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Breedsmic temperaments: Difference between revisions

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{{Technical data page}}
{{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|legend=1| -5 -1 -2 4 }}, [[ratio]]: 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.


The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that (49/40)(10/7) = 7/4 and (49/40)(10/7)<sup>2</sup> = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that (49/40)(10/7) = 7/4 and (49/40)(10/7)<sup>2</sup> = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
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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
: mapping generators: ~2, ~49/40


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464
* [[CTE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.4464{{c}}
: [[error map]]: {{val| 0.0000 +0.9379 -0.1531 -0.0224 }}
: [[error map]]: {{val| 0.0000 +0.9379 -0.1531 -0.0224 }}
* [[POTE]]: ~2 = 1200.0000, ~49/40 = 351.4774
* [[POTE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.4774{{c}}
: error map: {{val| 0.0000 +0.9999 +0.6221 +0.0307 }}
: error map: {{val| 0.0000 +0.9999 +0.6221 +0.0307 }}


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* [[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|unchanged-interval (eigenmonzo) 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
* CTE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4289{{c}}
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206
* POTE: ~2 = 1200.0000{{c}}, ~11/9 = 351.5206{{c}}


{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
* CTE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4331{{c}}
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734
* POTE: ~2 = 1200.0000{{c}}, ~11/9 = 351.5734{{c}}


{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}
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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
: mapping generators: ~99/70, ~400/231


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
* CTE: ~99/70 = 600.0000{{c}}, ~49/40 = 351.4722{{c}}
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047
* POTE: ~99/70 = 600.0000{{c}}, ~49/40 = 351.5047{{c}}


{{Optimal ET sequence|legend=0| 58, 140, 198 }}
{{Optimal ET sequence|legend=0| 58, 140, 198 }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
* CTE: ~99/70 = 600.0000{{c}}, ~49/40 = 351.4674{{c}}
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019
* POTE: ~99/70 = 600.0000{{c}}, ~49/40 = 351.5019{{c}}


{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
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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
: Mapping generators: ~2, ~243/220


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
* CTE: ~2 = 1200.0000{{c}}, ~243/220 = 175.7284{{c}}
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378
* POTE: ~2 = 1200.0000{{c}}, ~243/220 = 175.7378{{c}}


{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* CTE: ~2 = 1200.0000{{c}}, ~72/65 = 175.7412{{c}}
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470
* POTE: ~2 = 1200.0000{{c}}, ~72/65 = 175.7470{{c}}


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


: Mapping generators: ~2, ~256/245
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~256/245 = 77.191{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191


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


[[Badness]]: 0.012995
[[Badness]] (Smith): 0.012995


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}
Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}


Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~256/245 = 77.227{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}
{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171, 202 }}


Badness: 0.035576
Badness (Smith): 0.035576


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}
Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~117/112 = 77.203{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}
{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171 }}


Badness: 0.036876
Badness (Smith): 0.036876


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}
Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}


Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~68/65 = 77.201{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }}
{{Optimal ET sequence|legend=0| 31, 109eg, 140e, 171 }}


Badness: 0.027398
Badness (Smith): 0.027398


=== Tertia ===
=== Tertia ===
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Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}
Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/21 = 77.173{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }}
{{Optimal ET sequence|legend=0| 31, 109, 140, 171e, 311e }}


Badness: 0.030171
Badness (Smith): 0.030171


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}
Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/21 = 77.158{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}
{{Optimal ET sequence|legend=0| 31, 109, 140, 311e, 451ee }}


Badness: 0.028384
Badness (Smith): 0.028384


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}
Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/21 = 77.162{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}
{{Optimal ET sequence|legend=0| 31, 109g, 140, 311e, 451ee }}


Badness: 0.022416
Badness (Smith): 0.022416


=== Tertiaseptia ===
=== Tertiaseptia ===
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Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}
Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}


Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~256/245 = 77.169{{c}}


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
{{Optimal ET sequence|legend=0| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}


Badness: 0.056926
Badness (Smith): 0.056926


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}
Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~117/112 = 77.168{{c}}


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
{{Optimal ET sequence|legend=0| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}


Badness: 0.027474
Badness (Smith): 0.027474


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 1 3 2 3 -4 1 1 | 0 -22 5 -3 116 42 48 }}
Mapping: {{mapping| 1 3 2 3 -4 1 1 | 0 -22 5 -3 116 42 48 }}


Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~68/65 = 77.169{{c}}


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 140, 171, 311 }}


Badness: 0.018773
Badness (Smith): 0.018773


==== 19-limit ====
==== 19-limit ====
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Mapping: {{mapping| 1 3 2 3 -4 1 1 11 | 0 -22 5 -3 116 42 48 -105 }}
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 | 0 -22 5 -3 116 42 48 -105 }}


Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~68/65 = 77.169{{c}}


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
{{Optimal ET sequence|legend=0| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}


Badness: 0.017653
Badness (Smith): 0.017653


==== 23-limit ====
==== 23-limit ====
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Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 | 0 -22 5 -3 116 42 48 -105 117 }}
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 | 0 -22 5 -3 116 42 48 -105 117 }}


Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~23/22 = 77.168{{c}}


Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}
{{Optimal ET sequence|legend=0| 140, 311, 762g, 1073g, 1384cfgg }}


Badness: 0.015123
Badness (Smith): 0.015123


==== 29-limit ====
==== 29-limit ====
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Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 | 0 -22 5 -3 116 42 48 -105 117 60 }}
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 | 0 -22 5 -3 116 42 48 -105 117 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~23/22 = 77.167{{c}}


Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }}
{{Optimal ET sequence|legend=0| 140, 311, 762g, 1073g, 1384cfggj }}


Badness: 0.012181
Badness (Smith): 0.012181


==== 31-limit ====
==== 31-limit ====
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Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 | 0 -22 5 -3 116 42 48 -105 117 60 -94 }}
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 | 0 -22 5 -3 116 42 48 -105 117 60 -94 }}


Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~23/22 = 77.169{{c}}


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 140, 171, 311 }}


Badness: 0.012311
Badness (Smith): 0.012311


==== 37-limit ====
==== 37-limit ====
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Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 }}
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 }}


Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~23/22 = 77.170{{c}}


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 140, 171, 311 }}


Badness: 0.010949
Badness (Smith): 0.010949


==== 41-limit ====
==== 41-limit ====
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Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 6 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10 }}
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 6 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10 }}


Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~23/22 = 77.169{{c}}


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 140, 171, 311 }}


Badness: 0.009825
Badness (Smith): 0.009825


=== Hemitert ===
=== Hemitert ===
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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


: Mapping generators: ~2, ~45/44
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 38.596{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596
{{Optimal ET sequence|legend=0| 31, 280, 311, 342 }}


Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}
Badness (Smith): 0.015633
 
Badness: 0.015633


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 3 2 3 6 1 | 0 -44 10 -6 -79 84 }}
Mapping: {{mapping| 1 3 2 3 6 1 | 0 -44 10 -6 -79 84 }}


Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 38.588{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 964f, 1275f, 1586cff }}
{{Optimal ET sequence|legend=0| 31, 280, 311, 964f, 1275f, 1586cff }}


Badness: 0.033573
Badness (Smith): 0.033573


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 1 3 2 3 6 1 1 | 0 -44 10 -6 -79 84 96 }}
Mapping: {{mapping| 1 3 2 3 6 1 1 | 0 -44 10 -6 -79 84 96 }}


Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 38.589{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 653f, 964f }}
{{Optimal ET sequence|legend=0| 31, 280, 311, 653f, 964f }}


Badness: 0.025298
Badness (Smith): 0.025298


=== Semitert ===
=== Semitert ===
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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


: Mapping generators: ~99/70, ~256/245
Optimal tuning (POTE): ~99/70 = 1200.000{{c}}, ~256/245 = 77.193{{c}}


Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
{{Optimal ET sequence|legend=0| 62e, 140, 202, 342 }}


Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}
Badness (Smith): 0.025790
 
Badness: 0.025790


== Quasiorwell ==
== Quasiorwell ==
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 & 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 as expected, 270 remains an excellent tuning.
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.
Line 434: Line 428:


{{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


: Mapping generators: ~2, ~875/512
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1024/875 = 271.107{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107


{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}


[[Badness]]: 0.035832
[[Badness]] (Smith): 0.035832


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 31 0 9 53 | 0 -38 3 -8 -64 }}
Mapping: {{mapping| 1 31 0 9 53 | 0 -38 3 -8 -64 }}


Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~90/77 = 271.111{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }}
{{Optimal ET sequence|legend=0| 31, 208, 239, 270 }}


Badness: 0.017540
Badness (Smith): 0.017540


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 31 0 9 53 -59 | 0 -38 3 -8 -64 81 }}
Mapping: {{mapping| 1 31 0 9 53 -59 | 0 -38 3 -8 -64 81 }}


Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~90/77 = 271.107{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}
{{Optimal ET sequence|legend=0| 31, 239, 270, 571, 841, 1111 }}


Badness: 0.017921
Badness (Smith): 0.017921


== Neominor ==
== Neominor ==
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{{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
: Mapping generators: ~2, ~189/160


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[TE]]: ~2 = 1200.428, ~189/160 = 283.381
* [[TE]]: ~2 = 1200.428{{c}}, ~189/160 = 283.381{{c}}
* [[CTE]]: ~2 = 1200.000, ~189/160 = 283.247
* [[CTE]]: ~2 = 1200.000{{c}}, ~189/160 = 283.247{{c}}


{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
Line 496: Line 488:


Optimal tunings:  
Optimal tunings:  
* TE: ~2 = 1200.347, ~33/28 = 283.358
* TE: ~2 = 1200.347{{c}}, ~33/28 = 283.358{{c}}
* CTE: ~2 = 1200.000, ~33/28 = 283.247
* CTE: ~2 = 1200.000{{c}}, ~33/28 = 283.247{{c}}


Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }}
{{Optimal ET sequence|legend=0| 72, 161, 233, 305 }}


Badness (Sintel): 0.924
Badness (Sintel): 0.924
Line 511: Line 503:


Optimal tunings:  
Optimal tunings:  
* TE: ~2 = 1200.689, ~13/11 = 283.457
* TE: ~2 = 1200.689{{c}}, ~13/11 = 283.457{{c}}
* CTE: ~2 = 1200.000, ~13/11 = 283.233
* CTE: ~2 = 1200.000{{c}}, ~13/11 = 283.233{{c}}


Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233ff }}
{{Optimal ET sequence|legend=0| 72, 161f, 233ff }}


Badness (Sintel): 1.113
Badness (Sintel): 1.113
Line 526: Line 518:


Optimal tunings:  
Optimal tunings:  
* TE: ~2 = 1200.692, ~13/11 = 283.455
* TE: ~2 = 1200.692{{c}}, ~13/11 = 283.455{{c}}
* CTE: ~2 = 1200.000, ~13/11 = 283.229
* CTE: ~2 = 1200.000{{c}}, ~13/11 = 283.229{{c}}


Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233ff }}
{{Optimal ET sequence|legend=0| 72, 161f, 233ff }}


Badness (Sintel): 0.918
Badness (Sintel): 0.918
Line 541: Line 533:


{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}
{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}
: mapping generators: ~2, ~2187/1372


: Mapping generators: ~2, ~2187/1372
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~2744/2187 = 392.988{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988


{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}


[[Badness]]: 0.016736
[[Badness]] (Smith): 0.016736


=== 11-limit ===
=== 11-limit ===
Line 557: Line 548:
Mapping: {{mapping| 1 11 42 25 27 | 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 = 1200.000{{c}}, ~1372/1089 = 392.991{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


Badness: 0.052358
Badness (Smith): 0.052358


=== 13-limit ===
=== 13-limit ===
Line 570: Line 561:
Mapping: {{mapping| 1 11 42 25 27 38 | 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 = 1200.000{{c}}, ~180/143 = 392.989{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


Badness: 0.026974
Badness (Smith): 0.026974


=== 17-limit ===
=== 17-limit ===
Line 583: Line 574:
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}


Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~64/51 = 392.985{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


Badness: 0.023205
Badness (Smith): 0.023205


== Quinmite ==
== Quinmite ==
Line 597: Line 588:


{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}
{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}
: mapping generators: ~2, ~42/25


: Mapping generators: ~2, ~42/25
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25/21 = 302.997{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997


{{Optimal ET sequence|legend=1| 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]] (Smith): 0.037322


== Unthirds ==
== Unthirds ==
Line 616: Line 606:


{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}
{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}
: mapping generators: ~2, ~6125/3888


: Mapping generators: ~2, ~6125/3888
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3969/3125 = 416.717{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}


[[Badness]]: 0.075253
[[Badness]] (Smith): 0.075253


=== 11-limit ===
=== 11-limit ===
Line 632: Line 621:
Mapping: {{mapping| 1 29 33 25 25 | 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 = 1200.000{{c}}, ~14/11 = 416.718{{c}}


Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}
{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}


Badness: 0.022926
Badness (Smith): 0.022926


=== 13-limit ===
=== 13-limit ===
Line 645: Line 634:
Mapping: {{mapping| 1 29 33 25 25 99 | 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 = 1200.000{{c}}, ~14/11 = 416.716{{c}}


Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }}
{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}


Badness: 0.020888
Badness (Smith): 0.020888


== Newt ==
== Newt ==
Line 659: Line 648:


{{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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/40 = 351.113{{c}}


{{Optimal ET sequence|legend=1| 41, 147c, 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]] (Smith): 0.041878


=== 11-limit ===
=== 11-limit ===
Line 675: Line 663:
Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}
Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 351.115{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}
{{Optimal ET sequence|legend=0| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}


Badness: 0.019461
Badness (Smith): 0.019461


=== 13-limit ===
=== 13-limit ===
Line 688: Line 676:
Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 351.117{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}
{{Optimal ET sequence|legend=0| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}


Badness: 0.013830
Badness (Smith): 0.013830


=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
Line 701: Line 689:
Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
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 tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 351.117{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}
{{Optimal ET sequence|legend=0| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}


== Septidiasemi ==
== Septidiasemi ==
Line 715: Line 703:


{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}
{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}
: mpping generators: ~2, ~28/15


: Mapping generators: ~2, ~28/15
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~15/14 = 119.297{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}


[[Badness]]: 0.044115
[[Badness]] (Smith): 0.044115


=== Sedia ===
=== Sedia ===
The ''sedia'' temperament (10&amp;161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.
The ''sedia'' temperament (10 & 161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 733: Line 720:
Mapping: {{mapping| 1 25 -31 -8 62 | 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 = 1200.000{{c}}, ~15/14 = 119.279{{c}}


Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332 }}
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332 }}


Badness: 0.090687
Badness (Smith): 0.090687


==== 13-limit ====
==== 13-limit ====
Line 746: Line 733:
Mapping: {{mapping| 1 25 -31 -8 62 1 | 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 = 1200.000{{c}}, ~15/14 = 119.281{{c}}


Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 835eeff }}
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332, 835eeff }}


Badness: 0.045773
Badness (Smith): 0.045773


==== 17-limit ====
==== 17-limit ====
Line 759: Line 746:
Mapping: {{mapping| 1 25 -31 -8 62 1 23 | 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 = 1200.000{{c}}, ~15/14 = 119.281{{c}}


Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }}
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332, 503ef, 835eeff }}


Badness: 0.027322
Badness (Smith): 0.027322


== Maviloid ==
== Maviloid ==
Line 773: Line 760:


{{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


: Mapping generators: ~2, ~1296/875
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1296/875 = 678.810{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810


{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}


[[Badness]]: 0.057632
[[Badness]] (Smith): 0.057632


== Subneutral ==
== Subneutral ==
Line 790: Line 776:


{{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


: Mapping generators: ~2, ~57344/46875
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~57344/46875 = 348.301{{c}}
 
[[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]] (Smith): 0.045792


== Osiris ==
== Osiris ==
Line 807: Line 792:


{{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


: Mapping generators: ~2, ~2800/2187
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~2800/2187 = 428.066{{c}}
 
[[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]] (Smith): 0.028307


== Gorgik ==
== Gorgik ==
Line 822: Line 806:


{{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


: Mapping generators: ~2, ~8/7
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~8/7 = 227.512{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}


[[Badness]]: 0.158384
[[Badness]] (Smith): 0.158384


=== 11-limit ===
=== 11-limit ===
Line 838: Line 821:
Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}
Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~8/7 = 227.500{{c}}


Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}


Badness: 0.059260
Badness (Smith): 0.059260


=== 13-limit ===
=== 13-limit ===
Line 851: Line 834:
Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}
Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~8/7 = 227.493{{c}}


Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bcef, 211bccdeeff }}


Badness: 0.032205
Badness (Smith): 0.032205


== Fibo ==
== Fibo ==
Line 863: Line 846:


{{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


: Mapping generators: ~2, ~125/96
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~125/96 = 454.310{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310


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


Badness: 0.100511
Badness (Smith): 0.100511


=== 11-limit ===
=== 11-limit ===
Line 879: Line 861:
Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}
Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}


Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/77 = 454.318{{c}}


Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}


Badness: 0.056514
Badness (Smith): 0.056514


=== 13-limit ===
=== 13-limit ===
Line 892: Line 874:
Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}
Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 454.316{{c}}


Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}


Badness: 0.027429
Badness (Smith): 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 & 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 906: Line 888:


{{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


: Mapping generators: ~2, ~10/9
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10/9 = 186.343{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343


{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}


[[Badness]]: 0.125672
[[Badness]] (Smith): 0.125672


=== 11-limit ===
=== 11-limit ===
Line 922: Line 903:
Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}
Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 186.345{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }}
{{Optimal ET sequence|legend=0| 58, 103, 161, 425b, 586b, 747bc }}


Badness: 0.039962
Badness (Smith): 0.039962


=== 13-limit ===
=== 13-limit ===
Line 935: Line 916:
Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}
Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 186.347{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
{{Optimal ET sequence|legend=0| 58, 103, 161, 425b, 586bf }}


Badness: 0.021849
Badness (Smith): 0.021849


=== 17-limit ===
=== 17-limit ===
Line 948: Line 929:
Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}
Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 186.348{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
{{Optimal ET sequence|legend=0| 58, 103, 161, 425b, 586bf }}


Badness: 0.020295
Badness (Smith): 0.020295


== Catafourth ==
== Catafourth ==
Line 962: Line 943:


{{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


: Mapping generators: ~2, ~250/189
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~250/189 = 489.235{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235


{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}


Badness: 0.079579
Badness (Smith): 0.079579


=== 11-limit ===
=== 11-limit ===
Line 978: Line 958:
Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}
Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}


Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~250/189 = 489.252{{c}}


Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}
{{Optimal ET sequence|legend=0| 103, 130, 233, 363, 493e, 856be }}


Badness: 0.036785
Badness (Smith): 0.036785


=== 13-limit ===
=== 13-limit ===
Line 991: Line 971:
Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}
Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}


Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~65/49 = 489.256{{c}}


Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}
{{Optimal ET sequence|legend=0| 103, 130, 233, 363 }}


Badness: 0.021694
Badness (Smith): 0.021694


== Cotritone ==
== Cotritone ==
Line 1,003: Line 983:


{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}
{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}
: mappping generators: ~2, ~10/7


: Mappping generators: ~2, ~10/7
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 583.385{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385


{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}


[[Badness]]: 0.098322
[[Badness]] (Smith): 0.098322


=== 11-limit ===
=== 11-limit ===
Line 1,019: Line 998:
Mapping: {{mapping| 1 17 9 10 5 | 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 = 1200.000{{c}}, ~7/5 = 583.387{{c}}


Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=0| 35, 37, 72, 109, 181, 253 }}


Badness: 0.032225
Badness (Smith): 0.032225


=== 13-limit ===
=== 13-limit ===
Line 1,032: Line 1,011:
Mapping: {{mapping| 1 17 9 10 5 15 | 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 = 1200.000{{c}}, ~7/5 = 583.387{{c}}


Optimal ET sequence: {{Optimal ET sequence| 37, 72, 109, 181f }}
{{Optimal ET sequence|legend=0| 37, 72, 109, 181f }}


Badness: 0.028683
Badness (Smith): 0.028683


== Quasimoha ==
== Quasimoha ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasimoha]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,046: 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


: Mapping generators: ~2, ~49/40
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/40 = 348.603{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603


{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}


[[Badness]]: 0.110820
[[Badness]] (Smith): 0.110820


=== 11-limit ===
=== 11-limit ===
Line 1,062: Line 1,040:
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/9 = 348.639{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 86ce, 117ce, 148bce }}
{{Optimal ET sequence|legend=0| 31, 86ce, 117ce, 148bce }}


Badness: 0.046181
Badness (Smith): 0.046181


== Lockerbie ==
== Lockerbie ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #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.  
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.  


Line 1,081: Line 1,060:


{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
 
: mapping generators: ~2, ~3828125/2985984
: Mapping generators: ~2, ~3828125/2985984


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1071
* [[CTE]]: ~2 = 1200.0000{{c}}, ~3828125/2985984 = 431.1071{{c}}
: [[error map]]: {{val| 0.0000 -0.0270 +0.1502 -0.1120 }}
: [[error map]]: {{val| 0.0000 -0.0270 +0.1502 -0.1120 }}
* [[CWE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1072
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3828125/2985984 = 431.1072{{c}}
: error map: {{val| 0.0000 -0.0205 +0.1547 -0.1081 }}
: error map: {{val| 0.0000 -0.0205 +0.1547 -0.1081 }}


Line 1,102: Line 1,080:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
* CTE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1082{{c}}
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1078{{c}}


{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}
Line 1,117: Line 1,095:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~77/60 = 431.1085
* CTE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1085{{c}}
* CWE: ~2 = 1200.0000, ~77/60 = 431.1069
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1069{{c}}


{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}
Line 1,132: Line 1,110:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~77/60 = 431.107
* CTE: ~2 = 1200.000{{c}}, ~77/60 = 431.107{{c}}
* CWE: ~2 = 1200.000, ~77/60 = 431.108
* CWE: ~2 = 1200.000{{c}}, ~77/60 = 431.108{{c}}


{{Optimal ET sequence|legend=0| 103, 167, 270 }}
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
Line 1,147: Line 1,125:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~41/32 = 431.107
* CTE: ~2 = 1200.000{{c}}, ~41/32 = 431.107{{c}}
* CWE: ~2 = 1200.000, ~41/32 = 431.111
* CWE: ~2 = 1200.000{{c}}, ~41/32 = 431.111{{c}}


{{Optimal ET sequence|legend=0| 103, 167, 270 }}
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
Line 1,162: Line 1,140:


{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}
{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}
: mapping generators: ~2, ~7/4
: mapping generators: ~2, ~7/4


[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000{{c}}, ~7/4 = 970.690{{c}}


{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
Line 1,179: Line 1,156:


{{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 = 1200.0000{{c}}, ~675/448 = 709.9719{{c}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719


{{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]]: 0.202249
[[Badness]] (Smith): 0.202249


=== 11-limit ===
=== 11-limit ===
Line 1,195: Line 1,171:
Mapping: {{mapping| 1 43 -74 -25 36 | 0 -70 129 47 -55 }}
Mapping: {{mapping| 1 43 -74 -25 36 | 0 -70 129 47 -55 }}


Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720
Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~675/448 = 709.9720{{c}}


Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795 }}
{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795 }}


Badness: 0.052308
Badness (Smith): 0.052308


=== 13-limit ===
=== 13-limit ===
Line 1,208: Line 1,184:
Mapping: {{mapping| 1 43 -74 -25 36 25 | 0 -70 129 47 -55 -36 }}
Mapping: {{mapping| 1 43 -74 -25 36 25 | 0 -70 129 47 -55 -36 }}


Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723
Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~98/65 = 709.9723{{c}}


Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795f }}
{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795f }}


Badness: 0.032503
Badness (Smith): 0.032503


=== 17-limit ===
=== 17-limit ===
Line 1,221: Line 1,197:
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 | 0 -70 129 47 -55 -36 181 }}
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 | 0 -70 129 47 -55 -36 181 }}


Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~98/65 = 709.9722{{c}}


Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}


Badness: 0.020995
Badness (Smith): 0.020995


=== 19-limit ===
=== 19-limit ===
Line 1,234: Line 1,210:
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 -49 | 0 -70 129 47 -55 -36 181 90 }}
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 -49 | 0 -70 129 47 -55 -36 181 90 }}


Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~98/65 = 709.9722{{c}}


Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}


Badness: 0.013771
Badness (Smith): 0.013771


== Notes ==
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"