24576/24565: Difference between revisions

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== Temperaments ==

=== Rank 6 temperament ===

By tempering out this comma in the full [[17-limit]], the rank-6 '''mavka temperament''' is defined. You may find a list of good equal temperaments supporting it below.

~~=== Archagall ===~~

[[Subgroup]]: 2.3.5.7.11.13.17

~~By tempering it (S16/S17) out in the 2.75.85 subgroup, we have three~~ [[~~85/64~~]]~~'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone~~. This is a highly accurate temperament that could be considered to be encoding the "high accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 ~~temperament) in the following subgroup:~~

~~Subgroup~~: ~~2.75.85~~

[[Mapping]]:

~~Mapping:~~ [ ⟨ 1 ~~5 6 ] ⟨~~ 0 3 1 ] ⟩

[ ⟨ 1 0 1 0 0 0 4 ]

~~Some good (relative to their size) EDOs supporting it: 5, 17, 22, 39, 61, 83~~

⟨ 0 1 1 0 0 0 0 ]

~~=== Srutal archagall ===~~

⟨ 0 0 3 0 0 0 -1 ]

~~Named because this lower~~-accuracy temperament is also an extension of (the 5-limit) [[srutal]] temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating [[85/64]] with [[4/3]] by tempering their difference of S16 = [[256/255]]. Therefore it also tempers S17 = [[289/288]] and thus equates [[17/15]] with [[9/8]] due to tempering S16*S17. It could be described as the 10 & 12 temperament (with strong emphasis on [[12edo]] being the better tuning hence the list of EDOs mentioned) on the following prior-discussed subgroup:

~~Subgroup: 2.3.5.17~~

⟨ 0 0 0 1 0 0 0 ]

~~Mapping: [ ⟨ 2 3 5 8 ]~~ ⟨ 0 1 ~~-2 1~~ ] ⟩

⟨ 0 0 0 0 1 0 0 ]

~~Some good (relative to their size) EDOs supporting it: 12, 22, 34, 46, 56, 58, 80~~

⟨ 0 0 0 0 0 1 0 ] ⟩

=~~== Rank 6 temperament ===~~

{{Val list|legend=1| 46, 58, 80, 103, 137, 149, 159, 171, 183, 217, 296, 320, 342f, 354, 400, 422, 525, 571, 581, 742, 764, 935, 1084, 1106, 1323, 1506, 3593g, 3947eg, 5053fgg, 6559defgg, 8065cdefggg, 10152cdeffgggg }}.

~~By tempering it out in the full [[17-limit]]~~, ~~the rank-6 '''mavka temperament''' is defined~~, ~~to which extensions are also rank 6~~.

~~Subgroup:~~ 2.3.5~~.7.11.13.~~17

=== Archagall ===

By tempering it out in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. This is a highly accurate temperament that could be considered to be encoding the "high accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the following subgroup:

~~Mapping~~:

Subgroup: 2.75.85

[ ⟨ 1 0 1 ~~0 0 0 4~~ ]

Mapping: [ ⟨ 1 5 6 ] ⟨ 0 3 1 ] ⟩

~~⟨ 0 1 1 0 0 0 0 ]~~

Some good (relative to their size) EDOs supporting it: 5, 17, 22, 39, 61, 83

~~⟨ 0 0~~ 3 ~~0 0 0~~ -~~1 ]~~

=== Srutal archagall ===

Named because this lower-accuracy temperament is also an extension of (the 5-limit) [[srutal]] temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating [[85/64]] with [[4/3]] by tempering their difference of S16 = [[256/255]]. Therefore it also tempers S17 = [[289/288]] and thus equates [[17/15]] with [[9/8]] due to tempering S16*S17. It could be described as the 10 & 12 temperament (with strong emphasis on [[12edo]] being the better tuning hence the list of EDOs mentioned) on the following prior-discussed subgroup:

~~⟨ 0 0 0 1 0 0 0 ]~~

Subgroup: 2.3.5.17

~~⟨ 0 0 0 0 1 0 0 ]~~

Comma list: 256/255, 289/288

⟨ 0 ~~0 0 0 0~~ 1 0 ] ⟩

Mapping: [ ⟨ 2 3 5 8 ] ⟨ 0 1 -2 1 ] ⟩

~~{{Val list|legend=1| 46~~, 58, 80, ~~103~~, ~~137~~, ~~149~~, ~~159, 171, 183, 217, 296, 320, 342f, 354, 400, 422, 525, 571, 581, 742, 764, 935, 1084, 1106, 1323, 1506, 3593g, 3947eg, 5053fgg, 6559defgg, 8065cdefggg, 10152cdeffgggg }}.~~

Some good (relative to their size) EDOs supporting it: 12, 22, 34, 46, 56, 58, 80

[[Category:Mavka]]

@@ Line 10: / Line 10: @@
 == Temperaments ==
+=== Rank 6 temperament ===
+By tempering out this comma in the full [[17-limit]], the rank-6 '''mavka temperament''' is defined. You may find a list of good equal temperaments supporting it below.
-=== Archagall ===
+[[Subgroup]]: 2.3.5.7.11.13.17
-By tempering it (S16/S17) out in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. This is a highly accurate temperament that could be considered to be encoding the "high accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the following subgroup:
-Subgroup: 2.75.85
+[[Mapping]]:
-Mapping:  [ ⟨	1	5	6	]      ⟨	0	3	1	] ⟩
+[ ⟨	1	0	1	0	0	0	4	]
-Some good (relative to their size) EDOs supporting it: 5, 17, 22, 39, 61, 83
+⟨	0	1	1	0	0	0	0	]
-=== Srutal archagall ===
+⟨	0	0	3	0	0	0	-1	]
-Named because this lower-accuracy temperament is also an extension of (the 5-limit) [[srutal]] temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating [[85/64]] with [[4/3]] by tempering their difference of S16 = [[256/255]]. Therefore it also tempers S17 = [[289/288]] and thus equates [[17/15]] with [[9/8]] due to tempering S16*S17. It could be described as the 10 & 12 temperament (with strong emphasis on [[12edo]] being the better tuning hence the list of EDOs mentioned) on the following prior-discussed subgroup:
-Subgroup: 2.3.5.17
+⟨	0	0	0	1	0	0	0	]
-Mapping:  [ ⟨	2	3	5	8	]      ⟨	0	1	-2	1	] ⟩
+⟨	0	0	0	0	1	0	0	]
-Some good (relative to their size) EDOs supporting it: 12, 22, 34, 46, 56, 58, 80
+⟨	0	0	0	0	0	1	0	] ⟩
-=== Rank 6 temperament ===
+{{Val list|legend=1| 46, 58, 80, 103, 137, 149, 159, 171, 183, 217, 296, 320, 342f, 354, 400, 422, 525, 571, 581, 742, 764, 935, 1084, 1106, 1323, 1506, 3593g, 3947eg, 5053fgg, 6559defgg, 8065cdefggg, 10152cdeffgggg }}.
-By tempering it out in the full [[17-limit]], the rank-6 '''mavka temperament''' is defined, to which extensions are also rank 6.
-Subgroup: 2.3.5.7.11.13.17
+=== Archagall ===
+By tempering it out in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. This is a highly accurate temperament that could be considered to be encoding the "high accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the following subgroup:
-Mapping:
+Subgroup: 2.75.85
-[ ⟨	1	0	1	0	0	0	4	]
+Mapping:  [ ⟨	1	5	6	]      ⟨	0	3	1	] ⟩
-⟨	0	1	1	0	0	0	0	]
+Some good (relative to their size) EDOs supporting it: 5, 17, 22, 39, 61, 83
-⟨	0	0	3	0	0	0	-1	]
+=== Srutal archagall ===
+Named because this lower-accuracy temperament is also an extension of (the 5-limit) [[srutal]] temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating [[85/64]] with [[4/3]] by tempering their difference of S16 = [[256/255]]. Therefore it also tempers S17 = [[289/288]] and thus equates [[17/15]] with [[9/8]] due to tempering S16*S17. It could be described as the 10 & 12 temperament (with strong emphasis on [[12edo]] being the better tuning hence the list of EDOs mentioned) on the following prior-discussed subgroup:
-⟨	0	0	0	1	0	0	0	]
+Subgroup: 2.3.5.17
-⟨	0	0	0	0	1	0	0	]
+Comma list: 256/255, 289/288
-⟨	0	0	0	0	0	1	0	] ⟩
+Mapping:  [ ⟨	2	3	5	8	]      ⟨	0	1	-2	1	] ⟩
-{{Val list|legend=1| 46, 58, 80, 103, 137, 149, 159, 171, 183, 217, 296, 320, 342f, 354, 400, 422, 525, 571, 581, 742, 764, 935, 1084, 1106, 1323, 1506, 3593g, 3947eg, 5053fgg, 6559defgg, 8065cdefggg, 10152cdeffgggg }}.
+Some good (relative to their size) EDOs supporting it: 12, 22, 34, 46, 56, 58, 80
 [[Category:Mavka]]