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Archytas clan: Difference between revisions

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The '''archytas clan''' (or '''archy family''') tempers out the [[64/63|Archytas comma]], 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to [[81/80]] in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of [[22edo|22EDO]].
The '''archytas clan''' (or '''archy family''') tempers out the [[64/63|Archytas' comma]], 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to [[81/80]] in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3's and 7's quite sharp, such as those of [[22edo]].  
 
Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.
 
Discussed under their respective 5-limit families are:
* ''[[Mother]]'' → [[Father family #Mother]]
* ''[[Dominant]]'' → [[Meantone family #Dominant]]
* ''[[Augene]]'' → [[Augmented family #Augene]]
* [[Porcupine]] → [[Porcupine family #Septimal porcupine]]
* [[Pajara]] → [[Diaschismic family #Pajara]]
* ''[[Blacksmith]]'' → [[Limma family #Blacksmith]]
* ''[[Catalan]]'' → [[Kleismic family #Catalan]]
* ''[[Modus]]'' → [[Tetracot family #Modus]]
* ''[[Passion]]'' → [[Passion family #Septimal passion]]
* ''[[Immunized]]'' → [[Immunity family #Immunized]]
 
The rest are considered below.


== Archy ==
== Archy ==
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[[Comma list]]: 64/63
[[Comma list]]: 64/63


[[Sval]] [[mapping]]: [{{val| 1 0 6 }}, {{val| 0 1 -2 }}]
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}


Sval mapping generators: ~2, ~3
: sval mapping generators: ~2, ~3


Gencom mapping: [{{val| 1 1 0 4 }}, {{val| 0 1 0 -2 }}]
{{Mapping|legend=3| 1 1 0 4 | 0 1 0 -2 }}


[[Gencom]]: [2 3/2; 64/63]
: [[gencom]]: [2 3/2; 64/63]


[[Optimal tuning]] ([[POTE]]): ~3/2 = 709.321
[[Optimal tuning]] ([[POTE]]): ~3/2 = 709.321
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Scales: [[archy5]], [[archy7]], [[archy12]]
Scales: [[archy5]], [[archy7]], [[archy12]]
=== Overview to extensions ===
Adding [[50/49]] to the list of commas gives pajara, [[36/35]] gives dominant, [[16/15]] gives mother, [[126/125]] gives augene, [[28/27]] gives blacksmith, [[245/243]] gives superpyth, [[250/243]] gives porcupine, [[686/675]] gives beatles, [[360/343]] gives schism, [[3125/3087]] gives passion, [[2430/2401]] gives quasisuper, and [[4375/4374]] gives modus.
Discussed under their respective 5-limit families are:
* ''[[Mother]]'' → [[Father family #Mother|Father family]]
* ''[[Dominant]]'' → [[Meantone family #Dominant|Meantone family]]
* ''[[Augene]]'' → [[Augmented family #Augene|Augmented]]
* [[Porcupine]] → [[Porcupine family #Septimal porcupine|Porcupine family]]
* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Blacksmith]]'' → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
* ''[[Catalan]]'' → [[Kleismic family #Catalan|Kleismic family]]
* ''[[Modus]]'' → [[Tetracot family #Modus|Tetracot family]]
* ''[[Passion]]'' → [[Passion family #Septimal passion|Passion family]]
* ''[[Immunized]]'' → [[Immunity family #Immunized|Immunity family]]
The rest are considered below.


=== Supra ===
=== Supra ===
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Comma list: 64/63, 99/98
Comma list: 64/63, 99/98


Sval mapping: [{{val| 1 0 6 13 }}, {{val| 0 1 -2 -6 }}]
Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
 
Sval mapping generators: ~2, ~3


Gencom mapping: [{{val| 1 1 0 4 7 }}, {{val| 0 1 0 -2 -6 }}]
Gencom mapping: {{mapping| 1 1 0 4 7 | 0 1 0 -2 -6 }}


Gencom: [2 3/2; 64/63 99/98]
: gencom: [2 3/2; 64/63 99/98]


Optimal tuning (POTE): ~3/2 = 707.192
Optimal tuning (POTE): ~3/2 = 707.192
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Comma list: 64/63, 78/77, 99/98
Comma list: 64/63, 78/77, 99/98


Sval mapping: [{{val| 1 0 6 13 18 }}, {{val| 0 1 -2 -6 -9 }}]
Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}


Sval mapping generators: ~2, ~3
Gencom mapping: {{mapping| 1 1 0 4 7 9 | 0 1 0 -2 -6 -9 }}


Gencom mapping: [{{val| 1 1 0 4 7 9 }}, {{val| 0 1 0 -2 -6 -9 }}]
: gencom: [2 3/2; 64/63 78/77 99/98]
 
Gencom: [2 3/2; 64/63 78/77 99/98]


Optimal tuning (POTE): ~3/2 = 706.137
Optimal tuning (POTE): ~3/2 = 706.137
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Comma list: 64/63, 243/242
Comma list: 64/63, 243/242


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


Sval mapping generators: ~2, ~11/9
: sval mapping generators: ~2, ~11/9


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


Gencom: [2 3/2; 64/63 99/98]
: gencom: [2 3/2; 64/63 99/98]


Optimal tuning (POTE): ~11/9 = 353.958
Optimal tuning (POTE): ~11/9 = 353.958
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Comma list: 64/63, 78/77, 144/143
Comma list: 64/63, 78/77, 144/143


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


Sval mapping generators: ~2, ~11/9
Gencom mapping: {{mapping| 1 1 0 4 2 4 | 0 1 0 -4 5 -1 }}


Gencom mapping: [{{val| 1 1 0 4 2 4 }}, {{val| 0 1 0 -4 5 -1 }}]
: gencom: [2 3/2; 64/63 78/77 99/98]
 
Gencom: [2 3/2; 64/63 78/77 99/98]


Optimal tuning (POTE): ~11/9 = 353.775
Optimal tuning (POTE): ~11/9 = 353.775
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== Superpyth ==
== Superpyth ==
{{main| Superpyth }}
{{Main| Superpyth }}


In the 5-limit, superpyth tempers out 20480/19683 (sayo comma). This temperament has a fifth generator of ~3/2 = ~710¢ and 5/4 is equated to nine generators minus 5 octaves. A weak extension [[Jubilismic clan #Bipyth|bipyth]] (10cd&22) tempers out the same 5-limit comma as the superpyth, but with a half-octave period and the jubilisma (50/49) rather than the Archytas comma tempered out.
In the 5-limit, superpyth tempers out 20480/19683 (sayo comma). This temperament has a fifth generator of ~3/2 = ~710¢ and 5/4 is equated to nine generators minus 5 octaves. A weak extension [[Jubilismic clan #Bipyth|bipyth]] (10cd&22) tempers out the same 5-limit comma as the superpyth, but with a half-octave period and the jubilisma (50/49) rather than the Archytas comma tempered out.


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 20480/19683
[[Comma list]]: 20480/19683


[[Mapping]]: [{{val| 1 0 -12 }}, {{val| 0 1 9 }}]
{{Mapping|legend=1| 1 0 -12 | 0 1 9 }}


[[POTE generator]]: ~3/2 = 710.078
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 710.078


{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
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=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 245/243
[[Comma list]]: 64/63, 245/243


[[Mapping]]: [{{val| 1 0 -12 6 }}, {{val| 0 1 9 -2 }}]
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}


{{Multival|legend=1| 1 9 -2 12 -6 -30 }}
{{Multival|legend=1| 1 9 -2 12 -6 -30 }}


[[POTE generator]]: ~3/2 = 710.291
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 710.291


{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49 }}
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49 }}
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Comma list: 64/63, 100/99, 245/243
Comma list: 64/63, 100/99, 245/243


Mapping: [{{val| 1 0 -12 6 -22 }}, {{val| 0 1 9 -2 16 }}]
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}


POTE generator: ~3/2 = 710.175
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 710.175


{{Optimal ET sequence|legend=1| 22, 27e, 49 }}
{{Optimal ET sequence|legend=1| 22, 27e, 49 }}
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Comma list: 64/63, 78/77, 91/90, 100/99
Comma list: 64/63, 78/77, 91/90, 100/99


Mapping: [{{val| 1 0 -12 6 -22 -17 }}, {{val| 0 1 9 -2 16 13 }}]
Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}


POTE generator: ~3/2 = 710.479
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 710.479


{{Optimal ET sequence|legend=1| 22, 27e, 49, 76bcde }}
{{Optimal ET sequence|legend=1| 22, 27e, 49, 76bcde }}
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Comma list: 64/63, 100/99, 169/168, 245/243
Comma list: 64/63, 100/99, 169/168, 245/243


Mapping: [{{val| 1 1 -3 4 -6 4 }}, {{val| 0 2 18 -4 32 -1 }}]
Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}


POTE generator: ~16/13 = 355.036
Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 355.036


{{Optimal ET sequence|legend=1| 17e, 27e, 44, 71d }}
{{Optimal ET sequence|legend=1| 17e, 27e, 44, 71d }}
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Comma list: 55/54, 64/63, 99/98
Comma list: 55/54, 64/63, 99/98


Mapping: [{{val| 1 0 -12 6 13 }}, {{val| 0 1 9 -2 -6 }}]
Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}


POTE generator: ~3/2 = 709.495
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 709.495


{{Optimal ET sequence|legend=1| 5, 12c, 17, 22 }}
{{Optimal ET sequence|legend=1| 5, 12c, 17, 22 }}
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Comma list: 55/54, 64/63, 65/63, 99/98
Comma list: 55/54, 64/63, 65/63, 99/98


Mapping: [{{val| 1 0 -12 6 13 18 }}, {{val| 0 1 9 -2 -6 -9 }}]
Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}


POTE generator: ~3/2 = 708.703
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 708.703


{{Optimal ET sequence|legend=1| 17, 22, 83cdf }}
{{Optimal ET sequence|legend=1| 17, 22, 83cdf }}
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== Quasisuper ==
== Quasisuper ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 2430/2401
[[Comma list]]: 64/63, 2430/2401


[[Mapping]]: [{{val| 1 0 23 6 }}, {{val| 0 1 -13 -2 }}]
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}


{{Multival|legend=1| 1 -13 -2 -23 -6 32 }}
{{Multival|legend=1| 1 -13 -2 -23 -6 32 }}


[[POTE generator]]: ~3/2 = 708.328
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 708.328


{{Optimal ET sequence|legend=1| 17c, 22, 61d }}
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}
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Comma list: 64/63, 99/98, 121/120
Comma list: 64/63, 99/98, 121/120


Mapping: [{{val| 1 0 23 6 13 }}, {{val| 0 1 -13 -2 -6 }}]
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}


POTE generator: ~3/2 = 708.205
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 708.205


{{Optimal ET sequence|legend=1| 17c, 22, 39d, 61d }}
{{Optimal ET sequence|legend=1| 17c, 22, 39d, 61d }}
Line 244: Line 239:
Comma list: 64/63, 78/77, 91/90, 121/120
Comma list: 64/63, 78/77, 91/90, 121/120


Mapping: [{{val| 1 0 23 6 13 18 }}, {{val| 0 1 -13 -2 -6 -9 }}]
Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}


POTE generator: ~3/2 = 708.004
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 708.004


{{Optimal ET sequence|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
{{Optimal ET sequence|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
Line 257: Line 252:
Comma list: 55/54, 64/63, 2430/2401
Comma list: 55/54, 64/63, 2430/2401


Mapping: [{{val| 1 0 23 6 -22 }}, {{val| 0 1 -13 -2 16 }}]
Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}


POTE generator: ~3/2 = 709.021
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 709.021


{{Optimal ET sequence|legend=1| 5ce, 17ce, 22 }}
{{Optimal ET sequence|legend=1| 5ce, 17ce, 22 }}
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Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[The Biosphere #Oceanfront|oceanfront]] temperament, with mapping 5th harmonic to 14 fifths.
Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[The Biosphere #Oceanfront|oceanfront]] temperament, with mapping 5th harmonic to 14 fifths.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 6860/6561
[[Comma list]]: 64/63, 6860/6561


[[Mapping]]: [{{val| 1 0 -20 6 }}, {{val| 0 1 14 -2 }}]
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}


{{Multival|legend=1| 1 14 -2 20 -6 -44 }}
{{Multival|legend=1| 1 14 -2 20 -6 -44 }}


[[POTE generator]]: ~3/2 = 713.651
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 713.651


{{Optimal ET sequence|legend=1| 5, 32, 37 }}
{{Optimal ET sequence|legend=1| 5, 32, 37 }}
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Comma list: 55/54, 64/63, 2401/2376
Comma list: 55/54, 64/63, 2401/2376


Mapping: [{{val| 1 0 -20 6 21 }}, {{val| 0 1 14 -2 -11 }}]
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}


POTE generator: ~3/2 = 713.395
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 713.395


{{Optimal ET sequence|legend=1| 5, 32, 37 }}
{{Optimal ET sequence|legend=1| 5, 32, 37 }}
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Comma list: 55/54, 64/63, 91/90, 1573/1568
Comma list: 55/54, 64/63, 91/90, 1573/1568


Mapping: [{{val| 1 0 -20 6 21 -25 }}, {{val| 0 1 14 -2 -11 18 }}]
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}


POTE generator: ~3/2 = 713.500
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 713.500


{{Optimal ET sequence|legend=1| 5, 32, 37 }}
{{Optimal ET sequence|legend=1| 5, 32, 37 }}
Line 313: Line 308:
Comma list: 64/63, 100/99, 3773/3645
Comma list: 64/63, 100/99, 3773/3645


Mapping: [{{val| 1 0 -20 6 -38 }}, {{val| 0 1 14 -2 26 }}]
Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}


POTE generator: ~3/2 = 713.791
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 713.791


{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bce, 116bbce }}
{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bce, 116bbce }}
Line 326: Line 321:
Comma list: 64/63, 91/90, 100/99, 847/845
Comma list: 64/63, 91/90, 100/99, 847/845


Mapping: [{{val| 1 0 -20 6 -38 -25 }}, {{val| 0 1 14 -2 26 18 }}]
Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}


POTE generator: ~3/2 = 713.811
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 713.811


{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bcef, 116bbcef }}
{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bcef, 116bbcef }}
Line 337: Line 332:
{{see also|Schismatic family #Schism}}
{{see also|Schismatic family #Schism}}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 360/343
[[Comma list]]: 64/63, 360/343


[[Mapping]]: [{{val| 1 0 15 6 }}, {{val| 0 1 -8 -2 }}]
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}


{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}


[[POTE generator]]: ~3/2 = 701.556
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.556


{{Optimal ET sequence|legend=1| 12, 41d, 53d }}
{{Optimal ET sequence|legend=1| 12, 41d, 53d }}
Line 356: Line 351:
Comma list: 45/44, 64/63, 99/98
Comma list: 45/44, 64/63, 99/98


Mapping: [{{val| 1 0 15 6 13 }}, {{val| 0 1 -8 -2 -6 }}]
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


POTE generator: ~3/2 = 702.136
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.136


{{Optimal ET sequence|legend=1| 12, 29de, 41de }}
{{Optimal ET sequence|legend=1| 12, 29de, 41de }}
Line 367: Line 362:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Beatles]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Beatles]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 686/675
[[Comma list]]: 64/63, 686/675


[[Mapping]]: [{{val| 1 1 5 4 }}, {{val| 0 2 -9 -4 }}]
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}


{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}
{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}


[[POTE generator]]: ~49/40 = 355.904
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 355.904


{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
Line 389: Line 384:
Comma list: 64/63, 100/99, 686/675
Comma list: 64/63, 100/99, 686/675


Mapping: [{{val| 1 1 5 4 10 }}, {{val| 0 2 -9 -4 -22 }}]
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}


POTE generator: ~49/40 = 356.140
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 356.140


{{Optimal ET sequence|legend=1| 27e, 37, 64be, 91bcde }}
{{Optimal ET sequence|legend=1| 27e, 37, 64be, 91bcde }}
Line 402: Line 397:
Comma list: 64/63, 91/90, 100/99, 169/168
Comma list: 64/63, 91/90, 100/99, 169/168


Mapping: [{{val| 1 1 5 4 10 4 }}, {{val| 0 2 -9 -4 -22 -1 }}]
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}


POTE generator: ~16/13 = 356.229
Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 356.229


{{Optimal ET sequence|legend=1| 27e, 37, 64be }}
{{Optimal ET sequence|legend=1| 27e, 37, 64be }}
Line 415: Line 410:
Comma list: 56/55, 64/63, 540/539
Comma list: 56/55, 64/63, 540/539


Mapping: [{{val| 1 1 5 4 2 }}, {{val| 0 2 -9 -4 5 }}]
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}


POTE generator: ~11/9 = 355.419
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 355.419


{{Optimal ET sequence|legend=1| 10, 17c, 27e }}
{{Optimal ET sequence|legend=1| 10, 17c, 27e }}
Line 428: Line 423:
Comma list: 56/55, 64/63, 78/77, 91/90
Comma list: 56/55, 64/63, 78/77, 91/90


Mapping: [{{val| 1 1 5 4 2 4 }}, {{val| 0 2 -9 -4 5 -1 }}]
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}


POTE generator: ~11/9 = 355.456
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 355.456


{{Optimal ET sequence|legend=1| 10, 17c, 27e }}
{{Optimal ET sequence|legend=1| 10, 17c, 27e }}
Line 441: Line 436:
Comma list: 55/54, 64/63, 686/675
Comma list: 55/54, 64/63, 686/675


Mapping: [{{val|1 1 5 4 -1}}, {{val|0 2 -9 -4 15}}]
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15}}


POTE generator: ~49/40 = 356.710
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 356.710


{{Optimal ET sequence|legend=1| 10, 27, 37 }}
{{Optimal ET sequence|legend=1| 10, 27, 37 }}
Line 454: Line 449:
Comma list: 55/54, 64/63, 91/90, 169/168
Comma list: 55/54, 64/63, 91/90, 169/168


Mapping: [{{val|1 1 5 4 -1 4}}, {{val|0 2 -9 -4 15 -1}}]
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}


POTE generator: ~16/13 = 356.701
Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 356.701


{{Optimal ET sequence|legend=1| 10, 27, 37 }}
{{Optimal ET sequence|legend=1| 10, 27, 37 }}
Line 465: Line 460:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Fervor]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Fervor]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 9604/9375
[[Comma list]]: 64/63, 9604/9375


[[Mapping]]: [{{val| 1 4 -2 -2 }}, {{val| 0 -5 9 10 }}]
{{Mapping|legend=1| 1 4 -2 -2 | 0 -5 9 10 }}


{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}
{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}


[[POTE generator]]: ~7/5 = 577.776
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 577.776


{{Optimal ET sequence|legend=1| 2, 25, 27 }}
{{Optimal ET sequence|legend=1| 2, 25, 27 }}
Line 484: Line 479:
Comma list: 56/55, 64/63, 1350/1331
Comma list: 56/55, 64/63, 1350/1331


Mapping: [{{val| 1 4 -2 -2 3 }}, {{val| 0 -5 9 10 1 }}]
Mapping: {{mapping| 1 4 -2 -2 3 | 0 -5 9 10 1 }}


POTE generator: ~7/5 = 577.850
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 577.850


{{Optimal ET sequence|legend=1| 2, 25e, 27e }}
{{Optimal ET sequence|legend=1| 2, 25e, 27e }}
Line 497: Line 492:
Comma list: 56/55, 64/63, 78/77, 507/500
Comma list: 56/55, 64/63, 78/77, 507/500


Mapping: [{{val| 1 4 -2 -2 3 -4 }}, {{val| 0 -5 9 10 1 16 }}]
Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}


POTE generator: ~7/5 = 578.060
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 578.060


{{Optimal ET sequence|legend=1| 2f, 25ef, 27e }}
{{Optimal ET sequence|legend=1| 2f, 25ef, 27e }}
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: ''For the 5-limit version of this temperament, see [[High badness temperaments #Progress]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Progress]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 392/375
[[Comma list]]: 64/63, 392/375


[[Mapping]]: [{{val| 1 0 5 6 }}, {{val| 0 3 -5 -6 }}]
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}


{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}


[[POTE generator]]: ~7/5 = 562.122
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 562.122


{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 79bcc, 111bcc }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 79bcc, 111bcc }}
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Comma list: 56/55, 64/63, 77/75
Comma list: 56/55, 64/63, 77/75


Mapping: [{{val| 1 0 5 6 4 }}, {{val| 0 3 -5 -6 -1 }}]
Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}


POTE generator: ~7/5 = 562.085
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.085


{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 47bc, 79bcce }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 47bc, 79bcce }}
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Comma list: 56/55, 64/63, 66/65, 77/75
Comma list: 56/55, 64/63, 66/65, 77/75


Mapping: [{{val| 1 0 5 6 4 0 }}, {{val| 0 3 -5 -6 -1 7 }}]
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}


POTE generator: ~7/5 = 562.365
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.365


{{Optimal ET sequence|legend=1| 15, 17c, 32cf }}
{{Optimal ET sequence|legend=1| 15, 17c, 32cf }}
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Comma list: 26/25, 56/55, 64/63, 77/75
Comma list: 26/25, 56/55, 64/63, 77/75


Mapping: [{{val| 1 0 5 6 4 9 }}, {{val| 0 3 -5 -6 -1 -10 }}]
Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}


POTE generator: ~7/5 = 563.239
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 563.239


{{Optimal ET sequence|legend=1| 15f, 17c, 32c, 49c }}
{{Optimal ET sequence|legend=1| 15f, 17c, 32c, 49c }}
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== Sixix ==
== Sixix ==
:''See also: [[Dual-fifth temperaments#Dual-3 Sixix]]''
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma]]: 3125/2916
[[Comma list]]: 3125/2916


[[Mapping]]: [{{val| 1 3 4 }}, {{val| 0 -5 -6 }}]
{{Mapping|legend=1| 1 3 4 | 0 -5 -6 }}


[[POTE generator]]: ~6/5 = 338.365
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 338.365


{{Optimal ET sequence|legend=1| 7, 25, 32 }}
{{Optimal ET sequence|legend=1| 7, 25, 32 }}
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[[Comma list]]: 64/63, 3125/2916
[[Comma list]]: 64/63, 3125/2916


[[Mapping]]: [{{val| 1 3 4 0 }}, {{val| 0 -5 -6 10 }}]
{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}


{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}
{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}


[[POTE generator]]: ~6/5 = 337.442
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 337.442


{{Optimal ET sequence|legend=1| 7, 25, 32 }}
{{Optimal ET sequence|legend=1| 7, 25, 32 }}
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