User:TallKite/Temperament Template Proposal
The point is to standardize the mappings so that they are less confusing to newbies, and musically as useful as possible.
Rank-2 temperaments
Proposed changes to the pages for all rank-2 temperaments:
- If needed, change the mapping to mingen form, except use 3/2 not 4/3 (e.g. Diaschismic Family page should have ~3/2 not ~3/1)
- List equivalent ratios where appropriate, e.g. Pajara's mingen generator is ~16/15 = ~15/14 = ~21/20
- For every comma, add either the monzo or a link to a page that has the monzo
- Add the pergen and remove the wedgie
Proposed changes to the pages for fractional-octave temperaments:
- Add the period's ratio, this is very useful information
- Add the pergen mapping if different than the mingen mapping
The pergen mapping (short for canonical-pergen mapping) is the one which implies the canonical pergen. A mapping's implied pergen is found by 1) discarding all columns in the mapping which don't contain a pivot and 2) inverting the resulting square matrix. (See below for an exception to #1.) For example, the mingen mapping of 5-limit Srutal is [⟨2 3 5], ⟨0 1 -2]], which becomes [⟨2 3], ⟨0 1]], which inverts to [[1/2 0⟩ [-3/2 1⟩], which implies the pergen (P8/2, M2/2), which is a non-canonical pergen. The canonical pergen minimizes the splitting fractions and the cents of the multigen. Here it is (P8/2, P5) which is [[1/2 0⟩ [-1 1⟩], which inverts to [⟨2 2], ⟨0 1]]. This implies the mapping of [⟨2 2 7], ⟨0 1 -2]].
Examples of some of the proposed changes to the Diaschismic family page:
Srutal (12&34, aka diaschismic)
Subgroup: 2.3.5
Comma list: 2048/2025
Pergen: (P8/2, P5)
POTE Period: ~45/32 = 600¢
Mingen Mapping: [⟨2 3 5], ⟨0 1 -2]], POTE generator: ~16/15 = 104.898¢
Pergen Mapping: [⟨2 2 7], ⟨0 1 -2]], POTE generator: ~3/2 = 704.898¢
(etc.)
Srutal
Subgroup: 2.3.5.7
Comma list: 2048/2025, 4375/4374
Pergen: (P8/2, P5)
POTE Period: ~45/32 = 600¢
Mingen Mapping: [⟨2 3 5 3], ⟨0 1 -2 15]], POTE generator: ~16/15 = 104.814¢
Pergen Mapping: [⟨2 2 7 -12], ⟨0 1 -2 15]], POTE generator: ~3/2 = 704.814¢
(etc.)
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 896/891, 1331/1323
Pergen: (P8/2, P5)
POTE Period: ~45/32 = ~99/70 = 600¢
Mingen Mapping: [⟨2 3 5 3 5], ⟨0 1 -2 15 11]], POTE generator: ~16/15 = ~35/33 = 104.856¢
Pergen Mapping: [⟨2 2 7 -12 -6], ⟨0 1 -2 15 11]], POTE generator: ~3/2 = 704.856¢
(etc.)
Pajara
(retain the intro text of course)
Subgroup: 2.3.5.7
Comma list: 50/49, 64/63
Pergen: (P8/2, P5)
POTE Period: ~7/5 = 600¢
Mingen Mapping: [⟨2 3 5 6], ⟨0 1 -2 -2]], POTE generator: ~16/15 = ~15/14 = ~21/20 = 107.048¢
Pergen Mapping: [⟨2 2 7 8], ⟨0 1 -2 -2]], POTE generator: ~3/2 = 707.048¢
(etc.)
11-limit
Subgroup: 2.3.5.7.11
Comma list: 50/49, 64/63, 99/98
Pergen: (P8/2, P5)
POTE Period: ~7/5 = 600¢
Mingen Mapping: [⟨2 3 5 6 8], ⟨0 1 -2 -2 -6]], POTE generator: ~16/15 = ~15/14 = ~21/20 = 106.885¢
Pergen Mapping: [⟨2 2 7 8 14], ⟨0 1 -2 -2 -6]], POTE generator: ~3/2 = 706.885¢
(etc.)
Rank-3 temperaments
Since mingen mappings are impractical, HNF mappings are generally used instead. But these lead to musically awkward generators like 3/1 instead of the more compact 3/2. So I propose we use the canonical pergen to find the period and the first generator. Next examine the 3rd row of the mapping, which is always the same for all mappings except for possibly the sign. Find the smallest (closest to zero) nonzero number in the row. In case of a tie, use the first such occurrence. The column this number is in indicates the prime which defines the 2nd generator. If the 3rd row's entry is ±1, use the canonical comma for that prime as the 2nd generator (prime 5 = 81/80, prime 7 = 64/63, prime 11 = 33/32 or possibly 729/704, etc.). If it isn't, find the canonical multigen, analogous to rank-2 pergens,
Using the canonical comma mimics traditional microtonal notation, in which a higher-limit interval is written as a 3-limit interval inflected by one or more of these commas.
Two examples follow.
Proposed changes to Breed
The Breed temperament is 2.3.5.7 & 2401/2400. The HNF mapping is [⟨1 1 1 2], ⟨0 2 1 1], ⟨0 0 2 1]], with mapping generators ~2/1, ~49/40 and ~10/7. The first two ratios imply a pergen of (P8, P5/2), which is indeed canonical. The 3rd mapping-row has a 1 in prime 7's column, so the 3rd generator should be 7's canonical comma, 64/63. Thus the generator matrix should be [P8 P5/2 64/63]. The 3rd column (prime 5) is all zeros, so discard it to get a square matrix. Invert it to get a prime-5-less mapping matrix [⟨1 1 4], ⟨0 2 -4], ⟨0 0 -1]]. From the 3rd column we can deduce that the original mapping must be altered by adding the 3rd row twice to the 1st row and subtracting it five times from the 2nd row, and then changing the sign of the 3rd row. This yields [⟨1 1 5 4], ⟨0 2 -9 -4], ⟨0 0 -2 -1]]. (Perhaps someone can suggest a more direct algorithm in the discussions tab?)
Subgroup: 2.3.5.7
Comma list: 2401/2400
Pergen: (P8, P5/2, r1)
Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4], ⟨0 0 -2 -1]]
POTE generators: ~49/40 = 350.966¢, ~64/63 = 27.496¢
Proposed changes to Sengic
The Sengic temperament is 2.3.5.7 & 686/675. The HNF mapping is [⟨1 0 2 1], ⟨0 1 0 1], ⟨0 0 3 2]], with generators ~2, ~3/1 and ~15/14. The first two imply (P8, P12), which is non-canonical and must become (P8, P5). There are no ones in the 3rd row, and the smallest entry is the 2 in prime 7's column. Thus the 2nd generator must be of the form M/2, where M is a 2.3.7 multigen. Two ~15/14 generators approximates ~7/6 (a zo 3rd), thus the pergen should be (P8, P5, z3/2), which is [[1 0 0 0⟩ [-1 1 0 0⟩ [-1/2 -1/2 0 1/2⟩]. This implies a mapping of [⟨1 1 2 2], ⟨0 1 0 1], ⟨0 0 3 2]].
Subgroup: 2.3.5.7
Pergen: (P8, P5, z3/2)
Canonical Mapping: [⟨1 1 2 2], ⟨0 1 0 1], ⟨0 0 3 2]]
POTE generators: ~3/2 = 704.154¢, ~15/14 = 129.824¢
Possible additional proposal
Add the notations for higher primes, immediately after the pergen. For example in septimal meantone, 5/4 is a major 3rd and 7/4 is an augmented 6th. Thus the pergen line would be (P8, P5) M3 A6. For Pajara, it would be (P8, P5/2) M3 vm7. This would be enormously helpful to the musician/composer who isn't familiar with linear algebra. But it has the disadvantage that it requires settling on a specific notation for that temperament. There may be cases where that is difficult.