User:TallKite/Temperament Template Proposal: Difference between revisions
Created page with "(This is a work in progress) The point is to standardize the mappings so that they are less confusing to newbies, and musically as useful as possible. = Rank-2 temperaments..." |
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The point is to standardize the mappings so that they are less confusing to newbies, and musically as useful as possible. | The point is to standardize the mappings so that they are less confusing to newbies, and musically as useful as possible. | ||
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* If needed, correct the mapping to [[mingen]] form, except use 3/2 not 4/3 (e.g. Diaschismic Family page should have ~3/2 not ~3/1) | * If needed, correct 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 | * 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 | * Add the pergen and remove the wedgie | ||
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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 [{{val|1= 2 3 5}}, {{val|1= 0 1 -2 }}], which becomes [{{val|1= 2 3}}, {{val|1= 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 [{{val|1= 2 2}}, {{val|1= 0 1 }}]. This implies the mapping of [{{val|1= 2 2 7 }}, {{val|1= 0 1 -2 }}]. | 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 [{{val|1= 2 3 5}}, {{val|1= 0 1 -2 }}], which becomes [{{val|1= 2 3}}, {{val|1= 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 [{{val|1= 2 2}}, {{val|1= 0 1 }}]. This implies the mapping of [{{val|1= 2 2 7 }}, {{val|1= 0 1 -2 }}]. | ||
=== | === Examples of some of the proposed changes to the [[Diaschismic family]] page: === | ||
==Srutal (12&34, aka diaschismic)== | ==Srutal (12&34, aka diaschismic)== | ||
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[[POTE generator|POTE generators]]: ~3/2 = 704.154¢, ~15/14 = 129.824¢ | [[POTE generator|POTE generators]]: ~3/2 = 704.154¢, ~15/14 = 129.824¢ | ||
= Possible additional | = 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. | |||