EdXII

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Being the next most perfect perfect consonance after the octave, the twelfth is difficult to alter from the tritaveand and use as the base of a tuning of a scale. Here is a chart of why it is more logical to call the half octave an augmented fourth than a diminished fifth:

Octave 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192
Fifth 3 6

3/2

12 24 48 96 192 384 768 1536 3072 6144 12288
Fourth 4/3 8/3 16/3 32/3 64/3 128/3 256/3 512/3 1024/3 2048/3 4096/3