Breedsmic temperaments: Difference between revisions

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Hemififths tempers out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator, with [[99edo]] and [[140edo]] providing good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It may be called the 41 & 58 temperament. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}.

By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. [[99edo]] is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.

By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~49/40

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* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}

: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}

: [[Eigenmonzo basis|~~Eigenmonzo~~ (unchanged-interval) basis]]: 2.5

: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5

[[Algebraic generator]]: (2 + sqrt(2))/2

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Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}

: mapping generators: ~99/70, ~49/40

Optimal tuning (POTE): ~99/70 = 1\2, ~49/40 = 351.505

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Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}

: mapping generators: ~2, ~243/220

Optimal tuning (POTE): ~2 = 1\1, ~243/220 = 175.7378

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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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: mapping generators: ~2, ~256/245

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Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}

: mapping generators: ~2, ~45/44

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596

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Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}

: mapping generators: ~99/70, ~256/245

Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193

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: mapping generators: ~2, ~875/512

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Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[~~15/14 equal-step tuning|~~linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used its generator. It may be described as the 130 & 270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[quintosec]] temperament.

Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used as its generator. It may be described as the 130 & 270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[quintosec]] temperament.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~189/160

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== Emmthird ==

The generator for emmthird ~~temperament~~ is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

[[Subgroup]]: 2.3.5.7

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[[Comma list]]: 2401/2400, 14348907/14336000

: mapping generators: ~2, ~2187/1372

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988

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Comma list: 243/242, 441/440, 1792000/1771561

Mapping: {{mapping| 1 ~~-3 -17 -8 -8~~ | 0 14 59 33 35 }}

Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }}

Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991

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Comma list: 243/242, 364/363, 441/440, 2200/2197

Mapping: {{mapping| 1 ~~-3 -17 -8 -8 -13~~ | 0 14 59 33 35 51 }}

Mapping: {{mapping| 1 11 42 25 27 38 | 0 -14 -59 -33 -35 -51 }}

Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989

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[[Comma list]]: 2401/2400, 1959552/1953125

: mapping generators: ~2, ~42/25

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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997

{{Optimal ET sequence|legend=1| ~~95,~~ 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}

{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}

[[Badness]]: 0.037322

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[[Comma list]]: 2401/2400, 68359375/68024448

: mapping generators: ~2, ~6125/3888

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Comma list: 2401/2400, 3025/3024, 4000/3993

Mapping: {{mapping| 1 ~~-13 -14 -9 -8~~ | 0 42 47 34 33 }}

Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718

Badness: 0.022926

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Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400

Mapping: {{mapping| 1 ~~-13 -14 -9 -9 -47~~ | 0 42 47 34 33 146 }}

Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716

{{Optimal ET sequence|legend=1| 72, 311, 694, 1005c~~, 1699cd~~ }}

Badness: 0.020888

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: mapping generators: ~2, ~49/40

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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113

{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201~~, 6333bbcc~~ }}

{{Optimal ET sequence|legend=1| 41, …, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}

[[Badness]]: 0.041878

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Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115

{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 581, 851, 1121, 1972~~, 3093b, 4214b~~ }}

{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 581, 851, 1121, 1972 }}

Badness: 0.019461

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Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117

{{Optimal ET sequence|legend=1| 41, 229, 270, 581, 851, 2283b~~, 3134b~~ }}

Badness: 0.013830

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[[Comma list]]: 2401/2400, 2152828125/2147483648

: mapping generators: ~2, ~28/15

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Comma list: 243/242, 441/440, 939524096/935859375

Mapping: {{mapping| 1 -~~1 6 4~~ -3 | 0 26 -37 -12 65 }}

Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279

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Comma list: 243/242, 441/440, 2200/2197, 3584/3575

Mapping: {{mapping| 1 -1 ~~6 4 -3 4~~ | 0 26 -37 -12 65 -3 }}

Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281

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Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575

Mapping: {{mapping| 1 -1 ~~6 4 -3 4 2~~ | 0 26 -37 -12 65 -3 21 }}

Mapping: {{mapping| 1 25 -31 -8 62 1 23 | 0 -26 37 12 -65 3 -21 }}

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281

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: mapping generators: ~2, ~1296/875

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: mapping generators: ~2, ~57344/46875

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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301

{{Optimal ET sequence|legend=1| 31, 348, 379, 410, 441, 1354, 1795, 2236 }}

{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}

[[Badness]]: 0.045792

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: mapping generators: ~2, ~2800/2187

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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066

{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696~~, 6955dd~~ }}

{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}

[[Badness]]: 0.028307

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: mapping generators: ~2, ~8/7

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: mapping generators: ~2, ~125/96

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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310

{{Optimal ET sequence|legend=1| 37, 103, 140, 243, 383, 1009cd, 1392ccd }}

{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}

Badness: 0.100511

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Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318

Badness: 0.056514

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Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316

Badness: 0.027429

== Mintone ==

In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo|-3 11 -5 -1}} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 & 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.

In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo| -3 11 -5 -1 }} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 & 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~10/9

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: mapping generators: ~2, ~250/189

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[[Comma list]]: 2401/2400, 390625/387072

: mappping generators: ~2, ~10/7

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Comma list: 385/384, 1375/1372, 4000/3993

Mapping: {{mapping| 1 ~~-13 -4 -4 2~~ | 0 30 13 14 3 }}

Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387

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Comma list: 169/168, 364/363, 385/384, 625/624

Mapping: {{mapping| 1 ~~-13 -4 -4 2 -7~~ | 0 30 13 14 3 22 }}

Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387

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: mapping generators: ~2, ~49/40

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603

@@ Line 28: / Line 28: @@
 Hemififths tempers out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator, with [[99edo]] and [[140edo]] providing good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It may be called the 41 &amp; 58 temperament. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}.
-By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. [[99edo]] is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.
+By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.
 [[Subgroup]]: 2.3.5.7
@@ Line 35: / Line 35: @@
 {{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
+: mapping generators: ~2, ~49/40
 {{Multival|legend=1| 2 25 13 35 15 -40 }}
@@ Line 43: / Line 45: @@
 * [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
 : {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
-: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
 [[Algebraic generator]]: (2 + sqrt(2))/2
@@ Line 83: / Line 85: @@
 Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
+: mapping generators: ~99/70, ~49/40
 Optimal tuning (POTE): ~99/70 = 1\2, ~49/40 = 351.505
@@ Line 111: / Line 115: @@
 Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
+: mapping generators: ~2, ~243/220
 Optimal tuning (POTE): ~2 = 1\1, ~243/220 = 175.7378
@@ Line 134: / Line 140: @@
 {{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.
+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 141: / Line 147: @@
 {{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}
+: mapping generators: ~2, ~256/245
 {{Multival|legend=1| 22 -5 3 -59 -57 21 }}
@@ Line 351: / Line 359: @@
 Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}
+: mapping generators: ~2, ~45/44
 Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596
@@ Line 390: / Line 400: @@
 Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}
+: mapping generators: ~99/70, ~256/245
 Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
@@ Line 407: / Line 419: @@
 {{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}
+: mapping generators: ~2, ~875/512
 {{Multival|legend=1| 38 -3 8 -93 -94 27 }}
@@ Line 445: / Line 459: @@
 {{See also| Quintosec family #Decoid }}
-Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[15/14 equal-step tuning|linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used its generator. It may be described as the 130 &amp; 270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[quintosec]] temperament.
+Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used as its generator. It may be described as the 130 &amp; 270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[quintosec]] temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 497: / Line 511: @@
 {{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}
+: mapping generators: ~2, ~189/160
 {{Multival|legend=1| 6 41 22 51 18 -64 }}
@@ Line 533: / Line 549: @@
 == Emmthird ==
-The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
+The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
 [[Subgroup]]: 2.3.5.7
@@ Line 539: / Line 555: @@
 [[Comma list]]: 2401/2400, 14348907/14336000
-{{Mapping|legend=1| 1 -3 -17 -8 | 0 14 59 33 }}
+{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}
-{{Multival|legend=1|14 59 33 61 13 -89}}
+: mapping generators: ~2, ~2187/1372
+{{Multival|legend=1| 14 59 33 61 13 -8 9 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
@@ Line 554: / Line 572: @@
 Comma list: 243/242, 441/440, 1792000/1771561
-Mapping: {{mapping| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}
+Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }}
 Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991
@@ Line 567: / Line 585: @@
 Comma list: 243/242, 364/363, 441/440, 2200/2197
-Mapping: {{mapping| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}
+Mapping: {{mapping| 1 11 42 25 27 38 | 0 -14 -59 -33 -35 -51 }}
 Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989
@@ Line 595: / Line 613: @@
 [[Comma list]]: 2401/2400, 1959552/1953125
-{{Mapping|legend=1| 1 -7 -5 -3 | 0 34 29 23 }}
+{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}
+: mapping generators: ~2, ~42/25
 {{Multival|legend=1| 34 29 23 -33 -59 -28 }}
@@ Line 601: / Line 621: @@
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
-{{Optimal ET sequence|legend=1| 95, 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
+{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
 [[Badness]]: 0.037322
@@ Line 612: / Line 632: @@
 [[Comma list]]: 2401/2400, 68359375/68024448
-{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}
+{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}
+: mapping generators: ~2, ~6125/3888
 {{Multival|legend=1| 42 47 34 -23 -64 -53 }}
@@ Line 627: / Line 649: @@
 Comma list: 2401/2400, 3025/3024, 4000/3993
-Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}
+Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}
 Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718
-{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 1316c }}
+{{Optimal ET sequence|legend=1| 72, 167, 239, 311 }}
 Badness: 0.022926
@@ Line 640: / Line 662: @@
 Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
-Mapping: {{mapping| 1 -13 -14 -9 -9 -47 | 0 42 47 34 33 146 }}
+Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}
 Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716
-{{Optimal ET sequence|legend=1| 72, 311, 694, 1005c, 1699cd }}
+{{Optimal ET sequence|legend=1| 72, 239f, 311, 694, 1005c }}
 Badness: 0.020888
@@ Line 656: / Line 678: @@
 {{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}
+: mapping generators: ~2, ~49/40
 {{Multival|legend=1| 2 -57 -28 -95 -50 95 }}
@@ Line 661: / Line 685: @@
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
-{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bbcc }}
+{{Optimal ET sequence|legend=1| 41, …, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}
 [[Badness]]: 0.041878
@@ Line 674: / Line 698: @@
 Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115
-{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b }}
+{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 581, 851, 1121, 1972 }}
 Badness: 0.019461
@@ Line 687: / Line 711: @@
 Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
-{{Optimal ET sequence|legend=1| 41, 229, 270, 581, 851, 2283b, 3134b }}
+{{Optimal ET sequence|legend=1| 41, 229, 270, 581, 851, 2283b }}
 Badness: 0.013830
@@ Line 700: / Line 724: @@
 [[Comma list]]: 2401/2400, 2152828125/2147483648
-{{Mapping|legend=1| 1 -1 6 4 | 0 26 -37 -12 }}
+{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}
+: mapping generators: ~2, ~28/15
 {{Multival|legend=1| 26 -37 -12 -119 -92 76 }}
@@ Line 717: / Line 743: @@
 Comma list: 243/242, 441/440, 939524096/935859375
-Mapping: {{mapping| 1 -1 6 4 -3 | 0 26 -37 -12 65 }}
+Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}
 Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279
@@ Line 730: / Line 756: @@
 Comma list: 243/242, 441/440, 2200/2197, 3584/3575
-Mapping: {{mapping| 1 -1 6 4 -3 4 | 0 26 -37 -12 65 -3 }}
+Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }}
 Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
@@ Line 743: / Line 769: @@
 Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
-Mapping: {{mapping| 1 -1 6 4 -3 4 2 | 0 26 -37 -12 65 -3 21 }}
+Mapping: {{mapping| 1 25 -31 -8 62 1 23 | 0 -26 37 12 -65 3 -21 }}
 Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
@@ Line 759: / Line 785: @@
 {{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}
+: mapping generators: ~2, ~1296/875
 {{Multival|legend=1| 52 56 41 -32 -81 -62 }}
@@ Line 776: / Line 804: @@
 {{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}
+: mapping generators: ~2, ~57344/46875
 {{Multival|legend=1| 60 -8 11 -152 -151 48 }}
@@ Line 781: / Line 811: @@
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
-{{Optimal ET sequence|legend=1| 31, 348, 379, 410, 441, 1354, 1795, 2236 }}
+{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}
 [[Badness]]: 0.045792
@@ Line 793: / Line 823: @@
 {{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}
+: mapping generators: ~2, ~2800/2187
 {{Multival|legend=1| 32 86 51 62 -9 -123 }}
@@ Line 798: / Line 830: @@
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
-{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696, 6955dd }}
+{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}
 [[Badness]]: 0.028307
@@ Line 808: / Line 840: @@
 {{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}
+: mapping generators: ~2, ~8/7
 {{Multival|legend=1| 18 -7 1 -53 -49 22 }}
@@ Line 849: / Line 883: @@
 {{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}
+: mapping generators: ~2, ~125/96
 {{Multival|legend=1| 46 15 19 -83 -99 2 }}
@@ Line 854: / Line 890: @@
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
-{{Optimal ET sequence|legend=1| 37, 103, 140, 243, 383, 1009cd, 1392ccd }}
+{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
 Badness: 0.100511
@@ Line 867: / Line 903: @@
 Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318
-{{Optimal ET sequence|legend=1| 37, 103, 140, 243e }}
+{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243e }}
 Badness: 0.056514
@@ Line 880: / Line 916: @@
 Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316
-{{Optimal ET sequence|legend=1| 37, 103, 140, 243e }}
+{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243e }}
 Badness: 0.027429
 == Mintone ==
-In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo|-3 11 -5 -1}} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 &amp; 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.
+In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo| -3 11 -5 -1 }} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 &amp; 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.
 [[Subgroup]]: 2.3.5.7
@@ Line 892: / Line 928: @@
 {{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}
+: mapping generators: ~2, ~10/9
 {{Multival|legend=1| 22 43 27 17 -19 -58 }}
@@ Line 948: / Line 986: @@
 {{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}
+: mapping generators: ~2, ~250/189
 {{Multival|legend=1| 28 36 25 -8 -39 -43 }}
@@ Line 988: / Line 1,028: @@
 [[Comma list]]: 2401/2400, 390625/387072
-{{Mapping|legend=1| 1 -13 -4 -4 | 0 30 13 14 }}
+{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}
+: mappping generators: ~2, ~10/7
 {{Multival|legend=1| 30 13 14 -49 -62 -4 }}
@@ Line 1,003: / Line 1,045: @@
 Comma list: 385/384, 1375/1372, 4000/3993
-Mapping: {{mapping| 1 -13 -4 -4 2 | 0 30 13 14 3 }}
+Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}
 Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
@@ Line 1,016: / Line 1,058: @@
 Comma list: 169/168, 364/363, 385/384, 625/624
-Mapping: {{mapping| 1 -13 -4 -4 2 -7 | 0 30 13 14 3 22 }}
+Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }}
 Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
@@ Line 1,032: / Line 1,074: @@
 {{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
+: mapping generators: ~2, ~49/40
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603