Kite's thoughts on pergens: Difference between revisions

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In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes.

For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#~~Further Discussion-~~Notating multi-EDO pergens|multi-EDO pergens]] below).

For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.

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If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in lemba, where ^1 equals 64/63 minus 81/80.

Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[~~pergen~~#~~Further Discussion~~-~~Notating multi-EDO pergens~~|multi-EDO pergens]] pergens below.

Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below.

Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + 7c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic.

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 In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes.
-For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Further Discussion-Notating multi-EDO pergens|multi-EDO pergens]] below).
+For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).
 To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.
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 If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in lemba, where ^1 equals 64/63 minus 81/80.
-Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[pergen#Further Discussion-Notating multi-EDO pergens|multi-EDO pergens]] pergens below.
+Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below.
 Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + 7c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic.