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Kleismic family: Difference between revisions

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{{Mapping|legend=1| 1 0 1 | 0 6 5 }}
{{Mapping|legend=1| 1 0 1 | 0 6 5 }}
: mapping generators: ~2, ~6/5
: mapping generators: ~2, ~6/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~6/5 = 317.0308
* CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0308{{c}}
* POTE: ~2 = 1\1, ~6/5 = 317.007
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.007{{c}}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* [[5-odd-limit]] [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* 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, …, 882c }}


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


=== Overview to extensions ===
=== Overview to extensions ===
Line 47: Line 46:
{{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 = 1\1, ~6/5 = 316.473
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.473{{c}}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 56: Line 55:
{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd, 91dd }}
{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd, 91dd }}


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


=== 11-limit ===
=== 11-limit ===
Line 65: Line 64:
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 = 1\1, ~6/5 = 317.576
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.576{{c}}


Tuning ranges:  
Tuning ranges:  
Line 71: Line 70:
* 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=1| 4, 15, 19, 34 }}
{{Optimal ET sequence|legend=0| 4, 15, 19, 34 }}


Badness: 0.027410
Badness (Smith): 0.027410


==== 13-limit ====
==== 13-limit ====
Line 82: Line 81:
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 = 1\1, ~6/5 = 316.611
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.611{{c}}


Tuning ranges:  
Tuning ranges:  
Line 88: Line 87:
* 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=1| 4, 15f, 19, 53def, 72def }}
{{Optimal ET sequence|legend=0| 4, 15f, 19, 53def, 72def }}


Badness: 0.029749
Badness (Smith): 0.029749


==== Kema ====
==== Kema ====
Line 99: Line 98:
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 = 1\1, ~6/5 = 317.423
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.423{{c}}


Tuning ranges:  
Tuning ranges:  
Line 106: Line 105:
* 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=1| 15, 19, 34, 87ddee }}
{{Optimal ET sequence|legend=0| 15, 19, 34, 87ddee }}


Badness: 0.022749
Badness (Smith): 0.022749


==== Kumbaya ====
==== Kumbaya ====
Line 117: Line 116:
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 = 1\1, ~6/5 = 318.595
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 318.595{{c}}


{{Optimal ET sequence|legend=1| 4, 15, 19f, 34ff }}
{{Optimal ET sequence|legend=0| 4, 15, 19f, 34ff }}


Badness: 0.031633
Badness (Smith): 0.031633


=== Qeema ===
=== Qeema ===
Line 130: Line 129:
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 = 1\1, ~6/5 = 314.730
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 314.730{{c}}


{{Optimal ET sequence|legend=1| 4e, 19, 42bcd, 61bcdd }}
{{Optimal ET sequence|legend=0| 4e, 19, 42bcd, 61bcdd }}


Badness: 0.040056
Badness (Smith): 0.040056


==== 13-limit ====
==== 13-limit ====
Line 143: Line 142:
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 = 1\1, ~6/5 = 315.044
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.044{{c}}


{{Optimal ET sequence|legend=1| 4ef, 19 }}
{{Optimal ET sequence|legend=0| 4ef, 19 }}


Badness: 0.029419
Badness (Smith): 0.029419


=== Darjeeling ===
=== Darjeeling ===
Line 156: Line 155:
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 = 1\1, ~6/5 = 317.656
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.656{{c}}


{{Optimal ET sequence|legend=1| 15, 19e, 34e }}
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}


Badness: 0.027648
Badness (Smith): 0.027648


==== 13-limit ====
==== 13-limit ====
Line 169: Line 168:
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 = 1\1, ~6/5 = 317.298
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.298{{c}}


{{Optimal ET sequence|legend=1| 15, 19e, 34e, 53dee }}
{{Optimal ET sequence|legend=0| 15, 19e, 34e, 53dee }}


Badness: 0.021445
Badness (Smith): 0.021445


== Catalan ==
== Catalan ==
Line 182: Line 181:
{{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 = 1\1, ~6/5 = 318.267
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 318.267{{c}}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 190: Line 189:
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}


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


=== 11-limit ===
=== 11-limit ===
Line 199: Line 198:
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 = 1\1, ~6/5 = 318.282
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 318.282{{c}}


Tuning ranges:  
Tuning ranges:  
Line 205: Line 204:
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]


{{Optimal ET sequence|legend=1| 15, 34d, 49, 181bbcdddeee }}
{{Optimal ET sequence|legend=0| 15, 34d, 49, 181bbcdddeee }}


Badness: 0.036894
Badness (Smith): 0.036894


=== 13-limit ===
=== 13-limit ===
Line 216: Line 215:
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 = 1\1, ~6/5 = 317.9159
Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 317.9159{{c}}


{{Optimal ET sequence|legend=1| 15, 34d, 49f, 83def, 132bcddeefff }}
{{Optimal ET sequence|legend=0| 15, 34d, 49f, 83def, 132bcddeefff }}


Badness: 0.0263
Badness (Smith): 0.0263


== Catakleismic ==
== Catakleismic ==
Line 232: Line 231:
{{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 = 1\1, ~6/5 = 316.732
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.732{{c}}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 240: Line 239:
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}


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


==== 2.3.5.7.13 subgroup ====
==== 2.3.5.7.13 subgroup ====
The [[S-expression]]-based comma list of this temperament is {[[169/168|S13]], [[225/224|S15 = S25*S26*S27]], [[325/324|S10/S12 = S25*S26]](, [[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}.  
The [[S-expression]]-based comma list of this temperament is {[[169/168|S13]], [[225/224|S15 = S25*S26*S27]], [[325/324|S10/S12 = S25*S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}.  


Subgroup: 2.3.5.7.13
Subgroup: 2.3.5.7.13
Line 249: Line 248:
Comma list: 169/168, 225/224, 325/324
Comma list: 169/168, 225/224, 325/324


Sval 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 = 1\1, ~6/5 = 316.8865
Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 316.8865{{c}}


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 125f, 197f }}
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197f }}


Badness: 0.0118
Badness (Smith): 0.0118


=== 11-limit ===
=== 11-limit ===
Line 264: Line 263:
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 = 1\1, ~6/5 = 316.719
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.719{{c}}


Tuning ranges:  
Tuning ranges:  
Line 270: Line 269:
* 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=1| 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee }}
{{Optimal ET sequence|legend=0| 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee }}


Badness: 0.021849
Badness (Smith): 0.021849


==== 13-limit ====
==== 13-limit ====
Line 281: Line 280:
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 = 1\1, ~6/5 = 316.738
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.738{{c}}


Tuning ranges:  
Tuning ranges:  
Line 287: Line 286:
* 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=1| 19, 34de, 53, 72, 125f, 197ef, 269ceff }}
{{Optimal ET sequence|legend=0| 19, 34de, 53, 72, 125f, 197ef, 269ceff }}


Badness: 0.016883
Badness (Smith): 0.016883


=== Cataclysmic ===
=== Cataclysmic ===
Line 298: Line 297:
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 = 1\1, ~6/5 = 317.042
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.042{{c}}


{{Optimal ET sequence|legend=1| 19e, 34d, 53 }}
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}


Badness: 0.039954
Badness (Smith): 0.039954


==== 13-limit ====
==== 13-limit ====
Line 311: Line 310:
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 = 1\1, ~6/5 = 317.036
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.036{{c}}


{{Optimal ET sequence|legend=1| 19e, 34d, 53 }}
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}


Badness: 0.022555
Badness (Smith): 0.022555


=== Catalytic ===
=== Catalytic ===
Line 324: Line 323:
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 = 1\1, ~6/5 = 316.653
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.653{{c}}


{{Optimal ET sequence|legend=1| 19e, 53e, 72 }}
{{Optimal ET sequence|legend=0| 19e, 53e, 72 }}


Badness: 0.030422
Badness (Smith): 0.030422


==== 13-limit ====
==== 13-limit ====
Line 337: Line 336:
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 = 1\1, ~6/5 = 316.639
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.639{{c}}


{{Optimal ET sequence|legend=1| 19e, 53e, 72 }}
{{Optimal ET sequence|legend=0| 19e, 53e, 72 }}


Badness: 0.022337
Badness (Smith): 0.022337


=== Cataleptic ===
=== Cataleptic ===
Line 350: Line 349:
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 = 1\1, ~6/5 = 317.083
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.083{{c}}


{{Optimal ET sequence|legend=1| 19, 34d, 53e }}
{{Optimal ET sequence|legend=0| 19, 34d, 53e }}


Badness: 0.044335
Badness (Smith): 0.044335


==== 13-limit ====
==== 13-limit ====
Line 363: Line 362:
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 = 1\1, ~6/5 = 317.118
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.118{{c}}


{{Optimal ET sequence|legend=1| 19, 34d, 53e, 87dee }}
{{Optimal ET sequence|legend=0| 19, 34d, 53e, 87dee }}


Badness: 0.027343
Badness (Smith): 0.027343


=== Bikleismic ===
=== Bikleismic ===
Line 375: Line 374:


Mapping: {{mapping| 2 0 2 -6 -1 | 0 6 5 22 15 }}
Mapping: {{mapping| 2 0 2 -6 -1 | 0 6 5 22 15 }}
: mapping generators: ~99/70, ~6/5
: mapping generators: ~99/70, ~6/5


Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 316.721
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 316.721{{c}}


{{Optimal ET sequence|legend=1| 34d, 72, 322c, …, 610bcc }}
{{Optimal ET sequence|legend=0| 34d, 72, 322c, …, 610bcc }}


Badness: 0.029319
Badness (Smith): 0.029319


==== 13-limit ====
==== 13-limit ====
Line 391: Line 389:
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 = 1\2, ~6/5 = 316.726
Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~6/5 = 316.726{{c}}


{{Optimal ET sequence|legend=1| 34d, 72 }}
{{Optimal ET sequence|legend=0| 34d, 72 }}


Badness: 0.021814
Badness (Smith): 0.021814


==== 17-limit ====
==== 17-limit ====
Line 404: Line 402:
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 = 1\2, ~6/5 = 316.726
Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~6/5 = 316.726{{c}}


{{Optimal ET sequence|legend=1| 34d, 38df, 72 }}
{{Optimal ET sequence|legend=0| 34d, 38df, 72 }}


Badness: 0.015656
Badness (Smith): 0.015656


==== 19-limit ====
==== 19-limit ====
Line 417: Line 415:
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 = 1\2, ~6/5 = 316.726
Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~6/5 = 316.726{{c}}


{{Optimal ET sequence|legend=1| 34dh, 38df, 72 }}
{{Optimal ET sequence|legend=0| 34dh, 38df, 72 }}


Badness: 0.015771
Badness (Smith): 0.015771


== Countercata ==
== Countercata ==
Line 430: Line 428:
{{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 = 1\1, ~6/5 = 317.121
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 317.121{{c}}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 438: Line 436:
{{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, 806b }}


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


=== 11-limit ===
=== 11-limit ===
Line 447: Line 445:
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 = 1\1, ~6/5 = 317.162
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.162{{c}}


Tuning ranges:  
Tuning ranges:  
Line 453: Line 451:
* 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=1| 34, 53, 87, 140, 227, 367e, 507e }}
{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 227, 367e, 507e }}


Badness: 0.039770
Badness (Smith): 0.039770


=== 13-limit ===
=== 13-limit ===
Line 464: Line 462:
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 = 1\1, ~6/5 = 317.162
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.162{{c}}


Tuning ranges:  
Tuning ranges:  
Line 471: Line 469:
* 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=1| 34, 53, 87, 140, 367e, 507e }}
{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 367e, 507e }}


Badness: 0.020156
Badness (Smith): 0.020156


== Metakleismic ==
== Metakleismic ==
Line 482: Line 480:
{{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 = 1\1, ~6/5 = 317.314
[[Optimal tuning]] ([[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 }}


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


=== 11-limit ===
=== 11-limit ===
Line 495: Line 493:
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 = 1\1, ~6/5 = 317.311
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.311{{c}}


{{Optimal ET sequence|legend=1| 34d, 53d, 87, 121, 208 }}
{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}


Badness: 0.048570
Badness (Smith): 0.048570


=== 13-limit ===
=== 13-limit ===
Line 508: Line 506:
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 = 1\1, ~6/5 = 317.311
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.311{{c}}


{{Optimal ET sequence|legend=1| 34d, 53d, 87, 121, 208 }}
{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}


Badness: 0.024371
Badness (Smith): 0.024371


== Hemikleismic ==
== Hemikleismic ==
Line 521: Line 519:
{{Mapping|legend=1| 1 0 1 4 | 0 12 10 -9 }}
{{Mapping|legend=1| 1 0 1 4 | 0 12 10 -9 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/32 = 158.649
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~35/32 = 158.649{{c}}


{{Optimal ET sequence|legend=1| 15, 38, 53, 121 }}
{{Optimal ET sequence|legend=1| 15, 38, 53, 121 }}


[[Badness]]: 0.052054
[[Badness]] (Smith): 0.052054


=== 11-limit ===
=== 11-limit ===
Line 534: Line 532:
Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}
Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 158.677
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 158.677{{c}}


{{Optimal ET sequence|legend=1| 15, 38, 53, 68, 121e }}
{{Optimal ET sequence|legend=0| 15, 38, 53, 68, 121e }}


Badness: 0.038023
Badness (Smith): 0.038023


=== 13-limit ===
=== 13-limit ===
Line 547: Line 545:
Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}
Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 158.655
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 158.655{{c}}


{{Optimal ET sequence|legend=1| 15, 38f, 53, 121e }}
{{Optimal ET sequence|legend=0| 15, 38f, 53, 121e }}


Badness: 0.026005
Badness (Smith): 0.026005


== Clyde ==
== Clyde ==
Line 559: Line 557:


{{Mapping|legend=1| 1 6 6 12 | 0 -12 -10 -25 }}
{{Mapping|legend=1| 1 6 6 12 | 0 -12 -10 -25 }}
: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 441.335
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~9/7 = 441.335{{c}}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 12/25 0 0 -1/25 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 12/25 0 0 -1/25 }}
: {{monzo list| 1 0 0 0 | 6/25 0 0 12/25 | 6/5 0 0 2/5 | 0 0 0 1 }}
: {{monzo list| 1 0 0 0 | 6/25 0 0 12/25 | 6/5 0 0 2/5 | 0 0 0 1 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7


[[Algebraic generator]]: real root of 5''x''<sup>3</sup> - 6''x'' - 3, the Poussami generator. Approximately 441.309 [[cent]]s. Associated recurrence relationship quickly converges.
[[Algebraic generator]]: real root of 5''x''<sup>3</sup> - 6''x'' - 3, the Poussami generator. Approximately 441.309 [[cent]]s. Associated recurrence relationship quickly converges.
Line 573: Line 570:
{{Optimal ET sequence|legend=1| 19, 49, 68, 87, 155 }}
{{Optimal ET sequence|legend=1| 19, 49, 68, 87, 155 }}


[[Badness]]: 0.047261
[[Badness]] (Smith): 0.047261


=== 11-limit ===
=== 11-limit ===
Line 582: Line 579:
Mapping: {{mapping| 1 6 6 12 -5 | 0 -12 -10 -25 23 }}
Mapping: {{mapping| 1 6 6 12 -5 | 0 -12 -10 -25 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.355
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 441.355{{c}}


{{Optimal ET sequence|legend=1| 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd }}
{{Optimal ET sequence|legend=0| 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd }}


Badness: 0.047417
Badness (Smith): 0.047417


=== 13-limit ===
=== 13-limit ===
Line 595: Line 592:
Mapping: {{mapping| 1 6 6 12 -5 14 | 0 -12 -10 -25 23 -28 }}
Mapping: {{mapping| 1 6 6 12 -5 14 | 0 -12 -10 -25 23 -28 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.363
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 441.363{{c}}


{{Optimal ET sequence|legend=1| 19, 49ef, 68, 87, 503bdf, 590bdf }}
{{Optimal ET sequence|legend=0| 19, 49ef, 68, 87, 503bdf, 590bdf }}


Badness: 0.026842
Badness (Smith): 0.026842


== Tritikleismic ==
== Tritikleismic ==
Line 607: Line 604:


{{Mapping|legend=1| 3 0 3 10 | 0 6 5 -2 }}
{{Mapping|legend=1| 3 0 3 10 | 0 6 5 -2 }}
: mapping generators: ~63/50, ~6/5
: mapping generators: ~63/50, ~6/5


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~6/5 = 316.872 (~21/20 = 83.128)
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~6/5 = 316.872{{c}} (~21/20 = 83.128{{c}})


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~6/5 = {{monzo| 1/3 0 1/7 -1/7 }}
* [[7-odd-limit]]: ~6/5 = {{monzo| 1/3 0 1/7 -1/7 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 0 6/7 -6/7 }}, {{monzo| 8/3 0 5/7 -5/7 }}, {{monzo| 8/3 0 -2/7 2/7 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 0 6/7 -6/7 }}, {{monzo| 8/3 0 5/7 -5/7 }}, {{monzo| 8/3 0 -2/7 2/7 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~6/5 = {{monzo| 5/21 1/7 0 -1/14 }}
* [[9-odd-limit]]: ~6/5 = {{monzo| 5/21 1/7 0 -1/14 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 46/21 5/7 0 -5/14 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 46/21 5/7 0 -5/14 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 15, 42bc, 57, 72, 159, 231 }}
{{Optimal ET sequence|legend=1| 15, 42bc, 57, 72, 159, 231 }}


[[Badness]]: 0.056337
[[Badness]] (Smith): 0.056337


Music:  
Music:  
* ''[https://www.youtube.com/watch?v=vdjhC9i5KF4 Four Short Experiments in Octave Stretched 42edo (Dec 2024)]'' - [[Budjarn Lambeth]]
* ''[https://www.youtube.com/watch?v=vdjhC9i5KF4 Four Short Experiments in Octave Stretched 42edo (Dec 2024)]'' by [[Budjarn Lambeth]]


=== 11-limit ===
=== 11-limit ===
Line 634: Line 630:
Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}
Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~6/5 = 316.881 (~21/20 = 83.119)
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~6/5 = 316.881{{c}} (~21/20 = 83.119{{c}})


Minimax tuning:
Minimax tuning:
Line 641: Line 637:
: unchanged-interval (eigenmonzo) basis: 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7


{{Optimal ET sequence|legend=1| 15, 42bc, 57, 72, 159, 231 }}
{{Optimal ET sequence|legend=0| 15, 42bc, 57, 72, 159, 231 }}


Badness: 0.019333
Badness (Smith): 0.019333


=== 13-limit ===
=== 13-limit ===
Line 652: Line 648:
Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}
Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~6/5 = 316.9585 (~21/20 = 83.0415)
Optimal tuning (POTE): ~44/35 = 400.0000{{c}}, ~6/5 = 316.9585{{c}} (~21/20 = 83.0415{{c}})


{{Optimal ET sequence|legend=1| 72, 87, 159 }}
{{Optimal ET sequence|legend=0| 72, 87, 159 }}


Badness: 0.015652
Badness (Smith): 0.015652


=== 17-limit ===
=== 17-limit ===
Line 665: Line 661:
Mapping: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}
Mapping: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}


Optimal tuning (POTE): ~34/27 = 1\3, ~6/5 = 316.9082 (~21/20 = 83.0918)
Optimal tuning (POTE): ~34/27 = 400.0000{{c}}, ~6/5 = 316.9082{{c}} (~21/20 = 83.0918{{c}})


{{Optimal ET sequence|legend=1| 72, 159, 231f }}
{{Optimal ET sequence|legend=0| 72, 159, 231f }}


Badness: 0.013551
Badness (Smith): 0.013551


== Quadritikleismic ==
== Quadritikleismic ==
Line 677: Line 673:


{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
: mapping generators: ~25/21, ~6/5
: mapping generators: ~25/21, ~6/5


[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~6/5 = 316.9999 (~126/125 = 16.9999)
[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.0000{{c}}, ~6/5 = 316.9999{{c}} (~126/125 = 16.9999{{c}})


{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}


[[Badness]]: 0.039231
[[Badness]] (Smith): 0.039231


=== 11-limit ===
=== 11-limit ===
Line 693: Line 688:
Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9247 (~100/99 = 16.9247)
Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~6/5 = 316.9247{{c}} (~100/99 = 16.9247{{c}})


{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}


Badness: 0.023406
Badness (Smith): 0.023406


=== 13-limit ===
=== 13-limit ===
Line 706: Line 701:
Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9887 (~100/99 = 16.9887)
Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~6/5 = 316.9887{{c}} (~100/99 = 16.9887{{c}})


{{Optimal ET sequence|legend=1| 68, 72, 140, 212 }}
{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}


Badness: 0.018731
Badness (Smith): 0.018731


=== 17-limit ===
=== 17-limit ===
Line 719: Line 714:
Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}
Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9846 (~100/99 = 16.9846)
Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~6/5 = 316.9846{{c}} (~100/99 = 16.9846{{c}})


{{Optimal ET sequence|legend=1| 68, 72, 140, 212g }}
{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}


Badness: 0.012784
Badness (Smith): 0.012784


== Kleiboh ==
== Kleiboh ==
Line 731: Line 726:


{{Mapping|legend=1| 1 6 6 6 | 0 -18 -15 -13 }}
{{Mapping|legend=1| 1 6 6 6 | 0 -18 -15 -13 }}
: mapping generators: ~2, ~25/21
: mapping generators: ~2, ~25/21


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 294.303
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25/21 = 294.303{{c}}


{{Optimal ET sequence|legend=1| 49, 53, 314d }}
{{Optimal ET sequence|legend=1| 49, 53, 314d }}


[[Badness]]: 0.076460
[[Badness]] (Smith): 0.076460


=== 11-limit ===
=== 11-limit ===
Line 747: Line 741:
Mapping: {{mapping| 1 6 6 6 14 | 0 -18 -15 -13 -43 }}
Mapping: {{mapping| 1 6 6 6 14 | 0 -18 -15 -13 -43 }}


Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 294.181
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~25/21 = 294.181{{c}}


{{Optimal ET sequence|legend=1| 49, 53, 102d, 155d }}
{{Optimal ET sequence|legend=0| 49, 53, 102d, 155d }}


Badness: 0.052805
Badness (Smith): 0.052805


=== 13-limit ===
=== 13-limit ===
Line 760: Line 754:
Mapping: {{mapping| 1 6 6 6 14 14 | 0 -18 -15 -13 -43 -42 }}
Mapping: {{mapping| 1 6 6 6 14 14 | 0 -18 -15 -13 -43 -42 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 294.187
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/11 = 294.187{{c}}


{{Optimal ET sequence|legend=1| 49f, 53, 102df, 155d }}
{{Optimal ET sequence|legend=0| 49f, 53, 102df, 155d }}


Badness: 0.031074
Badness (Smith): 0.031074


== Marfifths ==
== Marfifths ==
The ''marfifths'' temperament (19&amp;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.
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
[[Subgroup]]: 2.3.5.7
Line 774: Line 768:


{{Mapping|legend=1| 1 -6 -4 -17 | 0 18 15 47 }}
{{Mapping|legend=1| 1 -6 -4 -17 | 0 18 15 47 }}
: mapping generators: ~2, ~75/5
: mapping generators: ~2, ~75/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/56 = 505.705
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~75/56 = 505.705{{c}}


{{Optimal ET sequence|legend=1| 19, …, 121, 140, 579, 719, 859bcd, 999bcd, 1858bbccdd }}
{{Optimal ET sequence|legend=1| 19, …, 121, 140, 579, 719, 859bcd, 999bcd, 1858bbccdd }}


[[Badness]]: 0.063448
[[Badness]] (Smith): 0.063448


=== 11-limit ===
=== 11-limit ===
Line 790: Line 783:
Mapping: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}
Mapping: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.684
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.684{{c}}


{{Optimal ET sequence|legend=1| 19, 121e, 140, 159, 299 }}
{{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}


Badness: 0.058902
Badness (Smith): 0.058902


==== 13-limit ====
==== 13-limit ====
Line 803: Line 796:
Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}
Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.686
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.686{{c}}


{{Optimal ET sequence|legend=1| 19, 121e, 140, 159, 299 }}
{{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}


Badness: 0.030082
Badness (Smith): 0.030082


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 818: Line 811:
Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}
Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.740
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.740{{c}}


{{Optimal ET sequence|legend=1| 19e, …, 121, 140, 261, 401 }}
{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}


Badness: 0.061172
Badness (Smith): 0.061172


==== 13-limit ====
==== 13-limit ====
Line 831: Line 824:
Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}
Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.740
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.740{{c}}


{{Optimal ET sequence|legend=1| 19e, …, 121, 140, 261, 401 }}
{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}


Badness: 0.028671
Badness (Smith): 0.028671


=== Marf ===
=== Marf ===
The ''marf'' temperament (19 &amp; 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.
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
Subgroup: 2.3.5.7.11
Line 846: Line 839:
Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}
Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.769
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.769{{c}}


{{Optimal ET sequence|legend=1| 19, 102d, 121 }}
{{Optimal ET sequence|legend=0| 19, 102d, 121 }}


Badness: 0.075112
Badness (Smith): 0.075112


==== 13-limit ====
==== 13-limit ====
Line 859: Line 852:
Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}
Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.771
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.771{{c}}


{{Optimal ET sequence|legend=1| 19, 102df, 121 }}
{{Optimal ET sequence|legend=0| 19, 102df, 121 }}


Badness: 0.038317
Badness (Smith): 0.038317


== Marthirds ==
== Marthirds ==
The ''marthirds'' temperament (19 &amp; 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.
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
[[Subgroup]]: 2.3.5.7
Line 873: Line 866:


{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
: mapping generators: ~2, ~56/45
: mapping generators: ~2, ~56/45


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 379.252
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~56/45 = 379.252{{c}}


{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}
{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}


[[Badness]]: 0.104253
[[Badness]] (Smith): 0.104253


=== 11-limit ===
=== 11-limit ===
Line 889: Line 881:
Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.257
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~56/45 = 379.257{{c}}


{{Optimal ET sequence|legend=1| 19e, …, 193, 212, 405, 617c, 1022cce }}
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c, 1022cce }}


Badness: 0.075624
Badness (Smith): 0.075624


=== 13-limit ===
=== 13-limit ===
Line 902: Line 894:
Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.256
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~56/45 = 379.256{{c}}


{{Optimal ET sequence|legend=1| 19e, …, 193, 212, 405f, 617cff }}
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}


Badness: 0.043728
Badness (Smith): 0.043728


== Quartkeenlig ==
== Quartkeenlig ==
Line 916: Line 908:


{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
: mapping generator: ~2, ~36/35
: mapping generator: ~2, ~36/35


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~36/35 = 52.8562
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~36/35 = 52.8562{{c}}


{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}


[[Badness]]: 0.146
[[Badness]] (Smith): 0.146


=== 11-limit ===
=== 11-limit ===
Line 932: Line 923:
Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}


Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 52.8524
Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 52.8524{{c}}


{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}


Badness: 0.0865
Badness (Smith): 0.0865


=== 13-limit ===
=== 13-limit ===
Line 945: Line 936:
Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}


Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 52.8562
Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 52.8562{{c}}


{{Optimal ET sequence|legend=1| 68, 159, 386d, 545ddf }}
{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}


Badness: 0.0477
Badness (Smith): 0.0477


== Novemkleismic ==
== Novemkleismic ==
Line 957: Line 948:


{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
: mapping generators: ~2592/2401, ~6/5
: mapping generators: ~2592/2401, ~6/5


[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 1\9, ~6/5 = 317.005 (~36/35 = 50.338)
[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 133.333{{c}}, ~6/5 = 317.005{{c}} (~36/35 = 50.338{{c}})


{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}


[[Badness]]: 0.193429
[[Badness]] (Smith): 0.193429


=== 11-limit ===
=== 11-limit ===
Line 973: Line 963:
Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}


Optimal tuning (POTE): ~250/231 = 1\9, ~6/5 = 317.010 (~36/35 = 50.343)
Optimal tuning (POTE): ~250/231 = 133.333{{c}}, ~6/5 = 317.010{{c}} (~36/35 = 50.343{{c}})


{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 882c }}
{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}


Badness: 0.051730
Badness (Smith): 0.051730


=== 13-limit ===
=== 13-limit ===
Line 986: Line 976:
Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}


Optimal tuning (POTE): ~250/231 = 1\9, ~6/5 = 317.086 (~36/35 = 50.419)
Optimal tuning (POTE): ~250/231 = 133.333{{c}}, ~6/5 = 317.086{{c}} (~36/35 = 50.419{{c}})


{{Optimal ET sequence|legend=1| 72, 189f, 261, 333, 738cf }}
{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}


Badness: 0.039072
Badness (Smith): 0.039072


== Sqrtphi ==
== Sqrtphi ==
Line 1,002: Line 992:


{{Mapping|legend=1| 1 12 11 16 | 0 -30 -25 -38 }}
{{Mapping|legend=1| 1 12 11 16 | 0 -30 -25 -38 }}
: mapping generators: ~2, 125/98
: mapping generators: ~2, 125/98


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/98 = 416.603
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~125/98 = 416.603{{c}}


{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}
{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}


[[Badness]]: 0.070378
[[Badness]] (Smith): 0.070378


=== 11-limit ===
=== 11-limit ===
Line 1,018: Line 1,007:
Mapping: {{mapping| 1 12 11 16 17 | 0 -30 -25 -38 -39 }}
Mapping: {{mapping| 1 12 11 16 17 | 0 -30 -25 -38 -39 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.604
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 416.604{{c}}


{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}
{{Optimal ET sequence|legend=0| 49, 72, 193, 265 }}


Badness: 0.025515
Badness (Smith): 0.025515


=== 13-limit ===
=== 13-limit ===
Line 1,031: Line 1,020:
Mapping: {{mapping| 1 12 11 16 17 28 | 0 -30 -25 -38 -39 -70 }}
Mapping: {{mapping| 1 12 11 16 17 28 | 0 -30 -25 -38 -39 -70 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 416.585{{c}}


{{Optimal ET sequence|legend=1| 49f, 72, 121, 193 }}
{{Optimal ET sequence|legend=0| 49f, 72, 121, 193 }}


Badness: 0.020040
Badness (Smith): 0.020040


=== 17-limit ===
=== 17-limit ===
Line 1,044: Line 1,033:
Mapping: {{mapping| 1 12 11 16 17 28 27 | 0 -30 -25 -38 -39 -70 -66 }}
Mapping: {{mapping| 1 12 11 16 17 28 27 | 0 -30 -25 -38 -39 -70 -66 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 416.585{{c}}


{{Optimal ET sequence|legend=1| 49fg, 72, 121, 193 }}
{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}


Badness: 0.013028
Badness (Smith): 0.013028


=== 19-limit ===
=== 19-limit ===
Line 1,057: Line 1,046:
Mapping: {{mapping| 1 12 11 16 17 28 27 -2 | 0 -30 -25 -38 -39 -70 -66 18 }}
Mapping: {{mapping| 1 12 11 16 17 28 27 -2 | 0 -30 -25 -38 -39 -70 -66 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.580
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 416.580{{c}}


{{Optimal ET sequence|legend=1| 49fg, 72, 121, 193 }}
{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}


Badness: 0.014748
Badness (Smith): 0.014748


; Scales
; Scales
Line 1,080: Line 1,069:
Comma list: 325/324, 625/624
Comma list: 325/324, 625/624


Sval mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 317.1110
* CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1110{{c}}
* POTE: ~2 = 1\1, ~6/5 = 317.0756
* POTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0756{{c}}


{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 140, 193, 246 }}
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 140, 193, 246 }}
Line 1,097: Line 1,086:
Comma list: 325/324, 625/624, [[481/480]], [[1600/1599]]
Comma list: 325/324, 625/624, [[481/480]], [[1600/1599]]


[[Mapping]]: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}
Subgroup-val mapping: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1200.165, ~6/5 = 317.113
* WE: ~2 = 1200.165{{c}}, ~6/5 = 317.113{{c}}
* CWE: 2, ~6/5 = 317.075
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 317.075{{c}}


Badness (Sintel): 0.223
Badness (Sintel): 0.223
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