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: mapping generators: ~2, ~5/4
: mapping generators: ~2, ~5/4


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 348.594
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1\1, ~5/4 = 348.594


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
* [[5-odd-limit]] [[diamond monotone]]: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [315.641, 386.314] (full comma to untempered)
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [315.641, 386.314] (full comma to untempered)
* 5-odd-limit diamond monotone and tradeoff: ~5/4 = [315.641, 386.314]


{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}
{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}
Line 29: Line 29:


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
The 2.3.5.11-subgroup extension is related to [[#Septimal dicot|septimal dicot]], [[#Sharp|sharp]], and [[#Dichotic|dichotic]].


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 25/24, 45/44
Comma list: 25/24, 45/44


[[Gencom]]: [2 5/4; 25/24 45/44]
Sval mapping: {{mapping| 1 1 2 2 | 0 2 1 5 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 2 0 2}}, {{val|0 2 1 0 5}}]
Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 2}}, {{val|0 2 1 5}}]
: gencom: [2 5/4; 25/24 45/44]


[[Tp tuning|POL2 generator]]: ~5/4 = 346.734
Optimal tunings:  
* POTE: ~2 = 1\1, ~5/4 = 346.734


{{Optimal ET sequence|legend=1| 3e, 4e, 7, 24c, 31c, 38cc, 45cce }}
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c, 38cc, 45cce }}


[[Tp tuning #T2 tuning|RMS error]]: 5.621 cents
RMS error: 5.621 cents


Related temperaments: [[Dicot family #Septimal dicot|dicot]], [[Dicot family #Sharp|sharp]], [[Dicot family #Dichotic|dichotic]]
==== 2.3.5.11.13 subgroup ====
 
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.11.13
Subgroup: 2.3.5.11.13


[[Comma list]]: 25/24, 40/39, 45/44
Comma list: 25/24, 40/39, 45/44


[[Gencom]]: [2 5/4; 25/24 40/39 45/44]
Gencom mapping]]: {{mapping| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 2 0 2 4}}, {{val|0 2 1 0 5 -1}}]
: gencom: [2 5/4; 25/24 40/39 45/44]


[[Mapping|Sval mapping]]: [{{val|1 1 2 2 4}}, {{val|0 2 1 5 -1}}]
Sval mapping: {{mapping| 1 1 2 2 4 | 0 2 1 5 -1 }}


[[Tp tuning|POL2 generator]]: ~5/4 = 350.526
Optimal tunings:  
* POTE: ~2 = 1\1, ~5/4 = 350.526


{{Optimal ET sequence|legend=1| 3e, 7, 17, 24c }}
{{Optimal ET sequence|legend=0| 3e, 7, 17, 24c }}


[[Tp tuning #T2 tuning|RMS error]]: 5.916 cents
RMS error: 5.916 cents


== Septimal dicot ==
== Septimal dicot ==
Line 74: Line 75:
{{Multival|legend=1| 2 1 3 -3 -1 4 }}
{{Multival|legend=1| 2 1 3 -3 -1 4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 336.381
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1\1, ~5/4 = 336.381


{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
Line 87: Line 89:
Mapping: {{mapping| 1 1 2 2 2 | 0 2 1 3 5 }}
Mapping: {{mapping| 1 1 2 2 2 | 0 2 1 3 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 342.125
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 342.125


{{Optimal ET sequence|legend=1| 3de, 4e, 7 }}
{{Optimal ET sequence|legend=0| 3de, 4e, 7 }}


Badness: 0.019854
Badness: 0.019854
Line 100: Line 103:
Mapping: {{mapping| 1 1 2 2 4 | 0 2 1 3 -2 }}
Mapping: {{mapping| 1 1 2 2 4 | 0 2 1 3 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 336.051
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 336.051


{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=0| 3d, 4, 7, 18bc, 25bccd }}


Badness: 0.027114
Badness: 0.027114
Line 113: Line 117:
Mapping: {{mapping| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}
Mapping: {{mapping| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 338.846
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 338.846


{{Optimal ET sequence|legend=1| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}
{{Optimal ET sequence|legend=0| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}


Badness: 0.023828
Badness: 0.023828
Line 128: Line 133:
{{Multival|legend=1| 2 1 -1 -3 -7 -5 }}
{{Multival|legend=1| 2 1 -1 -3 -7 -5 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 331.916
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1\1, ~5/4 = 331.916


{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
Line 141: Line 147:
Mapping: {{mapping| 1 1 2 3 4 | 0 2 1 -1 -2 }}
Mapping: {{mapping| 1 1 2 3 4 | 0 2 1 -1 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 337.532
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 337.532


{{Optimal ET sequence|legend=1| 3, 4, 7d }}
{{Optimal ET sequence|legend=0| 3, 4, 7d }}


Badness: 0.024988
Badness: 0.024988
Line 154: Line 161:
Mapping: {{mapping| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}
Mapping: {{mapping| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 341.023
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 341.023


{{Optimal ET sequence|legend=1| 3, 4, 7d }}
{{Optimal ET sequence|legend=0| 3, 4, 7d }}


Badness: 0.023420
Badness: 0.023420
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{{Mapping|legend=1| 1 1 2 1 | 0 2 1 6 }}
{{Mapping|legend=1| 1 1 2 1 | 0 2 1 6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 357.938
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1\1, ~5/4 = 357.938


{{Multival|legend=1| 2 1 6 -3 4 11 }}
{{Multival|legend=1| 2 1 6 -3 4 11 }}
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Mapping: {{mapping| 1 1 2 1 2 | 0 2 1 6 5 }}
Mapping: {{mapping| 1 1 2 1 2 | 0 2 1 6 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 356.106
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 356.106


{{Optimal ET sequence|legend=1| 3de, 7d, 10, 17d, 27cde }}
{{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d, 27cde }}


Badness: 0.022366
Badness: 0.022366
Line 197: Line 207:
{{Multival|legend=1| 2 1 -4 -3 -12 -12 }}
{{Multival|legend=1| 2 1 -4 -3 -12 -12 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 356.264
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1\1, ~5/4 = 356.264


{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}
{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}
Line 210: Line 221:
Mapping: {{mapping| 1 1 2 4 2 | 0 2 1 -4 5 }}
Mapping: {{mapping| 1 1 2 4 2 | 0 2 1 -4 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.262
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 354.262


{{Optimal ET sequence|legend=1| 7, 10, 17, 27ce, 44cce }}
{{Optimal ET sequence|legend=0| 7, 10, 17, 27ce, 44cce }}


Badness: 0.030680
Badness: 0.030680
Line 223: Line 235:
Mapping: {{mapping| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}
Mapping: {{mapping| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.365
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 354.365


{{Optimal ET sequence|legend=1| 7, 10, 17, 27ce, 44cce }}
{{Optimal ET sequence|legend=0| 7, 10, 17, 27ce, 44cce }}


Badness: 0.021674
Badness: 0.021674
Line 236: Line 249:
Mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -4 -2 }}
Mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -4 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.073
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 354.073


{{Optimal ET sequence|legend=1| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e, 17e }}


Badness: 0.031719
Badness: 0.031719
Line 249: Line 263:
Mapping: {{mapping| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}
Mapping: {{mapping| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.313
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 354.313


{{Optimal ET sequence|legend=1| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e, 17e }}


Badness: 0.022741
Badness: 0.022741
Line 262: Line 277:
Mapping: {{mapping| 1 1 2 4 5 | 0 2 1 -4 -5 }}
Mapping: {{mapping| 1 1 2 4 5 | 0 2 1 -4 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.659
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 360.659


{{Optimal ET sequence|legend=1| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness: 0.041361
Badness: 0.041361
Line 275: Line 291:
Mapping: {{mapping| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}
Mapping: {{mapping| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.646
Optimal tunings:
* POTE: ~2 = 1\1, ~5/4 = 360.646


{{Optimal ET sequence|legend=1| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness: 0.027938
Badness: 0.027938
Line 294: Line 311:
{{Multival|legend=1| 4 2 2 -6 -8 -1 }}
{{Multival|legend=1| 4 2 2 -6 -8 -1 }}


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~7/4 = 948.443 (~7/6 = 251.557)
[[Optimal tuning]]s:
* [[POTE]]: ~7/5 = 1\2, ~7/4 = 948.443 (~7/6 = 251.557)


{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
Line 301: Line 319:


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 308: Line 325:
Mapping: {{mapping| 2 0 3 4 -1 | 0 2 1 1 5 }}
Mapping: {{mapping| 2 0 3 4 -1 | 0 2 1 1 5 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 946.507 (~7/6 = 253.493)
Optimal tunings:
* POTE: ~7/5 = 1\2, ~7/4 = 946.507 (~7/6 = 253.493)


{{Optimal ET sequence|legend=1| 10, 14c, 24c, 38ccd, 52cccde }}
{{Optimal ET sequence|legend=0| 10, 14c, 24c, 38ccd, 52cccde }}


Badness: 0.026712
Badness: 0.026712
Line 321: Line 339:
Mapping: {{mapping| 2 0 3 4 -1 1| 0 2 1 1 5 4}}
Mapping: {{mapping| 2 0 3 4 -1 1| 0 2 1 1 5 4}}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 947.955 (~7/6 = 252.045)
Optimal tunings:
* POTE: ~7/5 = 1\2, ~7/4 = 947.955 (~7/6 = 252.045)


{{Optimal ET sequence|legend=1| 10, 14cf, 24cf}}
{{Optimal ET sequence|legend=0| 10, 14cf, 24cf}}


Badness: 0.021326
Badness: 0.021326
Line 334: Line 353:
Mapping: {{mapping| 2 0 3 4 10 | 0 2 1 1 -2 }}
Mapping: {{mapping| 2 0 3 4 10 | 0 2 1 1 -2 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 944.934 (~7/6 = 255.066)
Optimal tunings:
* POTE: ~7/5 = 1\2, ~7/4 = 944.934 (~7/6 = 255.066)


{{Optimal ET sequence|legend=1| 4, 10e, 14c }}
{{Optimal ET sequence|legend=0| 4, 10e, 14c }}


Badness: 0.031456
Badness: 0.031456
Line 347: Line 367:
Mapping: {{mapping| 2 0 3 4 7 | 0 2 1 1 0 }}
Mapping: {{mapping| 2 0 3 4 7 | 0 2 1 1 0 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 956.507 (~8/7 = 243.493)
Optimal tunings:
* POTE: ~7/5 = 1\2, ~7/4 = 956.507 (~8/7 = 243.493)


{{Optimal ET sequence|legend=1| 4, 6, 10 }}
{{Optimal ET sequence|legend=0| 4, 6, 10 }}


Badness: 0.032385
Badness: 0.032385
Line 364: Line 385:
{{Multival|legend=1| 4 2 9 -12 3 15 }}
{{Multival|legend=1| 4 2 9 -12 3 15 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 427.208
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1\1, ~9/7 = 427.208


{{Optimal ET sequence|legend=1| 3d, 14c, 45cc, 59bcccd }}
{{Optimal ET sequence|legend=1| 3d, 14c, 45cc, 59bcccd }}
Line 377: Line 399:
Mapping: {{mapping| 1 3 3 6 7 | 0 -4 -2 -9 -10 }}
Mapping: {{mapping| 1 3 3 6 7 | 0 -4 -2 -9 -10 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 427.273
Optimal tunings:
* POTE: ~2 = 1\1, ~9/7 = 427.273


{{Optimal ET sequence|legend=1| 3de, 14c, 45cce, 59bcccdee }}
{{Optimal ET sequence|legend=0| 3de, 14c, 45cce, 59bcccdee }}


Badness: 0.032957
Badness: 0.032957
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