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The Consistency of Cangwu Badness
x31eq
One of the desirable features about cangwu badness is that there’s a strong relationship between the best temperaments of different ranks. There are geometric reasons for this, but to see it take a look at the standard search the temperament finder does, 11-limit 5-cent target:

http://x31eq.com/cgi-bin/pregular.cgi?limit=11&error=5.0

The best equal temperaments are 31, 22, and 12.

The optimal rank 2 temperament is Orwell. It includes the two best equal temperaments: 31 and 22. The second best rank 2 class is Meantone. It includes the first and third best equal temperaments: 31 and 12. So the best three equal temperaments lead to the best two rank 2 classes. Also, the best four rank 2 classes all have the best equal temperament in common: 31.

The three best equal temperaments lead to the best rank 3 temperament: Minerva. The next best rank 3 class, Thrush, includes the first, second, and fourth best equal temperaments: 15, 31, and 12. The intersection between these two rank 3 classes is Meantone — only the second best rank 2 class, so the relationship isn’t perfect. Four of the top 3 rank 3 classes do include Orwell: Minerva, Marvel, and Zeus. The top 4 rank 3 classes include Orwell, Meantone, and Valentine as intersections -- the best three rank 2 classes. The next rank 3 class, Prodigy, also gives Miracle (with Marvel). So if you had been searching for rank 2 classes as intersections of the best rank 3 classes, the search would also have been efficient.

The top 6 rank 3 classes — and all but one of the top nine — include 31-equal, the best equal temperament.

All of this doesn’t mean the rank 3 badness is correct in terms of musical utility. All it shows is that the badness of the rank 3 classes is tied to that of the lower ranks. So, if the rank 2 badness makes sense, at least the rank 3 classes will represent unions that may allow for hyper-modulations from one rank 2 class to another.

11-limit rank 4 temperament classes are simple unison vectors. The best four here — 176:175, 441:440, 225:224, and 540:549 — define 31-equal. The subsets define:

176:175, 441:440, 225:224 — Meantone

176:175, 441:440, 540:539 — Myna

176:175, 225:224, 540:539 — Orwell

441:440, 225:224, 540:539 — Miracle

Not a perfect match for the rank 2 ordering but the best two still come out.

Add 100:99 to the mix and the equal temperaments 31, 12, 22, 27e, and 41 come out as subsets. These include the best three from the full search, and only 15 is missing from the top six (385:384) would give it.

I seed my searches with equal temperaments because that’s generally the most efficient way of doing it. Lower ranks are more interesting and you only need pairs of equal temperaments to find rank 2 classes. The number of unison vectors you need goes up as the number of primes rises. Also, equal temperaments are easy to find because you can iterate through octave divisions and the number of plausible mappings for each is low — especially when the badness criterion is strict. Cangwu badness guarantees that you can find the best n equal temperaments.

Cangwu badness makes higher rank searches more efficient by allowing you to discard poor looking intermediate results. The same trick would work if you started with unison vectors. In this case, the best six ratios would get you a long way. For rank 2 combinations, you’d be taking 4 combinations of 6 to give 15 options. But you needn’t look at all of those because some rank 3 results would have been discarded. The unison vectors should have been easy to find, as well, as they all correspond to ratios of two or three digit numbers. The best 10 are still this simple, and give you a larger search space: 210 possible rank 2 cases, but you wouldn’t need to look at all of them.

Then 3025:3024 comes in, so you do have to check fairly high, so I still think equal temperaments are better.

Cangwu badness provides an efficient way of finding good temperaments (by its own assessment). It’s most efficient at finding the best (all but one of the top twelve ratios in this search are unison vectors of 31) and less so for the mediocre ones (if you keep finding 31, it stops you finding other things).

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