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.