The biggest oddity would be that key signatures with up to nine sharps or flats would be distinct (rather than having C#/Db, F#/Gb, and B/Cb as pairs of sound-alike key signatures). –  supercat May 19 '13 at 17:51

Two points that may have not been completely answered.

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

    The anglo-saxon notation obscures the history a little. Tradition from church music led in Italy (then shortly after France and Spain) to naming notes of the reference major scale by conventional syllables: Ut Re Mi Fa Sol La Si (this corresponds to C D E F G A B) coming from the latin lyrics of a very well known piece of that time. The latter single-letter notation takes another starting point, but the reference character of the C major scale has persisted across Occidental countries even if you can find evidence of notations and keyboards using other notes as reference. One of the main influences has been the construction of keyboard instruments (notably the church organ). The current keyboard layout is a compromise between the typical width of the hands, playing the Ut (now mostly called Do or C) major scale easily and having access to all semitones and a few other things. Other designs have not been as successful.

    You also have to know that the theorization and standardization of music at least up to the 19th century was made under the patronage of the churches (orthodox, catholic, reformed, ...) pushing for uniformity. The nineteenth century has seen an even larger standardization and internationalization of tuning, music teaching and piano domination as reference and composition instrument. The last three centuries have progressively suppressed or put into oblivion most of the divergent traditions (as to scales, modes, tuning) in Europe. Nowadays, people learning about music are taught as an evidence the C major scale as a foundation of music theory and the minor scale and his variants is not always treated fairly.

  • Why is there a semitone between E & F and B & C and not elsewhere ?

    There are several scales/modes outside of the major scale, with a varying number of notes, where the semitones are not placed between the 3rd and 4th note and between the 7th and 8th. The three minor scales (harmonic, ascending, descending) for instance, but also dorian, phrygian, you can read an encyclopedia article about them.

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Excellent, I learnt a lot. Thanks. –  Chiron Dec 13 '11 at 23:56

It has to do with harmony. Notes clash the least when their frequencies match up. For example, a note and its octave match up every two cycles, or a 2/1 ratio. Other ratios that sound good are 3/2, 4/3, 5/3, 5/4, 6/5, and 8/5; these are called the basic consonant intervals. Intervals that clash are the dissonant intervals.

So why twelve notes?

The twelve-tone equal-tempered scale is the smallest equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation — within one percent — and contains more consonant intervals than dissonant intervals.

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

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I might be wrong, but I think the question is why just E and F, and B and C, rather than two other tones. Your answer is correct in the forming of the major scale, but it answers more to why there is a half tone between third and fourth, and seventh and octave (independently of the notes' name) . This is the same in C major. So your answer, plus why the choice of C major to define the name of the natural tones, would be complete I believe. –  Gauthier Apr 26 '11 at 19:58
    
@Gauthier I thought the question was more oriented towards the second part, and that "the note names are just tradition" (AFAIK) wasn't really interesting. –  Matthew Read Apr 26 '11 at 20:01

When two notes are played together, they sound pleasing only if their wave curves come together every few cycles. We call them harmonic sounding.

If the wave curves never come together, or don't do so within a few cycles, they sound discordant.

Wave curves will only come together if the two frequencies are multiples of each other. For example, if one frequency is 200 cycles per second and the other is 600 cycles per second, their sound curves will coincide exactly 3 times every second, and they will sound harmonic.

By dividing each octave into 12 intervals, you maximize the number of pleasingly sounding pairs of notes. That is because the number 12 is divisible by more small numbers than any other number less than 60. It is divisible by 1,2,3,4,and 6. The number 60 would allow more pleasing combinations (1,2,3,4,and 5), but it would be ridiculous to divide an octave into 60 intervals.

So in modern western music they use 12 intervals. That provides the maximum number of pleasingly sounding combinations to create harmony.

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I don't see why the divisors are important here. Because for example the equal tempered tritone has a frequency ratio 2^(6/12) which is one of the worst approximations (compared to just intonation) in the scale whereas the perfect fourth (2^(5/12)) is one of the best (see the link in Matthew's answer). Another little comment: If one frequency is 200Hz and another is 600Hz then, assuming they're synchronized, they will be in the same phase 200 times every second, i.e. every 3rd cycle of the faster one. –  nonpop Jun 20 '13 at 8:23
    
The frequencies don't need to be multiples of each other; they need to share a small common mutiple. See my answer here. –  Matthew Read Dec 9 '13 at 5:28

Great answer by @john Baldwin above. Jut wanted to add that these minimum divisions are also the most practical to use. Taking the case of singing for instance between one note say C and its higher Octave C, 7 intervals produce the most distinct sound, plus 5 sharps and flats = 12.

And then if we start dividing it further it slowly starts getting very fine sub harmonies for the human hearing to discern. And these 12 divisions then also repeat in the higher and lower octaves and so on.

The easiest to identify is 4 divisions which is a divisor of 12, which makes up a pentatonic scale with the higher note, and is why is easily enjoyable.

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This doesn't make a lot of sense to me. What do you mean by "distinct"? I would think that consonant intervals are less distinct that dissonant ones, for example, and the twelve tone scale is designed around consonant intervals. Sharps and flats aren't something you can disclude when counting intervals either, unless you're working within a particular key or harmonic theory or seomthing (and you haven't specified one). Finally, how can 7 intervals produce "the most distinct sound" if 4 (or rather 5) intervals are "the easiest to identify"? –  Matthew Read Dec 8 '13 at 4:53

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