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Kleismic family: Difference between revisions

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The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. Its monzo is {{monzo| -6 -5 6 }}, and flipping that yields {{multival| 6 5 -6 }} for the [[wedgie]]. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)<sup>5</sup> = 5/2 × 15625/15552. This 5-limit temperament (virtually a [[microtemperament]]) is commonly called '''hanson''', and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. Its [[monzo]] is {{monzo| -6 -5 6 }}, and flipping that yields {{multival| 6 5 -6 }} for the [[wedgie]]. This tells us the [[generator]] is a [[6/5|classical minor third (6/5)]], and that to get to the interval class of [[5/4|major thirds]] will require five of these, and so to get to [[3/2|fifths]] will require six. In fact, (6/5)<sup>5</sup> = 5/2 × 15625/15552. This 5-limit temperament (virtually a [[microtemperament]]) is commonly called '''hanson''', and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].


The second comma of the [[Normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 5120/5103, hemifamity, gives countercata, 6144/6125, the porwell comma, gives hemikleismic, 245/243, sensamagic, gives clyde, 1029/1024, the gamelisma, gives tritikleismic, and 2401/2400, the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3 octave period with minor third generator, and quadritikleismic a 1/4 octave period with the minor third generator.
The second comma of the [[Normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. [[875/864]], the keemic comma, gives keemun. [[4375/4374]], the ragisma, gives catakleismic. [[5120/5103]], hemifamity, gives countercata. [[6144/6125]], the porwell comma, gives hemikleismic. [[245/243]], sensamagic, gives clyde. [[1029/1024]], the gamelisma, gives tritikleismic. [[2401/2400]] the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.


== Hanson ==
== Hanson ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 15625/15552
[[Comma list]]: 15625/15552


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


[[POTE generator]]: ~6/5 = 317.007
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 317.007


[[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]
* 5-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263] (untempered to 1/5-comma)
* 5-odd-limit diamond monotone and tradeoff: ~6/5 = [315.641, 317.263]
* 5-odd-limit diamond monotone and tradeoff: ~6/5 = [315.641, 317.263]


Line 28: Line 28:


; Music:
; Music:
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3 Hanson Improv] by [[Petr Parizek]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3 ''Hanson Improv''] by [[Petr Parizek]]
* [http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3 In Hanson11] by [[Chris Vaisvil]]
* [http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3 ''In Hanson11''] by [[Chris Vaisvil]]


; Analysis and diagrams:
; Analysis and diagrams:
Line 41: Line 41:
Comma list: 325/324, 625/624
Comma list: 325/324, 625/624


Sval mapping: [{{val| 1 0 1 0 }}, {{val| 0 6 5 14 }}]
Sval mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}


POTE generator: ~6/5 = 317.0756
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.0756


{{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 52: Line 52:
{{Main| Keemun }}
{{Main| Keemun }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 126/125
[[Comma list]]: 49/48, 126/125


[[Mapping]]: [{{val| 1 0 1 2 }}, {{val| 0 6 5 3 }}]
{{Mapping|legend=1| 1 0 1 2 | 0 6 5 3 }}


{{Multival|legend=1| 6 5 3 -6 -12 -7 }}
{{Multival|legend=1| 6 5 3 -6 -12 -7 }}


[[POTE generator]]: ~6/5 = 316.473
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.473


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 78: Line 78:
Comma list: 49/48, 56/55, 100/99
Comma list: 49/48, 56/55, 100/99


Mapping: [{{val| 1 0 1 2 4 }}, {{val| 0 6 5 3 -2 }}]
Mapping: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}


POTE generator: ~6/5 = 317.576
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.576


Tuning ranges:  
Tuning ranges:  
Line 96: Line 96:
Comma list: 49/48, 56/55, 65/64, 100/99
Comma list: 49/48, 56/55, 65/64, 100/99


Mapping: [{{val| 1 0 1 2 4 5 }}, {{val| 0 6 5 3 -2 -5 }}]
Mapping: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}


POTE generator: ~6/5 = 316.611
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.611


Tuning ranges:  
Tuning ranges:  
Line 114: Line 114:
Comma list: 49/48, 56/55, 91/90, 100/99
Comma list: 49/48, 56/55, 91/90, 100/99


Mapping: [{{val| 1 0 1 2 4 0 }}, {{val| 0 6 5 3 -2 14 }}]
Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}


POTE generator: ~6/5 = 317.423
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.423


Tuning ranges:  
Tuning ranges:  
Line 134: Line 134:
Comma list: 40/39, 49/48, 56/55, 66/65
Comma list: 40/39, 49/48, 56/55, 66/65


Mapping: [{{val| 1 0 1 2 4 4 }}, {{val| 0 6 5 3 -2 -1 }}]
Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}


POTE generator: ~6/5 = 318.595
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 318.595


{{Optimal ET sequence|legend=1| 4, 15, 19f, 34ff }}
{{Optimal ET sequence|legend=1| 4, 15, 19f, 34ff }}
Line 147: Line 147:
Comma list: 45/44, 49/48, 126/125
Comma list: 45/44, 49/48, 126/125


Mapping: [{{val| 1 0 1 2 -1 }}, {{val| 0 6 5 3 17 }}]
Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}


POTE generator: ~6/5 = 314.730
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 314.730


{{Optimal ET sequence|legend=1| 4e, 19, 42bcd, 61bcdd }}
{{Optimal ET sequence|legend=1| 4e, 19, 42bcd, 61bcdd }}
Line 160: Line 160:
Comma list: 45/44, 49/48, 78/77, 126/125
Comma list: 45/44, 49/48, 78/77, 126/125


Mapping: [{{val| 1 0 1 2 -1 0 }}, {{val| 0 6 5 3 17 14 }}]
Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}


POTE generator: ~6/5 = 315.044
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.044


{{Optimal ET sequence|legend=1| 4ef, 19 }}
{{Optimal ET sequence|legend=1| 4ef, 19 }}
Line 173: Line 173:
Comma list: 49/48, 55/54, 77/75
Comma list: 49/48, 55/54, 77/75


Mapping: [{{val| 1 0 1 2 0 }}, {{val| 0 6 5 3 13 }}]
Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}


POTE generator: ~6/5 = 317.656
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.656


{{Optimal ET sequence|legend=1| 15, 19e, 34e }}
{{Optimal ET sequence|legend=1| 15, 19e, 34e }}
Line 186: Line 186:
Comma list: 49/48, 55/54, 66/65, 77/75
Comma list: 49/48, 55/54, 66/65, 77/75


Mapping: [{{val| 1 0 1 2 0 0 }}, {{val| 0 6 5 3 13 14 }}]
Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}


POTE generator: ~6/5 = 317.298
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.298


{{Optimal ET sequence|legend=1| 15, 19e, 34e, 53dee }}
{{Optimal ET sequence|legend=1| 15, 19e, 34e, 53dee }}
Line 195: Line 195:


== Catalan ==
== Catalan ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 15625/15552
[[Comma list]]: 64/63, 15625/15552


[[Mapping]]: [{{val| 1 0 1 6 }}, {{val| 0 6 5 -12 }}]
{{Mapping|legend=1| 1 0 1 6 | 0 6 5 -12 }}


{{Multival|legend=1| 6 5 -12 -6 -36 -42 }}
{{Multival|legend=1| 6 5 -12 -6 -36 -42 }}


[[POTE generator]]: ~6/5 = 318.267
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 318.267


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 219: Line 219:
Comma list: 64/63, 100/99, 1331/1323
Comma list: 64/63, 100/99, 1331/1323


Mapping: [{{val| 1 0 1 6 4 }}, {{val| 0 6 5 -12 -2 }}]
Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}


POTE generator: ~6/5 = 318.282
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 318.282


Tuning ranges:  
Tuning ranges:  
Line 240: Line 240:
[[Comma list]]: 225/224, 4375/4374
[[Comma list]]: 225/224, 4375/4374


[[Mapping]]: [{{val| 1 0 1 -3 }}, {{val| 0 6 5 22 }}]
{{Mapping|legend=1| 1 0 1 -3 | 0 6 5 22 }}


{{Multival|legend=1| 6 5 22 -6 18 37 }}
{{Multival|legend=1| 6 5 22 -6 18 37 }}


[[Optimal tuning]] ([[POTE]]): ~6/5 = 316.732
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.732


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 260: Line 260:
Comma list: 169/168, 225/224, 325/324
Comma list: 169/168, 225/224, 325/324


Sval mapping: [{{val| 1 0 1 -3 0 }}, {{val| 0 6 5 22 14 }}]
Sval mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}


Optimal tuning (CTE): ~6/5 = 316.8865
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 316.8865


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 125f, 197f }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 125f, 197f }}
Line 273: Line 273:
Comma list: 225/224, 385/384, 4375/4374
Comma list: 225/224, 385/384, 4375/4374


Mapping: [{{val| 1 0 1 -3 9 }}, {{val| 0 6 5 22 -21 }}]
Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}


Optimal tuning (POTE): ~6/5 = 316.719
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.719


Tuning ranges:  
Tuning ranges:  
Line 291: Line 291:
Comma list: 169/168, 225/224, 325/324, 385/384
Comma list: 169/168, 225/224, 325/324, 385/384


Mapping: [{{val| 1 0 1 -3 9 0 }}, {{val| 0 6 5 22 -21 14 }}]
Mapping: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}


Optimal tuning (POTE): ~6/5 = 316.738
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.738


Tuning ranges:  
Tuning ranges:  
Line 309: Line 309:
Comma list: 99/98, 176/175, 2200/2187
Comma list: 99/98, 176/175, 2200/2187


Mapping: [{{val| 1 0 1 -3 -5 }}, {{val| 0 6 5 22 32 }}]
Mapping: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}


Optimal tuning (POTE): ~6/5 = 317.042
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.042


{{Optimal ET sequence|legend=1| 19e, 34d, 53 }}
{{Optimal ET sequence|legend=1| 19e, 34d, 53 }}
Line 322: Line 322:
Comma list: 99/98, 169/168, 176/175, 275/273
Comma list: 99/98, 169/168, 176/175, 275/273


Mapping: [{{val| 1 0 1 -3 -5 0 }}, {{val| 0 6 5 22 32 14 }}]
Mapping: {{mapping| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}


Optimal tuning (POTE): ~6/5 = 317.036
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.036


{{Optimal ET sequence|legend=1| 19e, 34d, 53 }}
{{Optimal ET sequence|legend=1| 19e, 34d, 53 }}
Line 335: Line 335:
Comma list: 225/224, 441/440, 4375/4374
Comma list: 225/224, 441/440, 4375/4374


Mapping: [{{val| 1 0 1 -3 -10 }}, {{val| 0 6 5 22 51 }}]
Mapping: {{mapping| 1 0 1 -3 -10 | 0 6 5 22 51 }}


Optimal tuning (POTE): ~6/5 = 316.653
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.653


{{Optimal ET sequence|legend=1| 19e, 53e, 72 }}
{{Optimal ET sequence|legend=1| 19e, 53e, 72 }}
Line 348: Line 348:
Comma list: 169/168, 225/224, 325/324, 1716/1715
Comma list: 169/168, 225/224, 325/324, 1716/1715


Mapping: [{{val| 1 0 1 -3 -10 0 }}, {{val| 0 6 5 22 51 14 }}]
Mapping: {{mapping| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}


Optimal tuning (POTE): ~6/5 = 316.639
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.639


{{Optimal ET sequence|legend=1| 19e, 53e, 72 }}
{{Optimal ET sequence|legend=1| 19e, 53e, 72 }}
Line 361: Line 361:
Comma list: 100/99, 225/224, 864/847
Comma list: 100/99, 225/224, 864/847


Mapping: [{{val| 1 0 1 -3 4 }}, {{val| 0 6 5 22 -2 }}]
Mapping: {{mapping| 1 0 1 -3 4 | 0 6 5 22 -2 }}


Optimal tuning (POTE): ~6/5 = 317.083
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.083


{{Optimal ET sequence|legend=1| 19, 34d, 53e }}
{{Optimal ET sequence|legend=1| 19, 34d, 53e }}
Line 374: Line 374:
Comma list: 78/77, 100/99, 144/143, 676/675
Comma list: 78/77, 100/99, 144/143, 676/675


Mapping: [{{val| 1 0 1 -3 4 0 }}, {{val| 0 6 5 22 -2 14 }}]
Mapping: {{mapping| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}


Optimal tuning (POTE): ~6/5 = 317.118
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.118


{{Optimal ET sequence|legend=1| 19, 34d, 53e, 87dee }}
{{Optimal ET sequence|legend=1| 19, 34d, 53e, 87dee }}
Line 387: Line 387:
Comma list: 225/224, 243/242, 4375/4356
Comma list: 225/224, 243/242, 4375/4356


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


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


{{Optimal ET sequence|legend=1| 34d, 72, 322c, …, 610bcc }}
{{Optimal ET sequence|legend=1| 34d, 72, 322c, …, 610bcc }}
Line 400: Line 402:
Comma list: 169/168, 225/224, 243/242, 325/324
Comma list: 169/168, 225/224, 243/242, 325/324


Mapping: [{{val| 2 0 2 -6 -1 0 }}, {{val| 0 6 5 22 15 14 }}]
Mapping: {{mapping| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}


Optimal tuning (POTE): ~6/5 = 316.726
Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 316.726


{{Optimal ET sequence|legend=1| 34d, 72 }}
{{Optimal ET sequence|legend=1| 34d, 72 }}
Line 413: Line 415:
Comma list: 169/168, 221/220, 225/224, 243/242, 325/324
Comma list: 169/168, 221/220, 225/224, 243/242, 325/324


Mapping: [{{val| 2 0 2 -6 -1 0 5}}, {{val| 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): ~6/5 = 316.726
Optimal tuning (POTE): ~17/12 = 1\2, ~6/5 = 316.726


{{Optimal ET sequence|legend=1| 34d, 38df, 72 }}
{{Optimal ET sequence|legend=1| 34d, 38df, 72 }}
Line 426: Line 428:
Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324
Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324


Mapping: [{{val| 2 0 2 -6 -1 0 5 -1}}, {{val| 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): ~6/5 = 316.726
Optimal tuning (POTE): ~17/12 = 1\2, ~6/5 = 316.726


{{Optimal ET sequence|legend=1| 34dh, 38df, 72 }}
{{Optimal ET sequence|legend=1| 34dh, 38df, 72 }}
Line 435: Line 437:


== Countercata ==
== Countercata ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 15625/15552
[[Comma list]]: 5120/5103, 15625/15552


[[Mapping]]: [{{val| 1 0 1 11 }}, {{val| 0 6 5 -31 }}]
{{Mapping|legend=1| 1 0 1 11 | 0 6 5 -31 }}


{{Multival|legend=1| 6 5 -31 -6 -66 -86 }}
{{Multival|legend=1| 6 5 -31 -6 -66 -86 }}


[[POTE generator]]: ~6/5 = 317.121
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 317.121


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 459: Line 461:
Comma list: 385/384, 2200/2187, 3388/3375
Comma list: 385/384, 2200/2187, 3388/3375


Mapping: [{{val| 1 0 1 11 -5 }}, {{val| 0 6 5 -31 32 }}]
Mapping: {{mapping| 1 0 1 11 -5 | 0 6 5 -31 32 }}


POTE generator: ~6/5 = 317.162
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.162


Tuning ranges:  
Tuning ranges:  
Line 477: Line 479:
Comma list: 325/324, 352/351, 385/384, 625/624
Comma list: 325/324, 352/351, 385/384, 625/624


Mapping: [{{val| 1 0 1 11 -5 0 }}, {{val| 0 6 5 -31 32 14 }}]
Mapping: {{mapping| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}


POTE generator: ~6/5 = 317.162
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.162


Tuning ranges:  
Tuning ranges:  
Line 493: Line 495:


== Metakleismic ==
== Metakleismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552, 179200/177147
[[Comma list]]: 15625/15552, 179200/177147


[[Mapping]]: [{{val| 1 0 1 -12 }}, {{val| 0 6 5 56 }}]
{{Mapping|legend=1| 1 0 1 -12 | 0 6 5 56 }}


{{Multival|legend=1| 6 5 56 -6 72 116 }}
{{Multival|legend=1| 6 5 56 -6 72 116 }}


[[POTE generator]]: ~6/5 = 317.314
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 317.314


{{Optimal ET sequence|legend=1| 34d, 87, 121, 208 }}
{{Optimal ET sequence|legend=1| 34d, 87, 121, 208 }}
Line 512: Line 514:
Comma list: 896/891, 2200/2187, 14700/14641
Comma list: 896/891, 2200/2187, 14700/14641


Mapping: [{{val| 1 0 1 -12 -5 }}, {{val| 0 6 5 56 32 }}]
Mapping: {{mapping| 1 0 1 -12 -5 | 0 6 5 56 32 }}


POTE generator: ~6/5 = 317.311
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.311


{{Optimal ET sequence|legend=1| 34d, 53d, 87, 121, 208 }}
{{Optimal ET sequence|legend=1| 34d, 53d, 87, 121, 208 }}
Line 525: Line 527:
Comma list: 325/324, 352/351, 364/363, 625/624
Comma list: 325/324, 352/351, 364/363, 625/624


Mapping: [{{val| 1 0 1 -12 -5 0 }}, {{val| 0 6 5 56 32 14 }}]
Mapping: {{mapping| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}


POTE generator: ~6/5 = 317.311
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.311


{{Optimal ET sequence|legend=1| 34d, 53d, 87, 121, 208 }}
{{Optimal ET sequence|legend=1| 34d, 53d, 87, 121, 208 }}
Line 534: Line 536:


== Hemikleismic ==
== Hemikleismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4000/3969, 6144/6125
[[Comma list]]: 4000/3969, 6144/6125


[[Mapping]]: [{{val| 1 0 1 4 }}, {{val| 0 12 10 -9 }}]
{{Mapping|legend=1| 1 0 1 4 | 0 12 10 -9 }}


{{Multival|legend=1| 12 10 -9 -12 -48 -49 }}
{{Multival|legend=1| 12 10 -9 -12 -48 -49 }}


[[POTE generator]]: ~35/32 = 158.649
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/32 = 158.649


{{Optimal ET sequence|legend=1| 15, 38, 53, 121 }}
{{Optimal ET sequence|legend=1| 15, 38, 53, 121 }}
Line 553: Line 555:
Comma list: 121/120, 176/175, 4000/3969
Comma list: 121/120, 176/175, 4000/3969


Mapping: [{{val| 1 0 1 4 2 }}, {{val| 0 12 10 -9 11 }}]
Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}


POTE generator: ~11/10 = 158.677
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 158.677


{{Optimal ET sequence|legend=1| 15, 38, 53, 68, 121e }}
{{Optimal ET sequence|legend=1| 15, 38, 53, 68, 121e }}
Line 566: Line 568:
Comma list: 121/120, 176/175, 275/273, 325/324
Comma list: 121/120, 176/175, 275/273, 325/324


POTE generator: ~11/10 = 158.655
Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}


Mapping: [{{val| 1 0 1 4 2 0 }}, {{val| 0 12 10 -9 11 28 }}]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 158.655


{{Optimal ET sequence|legend=1| 15, 38f, 53, 121e }}
{{Optimal ET sequence|legend=1| 15, 38f, 53, 121e }}
Line 575: Line 577:


== Clyde ==
== Clyde ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 3136/3125
[[Comma list]]: 245/243, 3136/3125


[[Mapping]]: [{{val| 1 6 6 12 }}, {{val| 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


{{Multival|legend=1| 12 10 25 -12 6 30 }}
{{Multival|legend=1| 12 10 25 -12 6 30 }}


[[POTE generator]]: ~9/7 = 441.335
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 441.335


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 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| 1 0 0 0 }}, {{monzo| 6/25 0 0 12/25 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 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]]s (unchanged-intervals): 2, 7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) 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 603: Line 605:
Comma list: 245/243, 385/384, 3136/3125
Comma list: 245/243, 385/384, 3136/3125


Mapping: [{{val| 1 6 6 12 -5 }}, {{val| 0 -12 -10 -25 23 }}]
Mapping: {{mapping| 1 6 6 12 -5 | 0 -12 -10 -25 23 }}


POTE generator: ~9/7 = 441.355
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.355


{{Optimal ET sequence|legend=1| 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd }}
{{Optimal ET sequence|legend=1| 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd }}
Line 616: Line 618:
Comma list: 196/195, 245/243, 385/384, 625/624
Comma list: 196/195, 245/243, 385/384, 625/624


Mapping: [{{val| 1 6 6 12 -5 14 }}, {{val| 0 -12 -10 -25 23 -28 }}]
Mapping: {{mapping| 1 6 6 12 -5 14 | 0 -12 -10 -25 23 -28 }}


POTE generator: ~9/7 = 441.363
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.363


{{Optimal ET sequence|legend=1| 19, 49ef, 68, 87, 503bdf, 590bdf }}
{{Optimal ET sequence|legend=1| 19, 49ef, 68, 87, 503bdf, 590bdf }}
Line 625: Line 627:


== Tritikleismic ==
== Tritikleismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 15625/15552
[[Comma list]]: 1029/1024, 15625/15552


[[Mapping]]: [{{val| 3 0 3 10 }}, {{val| 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


{{Multival|legend=1| 18 15 -6 -18 -60 -56 }}
{{Multival|legend=1| 18 15 -6 -18 -60 -56 }}


[[POTE generator]]: ~6/5 = 316.872
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~6/5 = 316.872


[[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]]s (unchanged-intervals): 2, 7/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) 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]]s (unchanged-intervals): 2, 9/7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) 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 }}
Line 654: Line 656:
Comma list: 385/384, 441/440, 4000/3993
Comma list: 385/384, 441/440, 4000/3993


Mapping: [{{val| 3 0 3 10 8 }}, {{val| 0 6 5 -2 3 }}]
Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}


Mapping generators: ~44/35, ~6/5
Optimal tuning (POTE): ~44/35 = 1\3, ~6/5 = 316.881
 
POTE generator: ~6/5 = 316.881


Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~6/5 = {{monzo| 5/21 1/7 0 -1/14 }}
* 11-odd-limit: ~6/5 = {{monzo| 5/21 1/7 0 -1/14 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 46/21 5/7 0 -5/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 71/21 3/7 0 -3/14 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 46/21 5/7 0 -5/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 71/21 3/7 0 -3/14 0 }}]
: Eigenmonzos (unchanged-intervals): 2, 9/7
: eigenmonzo (unchanged-interval) 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 }}
Line 674: Line 674:
Comma list: 325/324, 364/363, 385/384, 625/624
Comma list: 325/324, 364/363, 385/384, 625/624


Mapping: [{{val| 3 0 3 10 8 0 }}, {{val| 0 6 5 -2 3 14 }}]
Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}
 
Mapping generators: ~44/35, ~6/5


POTE generator: ~6/5 = 316.9585
Optimal tuning (POTE): ~44/35 = 1\3, ~6/5 = 316.9585


{{Optimal ET sequence|legend=1| 72, 87, 159 }}
{{Optimal ET sequence|legend=1| 72, 87, 159 }}
Line 689: Line 687:
Comma list: 273/272, 325/324, 364/363, 375/374, 385/384
Comma list: 273/272, 325/324, 364/363, 375/374, 385/384


Mapping: [{{val| 3 0 3 10 8 0 -2 }}, {{val| 0 6 5 -2 3 14 18 }}]
Mapping: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}
 
Mapping generators: ~34/27, ~6/5


POTE generator: ~6/5 = 316.9082
Optimal tuning (POTE): ~34/27 = 1\3, ~6/5 = 316.9082


{{Optimal ET sequence|legend=1| 72, 159, 231f }}
{{Optimal ET sequence|legend=1| 72, 159, 231f }}
Line 700: Line 696:


== Quadritikleismic ==
== Quadritikleismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 15625/15552
[[Comma list]]: 2401/2400, 15625/15552


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


{{Multival|legend=1| 24 20 16 -24 -42 -19 }}
{{Multival|legend=1| 24 20 16 -24 -42 -19 }}


[[POTE generator]]: ~6/5 = 316.9999
[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~6/5 = 316.9999


{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
Line 719: Line 717:
Comma list: 385/384, 1375/1372, 6250/6237
Comma list: 385/384, 1375/1372, 6250/6237


Mapping: [{{val| 4 0 4 7 17 }}, {{val| 0 6 5 4 -3 }}]
Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}


POTE generator: ~6/5 = 316.9247
Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9247


{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
Line 732: Line 730:
Comma list: 325/324, 385/384, 625/624, 1375/1372
Comma list: 325/324, 385/384, 625/624, 1375/1372


Mapping: [{{val| 4 0 4 7 17 0 }}, {{val| 0 6 5 4 -3 14 }}]
Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}


POTE generator: ~6/5 = 316.9887
Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9887


{{Optimal ET sequence|legend=1| 68, 72, 140, 212 }}
{{Optimal ET sequence|legend=1| 68, 72, 140, 212 }}
Line 745: Line 743:
Comma list: 289/288, 325/324, 385/384, 442/441, 625/624
Comma list: 289/288, 325/324, 385/384, 442/441, 625/624


Mapping: [{{val| 4 0 4 7 17 0 10 }}, {{val| 0 6 5 4 -3 14 6 }}]
Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}


POTE generator: ~6/5 = 316.9846
Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9846


{{Optimal ET sequence|legend=1| 68, 72, 140, 212g }}
{{Optimal ET sequence|legend=1| 68, 72, 140, 212g }}
Line 754: Line 752:


== Kleiboh ==
== Kleiboh ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 3125/3087
[[Comma list]]: 1728/1715, 3125/3087


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


{{Multival|legend=1| 18 15 13 -18 -30 -12 }}
{{Multival|legend=1| 18 15 13 -18 -30 -12 }}


[[POTE generator]]: ~25/21 = 294.303
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 294.303


{{Optimal ET sequence|legend=1| 49, 53, 314d }}
{{Optimal ET sequence|legend=1| 49, 53, 314d }}
Line 773: Line 773:
Comma list: 176/175, 540/539, 3125/3087
Comma list: 176/175, 540/539, 3125/3087


Mapping: [{{val| 1 6 6 6 14 }}, {{val| 0 -18 -15 -13 -43 }}]
Mapping: {{mapping| 1 6 6 6 14 | 0 -18 -15 -13 -43 }}


POTE generator: ~25/21 = 294.181
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 294.181


{{Optimal ET sequence|legend=1| 49, 53, 102d, 155d }}
{{Optimal ET sequence|legend=1| 49, 53, 102d, 155d }}
Line 786: Line 786:
Comma list: 176/175, 275/273, 325/324, 540/539
Comma list: 176/175, 275/273, 325/324, 540/539


Mapping: [{{val| 1 6 6 6 14 14 }}, {{val| 0 -18 -15 -13 -43 -42 }}]
Mapping: {{mapping| 1 6 6 6 14 14 | 0 -18 -15 -13 -43 -42 }}


POTE generator: ~13/11 = 294.187
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 294.187


{{Optimal ET sequence|legend=1| 49f, 53, 102df, 155d }}
{{Optimal ET sequence|legend=1| 49f, 53, 102df, 155d }}
Line 797: Line 797:
The ''marfifths'' temperament (19&amp;140) tempers out the [[hemimage comma]], 10976/10935. It splits the interval of major tenth (~10/3) into three marvelous fifth ([[112/75]]) intervals, and uses it for a generator.
The ''marfifths'' temperament (19&amp;140) tempers out the [[hemimage comma]], 10976/10935. It splits the interval of major tenth (~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


[[Comma list]]: 10976/10935, 15625/15552
[[Comma list]]: 10976/10935, 15625/15552


[[Mapping]]: [{{val| 1 -6 -4 -17 }}, {{val| 0 18 15 47 }}]
{{Mapping|legend=1| 1 -6 -4 -17 | 0 18 15 47 }}
 
: mapping generators: ~2, ~75/56


{{Multival|legend=1| 18 15 47 -18 24 67 }}
{{Multival|legend=1| 18 15 47 -18 24 67 }}


[[POTE generator]]: ~75/56 = 505.705
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/56 = 505.705


{{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 }}
Line 816: Line 818:
Comma list: 385/384, 6250/6237, 10976/10935
Comma list: 385/384, 6250/6237, 10976/10935


Mapping: [{{val| 1 -6 -4 -17 22 }}, {{val| 0 18 15 47 -44 }}]
Mapping: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}


POTE generator: ~75/56 = 505.684
Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.684


{{Optimal ET sequence|legend=1| 19, 140, 159, 299 }}
{{Optimal ET sequence|legend=1| 19, 140, 159, 299 }}
Line 829: Line 831:
Comma list: 325/324, 385/384, 625/624, 10976/10935
Comma list: 325/324, 385/384, 625/624, 10976/10935


Mapping: [{{val| 1 -6 -4 -17 22 -14 }}, {{val| 0 18 15 47 -44 42 }}]
Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}


POTE generator: ~75/56 = 505.686
Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.686


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


Badness: 0.030082
Badness: 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 &amp; 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 844: Line 846:
Comma list: 1375/1372, 2200/2187, 5632/5625
Comma list: 1375/1372, 2200/2187, 5632/5625


Mapping: [{{val| 1 -6 -4 -17 -37 }}, {{val| 0 18 15 47 96 }}]
Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}


POTE generator: ~75/56 = 505.740
Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.740


{{Optimal ET sequence|legend=1|19e, 121, 140, 261, 401, 662bd }}
{{Optimal ET sequence|legend=1| 19e, 121, 140, 261, 401, 662bd }}


Badness: 0.061172
Badness: 0.061172
Line 857: Line 859:
Comma list: 325/324, 352/351, 625/624, 1375/1372
Comma list: 325/324, 352/351, 625/624, 1375/1372


Mapping: [{{val| 1 -6 -4 -17 -37 -14 }}, {{val| 0 18 15 47 96 42 }}]
Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}


POTE generator: ~75/56 = 505.740
Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.740


{{Optimal ET sequence|legend=1| 19e, 121, 140, 261, 401, 662bd }}
{{Optimal ET sequence|legend=1| 19e, 121, 140, 261, 401, 662bd }}
Line 866: Line 868:


=== 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 &amp; 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 872: Line 874:
Comma list: 540/539, 896/891, 15625/15552
Comma list: 540/539, 896/891, 15625/15552


Mapping: [{{val| 1 -6 -4 -17 14 }}, {{val| 0 18 15 47 -25 }}]
Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}


POTE generator: ~75/56 = 505.769
Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.769


{{Optimal ET sequence|legend=1| 19, 102d, 121 }}
{{Optimal ET sequence|legend=1| 19, 102d, 121 }}
Line 885: Line 887:
Comma list: 325/324, 540/539, 625/624, 896/891
Comma list: 325/324, 540/539, 625/624, 896/891


Mapping: [{{val| 1 -6 -4 -17 14 -14 }}, {{val| 0 18 15 47 -25 42 }}]
Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}


POTE generator: ~75/56 = 505.771
Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.771


{{Optimal ET sequence|legend=1| 19, 121 }}
{{Optimal ET sequence|legend=1| 19, 121 }}
Line 894: Line 896:


== Marthirds ==
== Marthirds ==
The ''marthirds'' temperament (19&amp;193) tempers out the sesquiquartisma (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 &amp; 193) tempers out the sesquiquartisma (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


[[Comma list]]: 15625/15552, 2460375/2458624
[[Comma list]]: 15625/15552, 2460375/2458624


[[Mapping]]: [{{val| 1 -6 -4 -19 }}, {{val| 0 24 20 69 }}]
{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}


{{Multival|legend=1| 24 20 69 -24 42 104 }}
{{Multival|legend=1| 24 20 69 -24 42 104 }}


[[POTE generator]]: ~56/45 = 379.252
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 379.252


{{Optimal ET sequence|legend=1| 19, 193, 212, 617c, 829c }}
{{Optimal ET sequence|legend=1| 19, 193, 212, 617c, 829c }}
Line 915: Line 917:
Comma list: 1375/1372, 15625/15552, 19712/19683
Comma list: 1375/1372, 15625/15552, 19712/19683


Mapping: [{{val| 1 -6 -4 -19 -43 }}, {{val| 0 24 20 69 147 }}]
Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}


POTE generator: ~56/45 = 379.257
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.257


{{Optimal ET sequence|legend=1| 19e, 193, 212, 405, 617c, 1022cce }}
{{Optimal ET sequence|legend=1| 19e, 193, 212, 405, 617c, 1022cce }}
Line 928: Line 930:
Comma list: 325/324, 625/624, 1375/1372, 19712/19683
Comma list: 325/324, 625/624, 1375/1372, 19712/19683


Mapping: [{{val| 1 -6 -4 -19 -43 -14 }}, {{val| 0 24 20 69 147 56 }}]
Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}


POTE generator: ~56/45 = 379.256
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.256


{{Optimal ET sequence|legend=1| 19e, 193, 212, 405f, 617cff }}
{{Optimal ET sequence|legend=1| 19e, 193, 212, 405f, 617cff }}
Line 943: Line 945:
[[Comma list]]: 15625/15552, 117649/116640
[[Comma list]]: 15625/15552, 117649/116640


[[Mapping]]: [{{val| 1 0 1 1 }}, {{val| 0 36 30 41 }}]
{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}


[[Optimal tuning]] ([[CTE]]): ~36/35 = 52.8562
: mapping generator: ~2, ~36/35
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~36/35 = 52.8562


{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
Line 956: Line 960:
Comma list: 385/384, 6250/6237, 67228/66825
Comma list: 385/384, 6250/6237, 67228/66825


Mapping: [{{val| 1 0 1 1 5 }}, {{val| 0 36 30 41 -35 }}]
Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}


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


{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
Line 965: Line 969:


== Novemkleismic ==
== Novemkleismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552, 40353607/40310784
[[Comma list]]: 15625/15552, 40353607/40310784


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


{{Multival|legend=1| 54 45 54 -54 -66 -1 }}
{{Multival|legend=1| 54 45 54 -54 -66 -1 }}


[[POTE generator]]: ~6/5 = 317.005
[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 1\9, ~6/5 = 317.005


{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
Line 984: Line 990:
Comma list: 1375/1372, 4000/3993, 15625/15552
Comma list: 1375/1372, 4000/3993, 15625/15552


Mapping: [{{val| 9 0 9 11 24 }}, {{val| 0 6 5 6 3 }}]
Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}


POTE generator: ~6/5 = 317.010
Optimal tuning (POTE): ~250/231 = 1\9, ~6/5 = 317.010


{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 882c }}
{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 882c }}
Line 997: Line 1,003:
Comma list: 325/324, 625/624, 1375/1372, 4000/3993
Comma list: 325/324, 625/624, 1375/1372, 4000/3993


Mapping: [{{val| 9 0 9 11 24 0 }}, {{val| 0 6 5 6 3 14 }}]
Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}


POTE generator: ~6/5 = 317.086
Optimal tuning (POTE): ~250/231 = 1\9, ~6/5 = 317.086


{{Optimal ET sequence|legend=1| 72, 261, 333, 738cf, 1071bcff }}
{{Optimal ET sequence|legend=1| 72, 261, 333, 738cf, 1071bcff }}
Line 1,006: Line 1,012:


== Sqrtphi ==
== Sqrtphi ==
{{main| Sqrtphi }}
{{Main| Sqrtphi }}


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552, 16875/16807
[[Comma list]]: 15625/15552, 16875/16807


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


[[POTE generator]]: ~125/98 = 416.603
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/98 = 416.603


{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}
{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}
Line 1,027: Line 1,035:
Comma list: 540/539, 1375/1372, 4375/4356
Comma list: 540/539, 1375/1372, 4375/4356


Mapping: [{{val| 1 12 11 16 17 }}, {{val| 0 -30 -25 -38 -39 }}]
Mapping: {{mapping| 1 12 11 16 17 | 0 -30 -25 -38 -39 }}


POTE generator: ~14/11 = 416.604
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.604


{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}
{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}
Line 1,040: Line 1,048:
Comma list: 325/324, 364/363, 625/624, 1375/1372
Comma list: 325/324, 364/363, 625/624, 1375/1372


Mapping: [{{val| 1 12 11 16 17 28 }}, {{val| 0 -30 -25 -38 -39 -70 }}]
Mapping: {{mapping| 1 12 11 16 17 28 | 0 -30 -25 -38 -39 -70 }}


POTE generator: ~14/11 = 416.585
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585


{{Optimal ET sequence|legend=1| 72, 121, 193 }}
{{Optimal ET sequence|legend=1| 72, 121, 193 }}
Line 1,053: Line 1,061:
Comma list: 325/324, 364/363, 375/374, 540/539, 595/594
Comma list: 325/324, 364/363, 375/374, 540/539, 595/594


Mapping: [{{val| 1 12 11 16 17 28 27 }}, {{val| 0 -30 -25 -38 -39 -70 -66 }}]
Mapping: {{mapping| 1 12 11 16 17 28 27 | 0 -30 -25 -38 -39 -70 -66 }}


POTE generator: ~14/11 = 416.585
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585


{{Optimal ET sequence|legend=1| 72, 121, 193 }}
{{Optimal ET sequence|legend=1| 72, 121, 193 }}
Line 1,066: Line 1,074:
Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594


Mapping: [{{val| 1 12 11 16 17 28 27 -2 }}, {{val| 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 }}


POTE generator: ~14/11 = 416.580
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.580


{{Optimal ET sequence|legend=1| 72, 121, 193 }}
{{Optimal ET sequence|legend=1| 72, 121, 193 }}
Line 1,080: Line 1,088:


; Music
; Music
* [http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3 Prelude for Piano in Square root of Phi Tuning] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3 ''Prelude for Piano in Square root of Phi Tuning''] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3 A Fight for Phi] by [[Vito Sicurella]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3 ''A Fight for Phi''] by [[Vito Sicurella]]


[[Category:Temperament families]]
[[Category:Temperament families]]
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