why are there 12 chromatic notes in the Western musical scale?

Question

Octaves come from doubling the number of cycles per second (and psychologically sound 'similar') but why is the octave range divided by twelve, and not, say ten to derive the chromatic scale? Is it just arbitary? Or is there
a natural/mathematical reason for tthis choice?

Answer 1

Mathematically it arises like this. Take two notes an octave apart. Take an intermediate note 4/3 times the lower one - that's 2/3 of the upper one, and it sounds nice with both of them. Take another note 3/4 of the upper one and 3/2 times the lower one, and that sounds nice too, because frequencies in simple proportions sound nice.

Then, on a logarithmic scale, it turns out that if we decide to call the interval between these two intermediate frequencies a "full tone", it makes the bigger intervals on either side of it approximately two and a half tones. You might say this could have turned out to be anything, but two and a half is what it is closest to, and that gives us 2.5 + 1 + 2.5 = 6 full tones in the octave, or 12 semitones. Bingo.

Further analysis boils all this down to the mathematical fact that 2^19 = 524288 is very close to 3^12 = 531441, and it's that exponent 12 which is the really fundamental reason for the 12-semitone scale. There are no smaller whole powers of 2 and 3 which are quite as close as 19 and 12.

Answer 2

Pythagoras was the one who discovered that musical intervals described by the ratios of small whole numbers (thirds, fourths, fifths, etc) made sounds that were pleasant to the ear.

For hundreds of years notes were made in these exact ratios and were called 'pure.' Unfortunately, though, there is no set of exact ratios which combine together in a way to land on an exact octave, or double the original frequency. That meant that music could not be transcribed into different keys and still sound the same.

Someone in the 17th century (more or less) made the lucky discovery that if an octave was divided into frequencies that were successive multiples of the 12th root of 2, all these notes were so close to the 'pure' frequencies that the human ear could not tell the difference, and suddenly compositions could be played in any key one desired - which means you could start *anywhere* on the 12-tone scale and the song would remain the same, just a little higher or lower.

There is no theory on why these 12th roots of 2 work so well; it is simply an amazing coincidence that has made modern music possible.

Answer 3

This actually has a complicated technical answer. Culturally, monks discovered the set of intervals that make up the chromatic scale.

Mathematically, these intervals provide for please resonances when combined (unpleasing resonances are the result of certain frequencies of dissonance that occurs).

There are also 1/4 tone scales that are used in Hindu and Muslim cultures that are interesting to the ear. Everyone is familiar with 1/4 tones that are present in some of the Muslim prayer tones or chants that are used.

Answer 4

From http://en.wikipedia.org/wiki/Equal_temperament the reason is "Twelve tone equal temperament took hold for a variety of reasons. It conveniently fit the existing keyboard design, and was a better approximation to just intonation than the nearby alternative equal temperaments. It permitted total harmonic freedom at the expense of just a little purity in every interval. This allowed greater expression through modulation, which became extremely important in the 19th century music of composers such as Chopin, Schumann, Liszt, and others."

Some more historical description is at http://www.yuvalnov.org/temperament/

Answer 5

Can you imagine the implication of a 13th note? It would change the statistics in the same way as a seventh number in the lottery. What would the 13th note be? The whole tone of music would be completely different. It would probably be as distinctly different as Chinese music is to western music.

Answer 6

There are loads of different scales and the Western Scale has only been in existence for a short time (relatively)
Yamaha in their early synths gave a huge variety of choice.
I guess it was for commercial reasons as usual, a bit like digital is today. Sterile and devoid of ambiance.
Cloth ears abound.