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. | 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 | 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). | ||
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. | 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. | 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 | 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. | ||
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. | 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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There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc. | There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc. | ||
==Notating | ==Notating multi-EDO pergens== | ||
A | A multi-EDO pergen is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section. | ||
A | A multi-EDO pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples: | ||
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The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15. | The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15. | ||
All | All multi-EDO pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Multi-EDO pergens are a small minority of rank-2 pergens. | ||
It's possible to have a fifth-8ve pergen with an independent 5th, but there will be very small intervals of about 20¢. Here are two such: | It's possible to have a fifth-8ve pergen with an independent 5th, but there will be very small intervals of about 20¢. Here are two such: | ||
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| style="text-align:center;" | 64/63 | | style="text-align:center;" | 64/63 | ||
|} | |} | ||
Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A | Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A multi-EDO pergen should never have an uninflected genchain. | ||
==Notating non-8ve and no-5ths pergens== | ==Notating non-8ve and no-5ths pergens== | ||
In | In multi-EDO pergens, the 5th is present but not independent. In no-5ths pergens, the 5th is not present, and the prime subgroup doesn't contain 3. | ||
In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. | In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. | ||
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For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1. | For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1. | ||
Multi-EDO pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (WWM3/5, d5) can optionally be replaced too. | |||
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To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24. 11/9 also works, it yields 243/242. | To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24. 11/9 also works, it yields 243/242. | ||
If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a | If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a multi-EDO pergen. | ||
If the 1st edo is 7edo, the pergen indicates the natural heptatonic notation for the 2nd edo, e.g. 12edo = unsplit, 15edo = third-4th and 17edo = half-5th. | If the 1st edo is 7edo, the pergen indicates the natural heptatonic notation for the 2nd edo, e.g. 12edo = unsplit, 15edo = third-4th and 17edo = half-5th. | ||