K*N subgroups
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2011-05-28 12:42:22 UTC.
- The original revision id was 232591164.
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Original Wikitext content:
For any [[Harmonic limits|prime limit]] p, EDO N and positive integer k, the p-limit k*N subgroup is the largest [[Just intonation subgroup ]] of the p-limit on which N-edo and k*N-edo approximate intervals to the same values using the mapping supplied by the [[patent val]] for k*N-edo. This also means they temper out the same commas. A procedure for finding the k*N subgroup is to take enough of the intervals of the p-limit which are mapped to a value divisible by k by the k*N patent val, and add to this set 2 and a basis for the commas of the k*N patent val, and then reduce this to a [[Normal lists|normal interval list]], giving the canonical list of generators for the subgroup. To get "enough" intervals, taking the [[Diamonds|diamond]] of the primes from 2 to p, which is the tonality diamond, usually seems to suffice, and taking the diamond of the diamond seems normally to be more than enough. For example, the 5-limit diamond is 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2. The intervals mapped to an even integer by <12 19 28| are 5/4, 8/5 and 2, and a basis for the 5-limit commas is 81/80 and 128/125. If we add 81/80 and 128/125 to the diamond and reduce to the normal list, we get 2.81.5 for the 2*6 subgroup, which is not correct; however the diamond of the diamond (or the diamond of the diamond of the diamond, etc.) gives the correct subgroup, 2.9.5. The diamond of the diamond of the diamond already contains a basis for the commas, and an alternative procedure would be to iterate the diamond construction until this happens.
Original HTML content:
<html><head><title>k*N subgroups</title></head><body>For any <a class="wiki_link" href="/Harmonic%20limits">prime limit</a> p, EDO N and positive integer k, the p-limit k*N subgroup is the largest <a class="wiki_link" href="/Just%20intonation%20subgroup">Just intonation subgroup</a> of the p-limit on which N-edo and k*N-edo approximate intervals to the same values using the mapping supplied by the <a class="wiki_link" href="/patent%20val">patent val</a> for k*N-edo. This also means they temper out the same commas. <br /> <br /> A procedure for finding the k*N subgroup is to take enough of the intervals of the p-limit which are mapped to a value divisible by k by the k*N patent val, and add to this set 2 and a basis for the commas of the k*N patent val, and then reduce this to a <a class="wiki_link" href="/Normal%20lists">normal interval list</a>, giving the canonical list of generators for the subgroup. To get "enough" intervals, taking the <a class="wiki_link" href="/Diamonds">diamond</a> of the primes from 2 to p, which is the tonality diamond, usually seems to suffice, and taking the diamond of the diamond seems normally to be more than enough.<br /> <br /> For example, the 5-limit diamond is 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2. The intervals mapped to an even integer by <12 19 28| are 5/4, 8/5 and 2, and a basis for the 5-limit commas is 81/80 and 128/125. If we add 81/80 and 128/125 to the diamond and reduce to the normal list, we get 2.81.5 for the 2*6 subgroup, which is not correct; however the diamond of the diamond (or the diamond of the diamond of the diamond, etc.) gives the correct subgroup, 2.9.5. The diamond of the diamond of the diamond already contains a basis for the commas, and an alternative procedure would be to iterate the diamond construction until this happens.</body></html>