EdXII

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