24576/24565: Difference between revisions

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== Temperaments ==

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

=== Mavka ===

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.

[[Subgroup]]: 2.3.5.7.11.13.17

[[Mapping]]:

[[Mapping]]:

[{{val| 1 0 1 0 0 0 4 }}

[ ⟨ 1 0 1 0 0 0 4 ]

{{val| 0 1 1 0 0 0 0 }}

{{val| 0 0 3 0 0 0 -1 }}

⟨ 0 1 1 0 0 0 0 ]

{{val| 0 0 0 1 0 0 0 }}

{{val| 0 0 0 0 1 0 0 }}

⟨ 0 0 3 0 0 0 -1 ]

{{val| 0 0 0 0 0 1 0 }}]

⟨ 0 0 0 1 0 0 0 ]

⟨ 0 0 0 0 1 0 0 ]

⟨ 0 0 0 0 0 1 0 ] ⟩

{{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 }}.

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Subgroup: 2.75.85

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

Comma list: {{monzo| 13 1 -3 }} = 24576/24565

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

Sval mapping: [{{val| 1 2 5 }}], {{val| 0 3 1 }}]

Sval mapping generators: ~2, ~85/32

Optimal GPV sequence: {{Val list| 5, 17, 22, 61, 83 }}

=== 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:

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) on the following prior-discussed subgroup:

Subgroup: 2.3.5.17

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Comma list: 256/255, 289/288

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

Sval mapping: [{{val| 2 0 11 5 }}], {{val| 0 1 -2 1 }}]

Sval mapping generators: ~17/12, ~3

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272

Optimal GPV sequence: {{Val list| 10, 12, 22, 34, 80, 114, 194bc }}

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

Badness: 0.00575

[[Category:Mavka]]

@@ Line 10: / Line 10: @@
 == Temperaments ==
-=== Rank 6 temperament ===
+=== Mavka ===
 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.
 [[Subgroup]]: 2.3.5.7.11.13.17
-[[Mapping]]:
+[[Mapping]]:<br>
+[{{val| 1 0 1 0 0 0 4 }}<br>
-[ ⟨	1	0	1	0	0	0	4	]
+{{val| 0 1 1 0 0 0 0 }}<br>
+{{val| 0 0 3 0 0 0 -1 }}<br>
-⟨	0	1	1	0	0	0	0	]
+{{val| 0 0 0 1 0 0 0 }}<br>
+{{val| 0 0 0 0 1 0 0 }}<br>
-⟨	0	0	3	0	0	0	-1	]
+{{val| 0 0 0 0 0 1 0 }}]
-⟨	0	0	0	1	0	0	0	]
-⟨	0	0	0	0	1	0	0	]
-⟨	0	0	0	0	0	1	0	] ⟩
 {{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 }}.
@@ Line 36: / Line 30: @@
 Subgroup: 2.75.85
-Mapping:  [ ⟨	1	5	6	]      ⟨	0	3	1	] ⟩
+Comma list: {{monzo| 13 1 -3 }} = 24576/24565
-Some good (relative to their size) EDOs supporting it: 5, 17, 22, 39, 61, 83
+Sval mapping: [{{val| 1 2 5 }}], {{val| 0 3 1 }}]
+Sval mapping generators: ~2, ~85/32
+Optimal GPV sequence: {{Val list| 5, 17, 22, 61, 83 }}
 === 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:
+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) on the following prior-discussed subgroup:
 Subgroup: 2.3.5.17
@@ Line 47: / Line 45: @@
 Comma list: 256/255, 289/288
-Mapping:  [ ⟨	2	3	5	8	]      ⟨	0	1	-2	1	] ⟩
+Sval mapping: [{{val| 2 0 11 5 }}], {{val| 0 1 -2 1 }}]
+Sval mapping generators: ~17/12, ~3
+Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272
+Optimal GPV sequence: {{Val list| 10, 12, 22, 34, 80, 114, 194bc }}
-Some good (relative to their size) EDOs supporting it: 12, 22, 34, 46, 56, 58, 80
+Badness: 0.00575
 [[Category:Mavka]]