Breedsmic temperaments: Difference between revisions

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= Tertiaseptal =

Aside from the breedsma, [[tertiaseptal]] tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152, the rainy comma. It can be described as the 31&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.

Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152, the rainy comma. It can be described as the 31&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.

Subgroup: 2.3.5.7

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POTE generator: ~22/21 = 77.162

Vals: {{Val list| 31, 109g, 140~~, 171e~~, 311e, 451ee }}

Vals: {{Val list| 31, 109g, 140, 311e, 451ee }}

Badness: 0.022416

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Badness: 0.015633

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095

Mapping: [{{val| 1 3 2 3 6 1 }}, {{val| 0 -44 10 -6 -79 84 }}]

POTE generator: ~45/44 = 38.588

Vals: {{Val list| 31, 280, 311, 964f, 1275f, 1586cff }}

Badness: 0.033573

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095

Mapping: [{{val| 1 3 2 3 6 1 1 }}, {{val| 0 -44 10 -6 -79 84 96 }}]

POTE generator: ~45/44 = 38.589

Vals: {{Val list| 31, 280, 311, 653f, 964f }}

Badness: 0.025298

= Harry =

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Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is <<12 34 20 30 ...||.

Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with <<12 34 20 30 52 ...|| as the octave wedgie. [[130edo]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.

Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with <<12 34 20 30 52 ...|| as the octave wedgie. [[130edo|130EDO]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.

Subgroup: 2.3.5.7

[[Comma list]]: 2401/2400, 19683/19600

[[Mapping]]: [{{val| 2 4 7 7 }}, {{val| 0 -6 -17 -10 }}]

~~Commas: 2401/2400, 19683/19600~~

[[POTE generator]]: ~21/20 = 83.156

~~Map: [<2 4 7 7~~|, ~~<~~0 ~~-6 -17 -10|]~~

[[Badness]]: 0.034077

~~Wedgie~~: ~~<<12 34 20 26 -~~2 ~~-49||~~

== 11-limit ==

Subgroup: 2.3.5.7.11

~~EDOs~~: ~~14c~~, 58, ~~72, 130, 202, 534, 938~~

Comma list: 243/242, 441/440, 4000/3993

~~Badness~~: 0~~.0341~~

Mapping: [{{val| 2 4 7 7 9 }}, {{val| 0 -6 -17 -10 -15 }}]

~~==11-limit==~~

POTE generator: ~21/20 = 83.167

~~Commas~~: ~~243~~/~~242, 441/440, 4000/3993~~

~~[[POTE_tuning~~|~~POTE generator]]: ~21/20 = 83.167~~

Vals: {{Val list| 14c, 58, 72, 130, 202 }}

~~Map~~: ~~[<2 4 7 7 9|, <~~0 ~~-6 -17 -10 -15|]~~

Badness: 0.015867

~~EDOs~~: ~~14c, 58, 72, 130, 202~~

== 13-limit ==

Subgroup: 2.3.5.7.11.13

~~Badness~~: ~~0.0159~~

Comma list: 243/242, 351/350, 441/440, 676/675

~~==13-limit==~~

Mapping: [{{val| 2 4 7 7 9 11 }}, {{val| 0 -6 -17 -10 -15 -26 }}]

~~Commas~~: ~~243/242~~, ~~351/350, 441/440, 676/675~~

POTE generator: ~21/20 = 83.116

~~Map~~: ~~[<2 4 7 7 9 11~~|, ~~<0 -6 -17 -10 -15 -26|]~~

Vals: {{Val list| 58, 72, 130, 332f, 462ef }}

~~EDOs~~: ~~58, 72, 130, 462~~

Badness: 0.013046

~~Badness~~: 0.~~0130~~

== 17-limit ==

Subgroup: 2.3.5.7.11.13.17

~~==17-limit==~~

Comma list: 221/220, 243/242, 289/288, 351/350, 441/440

~~Commas~~: 221/220, 243/242, 289/288, 351/350, 441/440

Mapping: [{{val| 2 4 7 7 9 11 9 }}, {{val| 0 -6 -17 -10 -15 -26 -6 }}]

POTE generator: ~21/20 = 83.168

~~Map~~: ~~[<2 4 7 7 9 11 9~~|~~, <0 -6 -17 -10 -15 -26 -6|]~~

Vals: {{Val list| 58, 72, 130, 202g }}

~~EDOs:~~ 58, 72, 130, 202g

Badness: 0.~~0127~~

Badness: 0.012657

=Quasiorwell=

= Quasiorwell =

In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1>. It has a wedgie <<38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.

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 = Tertiaseptal =
-Aside from the breedsma, [[tertiaseptal]] tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152, the rainy comma. It can be described as the 31&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.
+{{main|Tertiaseptal}}
+Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152, the rainy comma. It can be described as the 31&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.
 Subgroup: 2.3.5.7
@@ Line 189: / Line 191: @@
 POTE generator: ~22/21 = 77.162
-Vals: {{Val list| 31, 109g, 140, 171e, 311e, 451ee }}
+Vals: {{Val list| 31, 109g, 140, 311e, 451ee }}
 Badness: 0.022416
@@ Line 205: / Line 207: @@
 Badness: 0.015633
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
+Mapping: [{{val| 1 3 2 3 6 1 }}, {{val| 0 -44 10 -6 -79 84 }}]
+POTE generator: ~45/44 = 38.588
+Vals: {{Val list| 31, 280, 311, 964f, 1275f, 1586cff }}
+Badness: 0.033573
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
+Mapping: [{{val| 1 3 2 3 6 1 1 }}, {{val| 0 -44 10 -6 -79 84 96 }}]
+POTE generator: ~45/44 = 38.589
+Vals: {{Val list| 31, 280, 311, 653f, 964f }}
+Badness: 0.025298
 = Harry =
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 Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is &lt;&lt;12 34 20 30 ...||.
-Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with &lt;&lt;12 34 20 30 52 ...|| as the octave wedgie. [[130edo]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
+Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with &lt;&lt;12 34 20 30 52 ...|| as the octave wedgie. [[130edo|130EDO]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
+Subgroup: 2.3.5.7
+[[Comma list]]: 2401/2400, 19683/19600
+[[Mapping]]: [{{val| 2 4 7 7 }}, {{val| 0 -6 -17 -10 }}]
-Commas: 2401/2400, 19683/19600
+{{Multival|legend=1| 12 34 20 26 -2 -49 }}
 [[POTE generator]]: ~21/20 = 83.156
-Map: [&lt;2 4 7 7|, &lt;0 -6 -17 -10|]
+{{Val list|legend=1| 14c, 58, 72, 130, 202, 534, 736b, 938b }}
+[[Badness]]: 0.034077
-Wedgie: &lt;&lt;12 34 20 26 -2 -49||
+== 11-limit ==
+Subgroup: 2.3.5.7.11
-EDOs: 14c, 58, 72, 130, 202, 534, 938
+Comma list: 243/242, 441/440, 4000/3993
-Badness: 0.0341
+Mapping: [{{val| 2 4 7 7 9 }}, {{val| 0 -6 -17 -10 -15 }}]
-==11-limit==
+POTE generator: ~21/20 = 83.167
-Commas: 243/242, 441/440, 4000/3993
-[[POTE_tuning|POTE generator]]: ~21/20 = 83.167
+Vals: {{Val list| 14c, 58, 72, 130, 202 }}
-Map: [&lt;2 4 7 7 9|, &lt;0 -6 -17 -10 -15|]
+Badness: 0.015867
-EDOs: 14c, 58, 72, 130, 202
+== 13-limit ==
+Subgroup: 2.3.5.7.11.13
-Badness: 0.0159
+Comma list: 243/242, 351/350, 441/440, 676/675
-==13-limit==
+Mapping: [{{val| 2 4 7 7 9 11 }}, {{val| 0 -6 -17 -10 -15 -26 }}]
-Commas: 243/242, 351/350, 441/440, 676/675
 POTE generator: ~21/20 = 83.116
-Map: [&lt;2 4 7 7 9 11|, &lt;0 -6 -17 -10 -15 -26|]
+Vals: {{Val list| 58, 72, 130, 332f, 462ef }}
-EDOs: 58, 72, 130, 462
+Badness: 0.013046
-Badness: 0.0130
+== 17-limit ==
+Subgroup: 2.3.5.7.11.13.17
-==17-limit==
+Comma list: 221/220, 243/242, 289/288, 351/350, 441/440
-Commas: 221/220, 243/242, 289/288, 351/350, 441/440
+Mapping: [{{val| 2 4 7 7 9 11 9 }}, {{val| 0 -6 -17 -10 -15 -26 -6 }}]
 POTE generator: ~21/20 = 83.168
-Map: [&lt;2 4 7 7 9 11 9|, &lt;0 -6 -17 -10 -15 -26 -6|]
+Vals: {{Val list| 58, 72, 130, 202g }}
-EDOs: 58, 72, 130, 202g
-Badness: 0.0127
+Badness: 0.012657
-=Quasiorwell=
+= Quasiorwell =
 In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&gt;. It has a wedgie &lt;&lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.