Complexity spectrum

The complexity spectrum of a temperament is a sequence of q-odd-limit intervals between the unison and half an octave sorted by their temperamental complexity, where q is two less than the next prime after the prime limit of the temperament in question. In the case of rank-2 temperaments, the complexity is Graham complexity, but for higher limits we can use the octave-equivalent TE seminorm, which is proportional to Graham complexity in the rank-2 case, but is also valid for higher limits.

The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.

Here's the spectrum for 11-limit marvel:

5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11

You can see it favors 5 over 7 and 7 over 11; for how much we could stick in the actual numerical complexities, but you can see that 9/8 and 10/9 are more complex than some 7 and 11 limit intervals just from the above.

Here's the spectrum for 13-limit history, the temperament tempering out 364/363, 441/440 and 1001/1000 which is part of the Archipelago:

11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7

Even leaving aside the somewhat greater complexity and accuracy, it just will not taste the same.

External links

• Yahoo! Tuning Group | Spectrum of a temperament – Gene Ward Smith's original post