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 22:10:33 UTC.
- The original revision id was 232643896.
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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 the product m of the odd primes less than or equal to p, and then find the [[Euler genera|Euler genus]] Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the k*N patent val maps the interval to a number divisible by k, and then find the corresponding [[Normal lists|normal interval list]]. When two successive values i and i+1 lead to the same normal list, return that as the canonical list of generators for the k*N subgroup.
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 the product m of the odd primes less than or equal to p, and then find the <a class="wiki_link" href="/Euler%20genera">Euler genus</a> Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the k*N patent val maps the interval to a number divisible by k, and then find the corresponding <a class="wiki_link" href="/Normal%20lists">normal interval list</a>. When two successive values i and i+1 lead to the same normal list, return that as the canonical list of generators for the k*N subgroup.</body></html>