User:TallKite/Temperament Template Proposal - Revision history

TallKite: /* Rank-2 temperaments */

2022-05-26T18:15:48Z

Rank-2 temperaments

← Older revision		Revision as of 18:15, 26 May 2022
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	Proposed changes to the pages for all rank-2 temperaments:		Proposed changes to the pages for all rank-2 temperaments:

	* 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, 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		* 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		* For every comma, add either the monzo or a link to a page that has the monzo

TallKite at 19:45, 22 March 2022

2022-03-22T19:45:58Z

← Older revision		Revision as of 19:45, 22 March 2022
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	~~(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.		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:~~'''~~ ===		=== 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 ~~proposals (needs work)~~ =		= Possible additional proposal =
	1) 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.		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.

	~~2) Add the monzos for commas alongside the ratio. Or perhaps list the entire twin-square matrix, in a table.~~

	~~3) Add the color name for the commas~~.

TallKite: 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..."

2022-03-22T02:47:59Z

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..."

New page

(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 =
Proposed changes to the pages for all rank-2 temperaments:

* 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
* 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 [{{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)==
Subgroup: 2.3.5

[[Comma list]]: 2048/2025

[[Pergen]]: (P8/2, P5)

POTE Period: ~45/32 = 600¢

Mingen Mapping: [{{val|1= 2 3 5}}, {{val|1= 0 1 -2 }}], [[POTE generator]]: ~16/15 = 104.898¢

Pergen Mapping: [{{val|1= 2 2 7 }}, {{val|1= 0 1 -2 }}], [[POTE generator]]: ~3/2 = 704.898¢

[[Tuning ranges]]:

(etc.)
==Srutal==
{{see also|1= Srutal vs diaschismic }}
Subgroup: 2.3.5.7

[[Comma list]]: 2048/2025, 4375/4374

[[Pergen]]: (P8/2, P5)

POTE Period: ~45/32 = 600¢

Mingen [[Mapping]]: [{{val|1=2 3 5 3}}, {{val|1=0 1 -2 15}}], [[POTE generator]]: ~16/15 = 104.814¢

Pergen [[Mapping]]: [{{val|1=2 2 7 -12}}, {{val|1=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 family|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

[[Comma list]]: [[686/675]]

Pergen: (P8, P5, z3/2)

Canonical [[Mapping]]: [⟨1 1 2 2], ⟨0 1 0 1], ⟨0 0 3 2]]

[[POTE generator|POTE generators]]: ~3/2 = 704.154¢, ~15/14 = 129.824¢

= Possible additional proposals (needs work) =
1) 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.

2) Add the monzos for commas alongside the ratio. Or perhaps list the entire twin-square matrix, in a table.

3) Add the color name for the commas.