Kleismic family: Difference between revisions

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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0308{{c}}
* [[WE]]: ~2 = 1200.1659{{c}}, ~6/5 = 317.0504{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.007{{c}}
: [[error map]]: {{val| +0.166 +0.347 -0.896 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.0308{{c}}
: error map: {{val| 0.000 +0.230 -1.160 }}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0586{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.007{{c}} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 28: Line 32:
* 5-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263] (untempered to 1/5-comma)
* 5-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263] (untempered to 1/5-comma)


{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 458, 511c, …, 882c }}
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 458, 511c, …, 829c, 882c }}


[[Badness]] (Smith): 0.013234
[[Badness]] (Sintel): 0.310


=== Overview to extensions ===
=== Overview to extensions ===
Line 47: Line 51:
{{Mapping|legend=1| 1 0 1 -3 | 0 6 5 22 }}
{{Mapping|legend=1| 1 0 1 -3 | 0 6 5 22 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.732{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}
: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}
: error map: {{val| 0.000 -1.332 -2.461 +0.126 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.732{{c}} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 55: Line 64:
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}


[[Badness]] (Smith): 0.021501
[[Badness]] (Sintel): 0.544


==== 2.3.5.7.13 subgroup ====
==== 2.3.5.7.13 subgroup ====
Line 66: Line 75:
Subgroup-val mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
Subgroup-val mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 316.8865{{c}}
Optimal tunings:
* WE: ~2 = 1200.7838{{c}}, ~6/5 = 316.9478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~6/5 = 316.8865{{c}} -->


{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197f }}
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197f }}


Badness (Smith): 0.0118
Badness (Sintel): 0.410


=== 11-limit ===
=== 11-limit ===
Line 79: Line 91:
Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}
Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.719{{c}}
Optimal tunings:
* WE: ~2 = 1200.6524{{c}}, ~6/5 = 316.8911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7267{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 316.719{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 85: Line 100:
* 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]
* 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]


{{Optimal ET sequence|legend=0| 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee }}
{{Optimal ET sequence|legend=0| 19, 53, 72, 197e, 269ce, 341ce }}


Badness (Smith): 0.021849
Badness (Sintel): 0.722


==== 13-limit ====
==== 13-limit ====
Line 96: Line 111:
Mapping: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}
Mapping: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.738{{c}}
Optimal tunings:
* WE: ~2 = 1200.7982{{c}}, ~6/5 = 316.9482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7491{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 316.738{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 102: Line 120:
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]


{{Optimal ET sequence|legend=0| 19, 34de, 53, 72, 125f, 197ef, 269ceff }}
{{Optimal ET sequence|legend=0| 19, 53, 72, 125f, 197ef }}


Badness (Smith): 0.016883
Badness (Sintel): 0.698


=== Cataclysmic ===
=== Cataclysmic ===
Line 113: Line 131:
Mapping: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}
Mapping: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.042{{c}}
Optimal tunings:
* WE: ~2 = 1199.9590{{c}}, ~6/5 = 317.0315{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0403{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.042{{c}} -->


{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}


Badness (Smith): 0.039954
Badness (Sintel): 1.32


==== 13-limit ====
==== 13-limit ====
Line 126: Line 147:
Mapping: {{mapping| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}
Mapping: {{mapping| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.036{{c}}
Optimal tunings:
* WE: ~2 = 1200.0797{{c}}, ~6/5 = 317.0571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0400{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.036{{c}} -->


{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}


Badness (Smith): 0.022555
Badness (Sintel): 0.932


=== Catalytic ===
=== Catalytic ===
Line 139: Line 163:
Mapping: {{mapping| 1 0 1 -3 -10 | 0 6 5 22 51 }}
Mapping: {{mapping| 1 0 1 -3 -10 | 0 6 5 22 51 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.653{{c}}
Optimal tunings:
* WE: ~2 = 1200.8102{{c}}, ~6/5 = 316.8669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6768{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 316.653{{c}} -->


{{Optimal ET sequence|legend=0| 19e, 53e, 72 }}
{{Optimal ET sequence|legend=0| 19e, 53e, 72 }}


Badness (Smith): 0.030422
Badness (Sintel): 1.01


==== 13-limit ====
==== 13-limit ====
Line 152: Line 179:
Mapping: {{mapping| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}
Mapping: {{mapping| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.639{{c}}
Optimal tunings:
* WE: ~2 = 1201.0807{{c}}, ~6/5 = 316.9246{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6700{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 316.639{{c}} -->


{{Optimal ET sequence|legend=0| 19e, 53e, 72 }}
{{Optimal ET sequence|legend=0| 19e, 53e, 72, 307bcdeeffff }}


Badness (Smith): 0.022337
Badness (Sintel): 0.923


=== Cataleptic ===
=== Cataleptic ===
Line 165: Line 195:
Mapping: {{mapping| 1 0 1 -3 4 | 0 6 5 22 -2 }}
Mapping: {{mapping| 1 0 1 -3 4 | 0 6 5 22 -2 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.083{{c}}
Optimal tunings:
* WE: ~2 = 1198.6575{{c}}, ~6/5 = 316.7282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0257{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.083{{c}} -->


{{Optimal ET sequence|legend=0| 19, 34d, 53e }}
{{Optimal ET sequence|legend=0| 19, 34d, 53e }}


Badness (Smith): 0.044335
Badness (Sintel): 1.47


==== 13-limit ====
==== 13-limit ====
Line 178: Line 211:
Mapping: {{mapping| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}
Mapping: {{mapping| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.118{{c}}
Optimal tunings:
* WE: ~2 = 1198.8403{{c}}, ~6/5 = 316.8111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0652{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.118{{c}} -->


{{Optimal ET sequence|legend=0| 19, 34d, 53e, 87dee }}
{{Optimal ET sequence|legend=0| 19, 34d, 53e }}


Badness (Smith): 0.027343
Badness (Sintel): 1.13


=== Bikleismic ===
=== Bikleismic ===
Line 192: Line 228:
: mapping generators: ~99/70, ~6/5
: mapping generators: ~99/70, ~6/5


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 316.721{{c}}
Optimal tunings:
* WE: ~99/70 = 600.2674{{c}}, ~6/5 = 316.8624{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 316.7575{{c}}
<!-- * POTE: ~99/70 = 600.000{{c}}, ~6/5 = 316.721{{c}} -->


{{Optimal ET sequence|legend=0| 34d, 72, 322c, …, 610bcc }}
{{Optimal ET sequence|legend=0| 34d, 72, 322c, 394c }}


Badness (Smith): 0.029319
Badness (Sintel): 0.969


==== 13-limit ====
==== 13-limit ====
Line 205: Line 244:
Mapping: {{mapping| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}
Mapping: {{mapping| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}


Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~6/5 = 316.726{{c}}
Optimal tunings:
* WE: ~55/39 = 600.3582{{c}}, ~6/5 = 316.9152{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 316.7759{{c}}
<!-- * POTE: ~55/39 = 600.000{{c}}, ~6/5 = 316.726{{c}} -->


{{Optimal ET sequence|legend=0| 34d, 72 }}
{{Optimal ET sequence|legend=0| 34d, 72 }}


Badness (Smith): 0.021814
Badness (Sintel): 0.901


==== 17-limit ====
==== 17-limit ====
Line 218: Line 260:
Mapping: {{mapping| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}
Mapping: {{mapping| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}


Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~6/5 = 316.726{{c}}
Optimal tunings:
* WE: ~17/12 = 600.4210{{c}}, ~6/5 = 316.9282{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7578{{c}}
<!-- * POTE: ~17/12 = 600.000{{c}}, ~6/5 = 316.726{{c}} -->


{{Optimal ET sequence|legend=0| 34d, 38df, 72 }}
{{Optimal ET sequence|legend=0| 34d, 38df, 72 }}


Badness (Smith): 0.015656
Badness (Sintel): 0.798


==== 19-limit ====
==== 19-limit ====
Line 231: Line 276:
Mapping: {{mapping| 2 0 2 -6 -1 0 5 -1 | 0 6 5 22 15 14 6 18 }}
Mapping: {{mapping| 2 0 2 -6 -1 0 5 -1 | 0 6 5 22 15 14 6 18 }}


Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~6/5 = 316.726{{c}}
Optimal tunings:
* WE: ~17/12 = 600.3763{{c}}, ~6/5 = 316.8720{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7205{{c}}
<!-- * POTE: ~17/12 = 600.000{{c}}, ~6/5 = 316.726{{c}} -->


{{Optimal ET sequence|legend=0| 34dh, 38df, 72 }}
{{Optimal ET sequence|legend=0| 34dh, 38df, 72 }}


Badness (Smith): 0.015771
Badness (Sintel): 0.959


== Keemun ==
== Keemun ==
Line 246: Line 294:
{{Mapping|legend=1| 1 0 1 2 | 0 6 5 3 }}
{{Mapping|legend=1| 1 0 1 2 | 0 6 5 3 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.473{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.6235{{c}}, ~6/5 = 317.1646{{c}}
: [[error map]]: {{val| +2.624 +1.033 +2.133 -12.085 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.8293{{c}}
: error map: {{val| 0.000 -0.979 -2.167 -18.388 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.473{{c}} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 253: Line 306:
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [308.744, 322.942]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [308.744, 322.942]


{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd, 91dd }}
{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd }}


[[Badness]] (Smith): 0.027408
[[Badness]] (Sintel): 0.694


=== 11-limit ===
=== 11-limit ===
Line 264: Line 317:
Mapping: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}
Mapping: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.576{{c}}
Optimal tunings:
* WE: ~2 = 1199.7353{{c}}, ~6/5 = 317.5055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5546{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.576{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 270: Line 326:
* 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
* 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]


{{Optimal ET sequence|legend=0| 4, 15, 19, 34 }}
{{Optimal ET sequence|legend=0| 15, 19, 34 }}


Badness (Smith): 0.027410
Badness (Sintel): 0.906


==== 13-limit ====
==== 13-limit ====
Line 281: Line 337:
Mapping: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}
Mapping: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.611{{c}}
Optimal tunings:
* WE: ~2 = 1201.8360{{c}}, ~6/5 = 317.0958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6829{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 316.611{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 287: Line 346:
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]


{{Optimal ET sequence|legend=0| 4, 15f, 19, 53def, 72def }}
{{Optimal ET sequence|legend=0| 4, 15f, 19 }}


Badness (Smith): 0.029749
Badness (Sintel): 1.23


==== Kema ====
==== Kema ====
Line 298: Line 357:
Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.423{{c}}
Optimal tunings:
* WE: ~2 = 1199.7816{{c}}, ~6/5 = 317.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.4070{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.423{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 305: Line 367:
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]


{{Optimal ET sequence|legend=0| 15, 19, 34, 87ddee }}
{{Optimal ET sequence|legend=0| 15, 19, 34 }}


Badness (Smith): 0.022749
Badness (Sintel): 0.940


==== Kumbaya ====
==== Kumbaya ====
Line 316: Line 378:
Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}
Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 318.595{{c}}
Optimal tunings:
* WE: ~2 = 1196.7615{{c}}, ~6/5 = 317.7353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.4059{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 318.595{{c}} -->


{{Optimal ET sequence|legend=0| 4, 15, 19f, 34ff }}
{{Optimal ET sequence|legend=0| 4, 11b, 15 }}


Badness (Smith): 0.031633
Badness (Sintel): 1.31


=== Qeema ===
=== Qeema ===
Line 329: Line 394:
Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}
Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 314.730{{c}}
Optimal tunings:
* WE: ~2 = 1204.5534{{c}}, ~6/5 = 315.9247{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.1686{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 314.730{{c}} -->


{{Optimal ET sequence|legend=0| 4e, 19, 42bcd, 61bcdd }}
{{Optimal ET sequence|legend=0| 4e, 19, 42bcd }}


Badness (Smith): 0.040056
Badness (Sintel): 1.32


==== 13-limit ====
==== 13-limit ====
Line 342: Line 410:
Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}
Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.044{{c}}
Optimal tunings:
* WE: ~2 = 1204.4937{{c}}, ~6/5 = 316.2241{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.4748{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 315.044{{c}} -->


{{Optimal ET sequence|legend=0| 4ef, 19 }}
{{Optimal ET sequence|legend=0| 4ef, 19 }}


Badness (Smith): 0.029419
Badness (Sintel): 1.22


=== Darjeeling ===
=== Darjeeling ===
Line 355: Line 426:
Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}
Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.656{{c}}
Optimal tunings:
* WE: ~2 = 1201.6569{{c}}, ~6/5 = 318.0942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.8547{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.656{{c}} -->


{{Optimal ET sequence|legend=0| 15, 19e, 34e }}
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}


Badness (Smith): 0.027648
Badness (Sintel): 0.914


==== 13-limit ====
==== 13-limit ====
Line 368: Line 442:
Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}
Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.298{{c}}
Optimal tunings:
* WE: ~2 = 1201.9324{{c}}, ~6/5 = 317.8090{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5170{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.298{{c}} -->


{{Optimal ET sequence|legend=0| 15, 19e, 34e, 53dee }}
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}


Badness (Smith): 0.021445
Badness (Sintel): 0.886


== Catalan ==
== Catalan ==
Line 381: Line 458:
{{Mapping|legend=1| 1 0 1 6 | 0 6 5 -12 }}
{{Mapping|legend=1| 1 0 1 6 | 0 6 5 -12 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 318.267{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.1789{{c}}, ~6/5 = 317.5185{{c}}
: [[error map]]: {{val| -2.821 +3.156 -1.542 +4.025 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 318.2411{{c}}
: error map: {{val| 0.000 +7.492 +4.892 +12.281 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 318.267{{c}} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 389: Line 471:
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}


[[Badness]] (Smith): 0.094872
[[Badness]] (Sintel): 2.40


=== 11-limit ===
=== 11-limit ===
Line 398: Line 480:
Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}
Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 318.282{{c}}
Optimal tunings:
* WE: ~2 = 1197.0368{{c}}, ~6/5 = 317.4956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.2672{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 318.282{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 406: Line 491:
{{Optimal ET sequence|legend=0| 15, 34d, 49, 181bbcdddeee }}
{{Optimal ET sequence|legend=0| 15, 34d, 49, 181bbcdddeee }}


Badness (Smith): 0.036894
Badness (Sintel): 1.22


=== 13-limit ===
=== 13-limit ===
Line 415: Line 500:
Mapping: {{mapping| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}
Mapping: {{mapping| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 317.9159{{c}}
Optimal tunings:
* WE: ~2 = 1196.8961{{c}}, ~6/5 = 317.3837{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.1621{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.9159{{c}} -->


{{Optimal ET sequence|legend=0| 15, 34d, 49f, 83def, 132bcddeefff }}
{{Optimal ET sequence|legend=0| 15, 34d, 49f, 83def, 132bcddeefff }}


Badness (Smith): 0.0263
Badness (Sintel): 1.09


== Countercata ==
== Countercata ==
Line 428: Line 516:
{{Mapping|legend=1| 1 0 1 11 | 0 6 5 -31 }}
{{Mapping|legend=1| 1 0 1 11 | 0 6 5 -31 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 317.121{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9172{{c}}, ~6/5 = 317.0995{{c}}
: [[error map]]: {{val| -0.083 +0.642 -0.899 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.1220{{c}}
: error map: {{val| 0.000 +0.777 -0.704 +0.391 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 317.121{{c}} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 434: Line 527:
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]


{{Optimal ET sequence|legend=1| 19d, 34, 53, 87, 140, 333, 473, 806b }}
{{Optimal ET sequence|legend=1| 19d, 34, 53, 87, 140, 333, 473 }}


[[Badness]] (Smith): 0.052129
[[Badness]] (Sintel): 1.32


=== 11-limit ===
=== 11-limit ===
Line 445: Line 538:
Mapping: {{mapping| 1 0 1 11 -5 | 0 6 5 -31 32 }}
Mapping: {{mapping| 1 0 1 11 -5 | 0 6 5 -31 32 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.162{{c}}
Optimal tunings:
* WE: ~2 = 1200.0980{{c}}, ~6/5 = 317.1879{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1623{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.162{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 451: Line 547:
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]


{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 227, 367e, 507e }}
{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 227, 367e }}


Badness (Smith): 0.039770
Badness (Sintel): 1.31


=== 13-limit ===
=== 13-limit ===
Line 462: Line 558:
Mapping: {{mapping| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}
Mapping: {{mapping| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.162{{c}}
Optimal tunings:
* WE: ~2 = 1200.0936{{c}}, ~6/5 = 317.1864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1622{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.162{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 471: Line 570:
{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 367e, 507e }}
{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 367e, 507e }}


Badness (Smith): 0.020156
Badness (Sintel): 0.833


== Metakleismic ==
== Metakleismic ==
Line 480: Line 579:
{{Mapping|legend=1| 1 0 1 -12 | 0 6 5 56 }}
{{Mapping|legend=1| 1 0 1 -12 | 0 6 5 56 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 317.314{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5969{{c}}, ~6/5 = 317.2079{{c}}
: [[error map]]: {{val| -0.403 +1.292 -0.678 -0.349 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.3071{{c}}
: error map: {{val| 0.000 +1.887 +0.222 +0.370 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 317.314{{c}} -->


{{Optimal ET sequence|legend=1| 34d, 87, 121, 208 }}
{{Optimal ET sequence|legend=1| 34d, 87, 121, 208, 537b }}


[[Badness]] (Smith): 0.163519
[[Badness]] (Sintel): 4.14


=== 11-limit ===
=== 11-limit ===
Line 493: Line 597:
Mapping: {{mapping| 1 0 1 -12 -5 | 0 6 5 56 32 }}
Mapping: {{mapping| 1 0 1 -12 -5 | 0 6 5 56 32 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.311{{c}}
Optimal tunings:
* WE: ~2 = 1199.5425{{c}}, ~6/5 = 317.1901{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3020{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.311{{c}} -->


{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}
{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}


Badness (Smith): 0.048570
Badness (Sintel): 1.61


=== 13-limit ===
=== 13-limit ===
Line 506: Line 613:
Mapping: {{mapping| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}
Mapping: {{mapping| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.311{{c}}
Optimal tunings:
* WE: ~2 = 1199.5339{{c}}, ~6/5 = 317.1882{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3028{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.311{{c}} -->


{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}
{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}


Badness (Smith): 0.024371
Badness (Sintel): 1.01


== Hemikleismic ==
== Hemikleismic ==

Revision as of 16:55, 5 January 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 5-limit parent comma for the kleismic family is 15625/15552, the kleisma, which is the amount by which a stack of six classical minor thirds falls short of the 3rd harmonic.

Kleismic a.k.a. hanson

Main article: Kleismic

The generator of kleismic is a classical minor third, and to get to the interval class of major thirds requires five of these, and so to get to fifths requires six. In fact, (6/5)5 = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a microtemperament) is sometimes called hanson, and 14\53 is about perfect as a generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include 72edo, 87edo and 140edo.

Subgroup: 2.3.5

Comma list: 15625/15552

Mapping: [⟨1 0 1], ⟨0 6 5]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1200.1659 ¢, ~6/5 = 317.0504 ¢
error map: ⟨+0.166 +0.347 -0.896]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0308 ¢
error map: ⟨0.000 +0.230 -1.160]

Tuning ranges:

Optimal ET sequence: 15, 19, 34, 53, 458, 511c, …, 829c, 882c

Badness (Sintel): 0.310

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun. 4375/4374, the ragisma, gives catakleismic. 5120/5103, hemifamity, gives countercata. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun.

6144/6125, the porwell comma, gives hemikleismic. 245/243, sensamagic, gives clyde. 1029/1024, the gamelisma, gives tritikleismic. 2401/2400 the breedsma, gives quadritikleismic. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.

Catakleismic

Main article: Catakleismic

7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 4375/4374

Mapping: [⟨1 0 1 -3], ⟨0 6 5 22]]

Optimal tunings:

  • WE: ~2 = 1200.5965 ¢, ~6/5 = 316.8893 ¢
error map: ⟨+0.596 -0.619 -1.271 +0.948]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7705 ¢
error map: ⟨0.000 -1.332 -2.461 +0.126]

Tuning ranges:

Optimal ET sequence: 19, 34d, 53, 72, 197, 269c

Badness (Sintel): 0.544

2.3.5.7.13 subgroup

The S-expression-based comma list of this temperament is {S13, S15 = S25*S26*S27, S10/S12 = S25*S26, (S25, S26 = S13/S15, S27)}.

Subgroup: 2.3.5.7.13

Comma list: 169/168, 225/224, 325/324

Subgroup-val mapping: [⟨1 0 1 -3 0], ⟨0 6 5 22 14]]

Optimal tunings:

  • WE: ~2 = 1200.7838 ¢, ~6/5 = 316.9478 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7939 ¢

Optimal ET sequence: 19, 34d, 53, 72, 125f, 197f

Badness (Sintel): 0.410

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4375/4374

Mapping: [⟨1 0 1 -3 9], ⟨0 6 5 22 -21]]

Optimal tunings:

  • WE: ~2 = 1200.6524 ¢, ~6/5 = 316.8911 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7267 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone range: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
  • 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]

Optimal ET sequence: 19, 53, 72, 197e, 269ce, 341ce

Badness (Sintel): 0.722

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 385/384

Mapping: [⟨1 0 1 -3 9 0], ⟨0 6 5 22 -21 14]]

Optimal tunings:

  • WE: ~2 = 1200.7982 ¢, ~6/5 = 316.9482 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7491 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
  • 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]

Optimal ET sequence: 19, 53, 72, 125f, 197ef

Badness (Sintel): 0.698

Cataclysmic

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 2200/2187

Mapping: [⟨1 0 1 -3 -5], ⟨0 6 5 22 32]]

Optimal tunings:

  • WE: ~2 = 1199.9590 ¢, ~6/5 = 317.0315 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0403 ¢

Optimal ET sequence: 19e, 34d, 53

Badness (Sintel): 1.32

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 176/175, 275/273

Mapping: [⟨1 0 1 -3 -5 0], ⟨0 6 5 22 32 14]]

Optimal tunings:

  • WE: ~2 = 1200.0797 ¢, ~6/5 = 317.0571 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0400 ¢

Optimal ET sequence: 19e, 34d, 53

Badness (Sintel): 0.932

Catalytic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4375/4374

Mapping: [⟨1 0 1 -3 -10], ⟨0 6 5 22 51]]

Optimal tunings:

  • WE: ~2 = 1200.8102 ¢, ~6/5 = 316.8669 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.6768 ¢

Optimal ET sequence: 19e, 53e, 72

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 1716/1715

Mapping: [⟨1 0 1 -3 -10 0], ⟨0 6 5 22 51 14]]

Optimal tunings:

  • WE: ~2 = 1201.0807 ¢, ~6/5 = 316.9246 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.6700 ¢

Optimal ET sequence: 19e, 53e, 72, 307bcdeeffff

Badness (Sintel): 0.923

Cataleptic

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 864/847

Mapping: [⟨1 0 1 -3 4], ⟨0 6 5 22 -2]]

Optimal tunings:

  • WE: ~2 = 1198.6575 ¢, ~6/5 = 316.7282 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0257 ¢

Optimal ET sequence: 19, 34d, 53e

Badness (Sintel): 1.47

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 676/675

Mapping: [⟨1 0 1 -3 4 0], ⟨0 6 5 22 -2 14]]

Optimal tunings:

  • WE: ~2 = 1198.8403 ¢, ~6/5 = 316.8111 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0652 ¢

Optimal ET sequence: 19, 34d, 53e

Badness (Sintel): 1.13

Bikleismic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 4375/4356

Mapping: [⟨2 0 2 -6 -1], ⟨0 6 5 22 15]]

mapping generators: ~99/70, ~6/5

Optimal tunings:

  • WE: ~99/70 = 600.2674 ¢, ~6/5 = 316.8624 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~6/5 = 316.7575 ¢

Optimal ET sequence: 34d, 72, 322c, 394c

Badness (Sintel): 0.969

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 243/242, 325/324

Mapping: [⟨2 0 2 -6 -1 0], ⟨0 6 5 22 15 14]]

Optimal tunings:

  • WE: ~55/39 = 600.3582 ¢, ~6/5 = 316.9152 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~6/5 = 316.7759 ¢

Optimal ET sequence: 34d, 72

Badness (Sintel): 0.901

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 243/242, 325/324

Mapping: [⟨2 0 2 -6 -1 0 5], ⟨0 6 5 22 15 14 6]]

Optimal tunings:

  • WE: ~17/12 = 600.4210 ¢, ~6/5 = 316.9282 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~6/5 = 316.7578 ¢

Optimal ET sequence: 34d, 38df, 72

Badness (Sintel): 0.798

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324

Mapping: [⟨2 0 2 -6 -1 0 5 -1], ⟨0 6 5 22 15 14 6 18]]

Optimal tunings:

  • WE: ~17/12 = 600.3763 ¢, ~6/5 = 316.8720 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~6/5 = 316.7205 ¢

Optimal ET sequence: 34dh, 38df, 72

Badness (Sintel): 0.959

Keemun

Main article: Keemun

Subgroup: 2.3.5.7

Comma list: 49/48, 126/125

Mapping: [⟨1 0 1 2], ⟨0 6 5 3]]

Optimal tunings:

  • WE: ~2 = 1202.6235 ¢, ~6/5 = 317.1646 ¢
error map: ⟨+2.624 +1.033 +2.133 -12.085]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.8293 ¢
error map: ⟨0.000 -0.979 -2.167 -18.388]

Tuning ranges:

  • 7-odd-limit diamond monotone: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
  • 9-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
  • 7- and 9-odd-limit diamond tradeoff: ~6/5 = [308.744, 322.942]

Optimal ET sequence: 15, 19, 53d, 72dd

Badness (Sintel): 0.694

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 100/99

Mapping: [⟨1 0 1 2 4], ⟨0 6 5 3 -2]]

Optimal tunings:

  • WE: ~2 = 1199.7353 ¢, ~6/5 = 317.5055 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.5546 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
  • 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]

Optimal ET sequence: 15, 19, 34

Badness (Sintel): 0.906

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 65/64, 100/99

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

Optimal tunings:

  • WE: ~2 = 1201.8360 ¢, ~6/5 = 317.0958 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.6829 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
  • 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]

Optimal ET sequence: 4, 15f, 19

Badness (Sintel): 1.23

Kema

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 100/99

Mapping: [⟨1 0 1 2 4 0], ⟨0 6 5 3 -2 14]]

Optimal tunings:

  • WE: ~2 = 1199.7816 ¢, ~6/5 = 317.3653 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.4070 ¢

Tuning ranges:

  • 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
  • 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
  • 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]

Optimal ET sequence: 15, 19, 34

Badness (Sintel): 0.940

Kumbaya

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 49/48, 56/55, 66/65

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

Optimal tunings:

  • WE: ~2 = 1196.7615 ¢, ~6/5 = 317.7353 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 319.4059 ¢

Optimal ET sequence: 4, 11b, 15

Badness (Sintel): 1.31

Qeema

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 126/125

Mapping: [⟨1 0 1 2 -1], ⟨0 6 5 3 17]]

Optimal tunings:

  • WE: ~2 = 1204.5534 ¢, ~6/5 = 315.9247 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 315.1686 ¢

Optimal ET sequence: 4e, 19, 42bcd

Badness (Sintel): 1.32

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 78/77, 126/125

Mapping: [⟨1 0 1 2 -1 0], ⟨0 6 5 3 17 14]]

Optimal tunings:

  • WE: ~2 = 1204.4937 ¢, ~6/5 = 316.2241 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 315.4748 ¢

Optimal ET sequence: 4ef, 19

Badness (Sintel): 1.22

Darjeeling

Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 77/75

Mapping: [⟨1 0 1 2 0], ⟨0 6 5 3 13]]

Optimal tunings:

  • WE: ~2 = 1201.6569 ¢, ~6/5 = 318.0942 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.8547 ¢

Optimal ET sequence: 15, 19e, 34e

Badness (Sintel): 0.914

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 66/65, 77/75

Mapping: [⟨1 0 1 2 0 0], ⟨0 6 5 3 13 14]]

Optimal tunings:

  • WE: ~2 = 1201.9324 ¢, ~6/5 = 317.8090 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.5170 ¢

Optimal ET sequence: 15, 19e, 34e

Badness (Sintel): 0.886

Catalan

Subgroup: 2.3.5.7

Comma list: 64/63, 15625/15552

Mapping: [⟨1 0 1 6], ⟨0 6 5 -12]]

Optimal tunings:

  • WE: ~2 = 1197.1789 ¢, ~6/5 = 317.5185 ¢
error map: ⟨-2.821 +3.156 -1.542 +4.025]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 318.2411 ¢
error map: ⟨0.000 +7.492 +4.892 +12.281]

Tuning ranges:

Optimal ET sequence: 15, 34d, 49, 132bcdd, 181bbcddd

Badness (Sintel): 2.40

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 1331/1323

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

Optimal tunings:

  • WE: ~2 = 1197.0368 ¢, ~6/5 = 317.4956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 318.2672 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
  • 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]

Optimal ET sequence: 15, 34d, 49, 181bbcdddeee

Badness (Sintel): 1.22

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 144/143, 275/273

Mapping: [⟨1 0 1 6 4 0], ⟨0 6 5 -12 -2 14]]

Optimal tunings:

  • WE: ~2 = 1196.8961 ¢, ~6/5 = 317.3837 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 318.1621 ¢

Optimal ET sequence: 15, 34d, 49f, 83def, 132bcddeefff

Badness (Sintel): 1.09

Countercata

Subgroup: 2.3.5.7

Comma list: 5120/5103, 15625/15552

Mapping: [⟨1 0 1 11], ⟨0 6 5 -31]]

Optimal tunings:

  • WE: ~2 = 1199.9172 ¢, ~6/5 = 317.0995 ¢
error map: ⟨-0.083 +0.642 -0.899 +0.178]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.1220 ¢
error map: ⟨0.000 +0.777 -0.704 +0.391]

Tuning ranges:

Optimal ET sequence: 19d, 34, 53, 87, 140, 333, 473

Badness (Sintel): 1.32

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 3388/3375

Mapping: [⟨1 0 1 11 -5], ⟨0 6 5 -31 32]]

Optimal tunings:

  • WE: ~2 = 1200.0980 ¢, ~6/5 = 317.1879 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.1623 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
  • 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]

Optimal ET sequence: 34, 53, 87, 140, 227, 367e

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 625/624

Mapping: [⟨1 0 1 11 -5 0], ⟨0 6 5 -31 32 14]]

Optimal tunings:

  • WE: ~2 = 1200.0936 ¢, ~6/5 = 317.1864 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.1622 ¢

Tuning ranges:

  • 13-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
  • 15-odd-limit diamond monotone: ~6/5 = [316.981, 317.241] (14\53 to 23\87)
  • 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]

Optimal ET sequence: 34, 53, 87, 140, 367e, 507e

Badness (Sintel): 0.833

Metakleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 179200/177147

Mapping: [⟨1 0 1 -12], ⟨0 6 5 56]]

Optimal tunings:

  • WE: ~2 = 1199.5969 ¢, ~6/5 = 317.2079 ¢
error map: ⟨-0.403 +1.292 -0.678 -0.349]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.3071 ¢
error map: ⟨0.000 +1.887 +0.222 +0.370]

Optimal ET sequence: 34d, 87, 121, 208, 537b

Badness (Sintel): 4.14

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 14700/14641

Mapping: [⟨1 0 1 -12 -5], ⟨0 6 5 56 32]]

Optimal tunings:

  • WE: ~2 = 1199.5425 ¢, ~6/5 = 317.1901 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.3020 ¢

Optimal ET sequence: 34d, 53d, 87, 121, 208

Badness (Sintel): 1.61

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 625/624

Mapping: [⟨1 0 1 -12 -5 0], ⟨0 6 5 56 32 14]]

Optimal tunings:

  • WE: ~2 = 1199.5339 ¢, ~6/5 = 317.1882 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.3028 ¢

Optimal ET sequence: 34d, 53d, 87, 121, 208

Badness (Sintel): 1.01

Hemikleismic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 6144/6125

Mapping: [⟨1 0 1 4], ⟨0 12 10 -9]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~35/32 = 158.649 ¢

Optimal ET sequence: 15, 38, 53, 121

Badness (Smith): 0.052054

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 4000/3969

Mapping: [⟨1 0 1 4 2], ⟨0 12 10 -9 11]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 158.677 ¢

Optimal ET sequence: 15, 38, 53, 68, 121e

Badness (Smith): 0.038023

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 275/273, 325/324

Mapping: [⟨1 0 1 4 2 0], ⟨0 12 10 -9 11 28]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 158.655 ¢

Optimal ET sequence: 15, 38f, 53, 121e

Badness (Smith): 0.026005

Clyde

Subgroup: 2.3.5.7

Comma list: 245/243, 3136/3125

Mapping: [⟨1 6 6 12], ⟨0 -12 -10 -25]]

mapping generators: ~2, ~9/7

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~9/7 = 441.335 ¢

Minimax tuning:

[[1 0 0 0⟩, [6/25 0 0 12/25⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.7

Algebraic generator: real root of 5x3 - 6x - 3, the Poussami generator. Approximately 441.309 cents. Associated recurrence relationship quickly converges.

Optimal ET sequence: 19, 49, 68, 87, 155

Badness (Smith): 0.047261

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 3136/3125

Mapping: [⟨1 6 6 12 -5], ⟨0 -12 -10 -25 23]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~9/7 = 441.355 ¢

Optimal ET sequence: 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd

Badness (Smith): 0.047417

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 385/384, 625/624

Mapping: [⟨1 6 6 12 -5 14], ⟨0 -12 -10 -25 23 -28]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~9/7 = 441.363 ¢

Optimal ET sequence: 19, 49ef, 68, 87, 503bdf, 590bdf

Badness (Smith): 0.026842

Tritikleismic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 15625/15552

Mapping: [⟨3 0 3 10], ⟨0 6 5 -2]]

mapping generators: ~63/50, ~6/5

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~6/5 = 316.872 ¢ (~21/20 = 83.128 ¢)

Minimax tuning:

[[1 0 0 0⟩, [2 0 6/7 -6/7⟩, [8/3 0 5/7 -5/7⟩, [8/3 0 -2/7 2/7⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [46/21 5/7 0 -5/14⟩, [20/7 -2/7 0 1/7⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 15, 42bc, 57, 72, 159, 231

Badness (Smith): 0.056337

Music:

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 4000/3993

Mapping: [⟨3 0 3 10 8], ⟨0 6 5 -2 3]]

Optimal tuning (POTE): ~44/35 = 400.000 ¢, ~6/5 = 316.881 ¢ (~21/20 = 83.119 ¢)

Minimax tuning:

  • 11-odd-limit: ~6/5 = [5/21 1/7 0 -1/14⟩
[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [46/21 5/7 0 -5/14 0⟩, [20/7 -2/7 0 1/7 0⟩, [71/21 3/7 0 -3/14 0⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 15, 42bc, 57, 72, 159, 231

Badness (Smith): 0.019333

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 625/624

Mapping: [⟨3 0 3 10 8 0], ⟨0 6 5 -2 3 14]]

Optimal tuning (POTE): ~44/35 = 400.0000 ¢, ~6/5 = 316.9585 ¢ (~21/20 = 83.0415 ¢)

Optimal ET sequence: 72, 87, 159

Badness (Smith): 0.015652

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 375/374, 385/384

Mapping: [⟨3 0 3 10 8 0 -2], ⟨0 6 5 -2 3 14 18]]

Optimal tuning (POTE): ~34/27 = 400.0000 ¢, ~6/5 = 316.9082 ¢ (~21/20 = 83.0918 ¢)

Optimal ET sequence: 72, 159, 231f

Badness (Smith): 0.013551

Quadritikleismic

Subgroup: 2.3.5.7

Comma list: 2401/2400, 15625/15552

Mapping: [⟨4 0 4 7], ⟨0 6 5 4]]

mapping generators: ~25/21, ~6/5

Optimal tuning (POTE): ~25/21 = 300.0000 ¢, ~6/5 = 316.9999 ¢ (~126/125 = 16.9999 ¢)

Optimal ET sequence: 68, 72, 140, 212, 776cd, 988ccd, 1200ccd

Badness (Smith): 0.039231

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 6250/6237

Mapping: [⟨4 0 4 7 17], ⟨0 6 5 4 -3]]

Optimal tuning (POTE): ~25/21 = 300.0000 ¢, ~6/5 = 316.9247 ¢ (~100/99 = 16.9247 ¢)

Optimal ET sequence: 68, 72, 140, 212, 284, 496ce, 780ccdee

Badness (Smith): 0.023406

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 1375/1372

Mapping: [⟨4 0 4 7 17 0], ⟨0 6 5 4 -3 14]]

Optimal tuning (POTE): ~25/21 = 300.0000 ¢, ~6/5 = 316.9887 ¢ (~100/99 = 16.9887 ¢)

Optimal ET sequence: 68, 72, 140, 212

Badness (Smith): 0.018731

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 385/384, 442/441, 625/624

Mapping: [⟨4 0 4 7 17 0 10], ⟨0 6 5 4 -3 14 6]]

Optimal tuning (POTE): ~25/21 = 300.0000 ¢, ~6/5 = 316.9846 ¢ (~100/99 = 16.9846 ¢)

Optimal ET sequence: 68, 72, 140, 212g

Badness (Smith): 0.012784

Kleiboh

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3125/3087

Mapping: [⟨1 6 6 6], ⟨0 -18 -15 -13]]

mapping generators: ~2, ~25/21

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~25/21 = 294.303 ¢

Optimal ET sequence: 49, 53, 314d

Badness (Smith): 0.076460

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 3125/3087

Mapping: [⟨1 6 6 6 14], ⟨0 -18 -15 -13 -43]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~25/21 = 294.181 ¢

Optimal ET sequence: 49, 53, 102d, 155d

Badness (Smith): 0.052805

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 275/273, 325/324, 540/539

Mapping: [⟨1 6 6 6 14 14], ⟨0 -18 -15 -13 -43 -42]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~13/11 = 294.187 ¢

Optimal ET sequence: 49f, 53, 102df, 155d

Badness (Smith): 0.031074

Marfifths

The marfifths temperament (19 & 140) tempers out the hemimage comma, 10976/10935. It splits the interval of a major thirteenth (~10/3) into three marvelous fifth (112/75) intervals, and uses it for a generator.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 15625/15552

Mapping: [⟨1 -6 -4 -17], ⟨0 18 15 47]]

mapping generators: ~2, ~75/5

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~75/56 = 505.705 ¢

Optimal ET sequence: 19, …, 121, 140, 579, 719, 859bcd, 999bcd, 1858bbccdd

Badness (Smith): 0.063448

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237, 10976/10935

Mapping: [⟨1 -6 -4 -17 22], ⟨0 18 15 47 -44]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~75/56 = 505.684 ¢

Optimal ET sequence: 19, 121e, 140, 159, 299

Badness (Smith): 0.058902

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 10976/10935

Mapping: [⟨1 -6 -4 -17 22 -14], ⟨0 18 15 47 -44 42]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~75/56 = 505.686 ¢

Optimal ET sequence: 19, 121e, 140, 159, 299

Badness (Smith): 0.030082

Diatessic

The diatessic temperament (121 & 140) is closely related to the diatess tuning (generator: 505.727281 cents).

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2200/2187, 5632/5625

Mapping: [⟨1 -6 -4 -17 -37], ⟨0 18 15 47 96]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~75/56 = 505.740 ¢

Optimal ET sequence: 19e, …, 121, 140, 261, 401

Badness (Smith): 0.061172

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 625/624, 1375/1372

Mapping: [⟨1 -6 -4 -17 -37 -14], ⟨0 18 15 47 96 42]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~75/56 = 505.740 ¢

Optimal ET sequence: 19e, …, 121, 140, 261, 401

Badness (Smith): 0.028671

Marf

The marf temperament (19 & 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 15625/15552

Mapping: [⟨1 -6 -4 -17 14], ⟨0 18 15 47 -25]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~75/56 = 505.769 ¢

Optimal ET sequence: 19, 102d, 121

Badness (Smith): 0.075112

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 540/539, 625/624, 896/891

Mapping: [⟨1 -6 -4 -17 14 -14], ⟨0 18 15 47 -25 42]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~75/56 = 505.771 ¢

Optimal ET sequence: 19, 102df, 121

Badness (Smith): 0.038317

Marthirds

The marthirds temperament (19 & 193) tempers out the breeze comma (laquadru-atruyo comma), 2460375/2458624. It splits the interval of minor tenth (~12/5) into four marvelous major third (56/45) intervals, and uses it for a generator.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 2460375/2458624

Mapping: [⟨1 -6 -4 -19], ⟨0 24 20 69]]

mapping generators: ~2, ~56/45

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~56/45 = 379.252 ¢

Optimal ET sequence: 19, …, 193, 212, 617c, 829c

Badness (Smith): 0.104253

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 15625/15552, 19712/19683

Mapping: [⟨1 -6 -4 -19 -43], ⟨0 24 20 69 147]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~56/45 = 379.257 ¢

Optimal ET sequence: 19e, …, 193, 212, 405, 617c, 1022cce

Badness (Smith): 0.075624

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1375/1372, 19712/19683

Mapping: [⟨1 -6 -4 -19 -43 -14], ⟨0 24 20 69 147 56]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~56/45 = 379.256 ¢

Optimal ET sequence: 19e, …, 193, 212, 405f, 617cff

Badness (Smith): 0.043728

Quartkeenlig

Quartkeenlig uses a generator in the 11-limit that is 33/32~36/35 tempered together, and is called so because it tempers out the quartisma by virtue of five 33/32's being with 7/6, keenanisma, 385/384, tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). It can also be viewed as a regular temperament interpretation of stretched 23edo.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 117649/116640

Mapping: [⟨1 0 1 1], ⟨0 36 30 41]]

mapping generator: ~2, ~36/35

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~36/35 = 52.8562 ¢

Optimal ET sequence: 68, 91, 159, 386d, 545dd

Badness (Smith): 0.146

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237, 67228/66825

Mapping: [⟨1 0 1 1 5], ⟨0 36 30 41 -35]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~33/32 = 52.8524 ¢

Optimal ET sequence: 68, 91, 159, 386d, 545dd

Badness (Smith): 0.0865

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 16807/16731

Mapping: [⟨1 0 1 1 5 0], ⟨0 36 30 41 -35 84]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~33/32 = 52.8562 ¢

Optimal ET sequence: 68, 159, 386d, 545ddf

Badness (Smith): 0.0477

Novemkleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 40353607/40310784

Mapping: [⟨9 0 9 11], ⟨0 6 5 6]]

mapping generators: ~2592/2401, ~6/5

Optimal tuning (POTE): ~2592/2401 = 133.333 ¢, ~6/5 = 317.005 ¢ (~36/35 = 50.338 ¢)

Optimal ET sequence: 72, 261, 333, 405, 477c, 882c

Badness (Smith): 0.193429

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4000/3993, 15625/15552

Mapping: [⟨9 0 9 11 24], ⟨0 6 5 6 3]]

Optimal tuning (POTE): ~250/231 = 133.333 ¢, ~6/5 = 317.010 ¢ (~36/35 = 50.343 ¢)

Optimal ET sequence: 72, 261, 333, 405, 882c

Badness (Smith): 0.051730

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1375/1372, 4000/3993

Mapping: [⟨9 0 9 11 24 0], ⟨0 6 5 6 3 14]]

Optimal tuning (POTE): ~250/231 = 133.333 ¢, ~6/5 = 317.086 ¢ (~36/35 = 50.419 ¢)

Optimal ET sequence: 72, 189f, 261, 333, 738cf

Badness (Smith): 0.039072

Sqrtphi

Main article: Sqrtphi

The just value of sqrt (φ) is 416.545 cents.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 16875/16807

Mapping: [⟨1 12 11 16], ⟨0 -30 -25 -38]]

mapping generators: ~2, 125/98

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~125/98 = 416.603 ¢

Optimal ET sequence: 49, 72, 193, 265

Badness (Smith): 0.070378

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4375/4356

Mapping: [⟨1 12 11 16 17], ⟨0 -30 -25 -38 -39]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~14/11 = 416.604 ¢

Optimal ET sequence: 49, 72, 193, 265

Badness (Smith): 0.025515

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624, 1375/1372

Mapping: [⟨1 12 11 16 17 28], ⟨0 -30 -25 -38 -39 -70]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~14/11 = 416.585 ¢

Optimal ET sequence: 49f, 72, 121, 193

Badness (Smith): 0.020040

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 375/374, 540/539, 595/594

Mapping: [⟨1 12 11 16 17 28 27], ⟨0 -30 -25 -38 -39 -70 -66]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~14/11 = 416.585 ¢

Optimal ET sequence: 49fg, 72, 121, 193

Badness (Smith): 0.013028

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594

Mapping: [⟨1 12 11 16 17 28 27 -2], ⟨0 -30 -25 -38 -39 -70 -66 18]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~14/11 = 416.580 ¢

Optimal ET sequence: 49fg, 72, 121, 193

Badness (Smith): 0.014748

Subgroup extensions

Kleismic (2.3.5.13) a.k.a. cata

Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as 26/15. Notice 15625/15552 = (325/324)(625/624) and 325/324 = (625/624)(676/675). The S-expression-based comma list of the temperament is {S10/S12 = S25*S26, (S25,) S13/S15 = S26}. For the high-limit version of cata with a 1\5 period, see thunderclysmic.

Subgroup: 2.3.5.13

Comma list: 325/324, 625/624

Subgroup-val mapping: [⟨1 0 1 0], ⟨0 6 5 14]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~6/5 = 317.1110 ¢
  • POTE: ~2 = 1200.0000 ¢, ~6/5 = 317.0756 ¢

Optimal ET sequence: 15, 19, 34, 53, 140, 193, 246

Badness (Sintel): 0.131

2.3.5.13.37.41 subgroup

Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy.

Subgroup: 2.3.5.13.37.41

Comma list: 325/324, 625/624, 481/480, 1600/1599

Subgroup-val mapping: [⟨1 0 1 0 6 8], ⟨0 6 5 14 -3 -10]]

Optimal tunings:

  • WE: ~2 = 1200.165 ¢, ~6/5 = 317.113 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 317.075 ¢

Badness (Sintel): 0.223

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