Kite's color notation/Temperament names: Difference between revisions

Line 25:

Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is triyo & ru = y3&rT. There are three obvious ways to choose the comma set for multi-comma temperaments. The '''odd name''' minimizes the [[Odd limit|double odd limit]] of the comma set. The '''subgroup name''' minimizes the number and size of the primes used by each comma, but not their depth. Assuming no wa commas, the 1st comma is ya, the 2nd za, the 3rd ila, etc. The '''prime/odd name''' minimizes the prime-limit for each comma, and for each prime limit uses the comma of least double odd limit. The 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. (There is also the '''hermite name''', formed by hermite reduction, which minimizes the prime-limit of each comma, and for each prime limit, tends to minimize the color depth. Finally the '''LLL name''' uses Lenstra–Lenstra–Lovász reduction, and tends to minimize the integer limit of each comma.)

The odd name for 7-limit [[Pajara]] is rryy&rT, the subgroup name is sgg&rT, and the prime name is sgg&rryyT. Often the three names are identical, e.g. y3&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The subgroup name shows relationships between bicolored rank-2 temperaments better. The question of which name to use is not yet fully resolved.

The odd name for 7-limit [[Pajara]] is rryy&rT, the subgroup name is sgg&rT, and the prime/odd name is sgg&rryyT. Often the three names are identical, e.g. y3&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The subgroup name shows relationships between bicolored rank-2 temperaments better. The question of which name to use is not yet fully resolved.

If the comma is wa, an edo is implied. The temperament is named after the edo, not the wa comma. Tempering out the pyth comma and 225/224 makes 12edo&ryyT. If the comma(s) don't include every prime in the subgroup, some primes are untempered. These primes are added with '''plus''': [[Blackwood]] is 5edo+yT = 5-edo plus ya. The 2.3.5.7.11 subgroup with 81/80 tempered out is g+z1aT = gu plus zala.

@@ Line 25: / Line 25: @@
 Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is triyo & ru = y<sup>3</sup>&rT. There are three obvious ways to choose the comma set for multi-comma temperaments. The '''odd name''' minimizes the [[Odd limit|double odd limit]] of the comma set. The '''subgroup name''' minimizes the number and size of the primes used by each comma, but not their depth. Assuming no wa commas, the 1st comma is ya, the 2nd za, the 3rd ila, etc. The '''prime/odd name''' minimizes the prime-limit for each comma, and for each prime limit uses the comma of least double odd limit. The 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. (There is also the '''hermite name''', formed by hermite reduction, which minimizes the prime-limit of each comma, and for each prime limit, tends to minimize the color depth. Finally the '''LLL name''' uses Lenstra–Lenstra–Lovász reduction, and tends to minimize the integer limit of each comma.)
-The odd name for 7-limit [[Pajara]] is rryy&rT, the subgroup name is sgg&rT, and the prime name is sgg&rryyT. Often the three names are identical, e.g. y<sup>3</sup>&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The subgroup name shows relationships between bicolored rank-2 temperaments better. The question of which name to use is not yet fully resolved.
+The odd name for 7-limit [[Pajara]] is rryy&rT, the subgroup name is sgg&rT, and the prime/odd name is sgg&rryyT. Often the three names are identical, e.g. y<sup>3</sup>&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The subgroup name shows relationships between bicolored rank-2 temperaments better. The question of which name to use is not yet fully resolved.
 If the comma is wa, an edo is implied. The temperament is named after the edo, not the wa comma. Tempering out the pyth comma and 225/224 makes 12edo&ryyT. If the comma(s) don't include every prime in the subgroup, some primes are untempered. These primes are added with '''plus''': [[Blackwood]] is 5edo+yT = 5-edo plus ya. The 2.3.5.7.11 subgroup with 81/80 tempered out is g+z1aT = gu plus zala.