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frequencies increase exponentially

Here's 5 octaves of scale step 1s, using 1 as the base frequency:

octave freq power of 2
5 16 2^4
4 8 2^3
3 4 2^2
2 2 2^1
1 1 2^0

So we can use binary logarithms to linearize the frequencies. Here's two octaves:

step log(2) of freq
1 1
7 1.907
6 1.737
5 1.585
4 1.415
3 1.322
2 1.17
1 1
7 0.907
6 0.737
5 0.585
4 0.415
3 0.322
2 0.17
1 0

The spacing of the scale steps is clearer, for instance, when you use the binary logarithm of a frequency, rather than the frequency itself:

step log(2) of freq difference, in log(2), between steps
1 1
0.093
7 0.907
0.17
6 0.737
0.152
5 0.585
0.17
4 0.415
0.093
3 0.322
0.152
2 0.17
0.17
1 0
  • Now we can see that
    • the scale steps are fairly evenly spaced
    • the gaps between scale steps 1 and 2, and between 4 and 5, and between 6 and 7 are the same
    • the gaps between scale steps 2 and 3, and between 5 and 6 are the same
    • the gaps between scale steps 3 and 4, and between 7 and 1 (in the next octave) are the same