Spectrum of a temperament

One of the things I like to do when analyzing a temperament I am considering for use is to look at its complexity spectrum. By this I mean sorting the complexity of the intervals in the p-limit or other odd limit tonality diamond between the unison and half an octave. In the rank two case, the complexity is Graham complexity, but for higher limits we an use OE temperamental complexity, which is proportional to Graham complexity in the rank two 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 I 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:

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

Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> Here's the spectrum for 13-limit history, the temperament tempering out 364/363, 441/440 and 1001/1000:
>
> 11/10, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 13/12, 13/10, 6/5, 8/7, 12/11, 13/11, 9/8, 16/13, 10/9, 7/6, 11/9, 14/13, 9/7

I should have gone up to the 15-limit; this is an archipelago temperament, so we want to see 15/13:

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

Gene wrote:

>I should have gone up to the 15-limit; this is an archipelago
>temperament, so we want to see 15/13:
>
>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

Nice idea, but for me it would be more useful to see it in terms
of odd numbers, maybe with 2 thrown in. Or perhaps triads of
odd numbers. -Carl

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

If you look at the spectra of rank five temperaments stemming from a single 13-limit comma, usually 3 is the least complex, often followed by 5. Sometimes 5 is the least complex, followed by 3. Here are exceptions, which includes the island comma, 676/675:

169/168
16/13, 4/3, 14/13, 13/12, 5/4, 8/7, 13/10, 18/13, 16/15, 6/5, 15/13, 9/8, 7/6, 11/8, 7/5, 13/11, 12/11, 10/9, 15/14, 11/10, 9/7, 14/11, 15/11, 11/9

676/675
15/13, 4/3, 13/12, 5/4, 13/10, 16/15, 18/13, 16/13, 9/8, 6/5, 8/7, 7/6, 11/8, 7/5, 15/14, 12/11, 14/13, 10/9, 9/7, 11/10, 15/11, 13/11, 11/9, 14/11

6656/6655
11/8, 4/3, 12/11, 5/4, 13/11, 16/15, 6/5, 8/7, 14/11, 11/10, 9/8, 7/6, 11/9, 16/13, 15/11, 7/5, 13/12, 10/9, 15/14, 13/10, 9/7, 15/13,
14/13, 18/13

There are also some where 3 is the least complex, but then comes the higher limits:

364/363
4/3, 11/8, 5/4, 14/11, 13/11, 8/7, 16/15, 6/5, 12/11, 7/6, 11/10, 9/8, 16/13, 7/5, 13/12, 15/11, 15/14, 10/9, 9/7, 13/10, 11/9, 15/13,
18/13, 14/13

847/845
4/3, 13/11, 5/4, 11/8, 16/13, 8/7, 16/15, 6/5, 12/11, 13/12, 7/6, 11/10, 9/8, 14/13, 7/5, 15/11, 15/14, 13/10, 10/9, 11/9, 18/13, 9/7, 15/13, 14/11

10648/10647
4/3, 13/11, 11/8, 12/11, 5/4, 8/7, 16/15, 6/5, 14/11, 9/8, 16/13, 11/10, 11/9, 7/6, 15/11, 7/5, 13/12, 10/9, 13/10, 15/14, 9/7, 15/13, 14/13, 18/13

Similar things happen with the 11-limit:

121/120
11/8, 12/11, 11/10, 4/3, 5/4, 11/9, 6/5, 8/7, 14/11, 9/8, 7/6, 7/5, 10/9, 9/7

243/242
11/9, 4/3, 12/11, 9/8, 5/4, 6/5, 11/8, 8/7, 7/6, 10/9, 11/10, 9/7, 7/5, 14/11

4000/3993
11/10, 4/3, 5/4, 11/8, 6/5, 12/11, 8/7, 9/8, 7/6, 7/5, 14/11,
10/9, 11/9, 9/7

43923/43904
14/11, 4/3, 11/8, 5/4, 8/7, 12/11, 7/6, 6/5, 11/10, 9/8, 7/5, 9/7, 11/9, 10/9

Then there's:

1331/1323
4/3, 12/11, 14/11, 11/8, 5/4, 11/9, 8/7, 6/5, 9/8, 11/10, 7/6,
7/5, 10/9, 9/7