Breedsmic temperaments: Difference between revisions

Update keys
 
(26 intermediate revisions by 6 users not shown)
Line 1: Line 1:
{{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.


Line 12: Line 13:
* ''[[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]]'' (+2401/2400) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[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]]
Line 21: Line 22:
* ''[[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]]


Line 26: Line 29:
{{Main| Hemififths }}
{{Main| Hemififths }}


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}}.
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. [[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
[[Subgroup]]: 2.3.5.7
Line 36: Line 39:
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}


{{Multival|legend=1| 2 25 13 35 15 -40 }}
: mapping generators: ~2, ~49/40


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.477
[[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 (unchanged-interval) basis]]: 2.5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


[[Algebraic generator]]: (2 + sqrt(2))/2
[[Algebraic generator]]: (2 + sqrt(2))/2
Line 49: Line 56:
{{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 ===
Line 58: Line 65:
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 tuning (POTE): ~2 = 1\1, ~11/9 = 351.521
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206


{{Optimal ET sequence|legend=1| 17c, 41, 58, 99e }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}


Badness: 0.023498
Badness (Smith): 0.023498


==== 13-limit ====
==== 13-limit ====
Line 71: Line 80:
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 tuning (POTE): ~2 = 1\1, ~11/9 = 351.573
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734


{{Optimal ET sequence|legend=1| 17c, 41, 58, 99ef, 157eff }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}


Badness: 0.019090
Badness (Smith): 0.019090


=== Semihemi ===
=== Semihemi ===
Line 84: Line 95:
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~49/40 = 351.505
: mapping generators: ~99/70, ~400/231


{{Optimal ET sequence|legend=1| 58, 140, 198 }}
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 ====
Line 97: Line 112:
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 tuning (POTE): ~99/70 = 1\2, ~49/40 = 351.502
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019


{{Optimal ET sequence|legend=1| 58, 140, 198, 536f }}
{{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
Line 112: Line 129:
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}


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


{{Optimal ET sequence|legend=1| 41, 157, 198, 239, 676b, 915be }}
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 ====
Line 125: Line 146:
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 tuning (POTE): ~2 = 1\1, ~72/65 = 175.7470
Optimal tunings:
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470


{{Optimal ET sequence|legend=1| 41, 157, 198, 437f, 635bcff }}
{{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 &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, 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
Line 142: Line 165:
{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}
{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}


{{Multival|legend=1| 22 -5 3 -59 -57 21 }}
: 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
Line 159: Line 182:
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227


{{Optimal ET sequence|legend=1| 31, 109e, 140e, 171, 202 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}


Badness: 0.035576
Badness: 0.035576
Line 172: Line 195:
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203


{{Optimal ET sequence|legend=1| 31, 109e, 140e, 171 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}


Badness: 0.036876
Badness: 0.036876
Line 185: Line 208:
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201


{{Optimal ET sequence|legend=1| 31, 109eg, 140e, 171 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }}


Badness: 0.027398
Badness: 0.027398
Line 198: Line 221:
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173


{{Optimal ET sequence|legend=1| 31, 109, 140, 171e, 311e }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }}


Badness: 0.030171
Badness: 0.030171
Line 211: Line 234:
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158


{{Optimal ET sequence|legend=1| 31, 109, 140, 311e, 451ee }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}


Badness: 0.028384
Badness: 0.028384
Line 224: Line 247:
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162


{{Optimal ET sequence|legend=1| 31, 109g, 140, 311e, 451ee }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}


Badness: 0.022416
Badness: 0.022416
Line 237: Line 260:
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}


Badness: 0.056926
Badness: 0.056926
Line 250: Line 273:
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168


{{Optimal ET sequence|legend=1| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}


Badness: 0.027474
Badness: 0.027474
Line 263: Line 286:
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}


Badness: 0.018773
Badness: 0.018773
Line 276: Line 299:
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}


Badness: 0.017653
Badness: 0.017653
Line 289: Line 312:
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168


{{Optimal ET sequence|legend=1| 140, 311, 762g, 1073g, 1384cfgg }}
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}


Badness: 0.015123
Badness: 0.015123
Line 302: Line 325:
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167


{{Optimal ET sequence|legend=1| 140, 311, 762g, 1073g, 1384cfggj }}
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }}


Badness: 0.012181
Badness: 0.012181
Line 315: Line 338:
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
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|legend=1| 140, 171, 311 }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}


Badness: 0.010949
Badness: 0.010949
Line 341: Line 364:
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}


Badness: 0.009825
Badness: 0.009825
Line 351: Line 374:


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|legend=1| 31, 280, 311, 342 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}


Badness: 0.015633
Badness: 0.015633
Line 367: Line 392:
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588


{{Optimal ET sequence|legend=1| 31, 280, 311, 964f, 1275f, 1586cff }}
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 964f, 1275f, 1586cff }}


Badness: 0.033573
Badness: 0.033573
Line 380: Line 405:
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589


{{Optimal ET sequence|legend=1| 31, 280, 311, 653f, 964f }}
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 653f, 964f }}


Badness: 0.025298
Badness: 0.025298
Line 390: Line 415:


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|legend=1| 62e, 140, 202, 342 }}
Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}


Badness: 0.025790
Badness: 0.025790
Line 400: Line 427:
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 &amp; 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 &amp; 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 and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning.
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
Line 408: Line 435:
{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}
{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}


{{Multival|legend=1| 38 -3 8 -93 -94 27 }}
: 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
Line 425: Line 452:
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111


{{Optimal ET sequence|legend=1| 31, 208, 239, 270 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }}


Badness: 0.017540
Badness: 0.017540
Line 438: Line 465:
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107


{{Optimal ET sequence|legend=1| 31, 239, 270, 571, 841, 1111 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}


Badness: 0.017921
Badness: 0.017921
== Decoid ==
{{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.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 67108864/66976875
{{Mapping|legend=1| 10 0 47 36 | 0 2 -3 -1 }}
: mapping generators: ~15/14, ~8192/4725
{{Multival|legend=1| 20 -30 -10 -94 -72 61 }}
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099)
{{Optimal ET sequence|legend=1| 10, 120, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
[[Badness]]: 0.033902
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 5632/5625, 9801/9800
Mapping: {{mapping| 10 0 47 36 98 | 0 2 -3 -1 -8 }}
Optimal tuning (POTE): ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070)
{{Optimal ET sequence|legend=1| 10e, 130, 270, 670, 940, 1210, 2150c }}
Badness: 0.018735
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095
Mapping: {{mapping| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}
Optimal tuning (POTE): ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083)
{{Optimal ET sequence|legend=1| 10e, 130, 270, 940, 1210f, 1480cf }}
Badness: 0.013475


== 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 }}


{{Multival|legend=1| 6 41 22 51 18 -64 }}
: 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
Line 515: Line 495:
Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276
Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276


{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }}


Badness: 0.027959
Badness: 0.027959
Line 528: Line 508:
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294


{{Optimal ET sequence|legend=1| 72, 161f, 233f }}
Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233f }}


Badness: 0.026942
Badness: 0.026942


== Emmthird ==
== 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
[[Subgroup]]: 2.3.5.7
Line 539: Line 519:
[[Comma list]]: 2401/2400, 14348907/14336000
[[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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
Line 554: Line 534:
Comma list: 243/242, 441/440, 1792000/1771561
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
Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
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 -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
Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}


Badness: 0.026974
Badness: 0.026974
Line 584: Line 564:
Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985
Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
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 -7 -5 -3 | 0 34 29 23 }}
{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}


{{Multival|legend=1| 34 29 23 -33 -59 -28 }}
: 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| 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
[[Badness]]: 0.037322


== Unthirds ==
== Unthirds ==
The generator for unthirds temperament is undecimal major third, 14/11.
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 -13 -14 -9 | 0 42 47 34 }}
{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}


{{Multival|legend=1| 42 47 34 -23 -64 -53 }}
: 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 -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 tuning (POTE): ~2 = 1\1, ~14/11 = 416.718


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 1316c }}
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 -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 tuning (POTE): ~2 = 1\1, ~14/11 = 416.716


{{Optimal ET sequence|legend=1| 72, 311, 694, 1005c, 1699cd }}
Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }}


Badness: 0.020888
Badness: 0.020888


== Newt ==
== Newt ==
This temperament has a generator of neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]].
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 }}


{{Multival|legend=1| 2 -57 -28 -95 -50 95 }}
: 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, 6333bbcc }}
{{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|legend=1| 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b }}
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|legend=1| 41, 229, 270, 581, 851, 2283b, 3134b }}
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 -1 6 4 | 0 26 -37 -12 }}
{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}


{{Multival|legend=1| 26 -37 -12 -119 -92 76 }}
: 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 -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
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332 }}
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 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
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332, 835eeff }}
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 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
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332, 503ef, 835eeff }}
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 }}


{{Multival|legend=1| 52 56 41 -32 -81 -62 }}
: 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 }}


{{Multival|legend=1| 60 -8 11 -152 -151 48 }}
: 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 }}


{{Multival|legend=1| 32 86 51 62 -9 -123 }}
: 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, 6955dd }}
{{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 }}


{{Multival|legend=1| 18 -7 1 -53 -49 22 }}
: 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|legend=1| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
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|legend=1| 21, 37, 58, 153bcef, 211bccdeeff }}
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 }}


{{Multival|legend=1| 46 15 19 -83 -99 2 }}
: 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|legend=1| 37, 103, 140, 243e }}
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|legend=1| 37, 103, 140, 243e }}
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 &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
[[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 }}


{{Multival|legend=1| 22 43 27 17 -19 -58 }}
: 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|legend=1| 58, 103, 161, 425b, 586b, 747bc }}
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|legend=1| 58, 103, 161, 425b, 586bf }}
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|legend=1| 58, 103, 161, 425b, 586bf }}
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 }}


{{Multival|legend=1| 28 36 25 -8 -39 -43 }}
: 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|legend=1| 103, 130, 233, 363, 493e, 856be }}
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|legend=1| 103, 130, 233, 363 }}
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 -13 -4 -4 | 0 30 13 14 }}
{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}


{{Multival|legend=1| 30 13 14 -49 -62 -4 }}
: 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 -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
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387


{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
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 -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
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387


{{Optimal ET sequence|legend=1| 37, 72, 109, 181f }}
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|legend=1| 31, 86ce, 117ce, 148bce }}
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
: 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]]: 0.202
[[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|legend=1| 120, 191, 311, 742, 1053, 1795 }}
Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795 }}


Badness: 0.0523
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|legend=1| 120, 191, 311, 742, 1053, 1795f }}
Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795f }}


Badness: 0.0325
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|legend=1| 120g, 191g, 311, 431, 742, 1795f }}
Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}


Badness: 0.0325
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|legend=1| 120g, 191g, 311, 431, 742, 1795f }}
Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
 
Badness: 0.013771


Badness: 0.0138
== 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]]