Deriving music scale from harmonies

on August 21, 2011 in General

In the western world we have the predominance of the heptatonic scale.

I always wondered why the Major and the Minor scale are the dominating scales, whether one can derive this from some simple principle. It is kinda strange that there are 5 whole tone steps and 2 halftone steps:
Whole:Whole:Half:Whole:Whole:Whole:Half

One explanation is the use of the perfect 5th (Pythagorean tuning).
The interval of a perfect fifth, corresponds to a ratio of 3:2 of the frequencies. This ratio is experienced as quite harmonic.

The ratio of 2:1 is one octave.
So if we repeat perfect fifths and transpose them back into the octave, i.e. into values between 1 and 2 (by dividing by 2), we get the following values:

1. 3/2 = 1.5
2. 3/2 * 3/2 / 2 = 9/8 = 1.125
3. 3/2 * 3/2 * 3/2 / 2 = 27/16 = 1.6875
4. 3/2 * 3/2 * 3/2 * 3/2 / 4 = 81/64 = 1.265625
5. 3/2 * 3/2 * 3/2 * 3/2 * 3/2 / 4 = 243/128 = 1.8984375
6. (3/2)^6 / 8 = 729/512 = 1.423828125

When we order these values we get
0. < 2. < 4. < 6. < 1. < 3. < 5.
1 < 9/8 < 81/64 < 729/512 < 3/2 < 27/16 < 243/128 < 2
1 < 1.125 < 1.265625 < 1.423828125 < 1.5 < 1.6875 < 1.8984375 < 2

the multiplicative differences in each step are
9/8 / 1 = 9/8
81/64 / (9/8) = 9/8
729/512 / (81/64) = 9/8
3/2 / (729/512) = 256/243
27/16 / (3/2) = 9/8
243/128 /(27/16) = 9/8
2 / (243/128) = 256/243

Now 9/8 corresponds to a whole tone and 256/243=1.053497942386831... corresponds to a half tone.
So we have the characteristic pattern of 3 and 2 whole tones seperated by a half tone.

f=1, g=9/8, a=81/64, b=729/512, c=3/2, d=27/16, e=243/128, f=2
Or if we start with c=1, by multiplying with 2/3:
c=1, d=9/8, e=81/64, f=4/3, g=3/2, a=27/16, b=243/128, c=2

The following tones are
7. (3/2)^7 / 16 = 1.06787109375
8. (3/2)^8 / 16 = 1.601806640625
9. (3/2)^9 / 32 = 1.20135498046875
10. (3/2)^10 / 32 = 1.802032470703125
11. (3/2)^11 / 64 = 1.35152435302734375
12. (3/2)^12 / 128 = 1.0136432647705078125

The last value is called the Pythagorean comma.

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