Breed family: Difference between revisions

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Units & misc. cleanup
Switch to Sintel's badness, WE & CWE tunings
Line 17: Line 17:
: 49/40 length = 0.7858, 8/7 length = 1.1241
: 49/40 length = 0.7858, 8/7 length = 1.1241
: Angle (49/40, 8/7) = 107.367°
: Angle (49/40, 8/7) = 107.367°
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}}, ~10/7 = 617.6826{{c}}
: [[error map]]: {{val| +0.021 +0.010 +0.044 -0.130 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}}, ~10/7 = 617.6807{{c}}
: error map: {{val| 0.000 -0.006 +0.022 -0.171 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.5
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.021{{c}}, ~49/40 = 350.972{{c}}, ~10/7 = 617.683{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~49/40 = 350.974{{c}}, ~10/7 = 617.681{{c}}


{{Optimal ET sequence|legend=1| 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d }}
{{Optimal ET sequence|legend=1| 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d }}


[[Badness]] (Sintel): 0.067
[[Badness]] (Sintel): 0.0674


[[Projection pair]]: <code>3 2401/800</code> to 2.5.7
[[Projection pair]]: <code>3 2401/800</code> to 2.5.7
Line 46: Line 48:


{{Mapping|legend=1| 1 1 1 2 2 | 0 2 1 1 5 | 0 0 2 1 0 }}
{{Mapping|legend=1| 1 1 1 2 2 | 0 2 1 1 5 | 0 0 2 1 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0985{{c}}, ~11/9 = 350.5322{{c}}, ~10/7 = 617.8782{{c}}
: [[error map]]: {{val| +0.098 -0.792 +0.073 -0.218 +1.540 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5374{{c}}, ~10/7 = 617.8712{{c}}
: error map: {{val| 0.000 -0.880 -0.034 -0.417 +1.369 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11
* [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.098{{c}}, ~11/9 = 350.532{{c}}, ~10/7 = 617.878{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.537{{c}}, ~10/7 = 617.871{{c}}


{{Optimal ET sequence|legend=1| 27e, 31, 41, 58, 72, 130, 202 }}
{{Optimal ET sequence|legend=1| 27e, 31, 41, 58, 72, 130, 202 }}
Line 66: Line 70:


Mapping: {{mapping| 1 1 1 2 2 1 | 0 2 1 1 5 11 | 0 0 2 1 0 -1 }}
Mapping: {{mapping| 1 1 1 2 2 1 | 0 2 1 1 5 11 | 0 0 2 1 0 -1 }}
Optimal tunings:
* WE: ~2 = 1199.9775{{c}}, ~11/9 = 350.7113{{c}}, ~10/7 = 617.8170{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.7130{{c}}, ~10/7 = 617.8178{{c}}


Minimax tuning:
Minimax tuning:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/5.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/5.15/13
Optimal tunings:
* WE: ~2 = 1199.977{{c}}, ~11/9 = 350.711{{c}}, ~10/7 = 617.817{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 350.713{{c}}, ~10/7 = 617.818{{c}}


{{Optimal ET sequence|legend=0| 27eff, 31f, 41, 58, 72, 130, 243, 301e, 373e }}
{{Optimal ET sequence|legend=0| 27eff, 31f, 41, 58, 72, 130, 243, 301e, 373e }}
Line 85: Line 89:


Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 2 1 1 5 11 9 | 0 0 2 1 0 -1 -3 }}
Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 2 1 1 5 11 9 | 0 0 2 1 0 -1 -3 }}
Optimal tunings:
* WE: ~2 = 1200.0876{{c}}, ~11/9 = 350.7203{{c}}, ~10/7 = 617.5766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.7138{{c}}, ~10/7 = 617.5610{{c}}


Minimax tuning:
Minimax tuning:
* 17-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.17/9
* 17-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.17/9
Optimal tunings:
* WE: ~2 = 1200.088{{c}}, ~11/9 = 350.720{{c}}, ~10/7 = 617.577{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 350.714{{c}}, ~10/7 = 617.561{{c}}


{{Optimal ET sequence|legend=0| 27effg, 31fg, 41, 58, 72, 130, 171, 243 }}
{{Optimal ET sequence|legend=0| 27effg, 31fg, 41, 58, 72, 130, 171, 243 }}
Line 103: Line 107:


Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 4 0 1 10 23 12 | 0 0 2 1 0 -1 -1 }}
Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 4 0 1 10 23 12 | 0 0 2 1 0 -1 -1 }}
: Mapping generators: ~2, ~119/108, ~27/17
: Mapping generators: ~2, ~119/108, ~27/17


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1199.559{{c}}, ~119/108 = 175.353{{c}}, ~27/17 = 793.685{{c}}
* WE: ~2 = 1199.5594{{c}}, ~119/108 = 175.3533{{c}}, ~27/17 = 793.6847{{c}}
* CWE: ~2 = 1200.000{{c}}, ~119/108 = 175.371{{c}}, ~27/17 = 793.756{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~119/108 = 175.3707{{c}}, ~27/17 = 793.7559{{c}}


{{Optimal ET sequence|legend=0| 14cf, 27effg, 41, 89, 130g }}
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41, 75ce, 89, 116cefg, 130g }}


Badness (Sintel): 2.843
Badness (Sintel): 2.84


===== 19-limit =====
===== 19-limit =====
Line 122: Line 125:


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1199.733{{c}}, ~21/19 = 175.347{{c}}, ~19/12 = 793.779{{c}}
* WE: ~2 = 1199.7333{{c}}, ~21/19 = 175.3472{{c}}, ~19/12 = 793.7794{{c}}
* CWE: ~2 = 1200.000{{c}}, ~21/19 = 175.358{{c}}, ~19/12 = 793.819{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3584{{c}}, ~19/12 = 793.8194{{c}}


{{Optimal ET sequence|legend=0| 14cf, 27effg, 41, 89, 130g }}
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41, 75ce, 89, 116cefg, 130g }}


Badness (Sintel): 2.294
Badness (Sintel): 2.29


=== Jofur ===
=== Jofur ===
Line 137: Line 140:


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1198.953{{c}}, ~11/9 = 351.141{{c}}, ~10/7 = 618.349{{c}}
* WE: ~2 = 1198.9534{{c}}, ~11/9 = 351.1412{{c}}, ~10/7 = 618.3494{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 351.254{{c}}, ~10/7 = 618.593{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.2536{{c}}, ~10/7 = 618.5932{{c}}


{{Optimal ET sequence|legend=0| 27e, 31, 41, 58, 99ef, 157eff }}
{{Optimal ET sequence|legend=0| 27e, 31, 41, 58, 99ef, 157eff }}
Line 152: Line 155:


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1200.144{{c}}, ~11/9 = 350.435{{c}}, ~10/7 = 618.177{{c}}
* WE: ~2 = 1200.1435{{c}}, ~11/9 = 350.4354{{c}}, ~10/7 = 618.1775{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/9 = 350.440{{c}}, ~10/7 = 618.176{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4402{{c}}, ~10/7 = 618.1759{{c}}


{{Optimal ET sequence|legend=0| 27e, 31, 45ef, 58, 72, 103, 130, 233, 363 }}
{{Optimal ET sequence|legend=0| 27e, 31, 45ef, 58, 72, 103, 130, 233, 363 }}
Line 171: Line 174:
: 49/40 length = 0.756, 10/7 length = 0.819
: 49/40 length = 0.756, 10/7 length = 0.819
: Angle (49/40, 10/7) = 106.460 degrees
: Angle (49/40, 10/7) = 106.460 degrees
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4168{{c}}, ~49/40 = 350.8363{{c}}, ~10/7 = 617.2187{{c}}
: [[error map]]: {{val| +0.417 +0.134 -0.623 +0.063 -0.890 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.8369{{c}}, ~10/7 = 616.9907{{c}}
: error map: {{val| 0.000 -0.281 -1.495 -0.998 -2.290 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 23/10 3/10 2/5 0 -2/5 }}, {{monzo| 12/5 2/5 1/5 0 -1/5 }}, {{monzo| 23/10 3/10 -3/5 0 3/5 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 23/10 3/10 2/5 0 -2/5 }}, {{monzo| 12/5 2/5 1/5 0 -1/5 }}, {{monzo| 23/10 3/10 -3/5 0 3/5 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.417{{c}}, ~49/40 = 350.836{{c}}, ~10/7 = 617.219{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~49/40 = 350.837{{c}}, ~10/7 = 616.991{{c}}


{{Optimal ET sequence|legend=1| 27, 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee }}
{{Optimal ET sequence|legend=1| 27, 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee }}
Line 193: Line 198:


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1200.441{{c}}, ~49/40 = 350.809{{c}}, ~10/7 = 617.372{{c}}
* WE: ~2 = 1200.4413{{c}}, ~49/40 = 350.8092{{c}}, ~10/7 = 617.3718{{c}}
* CWE: ~2 = 1200.000{{c}}, ~49/40 = 350.805{{c}}, ~10/7 = 617.157{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.8046{{c}}, ~10/7 = 617.1572{{c}}


{{Optimal ET sequence|legend=0| 31, 68, 72, 103, 140, 212, 243e, 315ef, 455eef, 770cdeeeff }}
{{Optimal ET sequence|legend=0| 31, 68, 72, 103, 140, 212, 243e, 315ef, 455eef, 770cdeeeff }}
Line 206: Line 211:


{{Mapping|legend=1| 1 1 1 2 -3 | 0 2 1 1 8 | 0 0 2 1 8 }}
{{Mapping|legend=1| 1 1 1 2 -3 | 0 2 1 1 8 | 0 0 2 1 8 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9178{{c}}, ~49/40 = 350.0519{{c}}, ~10/7 = 617.8150{{c}}
: [[error map]]: {{val| -0.082 +0.067 +0.286 -0.123 -0.135 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.0615{{c}}, ~10/7 = 617.8571{{c}}
: error map: {{val| 0.000 +0.168 +0.462 +0.093 +0.032 }}


{{Optimal ET sequence|legend=1| 31, 68e, 99e, 130, 239, 270, 670, 940, 1210, 2150c }}
{{Optimal ET sequence|legend=1| 31, 68e, 99e, 130, 239, 270, 670, 940, 1210, 2150c }}


[[Badness]] (Smith): 0.640 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.769


=== 13-limit ===
=== 13-limit ===
Line 217: Line 228:


Mapping: {{mapping| 1 1 1 2 -3 7 | 0 2 1 1 8 -6 | 0 0 2 1 8 -3 }}
Mapping: {{mapping| 1 1 1 2 -3 7 | 0 2 1 1 8 -6 | 0 0 2 1 8 -3 }}
Optimal tunings:
* WE: ~2 = 1199.9787{{c}}, ~49/40 = 350.9976{{c}}, ~10/7 = 617.8701{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0039{{c}}, ~10/7 = 617.8795{{c}}


{{Optimal ET sequence|legend=0| 31, 78f, 99e, 109, 130, 239, 270, 571, 701, 841, 971, 1241 }}
{{Optimal ET sequence|legend=0| 31, 78f, 99e, 109, 130, 239, 270, 571, 701, 841, 971, 1241 }}


Badness (Smith): 0.830 × 10<sup>-3</sup>
Badness (Sintel): 0.776


== Lif ==
== Lif ==
Line 229: Line 244:
{{Mapping|legend=1| 1 1 1 2 8 | 0 2 1 1 -12 | 0 0 2 1 -2 }}
{{Mapping|legend=1| 1 1 1 2 8 | 0 2 1 1 -12 | 0 0 2 1 -2 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~49/40 = 351.0959{{c}}, ~10/7 = 617.6652{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9386{{c}}, ~49/40 = 350.0586{{c}}, ~10/7 = 617.7065{{c}}
: [[error map]]: {{val| -0.061 +0.101 +0.096 -0.184 +0.075 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.0806{{c}}, ~10/7 = 617.7228{{c}}
: error map: {{val| 0.000 +0.206 +0.213 -0.022 +0.269 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.0959{{c}}, ~10/7 = 617.6652{{c}} -->


{{Optimal ET sequence|legend=1| 41, 89, 130, 229, 270, 581, 670, 711, 981, 1251, 2232e }}
{{Optimal ET sequence|legend=1| 41, 89, 130, 229, 270, 581, 670, 711, 981, 1251, 2232e }}


[[Badness]] (Smith): 0.793 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.953


=== 13-limit ===
=== 13-limit ===
Line 242: Line 262:
Mapping: {{mapping| 1 1 1 2 8 7 | 0 2 1 1 -12 -6 | 0 0 2 1 -2 -3 }}
Mapping: {{mapping| 1 1 1 2 8 7 | 0 2 1 1 -12 -6 | 0 0 2 1 -2 -3 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~49/40 = 351.0960{{c}}, ~10/7 = 617.6533{{c}}
Optimal tunings:
* WE: ~2 = 1199.9597{{c}}, ~49/40 = 351.0718{{c}}, ~10/7 = 617.6543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0856{{c}}, ~10/7 = 617.6715{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0960{{c}}, ~10/7 = 617.6533{{c}} -->


{{Optimal ET sequence|legend=0| 41, 89, 99, 130, 270, 581, 711, 981, 1292, 1562 }}
{{Optimal ET sequence|legend=0| 41, 89, 99, 130, 270, 581, 711, 981, 1292, 1562 }}


Badness (Smith): 0.579 × 10<sup>-3</sup>
Badness (Sintel): 0.542


=== 2.3.5.7.11.13.19 subgroup ===
=== 2.3.5.7.11.13.19 subgroup ===
Line 253: Line 276:
Comma list: 1216/1215, 1729/1728, 2080/2079, 2401/2400
Comma list: 1216/1215, 1729/1728, 2080/2079, 2401/2400


Sval mapping: {{mapping| 1 1 1 2 8 7 0 | 0 2 1 1 -12 -6 11 | 0 0 2 1 -2 -3 2 }}
Subgroup-val mapping: {{mapping| 1 1 1 2 8 7 0 | 0 2 1 1 -12 -6 11 | 0 0 2 1 -2 -3 2 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~49/40 = 351.1007{{c}}, ~10/7 = 617.6501{{c}}
Optimal tunings:
* WE: ~2 = 1199.9752{{c}}, ~49/40 = 351.0908{{c}}, ~10/7 = 617.6473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0984{{c}}, ~10/7 = 617.6588{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1007{{c}}, ~10/7 = 617.6501{{c}} -->


{{Optimal ET sequence|legend=0| 41, 89, 130, 229, 270, 581, 851, 1562, 1832, 2413 }}
{{Optimal ET sequence|legend=0| 41, 89, 130, 229, 270, 581, 851, 1562, 1832, 2413 }}


Badness (Smith): 0.499 × 10<sup>-3</sup>
Badness (Sintel): 0.488


== Baldur ==
== Baldur ==
Line 268: Line 294:
{{Mapping|legend=1| 2 0 1 3 7 | 0 2 1 1 -2 | 0 0 2 1 3 }}
{{Mapping|legend=1| 2 0 1 3 7 | 0 2 1 1 -2 | 0 0 2 1 3 }}
: Mapping generators: ~99/70, ~343/198, ~10/7
: Mapping generators: ~99/70, ~343/198, ~10/7
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.0149{{c}}, ~343/198 = 950.9717{{c}}, ~10/7 = 617.7007{{c}}
: [[error map]]: {{val| +0.030 -0.012 +0.074 -0.109 -0.055 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~343/198 = 950.9565{{c}}, ~10/7 = 617.7017{{c}}
: error map: {{val| 0.000 -0.042 +0.046 -0.168 -0.126 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/4 0 1/2 1/2 -1/2 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 23/16 0 5/8 1/8 -1/8 }}, {{monzo| 23/16 0 5/8 -7/8 7/8 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/4 0 1/2 1/2 -1/2 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 23/16 0 5/8 1/8 -1/8 }}, {{monzo| 23/16 0 5/8 -7/8 7/8 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7
 
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.015{{c}}, ~343/198 = 950.972{{c}}, ~10/7 = 617.701{{c}}
* [[CWE]]: ~99/70 = 600.000{{c}}, ~343/198 = 950.956{{c}}, ~10/7 = 617.702{{c}}


{{Optimal ET sequence|legend=1| 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de }}
{{Optimal ET sequence|legend=1| 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de }}
Line 282: Line 310:
[[Badness]] (Sintel): 0.200
[[Badness]] (Sintel): 0.200


[[Projection pair]]s: ~2 = ~9801/4900, ~3 = ~117649/39204, ~7 = ~9801/1400, ~11 =  ~913517247483640899/83082326424002500 to 5.7/2.99/4
[[Projection pair]]s: <code>2 9801/4900, 3 117649/39204, 7 9801/1400, 11 913517247483640899/83082326424002500</code> to 5.7/2.99/4


=== Greenland ===
=== Greenland ===
Line 292: Line 320:


Optimal tunings:
Optimal tunings:
* WE: ~99/70 = 599.987{{c}}, ~26/15 = 951.054{{c}}, ~10/7 = 617.806{{c}}
* WE: ~99/70 = 599.9868{{c}}, ~26/15 = 951.0538{{c}}, ~10/7 = 617.8062{{c}}
* CWE: ~99/70 = 600.000{{c}}, ~26/15 = 951.069{{c}}, ~10/7 = 617.807{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/15 = 951.0689{{c}}, ~10/7 = 617.8067{{c}}


{{Optimal ET sequence|legend=0| 58, 72, 130, 198, 270, 940, 1210f }}
{{Optimal ET sequence|legend=0| 58, 72, 130, 198, 270, 940, 1210f }}
Line 311: Line 339:


Optimal tunings:
Optimal tunings:
* WE: ~17/12 = 600.112{{c}}, ~26/15 = 951.191{{c}}, ~10/7 = 617.610{{c}}
* WE: ~17/12 = 600.1118{{c}}, ~26/15 = 951.1907{{c}}, ~10/7 = 617.6100{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~26/15 = 951.063{{c}}, ~10/7 = 617.584{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.0626{{c}}, ~10/7 = 617.5842{{c}}


{{Optimal ET sequence|legend=0| 58, 72, 130, 140, 198, 270g }}
{{Optimal ET sequence|legend=0| 58, 72, 130, 140, 198, 212g, 270g, 342fg, 482fgg }}


Badness (Sintel): 0.720
Badness (Sintel): 0.720
Line 323: Line 351:
[[Comma list]]: 2401/2400, 3025/3024
[[Comma list]]: 2401/2400, 3025/3024


{{Mapping|legend=1| 1 1 3 3 2 | 0 2 3 2 1 | 0 0 -4 -2 3 }}
{{Mapping|legend=1| 1 1 -1 1 5 | 0 2 3 2 1 | 0 0 4 2 -3 }}
: Mapping generators: ~2, ~49/40, ~55/42
: Mapping generators: ~2, ~49/40, ~84/55
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0491{{c}}, ~49/40 = 350.9820{{c}}, ~84/55 = 733.3514{{c}}
: [[error map]]: {{val| +0.030 -0.012 +0.074 -0.109 -0.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9890{{c}}, ~84/55 = 733.3160{{c}}
: error map: {{val| 0.000 -0.042 +0.046 -0.168 -0.126 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]] eigenmonzos (unchanged-intervals): 2, 14/11, 4/3
* [[11-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3.11/7


{{Optimal ET sequence|legend=1| 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}
{{Optimal ET sequence|legend=1| 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}


[[Badness]] (Smith): 0.170 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.204


[[Projection pair]]s: ~3 = ~2401/800, ~5 = ~22880495169/4575312500, ~7 = ~1058841/151250, ~11 = ~33275/3024 to 2.49/5.77/3
[[Projection pair]]s: ~3 = ~2401/800, ~5 = ~22880495169/4575312500, ~7 = ~1058841/151250, ~11 = ~33275/3024 to 2.49/5.77/3
Line 340: Line 374:
Comma list: 2401/2400, 3025/3024, 4096/4095
Comma list: 2401/2400, 3025/3024, 4096/4095


Mapping: {{mapping| 1 1 3 3 2 4 | 0 2 3 2 1 -9 | 0 0 -4 -2 3 6 }}
Mapping: {{mapping| 1 1 -1 1 5 10 | 0 2 3 2 1 -9 | 0 0 4 2 -3 -6 }}
 
Optimal tunings:
* WE: ~2 = 1199.9870{{c}}, ~49/40 = 351.0502{{c}}, ~84/55 = 733.2969{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0534{{c}}, ~84/55 = 733.3055{{c}}


{{Optimal ET sequence|legend=0| 31, 41, 72f, 198f, 229, 239, 270, 571, 581, 851, 882, 1152, 1463, 1733, 2615 }}
{{Optimal ET sequence|legend=0| 31, 41, 72f, 198f, 229, 239, 270, 571, 581, 851, 882, 1152, 1463, 1733, 2615 }}


Badness (Smith): 0.855 × 10<sup>-3</sup>
Badness (Sintel): 0.800


Projection pairs: <code>3 2401/800, 5 22880495169/4575312500, 7 1058841/151250, 11 33275/3024, 13 1814078464000000000000000/139662717676432916098329</code> to 2.49/5.77/3
Projection pairs: <code>3 2401/800, 5 22880495169/4575312500, 7 1058841/151250, 11 33275/3024, 13 1814078464000000000000000/139662717676432916098329</code> to 2.49/5.77/3
Line 355: Line 393:
Comma list: 2080/2079, 2401/2400, 3025/3024
Comma list: 2080/2079, 2401/2400, 3025/3024


Mapping: {{mapping| 1 1 3 3 2 0 | 0 2 3 2 1 6 | 0 0 -4 -2 3 5 }}
Mapping: {{mapping| 1 1 -1 1 5 5 | 0 2 3 2 1 6 | 0 0 4 2 -3 -5 }}
 
Optimal tunings:
* WE: ~2 = 1199.9994{{c}}, ~49/40 = 351.0880{{c}}, ~84/55 = 733.2478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0880{{c}}, ~84/55 = 733.2482{{c}}


{{Optimal ET sequence|legend=0| 13cdf, 31f, 41, 72, 157, 185cf, 198, 270, 581, 851, 1504, 1774f, 2085, 2355f }}
{{Optimal ET sequence|legend=0| 13cdf, 31f, 41, 72, 157, 185cf, 198, 270, 581, 851, 1504, 1774f, 2085, 2355f }}


Badness (Smith): 0.581 × 10<sup>-3</sup>
Badness (Sintel): 0.543


==== 17-limit ====
==== 17-limit ====
Line 366: Line 408:
Comma list: 1225/1224, 2058/2057, 2080/2079, 2401/2400
Comma list: 1225/1224, 2058/2057, 2080/2079, 2401/2400


Mapping: {{mapping| 1 1 3 3 2 0 7 | 0 2 3 2 1 6 6 | 0 0 -4 -2 3 5 -12 }}
Mapping: {{mapping| 1 1 -1 1 5 5 -5 | 0 2 3 2 1 6 6 | 0 0 4 2 -3 -5 12 }}
 
Optimal tunings:
* WE: ~2 = 1200.0038{{c}}, ~49/40 = 351.0756{{c}}, ~84/55 = 733.2270{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0755{{c}}, ~84/55 = 733.2246{{c}}


{{Optimal ET sequence|legend=0| 41g, 72, 198g, 239f, 270, 311, 509, 581, 1234, 1815 }}
{{Optimal ET sequence|legend=0| 41g, 72, 198g, 239f, 270, 311, 509, 581, 1234, 1815 }}


Badness (Smith): 0.700 × 10<sup>-3</sup>
Badness (Sintel): 0.665


==== 19-limit ====
==== 19-limit ====
Line 377: Line 423:
Comma list: 1225/1224, 1540/1539, 2080/2079, 3136/3135, 4200/4199
Comma list: 1225/1224, 1540/1539, 2080/2079, 3136/3135, 4200/4199


Mapping: {{mapping| 1 1 3 3 2 0 7 6 | 0 2 3 2 1 6 6 -2 | 0 0 -4 -2 3 5 -12 -3 }}
Mapping: {{mapping| 1 1 -1 1 5 5 -5 3 | 0 2 3 2 1 6 6 -2 | 0 0 4 2 -3 -5 12 3 }}
: Mapping generators: ~2, ~49/40, ~55/42
 
Optimal tunings:
* WE: ~2 = 1200.0015{{c}}, ~49/40 = 351.0770{{c}}, ~84/55 = 733.2252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0768{{c}}, ~84/55 = 733.2243{{c}}


{{Optimal ET sequence|legend=0| 41g, 72, 198g, 239f, 270, 311, 581, 1234, 1815 }}
{{Optimal ET sequence|legend=0| 41g, 72, 198g, 239f, 270, 311, 581, 1234, 1815 }}


Badness (Smith): 0.692 × 10<sup>-3</sup>
Badness (Sintel): 0.712


=== Heimlaug ===
=== Heimlaug ===
Line 391: Line 440:
Comma list: 1001/1000, 1716/1715, 3025/3024
Comma list: 1001/1000, 1716/1715, 3025/3024


Mapping: {{mapping| 1 1 3 3 2 7 | 0 2 3 2 1 6 | 0 0 -4 -2 3 -13 }}
Mapping: {{mapping| 1 1 -1 1 5 -6 | 0 2 3 2 1 6 | 0 0 4 2 -3 13 }}


{{Optimal ET sequence|legend=0| 8bcef, 15bbccdeeff, 23bcf, 31, 72, 103, 167, 198, 270, 571, 643, 913f }}
Optimal tunings:
* WE: ~2 = 1200.0601{{c}}, ~49/40 = 350.9562{{c}}, ~84/55 = 733.4471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.9646{{c}}, ~84/55 = 733.4049{{c}}


Badness (Smith): 0.601 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 31, 72, 103, 167, 198, 270, 571, 643, 913f }}
 
Badness (Sintel): 0.562


==== 17-limit ====
==== 17-limit ====
Line 404: Line 457:
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 1716/1715
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 1716/1715


Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 2 3 2 1 6 6 | 0 0 -4 -2 3 -13 -12 }}
Mapping: {{mapping| 1 1 -1 1 5 -6 -5 | 0 2 3 2 1 6 6 | 0 0 4 2 -3 13 12 }}
: Mapping generators: ~2, ~49/40, ~17/13


{{Optimal ET sequence|legend=0| 8bcefg, 15bbccdeeffggg, 23bcfg, 31, 64be, 72, 103, 167, 198g, 239, 270 }}
Optimal tunings:
* WE: ~2 = 1200.0446{{c}}, ~49/40 = 350.9662{{c}}, ~26/17 = 733.3826{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.9724{{c}}, ~26/17 = 733.3515{{c}}


Badness (Smith): 0.829 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 31, 64be, 72, 103, 167, 198g, 239, 270 }}
 
Badness (Sintel): 0.789


== Vili ==
== Vili ==
Line 416: Line 472:
[[Comma list]]: 2401/2400, 391314/390625
[[Comma list]]: 2401/2400, 391314/390625


{{Mapping|legend=1| 1 1 5 4 10 | 0 2 3 2 6 | 0 0 -6 -3 -14 }}
{{Mapping|legend=1| 1 1 -1 1 -4 | 0 2 3 2 6 | 0 0 6 3 14 }}
: Mapping generators: ~2, ~49/40, ~175/132
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9872{{c}}, ~49/40 = 350.9740{{c}}, ~175/132 = 488.9418{{c}}
: [[error map]]: {{val| -0.013 -0.020 +0.272 -0.065 -0.238 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9728{{c}}, ~175/132 = 488.9475{{c}}
: error map: {{val| 0.000 -0.009 +0.290 -0.038 -0.216 }}


{{Optimal ET sequence|legend=1| 27e, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 670, 913, 1043, 1313, 1583 }}
{{Optimal ET sequence|legend=1| 27e, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 670, 913, 1043, 1313, 1583 }}


[[Badness]] (Smith): 1.26 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.51


=== 13-limit ===
=== 13-limit ===
Line 427: Line 490:
Comma list: 1001/1000, 1716/1715, 4225/4224
Comma list: 1001/1000, 1716/1715, 4225/4224


Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 3 2 6 1 | 0 0 -6 -3 -14 -1 }}
Mapping: {{mapping| 1 1 -1 1 -4 3 | 0 2 3 2 6 1 | 0 0 6 3 14 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.0597{{c}}, ~49/40 = 350.9419{{c}}, ~65/49 = 488.9741{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.9441{{c}}, ~65/49 = 488.9475{{c}}


{{Optimal ET sequence|legend=0| 27e, 37, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 913f }}
{{Optimal ET sequence|legend=0| 27e, 37, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 913f }}


Badness (Smith): 0.738 × 10<sup>-3</sup>
Badness (Sintel): 0.690


== Frigg ==
== Frigg ==
Line 438: Line 505:
[[Comma list]]: 2401/2400, 644204/643125
[[Comma list]]: 2401/2400, 644204/643125


{{Mapping|legend=1| 1 1 3 3 4 | 0 2 3 2 4 | 0 0 -10 -5 -11 }}
{{Mapping|legend=1| 1 1 -7 -2 -7 | 0 2 3 2 4 | 0 0 10 5 11 }}
: Mapping generators: ~2, ~49/40, ~88/49
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0183{{c}}, ~49/40 = 350.9854{{c}}, ~88/49 = 1013.3709{{c}}
: [[error map]]: {{val| +0.018 +0.034 +0.223 -0.037 -0.425 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9871{{c}}, ~88/49 = 1013.3556{{c}}
: error map: {{val| 0.000 +0.019 +0.203 -0.074 -0.458 }}


{{Optimal ET sequence|legend=1| 45e, 58, 103, 161, 212, 270, 643, 913, 1183e }}
{{Optimal ET sequence|legend=1| 45e, 58, 103, 161, 212, 270, 643, 913, 1183e }}


[[Badness]] (Smith): 1.79 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.15


=== 13-limit ===
=== 13-limit ===
Line 449: Line 523:
Comma list: 1001/1000, 1716/1715, 10648/10647
Comma list: 1001/1000, 1716/1715, 10648/10647


Mapping: {{mapping| 1 1 3 3 4 5 | 0 2 3 2 4 3 | 0 0 -10 -5 -11 -14 }}
Mapping: {{mapping| 1 1 -7 -2 -7 -9 | 0 2 3 2 4 3 | 0 0 10 5 11 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.0387{{c}}, ~49/40 = 350.9502{{c}}, ~70/39 = 1013.4065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.9526{{c}}, ~70/39 = 1013.3747{{c}}


{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161, 212, 270, 643, 913f, 1614ef }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161, 212, 270, 643, 913f, 1614ef }} <nowiki>*</nowiki>
<nowiki>*</nowiki> [[optimal patent val]]: [[1241edo|1241]]
<nowiki>*</nowiki> [[optimal patent val]]: [[1241edo|1241]]


Badness (Smith): 0.934 × 10<sup>-3</sup>
Badness (Sintel): 0.873


== Ennealimmic ==
== Ennealimmic ==
Line 466: Line 544:
: Mapping generators: ~27/25, ~5/3, ~11
: Mapping generators: ~27/25, ~5/3, ~11


{{Optimal ET sequence|legend=1| 27, 45, 72, 171, 198, 270, 342, 612, 954, 1323, 1395, 1665, 2007, 2277, 2619, 4284d, 6561dd }}
[[Optimal tuning]]s:
* [[WE]]: ~27/25 = 133.3357{{c}}, ~5/3 = 884.3288{{c}}, ~11/8 = 551.2529{{c}}
: [[error map]]: {{val| +0.021 +0.038 +0.009 -0.139 -0.000 }}
* [[CWE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3215{{c}}, ~11/8 = 551.2775{{c}}
: error map: {{val| 0.000 +0.021 -0.016 -0.183 -0.040 }}
 
{{Optimal ET sequence|legend=1| 72, …, 171, 198, 270, 342, 612, 954, 1323, 1395, 1665, 2007, 2277, 2619, 4284d, 6561dd, 6903dd }}


[[Badness]] (Smith): 0.275 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.330


=== 13-limit ===
=== 13-limit ===
Line 477: Line 561:
Mapping: {{mapping| 9 1 1 12 0 -31 | 0 2 3 2 0 5 | 0 0 0 0 1 1 }}
Mapping: {{mapping| 9 1 1 12 0 -31 | 0 2 3 2 0 5 | 0 0 0 0 1 1 }}


{{Optimal ET sequence|legend=0| 27, 45f, 54cff, 72, 171, 198, 270, 639, 711, 981, 1692e }}
Optimal tunings:
* WE: ~27/25 = 133.3276{{c}}, ~5/3 = 884.3660{{c}}, ~11/8 = 551.7221{{c}}
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3905{{c}}, ~11/8 = 551.7085{{c}}
 
{{Optimal ET sequence|legend=0| 72, …, 171, 198, 270, 639, 711, 981, 1692e }}


Badness (Smith): 0.755 × 10<sup>-3</sup>
Badness (Sintel): 0.706


[[Category:Temperament families]]
[[Category:Temperament families]]

Revision as of 14:23, 11 February 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The breed family of rank-3 temperaments are microtemperaments which temper out 2401/2400.

Breed

Breed has the same lattice structure as the 2.5.7 subgroup of JI. In terms of extension from the 5-limit, it splits 25/6 into four equal parts representing 10/7. While it is so accurate it hardly matters what is used to temper it, or whether it is tempered at all, the optimal patent val 2749et will certainly do the trick.

Subgroup: 2.3.5.7

Comma list: 2401/2400

Mapping: [⟨1 1 1 2], ⟨0 2 1 1], ⟨0 0 2 1]]

Mapping generators: ~2, ~49/40, ~10/7

Mapping to lattice: [⟨0 2 -1 0], ⟨0 0 -2 -1]]

Lattice basis:

49/40 length = 0.7858, 8/7 length = 1.1241
Angle (49/40, 8/7) = 107.367°

Optimal tunings:

  • WE: ~2 = 1200.0206 ¢, ~49/40 = 350.9724 ¢, ~10/7 = 617.6826 ¢
error map: ⟨+0.021 +0.010 +0.044 -0.130]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9743 ¢, ~10/7 = 617.6807 ¢
error map: ⟨0.000 -0.006 +0.022 -0.171]

Minimax tuning:

Optimal ET sequence: 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d

Badness (Sintel): 0.0674

Projection pair: 3 2401/800 to 2.5.7

Scales: breed11

Music

Jove

Main article: Jove

Jove (formerly known as wonder) tempers out 243/242 and 441/440. Jove converts breed into an 11-limit temperament via 441/440, which equates 49/40 with 11/9, and 243/242, which tells us 11/9 can serve as a neutral third. While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. 72, 130, 171 and 202 are good edos for jove.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440

Mapping: [⟨1 1 1 2 2], ⟨0 2 1 1 5], ⟨0 0 2 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.0985 ¢, ~11/9 = 350.5322 ¢, ~10/7 = 617.8782 ¢
error map: ⟨+0.098 -0.792 +0.073 -0.218 +1.540]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5374 ¢, ~10/7 = 617.8712 ¢
error map: ⟨0.000 -0.880 -0.034 -0.417 +1.369]

Minimax tuning:

Optimal ET sequence: 27e, 31, 41, 58, 72, 130, 202

Badness (Sintel): 0.290

Projection pairs: 3 242/81, 5 2200/441, 7 440/63, 11 644204/59049 to 2.7/5.11/9

Jovial

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 1 2 2 1], ⟨0 2 1 1 5 11], ⟨0 0 2 1 0 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9775 ¢, ~11/9 = 350.7113 ¢, ~10/7 = 617.8170 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.7130 ¢, ~10/7 = 617.8178 ¢

Minimax tuning:

  • 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
  • 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/5.15/13

Optimal ET sequence: 27eff, 31f, 41, 58, 72, 130, 243, 301e, 373e

Badness (Sintel): 0.583

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594

Mapping: [⟨1 1 1 2 2 1 3], ⟨0 2 1 1 5 11 9], ⟨0 0 2 1 0 -1 -3]]

Optimal tunings:

  • WE: ~2 = 1200.0876 ¢, ~11/9 = 350.7203 ¢, ~10/7 = 617.5766 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.7138 ¢, ~10/7 = 617.5610 ¢

Minimax tuning:

  • 17-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.17/9

Optimal ET sequence: 27effg, 31fg, 41, 58, 72, 130, 171, 243

Badness (Sintel): 0.704

Heartlandia

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 1452/1445

Mapping: [⟨1 1 1 2 2 1 3], ⟨0 4 0 1 10 23 12], ⟨0 0 2 1 0 -1 -1]]

Mapping generators: ~2, ~119/108, ~27/17

Optimal tunings:

  • WE: ~2 = 1199.5594 ¢, ~119/108 = 175.3533 ¢, ~27/17 = 793.6847 ¢
  • CWE: ~2 = 1200.0000 ¢, ~119/108 = 175.3707 ¢, ~27/17 = 793.7559 ¢

Optimal ET sequence: 14cf, 27effg, 41, 75ce, 89, 116cefg, 130g

Badness (Sintel): 2.84

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 171/170, 243/242, 324/323, 364/363, 441/440

Mapping: [⟨1 1 1 2 2 1 3 3], ⟨0 4 0 1 10 23 12 4], ⟨0 0 2 1 0 -1 -1 1]]

Optimal tunings:

  • WE: ~2 = 1199.7333 ¢, ~21/19 = 175.3472 ¢, ~19/12 = 793.7794 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.3584 ¢, ~19/12 = 793.8194 ¢

Optimal ET sequence: 14cf, 27effg, 41, 75ce, 89, 116cefg, 130g

Badness (Sintel): 2.29

Jofur

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242

Mapping: [⟨1 1 1 2 2 4], ⟨0 2 1 1 5 -1], ⟨0 0 2 1 0 0]]

Optimal tunings:

  • WE: ~2 = 1198.9534 ¢, ~11/9 = 351.1412 ¢, ~10/7 = 618.3494 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.2536 ¢, ~10/7 = 618.5932 ¢

Optimal ET sequence: 27e, 31, 41, 58, 99ef, 157eff

Badness (Sintel): 0.701

Jovis

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440

Mapping: [⟨1 1 1 2 2 2], ⟨0 2 1 1 5 -3], ⟨0 0 2 1 0 5]]

Optimal tunings:

  • WE: ~2 = 1200.1435 ¢, ~11/9 = 350.4354 ¢, ~10/7 = 618.1775 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.4402 ¢, ~10/7 = 618.1759 ¢

Optimal ET sequence: 27e, 31, 45ef, 58, 72, 103, 130, 233, 363

Badness (Sintel): 0.507

Agni

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372

Mapping: [⟨1 1 1 2 5], ⟨0 2 1 1 0], ⟨0 0 2 1 -3]]

Mapping to lattice: [⟨0 2 1 1 0], ⟨0 0 2 1 -3]]

Lattice basis:

49/40 length = 0.756, 10/7 length = 0.819
Angle (49/40, 10/7) = 106.460 degrees

Optimal tunings:

  • WE: ~2 = 1200.4168 ¢, ~49/40 = 350.8363 ¢, ~10/7 = 617.2187 ¢
error map: ⟨+0.417 +0.134 -0.623 +0.063 -0.890]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.8369 ¢, ~10/7 = 616.9907 ¢
error map: ⟨0.000 -0.281 -1.495 -0.998 -2.290]

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [23/10 3/10 2/5 0 -2/5⟩, [12/5 2/5 1/5 0 -1/5⟩, [23/10 3/10 -3/5 0 3/5⟩]
unchanged-interval (eigenmonzo) basis: 2.3.11/5

Optimal ET sequence: 27, 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee

Badness (Sintel): 0.593

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624, 1375/1372

Mapping: [⟨1 1 1 2 5 -1], ⟨0 2 1 1 0 2], ⟨0 0 2 1 -3 8]]

Optimal tunings:

  • WE: ~2 = 1200.4413 ¢, ~49/40 = 350.8092 ¢, ~10/7 = 617.3718 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.8046 ¢, ~10/7 = 617.1572 ¢

Optimal ET sequence: 31, 68, 72, 103, 140, 212, 243e, 315ef, 455eef, 770cdeeeff

Badness (Sintel): 0.863

Zisa

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 5632/5625

Mapping: [⟨1 1 1 2 -3], ⟨0 2 1 1 8], ⟨0 0 2 1 8]]

Optimal tunings:

  • WE: ~2 = 1199.9178 ¢, ~49/40 = 350.0519 ¢, ~10/7 = 617.8150 ¢
error map: ⟨-0.082 +0.067 +0.286 -0.123 -0.135]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.0615 ¢, ~10/7 = 617.8571 ¢
error map: ⟨0.000 +0.168 +0.462 +0.093 +0.032]

Optimal ET sequence: 31, 68e, 99e, 130, 239, 270, 670, 940, 1210, 2150c

Badness (Sintel): 0.769

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 4096/4095

Mapping: [⟨1 1 1 2 -3 7], ⟨0 2 1 1 8 -6], ⟨0 0 2 1 8 -3]]

Optimal tunings:

  • WE: ~2 = 1199.9787 ¢, ~49/40 = 350.9976 ¢, ~10/7 = 617.8701 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0039 ¢, ~10/7 = 617.8795 ¢

Optimal ET sequence: 31, 78f, 99e, 109, 130, 239, 270, 571, 701, 841, 971, 1241

Badness (Sintel): 0.776

Lif

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 131072/130977

Mapping: [⟨1 1 1 2 8], ⟨0 2 1 1 -12], ⟨0 0 2 1 -2]]

Optimal tunings:

  • WE: ~2 = 1199.9386 ¢, ~49/40 = 350.0586 ¢, ~10/7 = 617.7065 ¢
error map: ⟨-0.061 +0.101 +0.096 -0.184 +0.075]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0806 ¢, ~10/7 = 617.7228 ¢
error map: ⟨0.000 +0.206 +0.213 -0.022 +0.269]

Optimal ET sequence: 41, 89, 130, 229, 270, 581, 670, 711, 981, 1251, 2232e

Badness (Sintel): 0.953

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4096/4095

Mapping: [⟨1 1 1 2 8 7], ⟨0 2 1 1 -12 -6], ⟨0 0 2 1 -2 -3]]

Optimal tunings:

  • WE: ~2 = 1199.9597 ¢, ~49/40 = 351.0718 ¢, ~10/7 = 617.6543 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0856 ¢, ~10/7 = 617.6715 ¢

Optimal ET sequence: 41, 89, 99, 130, 270, 581, 711, 981, 1292, 1562

Badness (Sintel): 0.542

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1729/1728, 2080/2079, 2401/2400

Subgroup-val mapping: [⟨1 1 1 2 8 7 0], ⟨0 2 1 1 -12 -6 11], ⟨0 0 2 1 -2 -3 2]]

Optimal tunings:

  • WE: ~2 = 1199.9752 ¢, ~49/40 = 351.0908 ¢, ~10/7 = 617.6473 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0984 ¢, ~10/7 = 617.6588 ¢

Optimal ET sequence: 41, 89, 130, 229, 270, 581, 851, 1562, 1832, 2413

Badness (Sintel): 0.488

Baldur

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800

Mapping: [⟨2 0 1 3 7], ⟨0 2 1 1 -2], ⟨0 0 2 1 3]]

Mapping generators: ~99/70, ~343/198, ~10/7

Optimal tunings:

  • WE: ~99/70 = 600.0149 ¢, ~343/198 = 950.9717 ¢, ~10/7 = 617.7007 ¢
error map: ⟨+0.030 -0.012 +0.074 -0.109 -0.055]
  • CWE: ~99/70 = 600.0000 ¢, ~343/198 = 950.9565 ¢, ~10/7 = 617.7017 ¢
error map: ⟨0.000 -0.042 +0.046 -0.168 -0.126]

Minimax tuning:

[[1 0 0 0 0⟩, [3/4 0 1/2 1/2 -1/2⟩, [0 0 1 0 0⟩, [23/16 0 5/8 1/8 -1/8⟩, [23/16 0 5/8 -7/8 7/8⟩]
unchanged-interval (eigenmonzo) basis: 2.5.11/7

Optimal ET sequence: 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de

Badness (Sintel): 0.200

Projection pairs: 2 9801/4900, 3 117649/39204, 7 9801/1400, 11 913517247483640899/83082326424002500 to 5.7/2.99/4

Greenland

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715

Mapping: [⟨2 0 1 3 7 -1], ⟨0 2 1 1 -2 4], ⟨0 0 2 1 3 2]]

Optimal tunings:

  • WE: ~99/70 = 599.9868 ¢, ~26/15 = 951.0538 ¢, ~10/7 = 617.8062 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~26/15 = 951.0689 ¢, ~10/7 = 617.8067 ¢

Optimal ET sequence: 58, 72, 130, 198, 270, 940, 1210f

Badness (Sintel): 0.415

Complexity spectrum: 15/13, 7/5, 8/7, 7/6, 4/3, 15/14, 5/4, 18/13, 13/12, 14/13, 13/10, 6/5, 16/15, 11/10, 9/7, 9/8, 16/13, 10/9, 14/11, 11/8, 15/11, 12/11, 13/11, 11/9

Projection pairs: 2 19600/9801, 3 676/225, 5 10400/2079, 7 20384000/2910897, 11 19208000000000000/1750211597736459, 13 5026736/385875 to 10/7.200/99.26/15

Midnatssol

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 442/441, 561/560, 676/675

Mapping: [⟨2 0 1 3 7 -1 5], ⟨0 2 1 1 -2 4 2], ⟨0 0 2 1 3 2 0]]

Optimal tunings:

  • WE: ~17/12 = 600.1118 ¢, ~26/15 = 951.1907 ¢, ~10/7 = 617.6100 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~26/15 = 951.0626 ¢, ~10/7 = 617.5842 ¢

Optimal ET sequence: 58, 72, 130, 140, 198, 212g, 270g, 342fg, 482fgg

Badness (Sintel): 0.720

Freya

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024

Mapping: [⟨1 1 -1 1 5], ⟨0 2 3 2 1], ⟨0 0 4 2 -3]]

Mapping generators: ~2, ~49/40, ~84/55

Optimal tunings:

  • WE: ~2 = 1200.0491 ¢, ~49/40 = 350.9820 ¢, ~84/55 = 733.3514 ¢
error map: ⟨+0.030 -0.012 +0.074 -0.109 -0.055]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9890 ¢, ~84/55 = 733.3160 ¢
error map: ⟨0.000 -0.042 +0.046 -0.168 -0.126]

Minimax tuning:

Optimal ET sequence: 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee

Badness (Sintel): 0.204

Projection pairs: ~3 = ~2401/800, ~5 = ~22880495169/4575312500, ~7 = ~1058841/151250, ~11 = ~33275/3024 to 2.49/5.77/3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 3025/3024, 4096/4095

Mapping: [⟨1 1 -1 1 5 10], ⟨0 2 3 2 1 -9], ⟨0 0 4 2 -3 -6]]

Optimal tunings:

  • WE: ~2 = 1199.9870 ¢, ~49/40 = 351.0502 ¢, ~84/55 = 733.2969 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0534 ¢, ~84/55 = 733.3055 ¢

Optimal ET sequence: 31, 41, 72f, 198f, 229, 239, 270, 571, 581, 851, 882, 1152, 1463, 1733, 2615

Badness (Sintel): 0.800

Projection pairs: 3 2401/800, 5 22880495169/4575312500, 7 1058841/151250, 11 33275/3024, 13 1814078464000000000000000/139662717676432916098329 to 2.49/5.77/3

Eir

VIxen named this extension after a healer goddess or valkyrie from the Norse mythology, as it is an extension of freya with the ibnsinma that evokes associations with medicine.

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 3025/3024

Mapping: [⟨1 1 -1 1 5 5], ⟨0 2 3 2 1 6], ⟨0 0 4 2 -3 -5]]

Optimal tunings:

  • WE: ~2 = 1199.9994 ¢, ~49/40 = 351.0880 ¢, ~84/55 = 733.2478 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0880 ¢, ~84/55 = 733.2482 ¢

Optimal ET sequence: 13cdf, 31f, 41, 72, 157, 185cf, 198, 270, 581, 851, 1504, 1774f, 2085, 2355f

Badness (Sintel): 0.543

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1225/1224, 2058/2057, 2080/2079, 2401/2400

Mapping: [⟨1 1 -1 1 5 5 -5], ⟨0 2 3 2 1 6 6], ⟨0 0 4 2 -3 -5 12]]

Optimal tunings:

  • WE: ~2 = 1200.0038 ¢, ~49/40 = 351.0756 ¢, ~84/55 = 733.2270 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0755 ¢, ~84/55 = 733.2246 ¢

Optimal ET sequence: 41g, 72, 198g, 239f, 270, 311, 509, 581, 1234, 1815

Badness (Sintel): 0.665

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 1225/1224, 1540/1539, 2080/2079, 3136/3135, 4200/4199

Mapping: [⟨1 1 -1 1 5 5 -5 3], ⟨0 2 3 2 1 6 6 -2], ⟨0 0 4 2 -3 -5 12 3]]

Optimal tunings:

  • WE: ~2 = 1200.0015 ¢, ~49/40 = 351.0770 ¢, ~84/55 = 733.2252 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0768 ¢, ~84/55 = 733.2243 ¢

Optimal ET sequence: 41g, 72, 198g, 239f, 270, 311, 581, 1234, 1815

Badness (Sintel): 0.712

Heimlaug

VIxen named this extension after a völva (seeress) from the Gull-Þóris saga of Icelanders. It is an extension of freya with the fairytale comma and the ainisma, both adding to the mystical theme. The one of prophecy is bolstered by that this extension has benediction as a subtemperament.

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 3025/3024

Mapping: [⟨1 1 -1 1 5 -6], ⟨0 2 3 2 1 6], ⟨0 0 4 2 -3 13]]

Optimal tunings:

  • WE: ~2 = 1200.0601 ¢, ~49/40 = 350.9562 ¢, ~84/55 = 733.4471 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9646 ¢, ~84/55 = 733.4049 ¢

Optimal ET sequence: 31, 72, 103, 167, 198, 270, 571, 643, 913f

Badness (Sintel): 0.562

17-limit

Equave 10/7 and 16-note scales with that period are of interest.

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 1716/1715

Mapping: [⟨1 1 -1 1 5 -6 -5], ⟨0 2 3 2 1 6 6], ⟨0 0 4 2 -3 13 12]]

Optimal tunings:

  • WE: ~2 = 1200.0446 ¢, ~49/40 = 350.9662 ¢, ~26/17 = 733.3826 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9724 ¢, ~26/17 = 733.3515 ¢

Optimal ET sequence: 31, 64be, 72, 103, 167, 198g, 239, 270

Badness (Sintel): 0.789

Vili

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 391314/390625

Mapping: [⟨1 1 -1 1 -4], ⟨0 2 3 2 6], ⟨0 0 6 3 14]]

Mapping generators: ~2, ~49/40, ~175/132

Optimal tunings:

  • WE: ~2 = 1199.9872 ¢, ~49/40 = 350.9740 ¢, ~175/132 = 488.9418 ¢
error map: ⟨-0.013 -0.020 +0.272 -0.065 -0.238]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9728 ¢, ~175/132 = 488.9475 ¢
error map: ⟨0.000 -0.009 +0.290 -0.038 -0.216]

Optimal ET sequence: 27e, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 670, 913, 1043, 1313, 1583

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 4225/4224

Mapping: [⟨1 1 -1 1 -4 3], ⟨0 2 3 2 6 1], ⟨0 0 6 3 14 1]]

Optimal tunings:

  • WE: ~2 = 1200.0597 ¢, ~49/40 = 350.9419 ¢, ~65/49 = 488.9741 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9441 ¢, ~65/49 = 488.9475 ¢

Optimal ET sequence: 27e, 37, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 913f

Badness (Sintel): 0.690

Frigg

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 644204/643125

Mapping: [⟨1 1 -7 -2 -7], ⟨0 2 3 2 4], ⟨0 0 10 5 11]]

Mapping generators: ~2, ~49/40, ~88/49

Optimal tunings:

  • WE: ~2 = 1200.0183 ¢, ~49/40 = 350.9854 ¢, ~88/49 = 1013.3709 ¢
error map: ⟨+0.018 +0.034 +0.223 -0.037 -0.425]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9871 ¢, ~88/49 = 1013.3556 ¢
error map: ⟨0.000 +0.019 +0.203 -0.074 -0.458]

Optimal ET sequence: 45e, 58, 103, 161, 212, 270, 643, 913, 1183e

Badness (Sintel): 2.15

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 10648/10647

Mapping: [⟨1 1 -7 -2 -7 -9], ⟨0 2 3 2 4 3], ⟨0 0 10 5 11 14]]

Optimal tunings:

  • WE: ~2 = 1200.0387 ¢, ~49/40 = 350.9502 ¢, ~70/39 = 1013.4065 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9526 ¢, ~70/39 = 1013.3747 ¢

Optimal ET sequence: 45ef, 58, 103, 161, 212, 270, 643, 913f, 1614ef * * optimal patent val: 1241

Badness (Sintel): 0.873

Ennealimmic

Not to be confused with Ennealimnic.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374

Mapping: [⟨9 1 1 12 0], ⟨0 2 3 2 0], ⟨0 0 0 0 1]]

Mapping generators: ~27/25, ~5/3, ~11

Optimal tunings:

  • WE: ~27/25 = 133.3357 ¢, ~5/3 = 884.3288 ¢, ~11/8 = 551.2529 ¢
error map: ⟨+0.021 +0.038 +0.009 -0.139 -0.000]
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3215 ¢, ~11/8 = 551.2775 ¢
error map: ⟨0.000 +0.021 -0.016 -0.183 -0.040]

Optimal ET sequence: 72, …, 171, 198, 270, 342, 612, 954, 1323, 1395, 1665, 2007, 2277, 2619, 4284d, 6561dd, 6903dd

Badness (Sintel): 0.330

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4375/4374

Mapping: [⟨9 1 1 12 0 -31], ⟨0 2 3 2 0 5], ⟨0 0 0 0 1 1]]

Optimal tunings:

  • WE: ~27/25 = 133.3276 ¢, ~5/3 = 884.3660 ¢, ~11/8 = 551.7221 ¢
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3905 ¢, ~11/8 = 551.7085 ¢

Optimal ET sequence: 72, …, 171, 198, 270, 639, 711, 981, 1692e

Badness (Sintel): 0.706

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