K*N subgroups
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-05-28 12:50:44 UTC.
- The original revision id was 232592028.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
For any [[Harmonic limit|prime limit]] p, EDO N and positive integer k, the p-limit k*N subgroup is the largest [[Just intonation subgroups|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, take the [[Diamonds|diamond]] of the primes from 2 to p, which is the tonality diamond, then the diamond of the diamond, and so forth, until the iterated diamond construction includes a basis for the commas of the k*N patent val. 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 contains a basis for the commas, and therefore the procedure described above would give the correct answer.
Original HTML content:
<html><head><title>k*N subgroups</title></head><body>For any <a class="wiki_link" href="/Harmonic%20limit">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%20subgroups">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, take the <a class="wiki_link" href="/Diamonds">diamond</a> of the primes from 2 to p, which is the tonality diamond, then the diamond of the diamond, and so forth, until the iterated diamond construction includes a basis for the commas of the k*N patent val.<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 contains a basis for the commas, and therefore the procedure described above would give the correct answer.</body></html>