Breedsmic temperaments: Difference between revisions
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{{Technical data page}} | |||
This page discusses miscellaneous [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[breedsma]], {{monzo| -5 -1 -2 4 }} = 2401/2400. This is the amount by which two [[49/40]] intervals exceed [[3/2]], and by which two [[60/49]] intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system ([[12edo]], for example) which does not possess a neutral third cannot be tempering out the breedsma. | This page discusses miscellaneous [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[breedsma]], {{monzo| -5 -1 -2 4 }} = 2401/2400. This is the amount by which two [[49/40]] intervals exceed [[3/2]], and by which two [[60/49]] intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system ([[12edo]], for example) which does not possess a neutral third cannot be tempering out the breedsma. | ||
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* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]] | * ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]] | ||
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]] | * ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]] | ||
* ''[[Quadrasruta]]'' (+ | * ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]] | ||
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]] | * ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]] | ||
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]] | * ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]] | ||
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* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]] | * ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]] | ||
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]] | * ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]] | ||
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]] | |||
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]] | |||
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]] | * ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]] | ||
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{{Main| Hemififths }} | {{Main| Hemififths }} | ||
Hemififths | Hemififths may be described as the {{nowrap| 41 & 58 }} temperament, tempering out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator; its [[ploidacot]] is dicot. [[99edo]] and [[140edo]] provides 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 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. | 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 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }} | {{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }} | ||
: mapping generators: ~2, ~49/40 | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464 | |||
: [[error map]]: {{val| 0.0000 +0.9379 -0.1531 -0.0224 }} | |||
* [[POTE]]: ~2 = 1200.0000, ~49/40 = 351.4774 | |||
: error map: {{val| 0.0000 +0.9999 +0.6221 +0.0307 }} | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }} | * [[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 }} | : {{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 basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
[[Algebraic generator]]: (2 + sqrt(2))/2 | [[Algebraic generator]]: (2 + sqrt(2))/2 | ||
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{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }} | {{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }} | ||
[[Badness]]: 0.022243 | [[Badness]] (Smith): 0.022243 | ||
=== 11-limit === | === 11-limit === | ||
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Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }} | Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }} | ||
Optimal | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289 | |||
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206 | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }} | ||
Badness: 0.023498 | Badness (Smith): 0.023498 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
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Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }} | Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }} | ||
Optimal | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331 | |||
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734 | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }} | ||
Badness: 0.019090 | Badness (Smith): 0.019090 | ||
=== Semihemi === | === Semihemi === | ||
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Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }} | Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }} | ||
: mapping generators: ~99/70, ~400/231 | |||
Optimal tunings: | |||
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722 | |||
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047 | |||
Badness: 0.042487 | {{Optimal ET sequence|legend=0| 58, 140, 198 }} | ||
Badness (Smith): 0.042487 | |||
==== 13-limit ==== | ==== 13-limit ==== | ||
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Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }} | Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }} | ||
Optimal | Optimal tunings: | ||
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674 | |||
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019 | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 58, 140, 198, 536f }} | ||
Badness: 0.021188 | Badness (Smith): 0.021188 | ||
=== Quadrafifths === | === Quadrafifths === | ||
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense. | This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }} | Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }} | ||
: Mapping generators: ~2, ~243/220 | |||
Optimal tunings: | |||
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284 | |||
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378 | |||
Badness: 0.040170 | {{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }} | ||
Badness (Smith): 0.040170 | |||
==== 13-limit ==== | ==== 13-limit ==== | ||
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Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }} | Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }} | ||
Optimal | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412 | |||
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470 | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }} | ||
Badness: 0.031144 | Badness (Smith): 0.031144 | ||
== Tertiaseptal == | == Tertiaseptal == | ||
{{Main| Tertiaseptal }} | {{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 & 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 | 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, although 171edo - [[31edo]] = [[140edo]] also makes sense, and in very high limits 140edo + 171edo = [[311edo]] is especially notable. 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 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }} | {{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }} | ||
: Mapping generators: ~2, ~256/245 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227 | Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }} | ||
Badness: 0.035576 | Badness: 0.035576 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203 | Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }} | ||
Badness: 0.036876 | Badness: 0.036876 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201 | Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }} | ||
Badness: 0.027398 | Badness: 0.027398 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173 | Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }} | ||
Badness: 0.030171 | Badness: 0.030171 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158 | Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }} | ||
Badness: 0.028384 | Badness: 0.028384 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162 | Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }} | ||
Badness: 0.022416 | Badness: 0.022416 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169 | Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }} | ||
Badness: 0.056926 | Badness: 0.056926 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168 | Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }} | ||
Badness: 0.027474 | Badness: 0.027474 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169 | Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }} | ||
Badness: 0.018773 | Badness: 0.018773 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169 | Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }} | ||
Badness: 0.017653 | Badness: 0.017653 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168 | Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }} | ||
Badness: 0.015123 | Badness: 0.015123 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167 | Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }} | ||
Badness: 0.012181 | Badness: 0.012181 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169 | Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }} | ||
Badness: 0.012311 | Badness: 0.012311 | ||
Line 328: | Line 351: | ||
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170 | Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }} | ||
Badness: 0.010949 | Badness: 0.010949 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169 | Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }} | ||
Badness: 0.009825 | Badness: 0.009825 | ||
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Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }} | 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 | Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }} | ||
Badness: 0.015633 | Badness: 0.015633 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588 | Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 964f, 1275f, 1586cff }} | ||
Badness: 0.033573 | Badness: 0.033573 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589 | Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 653f, 964f }} | ||
Badness: 0.025298 | Badness: 0.025298 | ||
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Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }} | 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 | Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }} | ||
Badness: 0.025790 | Badness: 0.025790 | ||
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In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = {{monzo| 22 -1 -10 1 }}. 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)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths. | In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = {{monzo| 22 -1 -10 1 }}. 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)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths. | ||
Adding 3025/3024 extends to the 11-limit | Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }} | {{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }} | ||
: Mapping generators: ~2, ~875/512 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111 | Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }} | ||
Badness: 0.017540 | Badness: 0.017540 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107 | Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }} | ||
Badness: 0.017921 | Badness: 0.017921 | ||
== Neominor == | == Neominor == | ||
Line 498: | Line 478: | ||
{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }} | {{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }} | ||
: Mapping generators: ~2, ~189/160 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276 | Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }} | ||
Badness: 0.027959 | Badness: 0.027959 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294 | Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233f }} | ||
Badness: 0.026942 | Badness: 0.026942 | ||
== Emmthird == | == Emmthird == | ||
The generator for emmthird | The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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[[Comma list]]: 2401/2400, 14348907/14336000 | [[Comma list]]: 2401/2400, 14348907/14336000 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }} | ||
: Mapping generators: ~2, ~2187/1372 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988 | ||
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Comma list: 243/242, 441/440, 1792000/1771561 | Comma list: 243/242, 441/440, 1792000/1771561 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }} | ||
Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991 | Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }} | ||
Badness: 0.052358 | Badness: 0.052358 | ||
Line 567: | Line 547: | ||
Comma list: 243/242, 364/363, 441/440, 2200/2197 | Comma list: 243/242, 364/363, 441/440, 2200/2197 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 11 42 25 27 38 | 0 -14 -59 -33 -35 -51 }} | ||
Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989 | Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }} | ||
Badness: 0.026974 | Badness: 0.026974 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985 | Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }} | ||
Badness: 0.023205 | Badness: 0.023205 | ||
Line 595: | Line 575: | ||
[[Comma list]]: 2401/2400, 1959552/1953125 | [[Comma list]]: 2401/2400, 1959552/1953125 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }} | ||
: Mapping generators: ~2, ~42/25 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }} | ||
[[Badness]]: 0.037322 | [[Badness]]: 0.037322 | ||
== Unthirds == | == Unthirds == | ||
Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however. | |||
The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's). | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
Line 612: | Line 594: | ||
[[Comma list]]: 2401/2400, 68359375/68024448 | [[Comma list]]: 2401/2400, 68359375/68024448 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }} | ||
: Mapping generators: ~2, ~6125/3888 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717 | ||
Line 627: | Line 609: | ||
Comma list: 2401/2400, 3025/3024, 4000/3993 | Comma list: 2401/2400, 3025/3024, 4000/3993 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }} | ||
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718 | Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }} | ||
Badness: 0.022926 | Badness: 0.022926 | ||
Line 640: | Line 622: | ||
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400 | Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400 | ||
Mapping: {{mapping| 1 | 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 tuning (POTE): ~2 = 1\1, ~14/11 = 416.716 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }} | ||
Badness: 0.020888 | Badness: 0.020888 | ||
== Newt == | == Newt == | ||
Newt has a generator of a neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]]. It can be described as the 41 & 270 temperament, and extends naturally to the no-17 19-limit, a.k.a. '''neonewt'''. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] work much better. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
Line 657: | Line 639: | ||
{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }} | {{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }} | ||
: mapping generators: ~2, ~49/40 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113 | ||
{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 | {{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }} | ||
[[Badness]]: 0.041878 | [[Badness]]: 0.041878 | ||
Line 674: | Line 656: | ||
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115 | Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }} | ||
Badness: 0.019461 | Badness: 0.019461 | ||
Line 687: | Line 669: | ||
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117 | Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }} | ||
Badness: 0.013830 | Badness: 0.013830 | ||
=== 2.3.5.7.11.13.19 subgroup (neonewt) === | |||
Subgroup: 2.3.5.7.11.13.19 | |||
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400 | |||
Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }} | |||
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117 | |||
Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }} | |||
== Septidiasemi == | == Septidiasemi == | ||
Line 700: | Line 693: | ||
[[Comma list]]: 2401/2400, 2152828125/2147483648 | [[Comma list]]: 2401/2400, 2152828125/2147483648 | ||
{{Mapping|legend=1| 1 - | {{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }} | ||
: Mapping generators: ~2, ~28/15 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297 | ||
Line 717: | Line 710: | ||
Comma list: 243/242, 441/440, 939524096/935859375 | Comma list: 243/242, 441/440, 939524096/935859375 | ||
Mapping: {{mapping| 1 - | Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }} | ||
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279 | Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332 }} | ||
Badness: 0.090687 | Badness: 0.090687 | ||
Line 730: | Line 723: | ||
Comma list: 243/242, 441/440, 2200/2197, 3584/3575 | Comma list: 243/242, 441/440, 2200/2197, 3584/3575 | ||
Mapping: {{mapping| 1 -1 | Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }} | ||
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281 | Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 835eeff }} | ||
Badness: 0.045773 | Badness: 0.045773 | ||
Line 743: | Line 736: | ||
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575 | Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575 | ||
Mapping: {{mapping| 1 -1 | 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 | Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }} | ||
Badness: 0.027322 | Badness: 0.027322 | ||
Line 760: | Line 753: | ||
{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }} | {{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }} | ||
: Mapping generators: ~2, ~1296/875 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810 | ||
Line 777: | Line 770: | ||
{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }} | {{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }} | ||
: Mapping generators: ~2, ~57344/46875 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301 | [[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 | [[Badness]]: 0.045792 | ||
Line 794: | Line 787: | ||
{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }} | {{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }} | ||
: Mapping generators: ~2, ~2800/2187 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066 | ||
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 | {{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }} | ||
[[Badness]]: 0.028307 | [[Badness]]: 0.028307 | ||
Line 809: | Line 802: | ||
{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }} | {{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }} | ||
: Mapping generators: ~2, ~8/7 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512 | ||
Line 826: | Line 819: | ||
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500 | Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }} | ||
Badness: 0.059260 | Badness: 0.059260 | ||
Line 839: | Line 832: | ||
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493 | Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }} | ||
Badness: 0.032205 | Badness: 0.032205 | ||
Line 850: | Line 843: | ||
{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }} | {{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }} | ||
: Mapping generators: ~2, ~125/96 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310 | [[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 | Badness: 0.100511 | ||
Line 867: | Line 860: | ||
Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318 | Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }} | ||
Badness: 0.056514 | Badness: 0.056514 | ||
Line 880: | Line 873: | ||
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316 | Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }} | ||
Badness: 0.027429 | Badness: 0.027429 | ||
== Mintone == | == 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 | [[Subgroup]]: 2.3.5.7 | ||
Line 893: | Line 886: | ||
{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }} | {{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }} | ||
: Mapping generators: ~2, ~10/9 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343 | ||
Line 910: | Line 903: | ||
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345 | Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }} | ||
Badness: 0.039962 | Badness: 0.039962 | ||
Line 923: | Line 916: | ||
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347 | Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }} | ||
Badness: 0.021849 | Badness: 0.021849 | ||
Line 936: | Line 929: | ||
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348 | Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }} | ||
Badness: 0.020295 | Badness: 0.020295 | ||
Line 949: | Line 942: | ||
{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }} | {{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }} | ||
: Mapping generators: ~2, ~250/189 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235 | ||
Line 966: | Line 959: | ||
Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252 | Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }} | ||
Badness: 0.036785 | Badness: 0.036785 | ||
Line 979: | Line 972: | ||
Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256 | Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }} | ||
Badness: 0.021694 | Badness: 0.021694 | ||
Line 988: | Line 981: | ||
[[Comma list]]: 2401/2400, 390625/387072 | [[Comma list]]: 2401/2400, 390625/387072 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }} | ||
: Mappping generators: ~2, ~10/7 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385 | ||
Line 1,003: | Line 996: | ||
Comma list: 385/384, 1375/1372, 4000/3993 | Comma list: 385/384, 1375/1372, 4000/3993 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }} | ||
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387 | Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }} | ||
Badness: 0.032225 | Badness: 0.032225 | ||
Line 1,016: | Line 1,009: | ||
Comma list: 169/168, 364/363, 385/384, 625/624 | Comma list: 169/168, 364/363, 385/384, 625/624 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }} | ||
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387 | Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 37, 72, 109, 181f }} | ||
Badness: 0.028683 | Badness: 0.028683 | ||
Line 1,032: | Line 1,025: | ||
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }} | {{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 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603 | ||
Line 1,048: | Line 1,043: | ||
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639 | Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 31, 86ce, 117ce, 148bce }} | ||
Badness: 0.046181 | Badness: 0.046181 | ||
== Lockerbie == | |||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].'' | |||
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[120/77]] or [[77/60]]. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well. | |||
The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270. | |||
Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }} | |||
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }} | |||
: Mapping generators: ~2, ~3828125/2985984 | |||
[[Optimal tuning]]s: | |||
* [[CTE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1071 | |||
: [[error map]]: {{val| 0.0000 -0.0270 +0.1502 -0.1120 }} | |||
* [[CWE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1072 | |||
: error map: {{val| 0.0000 -0.0205 +0.1547 -0.1081 }} | |||
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913 }} | |||
[[Badness]] (Smith): 0.0597 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 2401/2400, 3025/3024, 766656/765625 | |||
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082 | |||
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078 | |||
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }} | |||
Badness (Smith): 0.0262 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224 | |||
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1200.0000, ~77/60 = 431.1085 | |||
* CWE: ~2 = 1200.0000, ~77/60 = 431.1069 | |||
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }} | |||
Badness (Smith): 0.0160 | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224 | |||
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1200.000, ~77/60 = 431.107 | |||
* CWE: ~2 = 1200.000, ~77/60 = 431.108 | |||
{{Optimal ET sequence|legend=0| 103, 167, 270 }} | |||
Badness (Smith): 0.0210 | |||
=== 2.3.5.7.11.13.17.41 subgroup === | |||
Subgroup: 2.3.5.7.11.13.17.41 | |||
Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224 | |||
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1200.000, ~41/32 = 431.107 | |||
* CWE: ~2 = 1200.000, ~41/32 = 431.111 | |||
{{Optimal ET sequence|legend=0| 103, 167, 270 }} | |||
== Hemigoldis == | |||
: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].'' | |||
Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 2401/2400, 549755813888/533935546875 | |||
{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }} | |||
: mapping generators: ~2, ~7/4 | |||
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690 | |||
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }} | |||
[[Badness]] (Sintel): 4.40 | |||
== Surmarvelpyth == | == Surmarvelpyth == | ||
Line 1,061: | Line 1,159: | ||
{{Mapping|legend=1| 1 43 -74 -25 | 0 -70 129 47 }} | {{Mapping|legend=1| 1 43 -74 -25 | 0 -70 129 47 }} | ||
: | : Mapping generators: ~2, ~675/448 | ||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719 | [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719 | ||
Line 1,067: | Line 1,165: | ||
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }} | {{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }} | ||
[[Badness]]: 0. | [[Badness]]: 0.202249 | ||
=== 11-limit === | === 11-limit === | ||
Line 1,078: | Line 1,176: | ||
Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720 | Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795 }} | ||
Badness: 0. | Badness: 0.052308 | ||
=== 13-limit === | === 13-limit === | ||
Line 1,091: | Line 1,189: | ||
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723 | Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795f }} | ||
Badness: 0. | Badness: 0.032503 | ||
=== 17-limit === | === 17-limit === | ||
Line 1,104: | Line 1,202: | ||
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722 | Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }} | ||
Badness: 0. | Badness: 0.020995 | ||
=== 19-limit === | === 19-limit === | ||
Line 1,117: | Line 1,215: | ||
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722 | Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }} | ||
Badness: 0.013771 | |||
== Notes == | |||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Breedsmic temperaments| ]] <!-- main article --> | [[Category:Breedsmic temperaments| ]] <!-- main article --> | ||
[[Category:Breed| ]] <!-- key article --> | [[Category:Breed| ]] <!-- key article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] |