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* ''[[Fourfives]]'', {245/243, 235298/234375} → [[Fifive family #Fourfives|Fifive family]]
* ''[[Fourfives]]'', {245/243, 235298/234375} → [[Fifive family #Fourfives|Fifive family]]


Tempering out 245/243 alone in the full 7-limit leads to a [[Planar temperament|rank-3 temperament]], [[sensamagic]], for which [[283edo|283EDO]] is the [[optimal patent val]].
Tempering out 245/243 alone in the full 7-limit leads to a [[Planar temperament|rank-3 temperament]], [[sensamagic]], for which [[283edo]] is the [[optimal patent val]].


== BPS ==
== BPS ==
The ''BPS'', for ''Bohlen–Pierce–Stearns'', is the 3.5.7 subgroup temperament tempering out 245/243. This subgroup temperament was formerly called as ''lambda temperament'', which was named after [[4L 5s (tritave-equivalent)|lambda scale]].
The ''BPS'', for ''Bohlen–Pierce–Stearns'', is the 3.5.7 subgroup temperament tempering out 245/243. This subgroup temperament was formerly called the ''lambda'' temperament, which was named after the [[4L 5s (tritave-equivalent)|lambda scale]].


Subgroup: 3.5.7
[[Subgroup]]: 3.5.7


[[Comma list]]: 245/243
[[Comma list]]: 245/243
Line 30: Line 30:
Sval mapping generators: ~3, ~9/7
Sval mapping generators: ~3, ~9/7


[[POTE generator]]: ~9/7 = 440.4881
[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~9/7 = 440.4881


[[Optimal GPV sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
[[Optimal GPV sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
Line 51: Line 51:
{{Multival|legend=1| 7 9 13 -2 1 5 }}
{{Multival|legend=1| 7 9 13 -2 1 5 }}


[[POTE generator]]: ~9/7 = 443.383
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 443.383


{{Val list|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}
{{Val list|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}
Line 68: Line 68:
Gencom: [2 9/7; 91/90 126/125 169/168]
Gencom: [2 9/7; 91/90 126/125 169/168]


POTE generator: ~9/7 = 443.322
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.322


Optimal GPV sequence: {{Val list| 19, 27, 46, 111de, 157de }}
Optimal GPV sequence: {{Val list| 19, 27, 46, 111de, 157de }}
Line 79: Line 79:
Mapping: [{{val| 1 -1 -1 -2 9 }}, {{val| 0 7 9 13 -15 }}]
Mapping: [{{val| 1 -1 -1 -2 9 }}, {{val| 0 7 9 13 -15 }}]


POTE generator: ~9/7 = 443.294
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.294


Optimal GPV sequence: {{Val list| 19, 27, 46, 111d, 157d, 268cdd }}
Optimal GPV sequence: {{Val list| 19, 27, 46, 111d, 157d, 268cdd }}
Line 92: Line 92:
Mapping: [{{val| 1 -1 -1 -2 9 0 }}, {{val| 0 7 9 13 -15 10 }}]
Mapping: [{{val| 1 -1 -1 -2 9 0 }}, {{val| 0 7 9 13 -15 10 }}]


POTE generator: ~9/7 = 443.321
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.321


Optimal GPV sequence: {{Val list| 19, 27, 46, 111df, 157df }}
Optimal GPV sequence: {{Val list| 19, 27, 46, 111df, 157df }}
Line 105: Line 105:
Mapping: [{{val| 1 -1 -1 -2 9 0 10 }}, {{val| 0 7 9 13 -15 10 -16 }}]
Mapping: [{{val| 1 -1 -1 -2 9 0 10 }}, {{val| 0 7 9 13 -15 10 -16 }}]


POTE generator: ~9/7 = 443.365
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.365


Optimal GPV sequence: {{Val list| 19, 27, 46, 157df, 203cdff, 249cddff }}
Optimal GPV sequence: {{Val list| 19, 27, 46, 157df, 203cdff, 249cddff }}
Line 118: Line 118:
Mapping: [{{val| 1 -1 -1 -2 2 }}, {{val| 0 7 9 13 4 }}]
Mapping: [{{val| 1 -1 -1 -2 2 }}, {{val| 0 7 9 13 4 }}]


POTE generator: ~9/7 = 443.962
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.962


Optimal GPV sequence: {{Val list| 8d, 19, 27e, 73ee }}
Optimal GPV sequence: {{Val list| 8d, 19, 27e, 73ee }}
Line 131: Line 131:
Mapping: [{{val| 1 -1 -1 -2 2 0 }}, {{val| 0 7 9 13 4 10 }}]
Mapping: [{{val| 1 -1 -1 -2 2 0 }}, {{val| 0 7 9 13 4 10 }}]


POTE generator: ~9/7 = 443.945
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.945


Optimal GPV sequence: {{Val list| 19, 27e, 46e, 73ee }}
Optimal GPV sequence: {{Val list| 19, 27e, 46e, 73ee }}
Line 144: Line 144:
Mapping: [{{val| 1 -1 -1 -2 -8 }}, {{val| 0 7 9 13 31 }}]
Mapping: [{{val| 1 -1 -1 -2 -8 }}, {{val| 0 7 9 13 31 }}]


POTE generator: ~9/7 = 443.626
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.626


Optimal GPV sequence: {{Val list| 19e, 27e, 46, 119c, 165c }}
Optimal GPV sequence: {{Val list| 19e, 27e, 46, 119c, 165c }}
Line 157: Line 157:
Mapping: [{{val| 1 -1 -1 -2 -8 0 }}, {{val| 0 7 9 13 31 10 }}]
Mapping: [{{val| 1 -1 -1 -2 -8 0 }}, {{val| 0 7 9 13 31 10 }}]


POTE generator: ~9/7 = 443.559
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.559


Optimal GPV sequence: {{Val list| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
Optimal GPV sequence: {{Val list| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
Line 170: Line 170:
Mapping: [{{val| 1 -1 -1 -2 -8 0 -7 }}, {{val| 0 7 9 13 31 10 30 }}]
Mapping: [{{val| 1 -1 -1 -2 -8 0 -7 }}, {{val| 0 7 9 13 31 10 30 }}]


POTE generator: ~9/7 = 443.551
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.551


Optimal GPV sequence: {{Val list| 19eg, 27eg, 46 }}
Optimal GPV sequence: {{Val list| 19eg, 27eg, 46 }}
Line 183: Line 183:
Mapping: [{{val| 1 -1 -1 -2 -1 }}, {{val| 0 7 9 13 12 }}]
Mapping: [{{val| 1 -1 -1 -2 -1 }}, {{val| 0 7 9 13 12 }}]


POTE generator: ~9/7 = 443.518
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.518


Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}
Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}
Line 196: Line 196:
Mapping: [{{val| 1 -1 -1 -2 -1 0 }}, {{val| 0 7 9 13 12 11 }}]
Mapping: [{{val| 1 -1 -1 -2 -1 0 }}, {{val| 0 7 9 13 12 11 }}]


POTE generator: ~9/7 = 443.506
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.506


Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}
Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}
Line 209: Line 209:
Mapping: [{{val| 1 -1 -1 -2 -3 }}, {{val| 0 14 18 26 35 }}]
Mapping: [{{val| 1 -1 -1 -2 -3 }}, {{val| 0 14 18 26 35 }}]


POTE generator: ~25/22 = 221.605
Optimal tuning (POTE): ~2 = 1\1, ~25/22 = 221.605


Optimal GPV sequence: {{Val list| 27e, 38d, 65, 157de, 222cde }}
Optimal GPV sequence: {{Val list| 27e, 38d, 65, 157de, 222cde }}
Line 222: Line 222:
Mapping: [{{val| 1 -1 -1 -2 -3 0 }}, {{val| 0 14 18 26 35 30 }}]
Mapping: [{{val| 1 -1 -1 -2 -3 0 }}, {{val| 0 14 18 26 35 30 }}]


POTE generator: ~25/22 = 221.556
Optimal tuning (POTE): ~2 = 1\1, ~25/22 = 221.556


Optimal GPV sequence: {{Val list| 27e, 38df, 65f }}
Optimal GPV sequence: {{Val list| 27e, 38df, 65f }}
Line 235: Line 235:
Mapping: [{{val| 2 5 7 9 9 }}, {{val| 0 -7 -9 -13 -8 }}]
Mapping: [{{val| 2 5 7 9 9 }}, {{val| 0 -7 -9 -13 -8 }}]


POTE generator: ~11/10 = 156.692
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 156.692


Optimal GPV sequence: {{Val list| 8d, …, 38d, 46, 176dde, 222cdde, 268cddee }}
Optimal GPV sequence: {{Val list| 8d, …, 38d, 46, 176dde, 222cdde, 268cddee }}
Line 248: Line 248:
Mapping: [{{val| 2 5 7 9 9 10 }}, {{val| 0 -7 -9 -13 -8 -10 }}]
Mapping: [{{val| 2 5 7 9 9 10 }}, {{val| 0 -7 -9 -13 -8 -10 }}]


POTE generator: ~11/10 = 156.725
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 156.725


Optimal GPV sequence: {{Val list| 8d, …, 38df, 46 }}
Optimal GPV sequence: {{Val list| 8d, …, 38df, 46 }}
Line 256: Line 256:
== Bohpier ==
== Bohpier ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Bohpier]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Bohpier]].''
{{main|Bohpier}}
{{Main| Bohpier }}


'''[[Bohpier]]''' is named after its [[Relationship between Bohlen-Pierce and octave-ful temperaments|interesting relationship with the non-octave Bohlen-Pierce equal temperament]].
'''[[Bohpier]]''' is named after its [[Relationship between Bohlen-Pierce and octave-ful temperaments|interesting relationship with the non-octave Bohlen-Pierce equal temperament]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 3125/3087
[[Comma list]]: 245/243, 3125/3087
Line 268: Line 268:
{{Multival|legend=1| 13 19 23 0 0 0 }}
{{Multival|legend=1| 13 19 23 0 0 0 }}


[[POTE generator]]: ~27/25 = 146.474
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 146.474


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7-odd-limit: ~27/25 = {{monzo| 0 0 1/19 }}
* 7-odd-limit: ~27/25 = {{monzo| 0 0 1/19 }}
: Eigenmonzos (unchanged intervals): 2, 5/4
: [[Eigenmonzo basis]]: 2.5
* 9-odd-limit: ~27/25 = {{monzo| 0 1/13 }}
* 9-odd-limit: ~27/25 = {{monzo| 0 1/13 }}
: Eigenmonzos (unchanged intervals): 2, 4/3
: [[Eigenmonzo basis]]: 2.3


{{Val list|legend=1| 41, 131, 172, 213c }}
{{Val list|legend=1| 41, 131, 172, 213c }}
Line 287: Line 287:
Mapping: [{{val| 1 0 0 0 2 }}, {{val| 0 13 19 23 12 }}]
Mapping: [{{val| 1 0 0 0 2 }}, {{val| 0 13 19 23 12 }}]


POTE generator: ~12/11 = 146.545
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.545


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~12/11 = {{monzo| 1/7 1/7 0 0 -1/14 }}
* 11-odd-limit: ~12/11 = {{monzo| 1/7 1/7 0 0 -1/14 }}
: Eigenmonzos (unchanged intervals): 2, 11/9
: Eigenmonzo basis: 2.11/9


Optimal GPV sequence: {{Val list| 41, 90e, 131e }}
Optimal GPV sequence: {{Val list| 41, 90e, 131e }}
Line 304: Line 304:
Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 13 19 23 12 14 }}]
Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 13 19 23 12 14 }}]


POTE generator: ~12/11 = 146.603
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.603


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}
* 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}
: Eigenmonzos (unchanged intervals): 2, 5/4
: Eigenmonzo basis: 2.5


Optimal GPV sequence: {{Val list| 41, 90ef, 131ef, 221bdeff }}
Optimal GPV sequence: {{Val list| 41, 90ef, 131ef, 221bdeff }}
Line 328: Line 328:
Mapping: [{{val| 1 0 0 0 0 }}, {{val| 0 39 57 69 85 }}]
Mapping: [{{val| 1 0 0 0 0 }}, {{val| 0 39 57 69 85 }}]


POTE generator: ~77/75 = 48.828
Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.828


Optimal GPV sequence: {{Val list| 49, 123ce, 172 }}
Optimal GPV sequence: {{Val list| 49, 123ce, 172 }}
Line 341: Line 341:
Mapping: [{{val| 1 0 0 0 0 0 }}, {{val| 0 39 57 69 85 91 }}]
Mapping: [{{val| 1 0 0 0 0 0 }}, {{val| 0 39 57 69 85 91 }}]


POTE generator: ~77/75 = 48.822
Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.822


Optimal GPV sequence: {{Val list| 49f, 123ce, 172f, 295ce, 467bccef }}
Optimal GPV sequence: {{Val list| 49f, 123ce, 172f, 295ce, 467bccef }}
Line 348: Line 348:


== Escaped ==
== Escaped ==
{{see also| Escapade family #Escaped }}
{{See also| Escapade family #Escaped }}


This temperament is also called as "sensa" because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with 19e&27 temperament (sensi extension).''
This temperament is also known as "sensa" because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with the 19e & 27 temperament (sensi extension).''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 65625/65536
[[Comma list]]: 245/243, 65625/65536
Line 360: Line 360:
{{Multival|legend=1| 9 -7 26 -32 16 80 }}
{{Multival|legend=1| 9 -7 26 -32 16 80 }}


[[POTE generator]]: ~28/27 = 55.122
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 55.122


{{Val list|legend=1| 22, 65, 87, 196, 283 }}
{{Val list|legend=1| 22, 65, 87, 196, 283 }}
Line 373: Line 373:
Mapping: [{{val| 1 2 2 4 3 }}, {{val| 0 -9 7 -26 10 }}]
Mapping: [{{val| 1 2 2 4 3 }}, {{val| 0 -9 7 -26 10 }}]


POTE generator: ~28/27 = 55.126
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.126


Optimal GPV sequence: {{Val list| 22, 65, 87, 196, 283 }}
Optimal GPV sequence: {{Val list| 22, 65, 87, 196, 283 }}
Line 386: Line 386:
Mapping: [{{val| 1 2 2 4 3 2 }}, {{val| 0 -9 7 -26 10 37 }}]
Mapping: [{{val| 1 2 2 4 3 2 }}, {{val| 0 -9 7 -26 10 37 }}]


POTE generator: ~28/27 = 55.138
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138


Optimal GPV sequence: {{Val list| 22, 65, 87, 283 }}
Optimal GPV sequence: {{Val list| 22, 65, 87, 283 }}
Line 393: Line 393:


== Salsa ==
== Salsa ==
{{see also| Schismatic family }}
{{See also| Schismatic family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 32805/32768
[[Comma list]]: 245/243, 32805/32768
Line 403: Line 403:
{{Multival|legend=1| 2 -16 13 -30 15 75 }}
{{Multival|legend=1| 2 -16 13 -30 15 75 }}


[[POTE generator]]: ~128/105 = 351.049
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 351.049


{{Val list|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
{{Val list|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
Line 416: Line 416:
Mapping: [{{val| 1 1 7 -1 2 }}, {{val| 0 2 -16 13 5 }}]
Mapping: [{{val| 1 1 7 -1 2 }}, {{val| 0 2 -16 13 5 }}]


POTE generator: ~11/9 = 351.014
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014


Optimal GPV sequence: {{Val list| 17, 24, 41, 106d, 147d }}
Optimal GPV sequence: {{Val list| 17, 24, 41, 106d, 147d }}
Line 429: Line 429:
Mapping: [{{val| 1 1 7 -1 2 4 }}, {{val| 0 2 -16 13 5 -1 }}]
Mapping: [{{val| 1 1 7 -1 2 4 }}, {{val| 0 2 -16 13 5 -1 }}]


POTE generator: ~11/9 = 351.025
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.025


Optimal GPV sequence: {{Val list| 17, 24, 41, 106df, 147df }}
Optimal GPV sequence: {{Val list| 17, 24, 41, 106df, 147df }}
Line 436: Line 436:


== Pycnic ==
== Pycnic ==
{{see also| High badness temperaments #Stump }}
{{See also| High badness temperaments #Stump }}


The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has MOS of size 9, 11, 13, 15, 17... which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.
The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has [[mos]] of size 9, 11, 13, 15, 17… which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 525/512
[[Comma list]]: 245/243, 525/512
Line 448: Line 448:
{{Multival|legend=1| 3 -7 11 -18 9 45 }}
{{Multival|legend=1| 3 -7 11 -18 9 45 }}


[[POTE generator]]: ~45/32 = 567.720
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 567.720


{{Val list|legend=1| 17, 19, 55c, 74cd, 93cdd }}
{{Val list|legend=1| 17, 19, 55c, 74cd, 93cdd }}
Line 455: Line 455:


== Cohemiripple ==
== Cohemiripple ==
{{see also| Ripple family }}
{{See also| Ripple family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 1323/1250
[[Comma list]]: 245/243, 1323/1250
Line 465: Line 465:
{{Multival|legend=1| 10 16 17 2 -1 -5 }}
{{Multival|legend=1| 10 16 17 2 -1 -5 }}


[[POTE generator]]: ~7/5 = 549.944
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 549.944


{{Val list|legend=1| 11cd, 13cd, 24 }}
{{Val list|legend=1| 11cd, 13cd, 24 }}
Line 478: Line 478:
Mapping: [{{val| 1 -3 -5 -5 -8 }}, {{val| 0 10 16 17 25 }}]
Mapping: [{{val| 1 -3 -5 -5 -8 }}, {{val| 0 10 16 17 25 }}]


POTE generator: ~7/5 = 549.945
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.945


Optimal GPV sequence: {{Val list| 11cdee, 13cdee, 24 }}
Optimal GPV sequence: {{Val list| 11cdee, 13cdee, 24 }}
Line 491: Line 491:
Mapping: [{{val| 1 -3 -5 -5 -8 -5 }}, {{val| 0 -10 -16 -17 -25 -19 }}]
Mapping: [{{val| 1 -3 -5 -5 -8 -5 }}, {{val| 0 -10 -16 -17 -25 -19 }}]


POTE generator: ~7/5 = 549.958
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.958


Optimal GPV sequence: {{Val list| 11cdeef, 13cdeef, 24 }}
Optimal GPV sequence: {{Val list| 11cdeef, 13cdeef, 24 }}
Line 498: Line 498:


== Superthird ==
== Superthird ==
{{see also| Shibboleth family }}
{{See also| Shibboleth family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 78125/76832
[[Comma list]]: 245/243, 78125/76832
Line 508: Line 508:
{{Multival|legend=1| 18 20 35 -10 5 25 }}
{{Multival|legend=1| 18 20 35 -10 5 25 }}


[[POTE generator]]: ~9/7 = 439.076
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 439.076


{{Val list|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
{{Val list|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
Line 521: Line 521:
Mapping: [{{val| 1 -5 -5 -10 2 }}, {{val| 0 18 20 35 4 }}]
Mapping: [{{val| 1 -5 -5 -10 2 }}, {{val| 0 18 20 35 4 }}]


POTE generator: ~9/7 = 439.152
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.152


Optimal GPV sequence: {{Val list| 11cd, 30d, 41, 153be, 194be, 235bcee }}
Optimal GPV sequence: {{Val list| 11cd, 30d, 41, 153be, 194be, 235bcee }}
Line 534: Line 534:
Mapping: [{{val| 1 -5 -5 -10 2 -8 }}, {{val| 0 18 20 35 4 32 }}]
Mapping: [{{val| 1 -5 -5 -10 2 -8 }}, {{val| 0 18 20 35 4 32 }}]


POTE generator: ~9/7 = 439.119
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.119


Optimal GPV sequence: {{Val list| 11cdf, 30df, 41 }}
Optimal GPV sequence: {{Val list| 11cdf, 30df, 41 }}
Line 541: Line 541:


== Superenneadecal ==
== Superenneadecal ==
Superenneadecal is a cousin of [[enneadecal]] but sharper fifth is used to temper 245/243.
Superenneadecal is a cousin of [[enneadecal]] but sharper fifth is used to temper 245/243.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 395136/390625
[[Comma list]]: 245/243, 395136/390625


Mapping: [⟨19 0 14 -7], ⟨0 1 1 2]]
[[Mapping]]: [{{val| 19 0 14 -7 }}, {{val| 0 1 1 2 }}]


{{Val list|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
{{Val list|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
Line 555: Line 554:
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 245/243, 2560/2541, 3773/3750
Comma list: 245/243, 2560/2541, 3773/3750


Mapping: [⟨19 0 14 -7 96], ⟨0 1 1 2 -1]]
Mapping: [{{val| 19 0 14 -7 96 }}, {{val| 0 1 1 2 -1 }}]


{{Val list|legend=1| 19, 76bcd, 95, 114e }}
{{Val list|legend=1| 19, 76bcd, 95, 114e }}
Line 564: Line 563:
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 196/195, 245/243, 832/825, 1001/1000
Comma list: 196/195, 245/243, 832/825, 1001/1000


Mapping: [⟨19 0 14 -7 96 10], ⟨0 1 1 2 -1 2]]
Mapping: [{{val| 19 0 14 -7 96 10 }}, {{val| 0 1 1 2 -1 2 }}]


{{Val list|legend=1| 19, 76bcdf, 95, 114e }}
{{Val list|legend=1| 19, 76bcdf, 95, 114e }}
Line 573: Line 572:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Magus]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Magus]].''


Magus temperament tempers out 50331648/48828125 (salegu) in the 5-limit. This temperament can be described as 46&49 temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). Alternative extension [[Starling temperaments #Amigo|amigo]] (43&46) tempers out the same 5-limit comma as the magus, but with the [[126/125|starling comma]] (126/125) rather than the sensamagic tempered out.
Magus temperament tempers out 50331648/48828125 (salegu) in the 5-limit. This temperament can be described as 46 & 49 temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). The alternative extension [[Starling temperaments #Amigo|amigo]] (43 & 46) tempers out the same 5-limit comma as the magus, but with the [[126/125|starling comma]] (126/125) rather than the sensamagic tempered out.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 28672/28125
[[Comma list]]: 245/243, 28672/28125
Line 583: Line 582:
{{Multival|legend=1| 11 1 27 -24 12 60 }}
{{Multival|legend=1| 11 1 27 -24 12 60 }}


[[POTE generator]]: ~5/4 = 391.465
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.465


{{Val list|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
{{Val list|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
Line 596: Line 595:
Mapping: [{{val| 1 -2 2 -6 -6 }}, {{val| 0 11 1 27 29 }}]
Mapping: [{{val| 1 -2 2 -6 -6 }}, {{val| 0 11 1 27 29 }}]


POTE generator: ~5/4 = 391.503
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.503


Optimal GPV sequence: {{Val list| 46, 95, 141bc }}
Optimal GPV sequence: {{Val list| 46, 95, 141bc }}
Line 609: Line 608:
Mapping: [{{val| 1 -2 2 -6 -6 5 }}, {{val| 0 11 1 27 29 -4 }}]
Mapping: [{{val| 1 -2 2 -6 -6 5 }}, {{val| 0 11 1 27 29 -4 }}]


POTE generator: ~5/4 = 391.366
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.366


Optimal GPV sequence: {{Val list| 46, 233bcff, 279bccff }}
Optimal GPV sequence: {{Val list| 46, 233bcff, 279bccff }}
Line 618: Line 617:
''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''
''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 2097152/2066715
[[Comma list]]: 245/243, 2097152/2066715
Line 624: Line 623:
[[Mapping]]: [{{val| 1 1 17 -6 }}, {{val| 0 1 -25 15 }}]
[[Mapping]]: [{{val| 1 1 17 -6 }}, {{val| 0 1 -25 15 }}]


[[POTE generator]]: ~3/2 = 704.536
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.536


{{Val list|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
{{Val list|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
Line 637: Line 636:
Mapping: [{{val| 1 1 17 -6 -3 }}, {{val| 0 1 -25 15 11 }}]
Mapping: [{{val| 1 1 17 -6 -3 }}, {{val| 0 1 -25 15 11 }}]


POTE generator: ~3/2 = 704.554
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.554


Optimal GPV sequence: {{Val list| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
Optimal GPV sequence: {{Val list| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
Line 650: Line 649:
Mapping: [{{val| 1 1 17 -6 -3 -1 }}, {{val| 0 1 -25 15 11 8 }}]
Mapping: [{{val| 1 1 17 -6 -3 -1 }}, {{val| 0 1 -25 15 11 8 }}]


POTE generator: ~3/2 = 704.571
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.571


Optimal GPV sequence: {{Val list| 17, 29c, 46, 63, 109 }}
Optimal GPV sequence: {{Val list| 17, 29c, 46, 63, 109 }}
Line 663: Line 662:
Mapping: [{{val| 1 1 17 -6 -3 -1 -10 }}, {{val| 0 1 -25 15 11 8 24 }}]
Mapping: [{{val| 1 1 17 -6 -3 -1 -10 }}, {{val| 0 1 -25 15 11 8 24 }}]


POTE generator: ~3/2 = 704.540
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.540


Optimal GPV sequence: {{Val list| 17g, 29cg, 46, 109, 155f, 264bfg }}
Optimal GPV sequence: {{Val list| 17g, 29cg, 46, 109, 155f, 264bfg }}
Line 676: Line 675:
Mapping: [{{val| 1 1 17 -6 -3 -1 17 }}, {{val| 0 1 -25 15 11 8 -22 }}]
Mapping: [{{val| 1 1 17 -6 -3 -1 17 }}, {{val| 0 1 -25 15 11 8 -22 }}]


POTE generator: ~3/2 = 704.537
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.537


Optimal GPV sequence: {{Val list| 17, 29c, 46, 109g, 155fg, 264bfgg }}
Optimal GPV sequence: {{Val list| 17, 29c, 46, 109g, 155fg, 264bfgg }}
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