K*N subgroups: Difference between revisions
Wikispaces>genewardsmith **Imported revision 232650890 - Original comment: ** |
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{{todo|explain its xenharmonic value}} | |||
For any [[Harmonic_limit|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 which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the [[ | A procedure for finding the ''k''*''N'' subgroup which seems to suffice is to take the product ''m'' of the odd primes less than or equal to ''p'', and then find the [[Euler genus]] Eu(''m''<sup>''i''</sup>) for integers 1, 2, and so forth. Here Eu(''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 {{nowrap|''i'' + 1}} lead to the same normal list, add to that a basis for the commas of the ''k''*''N'' patent val, and return the normal interval list for that. | ||
For example, to find the 7-limit 2*6 subgroup we first find {{nowrap|''m'' {{=}} 3 × 5 × 7 {{=}} 105}}. The subgroup of Eu(105) consisting of those intervals mapped to an even number by {{vector|12 19 28 34}} is 2.5.7. The subgroup of Eu(105<sup>2</sup>) is 2.9.5.7, not the same. However, the subgroup of Eu(105<sup>3</sup>) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime. | |||
[[Category:Algorithms]] | |||
[[Category:Math]] | |||
[[Category:Regular temperament theory]] | |||
Latest revision as of 07:58, 31 March 2025
For any 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 which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the Euler genus Eu(mi) for integers 1, 2, and so forth. Here Eu(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 interval list. When two successive values i and i + 1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.
For example, to find the 7-limit 2*6 subgroup we first find m = 3 × 5 × 7 = 105. The subgroup of Eu(105) consisting of those intervals mapped to an even number by [12 19 28 34⟩ is 2.5.7. The subgroup of Eu(1052) is 2.9.5.7, not the same. However, the subgroup of Eu(1053) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime.