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Minimal diamond-supporting 11-limit semistandard vals

🔗Gene Ward Smith <gwsmith@svpal.org>

6/29/2005 1:21:49 PM

The smallest 11-limit semistandard vals which distinguish all 29 notes
of the 11-limit diamond are

v51 = <51 81 118 143 177|

and

v52 = <52 82 121 146 179|

These both support miracle, and wedging them together and reducing
gives the wedgie for miracle. Next up is the standard 58 val, which of
course does not support miracle.

The TM basis for v51 is {121/120,225/224,243/242,385/384}, which is
the TM basis for miracle, plus 121/120. We also get it by adding
250/243 to miracle, but v51 does not support the usual version of
11-limit porcupine. Putting together {225/224,250/243,385/384} gives
another version of porcupine, <<3 5 16 -18 1 17 -39 23 -59 -106||,
which it does support, and which tuned to 22-edo is the same thing.

The TM basis for v52 is {225/224,243/242,363/350,385/384}, which is
miracle plus 363/350. In the 7-limit, it is miracle plus 648/625, and
it supports an exotic version of diminished which comes from
{225/224,385/384,648/625}.

Detempering either v51 or v52 will give an 11-limit scale containing a
diamond, which might be of interest.