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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~49/40 = 351.4464
* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464
* [[POTE]]: ~2 = 1\1, ~49/40 = 351.477
: [[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
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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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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 351.4289
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
* POTE: ~2 = 1\1, ~11/9 = 351.521
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206


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


Badness: 0.023498
Badness (Smith): 0.023498


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 351.4331
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
* POTE: ~2 = 1\1, ~11/9 = 351.573
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734


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


Badness: 0.019090
Badness (Smith): 0.019090


=== Semihemi ===
=== Semihemi ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 1\2, ~49/40 = 351.4722
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
* POTE: ~99/70 = 1\2, ~49/40 = 351.505
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047


Optimal ET sequence: {{Optimal ET sequence| 58, 140, 198 }}
{{Optimal ET sequence|legend=0| 58, 140, 198 }}


Badness: 0.042487
Badness (Smith): 0.042487


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 1\2, ~49/40 = 351.4674
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
* POTE: ~99/70 = 1\2, ~49/40 = 351.502
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019


Optimal ET sequence: {{Optimal ET sequence| 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
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~243/220 = 175.7284
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
* POTE: ~2 = 1\1, ~243/220 = 175.7378
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378


Optimal ET sequence: {{Optimal ET sequence| 41, 157, 198, 239, 676b, 915be }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}


Badness: 0.040170
Badness (Smith): 0.040170


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~72/65 = 175.7412
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* POTE: ~2 = 1\1, ~72/65 = 175.7470
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470


Optimal ET sequence: {{Optimal ET sequence| 41, 157, 198, 437f, 635bcff }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}


Badness: 0.031144
Badness (Smith): 0.031144


== Tertiaseptal ==
== Tertiaseptal ==
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: Mapping generators: ~2, ~256/245
: Mapping generators: ~2, ~256/245
{{Multival|legend=1| 22 -5 3 -59 -57 21 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
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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 &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
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: Mapping generators: ~2, ~875/512
: Mapping generators: ~2, ~875/512
{{Multival|legend=1| 38 -3 8 -93 -94 27 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
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Badness: 0.017921
Badness: 0.017921
== Decoid ==
{{See also| Quintosec family #Decoid }}
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
[[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: {{Optimal ET sequence| 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: {{Optimal ET sequence| 130, 270, 940, 1210f, 1480cf }}
Badness: 0.013475


== Neominor ==
== Neominor ==
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: Mapping generators: ~2, ~189/160
: Mapping generators: ~2, ~189/160
{{Multival|legend=1| 6 41 22 51 18 -64 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280
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: Mapping generators: ~2, ~2187/1372
: 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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
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: Mapping generators: ~2, ~42/25
: Mapping generators: ~2, ~42/25
{{Multival|legend=1| 34 29 23 -33 -59 -28 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
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== 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
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: Mapping generators: ~2, ~6125/3888
: Mapping generators: ~2, ~6125/3888
{{Multival|legend=1| 42 47 34 -23 -64 -53 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
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== Newt ==
== Newt ==
This temperament has a generator of neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]]. A notable tuning of newt not shown here is [[311edo]] with great consistency in the 41-limit.
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
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{{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
: mapping generators: ~2, ~49/40
 
{{Multival|legend=1| 2 -57 -28 -95 -50 95 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
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Badness: 0.013830
Badness: 0.013830


==== Neoneut ====
=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
'''Neonewt''' is a remarkable subgroup extension with a prime harmonic of 19.
 
Subgroup: 2.3.5.7.11.13.19
Subgroup: 2.3.5.7.11.13.19


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: Mapping generators: ~2, ~28/15
: Mapping generators: ~2, ~28/15
{{Multival|legend=1| 26 -37 -12 -119 -92 76 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
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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|l 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
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: Mapping generators: ~2, ~1296/875
: Mapping generators: ~2, ~1296/875
{{Multival|legend=1| 52 56 41 -32 -81 -62 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
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: Mapping generators: ~2, ~57344/46875
: Mapping generators: ~2, ~57344/46875
{{Multival|legend=1| 60 -8 11 -152 -151 48 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
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: Mapping generators: ~2, ~2800/2187
: Mapping generators: ~2, ~2800/2187
{{Multival|legend=1| 32 86 51 62 -9 -123 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
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: Mapping generators: ~2, ~8/7
: Mapping generators: ~2, ~8/7
{{Multival|legend=1| 18 -7 1 -53 -49 22 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
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: Mapping generators: ~2, ~125/96
: Mapping generators: ~2, ~125/96
{{Multival|legend=1| 46 15 19 -83 -99 2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
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: Mapping generators: ~2, ~10/9
: Mapping generators: ~2, ~10/9
{{Multival|legend=1| 22 43 27 17 -19 -58 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
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: Mapping generators: ~2, ~250/189
: Mapping generators: ~2, ~250/189
{{Multival|legend=1| 28 36 25 -8 -39 -43 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
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: Mappping generators: ~2, ~10/7
: Mappping generators: ~2, ~10/7
{{Multival|legend=1| 30 13 14 -49 -62 -4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
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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 1,192: Line 1,218:


Badness: 0.013771
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]]
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