Xenharmonic Wiki

Archytas clan: Difference between revisions

← Older edit
VisualWikitext
Inthar (talk | contribs)
autopatrolled
15,004 edits
m →Archy: schism isn't an exo
 
(71 intermediate revisions by 14 users not shown)
Line 1: Line 1:
The '''archytas clan''' 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]].  
{{Technical data page}}
The '''archytas clan''' (or '''archy family''') [[tempering out|tempers out]] the [[64/63|Archytas' comma]], 64/63. This means a stack of two [[3/2]] fifths [[octave reduction|octave-reduced]] equals a whole tone of [[8/7]][[~]][[9/8]] tempered together; two of these tones or equivalently four stacked fifths octave-reduced equal a [[9/7]] major third. Note the similarity in function to [[81/80]] in meantone, where four stacked fifths octave-reduced equal a [[5/4]] major third. This leads to tunings with 3's and 7's quite sharp, such as those of [[22edo]], [[27edo]], or [[49edo]].  


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.  
This article focuses on rank-2 temperaments. See [[Archytas family]] for the [[rank-3 temperament]] resulting from tempering out 64/63 alone in the full 7-limit.  


Discussed under their respective 5-limit families are [[Father family #Mother|mother]], [[Meantone family #Dominant|dominant]], [[Augmented family #Augene|augene]], [[Porcupine family|porcupine]], [[Diaschismic family #Pajara|pajara]], [[Kleismic family #Catalan|catalan]], [[Tetracot family #Modus|modus]], and [[Immunity family #Immunized|immunized]]. The rest are considered below.
== Archy ==
{{Main| Superpyth }}


= Archy =
[[Subgroup]]: 2.3.7


Subgroup: 2.3.7
[[Comma list]]: 64/63


[[Comma list]]: 64/63
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}


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


Sval mapping generators: ~2, ~3
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}


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


[[Gencom]]: [2 3/2; 64/63]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}
: [[error map]]: {{val| -3.045 +2.522 +3.952 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}
: error map: {{val| 0.000 +7.435 +12.394 }}


[[POTE generator]]: ~3/2 = 709.321
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}


{{Val list|legend=1| 2, 3, 5, 12, 17, 22 }}
[[Badness]] (Sintel): 0.159


Scales: [[archy5]], [[archy7]], [[archy12]]
Scales: [[archy5]], [[archy7]], [[archy12]]


== Supra ==
=== Overview to extensions ===
==== 7-limit extensions ====
The second comma in the comma list defines which [[7-limit]] family member we are looking at:
* [[#Schism|Schism]] adds 360/343, for a tuning around [[12edo]];
* Dominant adds [[36/35]], for a tuning between [[12edo]] and [[17edo|17c-edo]];
* [[#Quasisuper|Quasisuper]] adds [[2430/2401]], for a tuning between 17c-edo and [[22edo]];
* [[#Superpyth|Superpyth]] adds [[245/243]], for a tuning between 22edo and [[27edo]];
* [[#Quasiultra|Quasiultra]] adds 33614/32805, for a tuning between 27edo and [[32edo]];
* [[#Ultrapyth|Ultrapyth]] adds 6860/6561, for a tuning sharp of 32edo;
* Mother adds [[16/15]], for an exotemperament well tuned around [[5edo]].
 
These all use the same generators as archy.
 
[[686/675]] gives beatles, splitting the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a semioctave period. [[392/375]] gives progress, splitting the twelfth in three. [[250/243]] gives porcupine, splitting the fourth in three. [[126/125]] gives augene with a 1/3-octave period. [[4375/4374]] gives modus, splitting the fifth in four. [[3125/3024]] gives brightstone. [[9604/9375]] gives fervor. [[3125/2916]] gives sixix. [[3125/3087]] gives passion. Those split the generator in five in various ways. [[28/27]] gives blacksmith with a 1/5-octave period. Finally, [[15625/15552]] gives catalan, splitting the twelfth in six.
 
Temperaments discussed elsewhere are:
* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Augene]] (+126/125) → [[Augmented family #Augene|Augmented family]]
* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* ''[[Modus]]'' (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
 
Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.
 
==== Subgroup extensions ====
Omitting prime 5, archy can be extended to the 2.3.7.11 subgroup by identifying 11/8 as a diminished fourth (C–Gb). This is called supra, given right below. Discussed elsewhere is [[suhajira]] of the [[neutral clan #Suhajira|neutral clan]].


=== Supra ===
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


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: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}


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


Gencom: [2 3/2; 64/63 99/98]
Optimal tunings:  
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}


POTE generator: ~3/2 = 707.192
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}


{{Val list|legend=1| 5, 12, 17, 39d, 56d }}
Badness (Sintel): 0.352


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]


=== Supraphon ===
==== Supraphon ====
 
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


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 0 0 6 13 18 | 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; 64/63 78/77 99/98]


Gencom: [2 3/2; 64/63 78/77 99/98]
Optimal tunings:  
* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}


POTE generator: ~3/2 = 706.137
{{Optimal ET sequence|legend=0| 12f, 17 }}


{{Val list|legend=1| 12f, 17 }}
Badness (Sintel): 0.498


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]


== Suhajira ==
== Superpyth ==
{{Main| Superpyth }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


Subgroup: 2.3.7.11
Superpyth, virtually the canonical extension, adds [[245/243]] and [[1728/1715]] to the comma list and can be described as {{nowrap| 22 & 27 }}. ~5/4 is found at +9 generator steps, as an augmented second (C–D#). 49edo remains an obvious tuning choice.  


Comma list: 64/63, 243/242
[[Subgroup]]: 2.3.5.7


Sval mapping: [{{val| 1 1 4 2 }}, {{val| 0 2 -4 5 }}]
[[Comma list]]: 64/63, 245/243


Sval mapping generators: ~2, ~3
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}


Gencom mapping: [{{val| 1 1 0 4 2 }}, {{val| 0 1 0 -4 5 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.0549{{c}}, ~3/2 = 708.5478{{c}}
: [[error map]]: {{val| -2.945 +3.648 -0.548 +2.298 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1193{{c}}
: error map: {{val| 0.000 +8.164 +4.760 +10.935 }}


Gencom: [2 3/2; 64/63 99/98]
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}


POTE generator: ~11/9 = 353.958
[[Badness]] (Sintel): 0.818


{{Val list|legend=1| 7, 10, 17, 44e, 61de }}
=== 11-limit ===
The canonical extension to the 13-limit finds the ~11/8 at +16 generator steps, as a double-augmented second (C–Dx) and finds the ~13/8 at +13 generator steps, as a double-augmented fourth (C–Fx).


Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
Subgroup: 2.3.5.7.11


=== 2.3.7.11.13 ===
Comma list: 64/63, 100/99, 245/243


Subgroup: 2.3.7.11.13
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}


Comma list: 64/63, 78/77, 144/143
Optimal tunings:  
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


Sval mapping: [{{val| 1 1 4 2 4 }}, {{val| 0 2 -4 5 -1 }}]
{{Optimal ET sequence|legend=0| 22, 27e, 49 }}


Sval mapping generators: ~2, ~3
Badness (Sintel): 0.826


Gencom mapping: [{{val| 1 1 0 4 2 4 }}, {{val| 0 1 0 -4 5 -1 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Gencom: [2 3/2; 64/63 78/77 99/98]
Comma list: 64/63, 78/77, 91/90, 100/99
 
POTE generator: ~11/9 = 353.775
 
{{Val list|legend=1| 7, 10, 17, 44e, 61de }}
 
Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
 
= Superpyth =
{{main| Superpyth }}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 64/63, 245/243


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


{{Multival|legend=1| 1 9 -2 12 -6 -30 }}
Optimal tunings:
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}


[[POTE generator]]: ~3/2 = 710.291
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}


{{Val list|legend=1| 5, 17, 22, 27, 49 }}
Badness (Sintel): 1.02
 
[[Badness]]: 0.0323
 
== 11-limit ==
 
Subgroup: 2.3.5.7.11
 
Comma list: 64/63, 100/99, 245/243
 
Mapping: [{{val| 1 0 -12 6 -22 }}, {{val| 0 1 9 -2 16 }}]
 
POTE generator: ~3/2 = 710.175
 
{{Val list|legend=1| 22, 27e, 49 }}
 
Badness: 0.0250
 
=== 13-limit ===


==== Thomas ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 64/63, 78/77, 91/90, 100/99
Comma list: 64/63, 100/99, 169/168, 245/243


Mapping: [{{val| 1 0 -12 6 -22 -17 }}, {{val| 0 1 9 -2 16 13 }}]
Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}


POTE generator: ~3/2 = 710.479
Optimal tunings:
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


{{Val list|legend=1| 22, 27e, 49, 76bcde }}
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}


Badness: 0.0247
Badness (Sintel): 2.03


== Suprapyth ==
=== Suprapyth ===
Suprapyth finds the ~11/8 at the diminished fifth (C–Gb), and finds the ~13/8 at the diminished seventh (C–Bbb).


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 156: Line 183:
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


{{Val list|legend=1| 17, 22 }}
Optimal tunings:
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


Badness: 0.0328
{{Optimal ET sequence|legend=0| 5, 17, 22 }}


=== 13-limit ===
Badness (Sintel): 1.08


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 tunings:  
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}


{{Val list|legend=1| 17, 22, 83cdf }}
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}


Badness: 0.0363
Badness (Sintel): 1.50


= Quasisuper =
== Quasisuper ==
Quasisuper can be described as {{nowrap| 17c & 22 }}, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–Gbb).


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 -2 -6 32 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}
: [[error map]]: {{val| -3.017 +1.486 -0.435 +6.190 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 708.3716{{c}}
: error map: {{val| 0.000 +6.417 +4.855 +14.431 }}


[[POTE generator]]: ~3/2 = 708.328
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}


{{Val list|legend=1| 17c, 22, 61d }}
[[Badness]] (Sintel): 1.61


[[Badness]]: 0.0638
=== Quasisupra ===
 
== Quasisupra ==
Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament [[supra]], with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).
Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament [[supra]], with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).


Line 201: Line 234:
Comma list: 64/63, 99/98, 121/120
Comma list: 64/63, 99/98, 121/120


Mapping: [{{val| 1 2 -3 2 1 }}, {{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 tunings:  
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


{{Val list|legend=1| 17c, 22, 39d, 61d }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}


Badness: 0.0322
Badness (Sintel): 1.06
 
=== 13-limit ===


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


{{Val list|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
Optimal tunings:
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


Badness: 0.0302
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


== Quasisoup ==
Badness (Sintel): 1.25


=== Quasisoup ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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 tunings:  
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


{{Val list|legend=1| 22 }}
{{Optimal ET sequence|legend=0| 22 }}


Badness: 0.0835
Badness (Sintel): 2.76


= Schism =
== Ultrapyth ==
{{see also|Schismatic family #Schism}}
{{Main| Ultrapyth }}


Subgroup: 2.3.5.7
Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[the Biosphere #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–Cx).


[[Comma list]]: 64/63, 360/343
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 0 15 6 }}, {{val| 0 1 -8 -2 }}]
[[Comma list]]: 64/63, 6860/6561


{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}


[[POTE generator]]: ~3/2 = 701.556
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.2673{{c}}, ~3/2 = 712.0258{{c}}
: [[error map]]: {{val| -2.733 +7.338 -1.557 -3.808 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5430{{c}}
: error map: {{val| 0.000 +11.588 +3.288 +4.088 }}


{{Val list|legend=1| 12, 41d, 53d }}
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}


[[Badness]]: 0.0566
[[Badness]] (Sintel): 2.74
 
== 11-limit ==


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


Comma list: 45/44, 64/63, 99/98
Comma list: 55/54, 64/63, 2401/2376


Mapping: [{{val| 1 0 15 6 13 }}, {{val| 0 1 -8 -2 -6 }}]
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}


POTE generator: ~3/2 = 702.136
Optimal tunings:  
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


{{Val list|legend=1| 12, 29de, 41de }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Badness: 0.0375
Badness (Sintel): 2.26


= Blacksmith =
==== 13-limit ====
 
Subgroup: 2.3.5.7.11.13
[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]
 
== 5-limit (blackwood) ==
 
Subgroup: 2.3.5
 
Comma list: 256/243
 
Mapping: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
 
Mapping generators: ~9/8, ~5
 
POTE generator: ~5/4 = 399.594
 
{{Val list|legend=1| 5, 10, 15 }}
 
Badness: 0.0638
 
== 7-limit ==
 
Subgroup: 2.3.5.7
 
Comma list: 28/27, 49/48
 
Mapping: [{{val| 5 8 0 14 }}, {{val| 0 0 1 0 }}]
 
Mapping generators: ~7/6, ~5


{{Multival|legend=1| 0 5 0 8 0 -14 }}
Comma list: 55/54, 64/63, 91/90, 1573/1568


POTE generator: ~5/4 = 392.767
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}


{{Val list|legend=1| 5, 10, 15, 40b, 55b }}
Optimal tunings:
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


Badness: 0.0256
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


== 11-limit ==
Badness (Sintel): 2.03


=== Ultramarine ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 28/27, 49/48, 55/54
Comma list: 64/63, 100/99, 3773/3645
 
Mapping: [{{val| 5 8 0 14 29 }}, {{val| 0 0 1 0 -1 }}]


POTE generator: ~5/4 = 394.948
Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}


{{Val list|legend=1| 5, 10, 15, 40be, 55be, 70bde, 85bcde}}
Optimal tunings:
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


Badness: 0.0246
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}


=== 13-limit ===
Badness (Sintel): 2.58


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 28/27, 40/39, 49/48, 55/54
Comma list: 64/63, 91/90, 100/99, 847/845


Mapping: [{{val| 5 8 0 14 29 7 }}, {{val| 0 0 1 0 -1 1 }}]
Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}


POTE generator: ~5/4 = 391.0367
Optimal tunings:  
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}


{{Val list|legend=1| 5, 10, 15, 25e, 40bef}}
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}


Badness: 0.0205
Badness (Sintel): 1.89


== Farrier ==
== Quasiultra ==
Quasiultra is to ultrapyth what quasisuper is to superpyth. It is the {{nowrap| 27 & 32 }} temperament, mapping the ~5/4 to -18 fifths as a double diminished sixth (C–Abbb).


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 28/27, 49/48, 77/75
[[Comma list]]: 64/63, 33614/32805


Mapping: [{{val| 5 8 0 14 -6 }}, {{val| 0 0 1 0 2 }}]
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}


POTE generator: ~5/4 = 398.070
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.9257{{c}}, ~3/2 = 709.6211{{c}}
: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5429{{c}}
: error map: {{val| 0.000 +9.588 +5.914 +8.088 }}


{{Val list|legend=1| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}


Badness: 0.0292
[[Badness]] (Sintel): 3.34


=== 13-limit ===
== Schism ==
{{See also| Schismatic family #Schism }}


Subgroup: 2.3.5.7.11.13
Schism tempers out the [[schisma]], mapping the ~5/4 to -8 fifths as a diminished fourth (C–Fb) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53dd val) can be used.  


Comma list: 28/27, 40/39, 49/48, 66/65
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 5 8 0 14 -6 7 }}, {{val| 0 0 1 0 2 1 }}]
[[Comma list]]: 64/63, 360/343


POTE generator: ~5/4 = 396.812
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}


{{Val list|legend=1| 5e, 10e, 15 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}
: [[error map]]: {{val| -2.640 -4.583 -4.896 +20.588 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7376{{c}}
: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}


Badness: 0.0223
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}


== Ferrum ==
[[Badness]] (Sintel): 1.43


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


Comma list: 28/27, 35/33, 49/48
Comma list: 45/44, 64/63, 99/98


Mapping: [{{val| 5 8 0 14 6 }}, {{val| 0 0 1 0 1 }}]
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


POTE generator: ~5/4 = 374.763
Optimal tunings:  
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


{{Val list|legend=1| 5e, 10 }}
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}


Badness: 0.0309
Badness (Sintel): 1.24


= Beatles =
== Beatles ==
== 5-limit ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''


Subgroup: 2.3.5
Beatles tempers out 686/675, which may also be characterized by saying it tempers out [[2401/2400]]. It may be described as the {{nowrap| 10 & 17c }} temperament. It splits the fifth into two neutral-third generators of 49/40~60/49; its [[ploidacot]] is dicot. 5/4 may be found at -9 generator steps, as a semidiminished fourth (C–Fd). 27edo is an obvious tuning, though 17c-edo and 37edo are among the possibilities.


Comma list: 524288/492075
Beatles extends easily to the no-11 13-limit, as the generator can be interpreted as ~16/13, tempering out 91/90, 169/168, and 196/195.


Mapping: [{{val| 1 1 5 }}, {{val| 0 2 -9 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~512/405 = 355.930
[[Comma list]]: 64/63, 686/675


{{Val list|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}


Badness: 0.3585
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}
: [[error map]]: {{val| -3.376 +4.475 +2.682 -1.940 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}
: error map: {{val| 0.000 +10.209 +8.949 +6.847 }}


== 7-limit ==
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 1.16
 
Comma list: 64/63, 686/675
 
Mapping: [{{val| 1 1 5 4 }}, {{val| 0 2 -9 -4 }}]
 
{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}
 
POTE generator: ~49/40 = 355.904
 
{{Val list|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
 
Badness: 0.0459


; Music
; Music
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 Beatles Improv] by Herman Miller
* [https://web.archive.org/web/20201127013829/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 ''Beatles Improv''] by [[Herman Miller]]
 
== 11-limit ==


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


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


{{Val list|legend=1| 27e, 37, 64be, 91bcde }}
Optimal tunings:
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


Badness: 0.0456
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}


=== 13-limit ===
Badness (Sintel): 1.51


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


{{Val list|legend=1| 27e, 37, 64be }}
Optimal tunings:
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


Badness: 0.0302
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}


== Ringo ==
Badness (Sintel): 1.25


=== Ringo ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


{{Val list|legend=1| 10, 17c, 27e }}
Optimal tunings:
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


Badness: 0.0329
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


=== 13-limit ===
Badness (Sintel): 1.09


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 }}
 
Optimal tunings:
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}
 
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


POTE generator: ~11/9 = 355.456
Badness (Sintel): 0.935


{{Val list|legend=1| 10, 17c, 27e }}
=== Beetle ===
Subgroup: 2.3.5.7.11


Badness: 0.0226
Comma list: 55/54, 64/63, 686/675


= Passion =
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}
== 5-limit ==


Subgroup: 2.3.5
Optimal tunings:  
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


Comma list: 262144/253125
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Mapping: [{{val| 1 2 2 }}, {{val| 0 -5 4 }}]
Badness (Sintel): 1.92


Mapping generators: 2, 16/15
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~16/15 = 98.670
Comma list: 55/54, 64/63, 91/90, 169/168


{{Val list|legend=1| 12, 49, 61, 73 }}
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}


Badness: 0.1686
Optimal tunings:  
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.40


Comma list: 64/63, 3125/3087
== Progress ==
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''


Mapping: [{{val| 1 2 2 2 }}, {{val| 0 -5 4 10 }}]
Progress tempers out 392/375 and may be described as {{nowrap| 15 & 17c }}. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 32c-edo gives an obvious tuning.


Mapping generators: 2, 16/15
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 5 -4 -10 -18 -30 -12 }}
[[Comma list]]: 64/63, 392/375


POTE generator: ~16/15 = 98.153
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}


{{Val list|legend=1| 12, 37, 49, 110bcd }}
: mapping generators: ~2, ~10/7


Badness: 0.0623
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}
: [[error map]]: {{val| -4.862 +3.925 +12.908 -9.759 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 638.0791{{c}}
: error map: {{val| 0.000 +12.282 +23.291 +2.700 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


[[Badness]] (Sintel): 1.68
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 64/63, 100/99, 1375/1372
Comma list: 56/55, 64/63, 77/75
 
Mapping: [{{val| 1 2 2 2 2 }}, {{val| 0 -5 4 10 18 }}]


POTE generator: ~16/15 = 98.019
Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}


{{Val list|legend=1| 12, 37, 49 }}
Optimal tunings:
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}


Badness: 0.0408
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}


== 13-limit ==
Badness (Sintel): 1.03


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 64/63, 100/99, 196/195, 275/273
Comma list: 56/55, 64/63, 66/65, 77/75


Mapping: [{{val| 1 2 2 2 2 2 }}, {{val| 0 -5 4 10 18 21 }}]
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}


POTE generator: ~16/15 = 97.910
Optimal tunings:  
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}


{{Val list|legend=1| 12f, 25f, 37, 49f }}
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}


Badness: 0.0309
Badness (Sintel): 1.08


= Fervor =
==== Progressive ====
== 5-limit ==
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5
Comma list: 26/25, 56/55, 64/63, 77/75


Comma list: 67108864/61509375
Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}


Mapping: [{{val| 1 4 -2 }}, {{val| 0 -5 9 }}]
Optimal tunings:  
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}


POTE generator: ~64/45 = 577.705
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}


{{Val list|legend=1| 25, 27 }}
Badness (Sintel): 1.35


Badness: 0.8526
== Fervor ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''


== 7-limit ==
Fervor tempers out 9704/9375 and may be described as {{nowrap| 25 & 27 }}. It splits the 6th harmonic into five generators of ~10/7; its ploidacot is beta-pentacot. 27edo is about as accurate as it can be tuned.


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 -1 7 8 | 0 5 -9 -10 }}


{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}
: mapping generators: ~2, ~10/7


POTE generator: ~7/5 = 577.777
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}
: [[error map]]: {{val| -3.726 +3.230 +4.980 -1.550 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 622.3179{{c}}
: error map: {{val| 0.000 +9.634 +12.826 +7.996 }}


{{Val list|legend=1| 25, 27 }}
{{Optimal ET sequence|legend=1| 2, 25, 27 }}


Badness: 0.1085
[[Badness]] (Sintel): 2.74
 
== 11-limit ==


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


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 -1 7 8 4 | 0 5 -9 -10 -1 }}


POTE generator: ~7/5 = 577.850
Optimal tunings:  
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}


{{Val list|legend=1| 25e, 27e }}
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}


Badness: 0.0521
Badness (Sintel): 1.72
 
== 13-limit ==


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}
 
POTE generator: ~7/5 = 578.060
 
{{Val list|legend=1| 27e }}
 
Badness: 0.0397
 
= Progress =
== 5-limit ==
 
Subgroup: 2.3.5


Comma list: 32768/30375
Optimal tunings:  
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}


Mapping: [{{val| 1 0 5 }}, {{val| 0 3 -5 }}]
{{Optimal ET sequence|legend=0| 2f, 27e }}


POTE generator: ~64/45 = 561.264
Badness (Sintel): 1.64


{{Val list|legend=1| 15, 32c, 47bc, 62bc }}
== Sixix ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sixix (5-limit)]].''
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}


Badness: 0.2461
Sixix tempers out 3125/2916 and may be described as {{nowrap| 25 & 32 }}. It is related to the [[kleismic family]] in a way similar to the one between [[meantone]] and [[mavila]]. In both cases the generator is nominally a 6/5 and the complexity to generate major and minor chords is the same, but in sixix it is tuned extremely sharply, to the point where the 3rd and 5th harmonics are reached by going down instead of up, inverting the logic of chord construction. Its ploidacot is gamma-pentacot.  


== 7-limit ==
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 64/63, 3125/2916


Comma list: 64/63, 392/375
{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}


Mapping: [{{val| 1 0 5 6 }}, {{val| 0 3 -5 -6 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}
: [[error map]]: {{val| -1.097 +9.086 -13.503 +2.508 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 337.4588{{c}}
: error map: {{val| 0.000 +10.751 -11.066 +5.762 }}


{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}


POTE generator: ~7/5 = 562.122
[[Badness]] (Sintel): 4.02
 
{{Val list|legend=1| 15, 32c, 79bcc, 111bcc }}
 
Badness: 0.0664
 
== 11-limit ==


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


Comma list: 56/55, 64/63, 77/75
Comma list: 55/54, 64/63, 125/121


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


POTE generator: ~7/5 = 562.085
Optimal tunings:  
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}


{{Val list|legend=1| 15, 32c, 47bc, 79bcce }}
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}


Badness: 0.0310
Badness (Sintel): 2.34


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 64/63, 66/65, 77/75
Comma list: 40/39, 55/54, 64/63, 125/121
 
Mapping: [{{val| 1 0 5 6 4 0 }}, {{val| 0 3 -5 -6 -1 7 }}]
 
POTE generator: ~7/5 = 562.365
 
{{Val list|legend=1| 15, 17c, 32cf }}
 
Badness: 0.0262
 
=== Progressive ===
 
Subgroup: 2.3.5.7.11.13
 
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 }}]
 
POTE generator: ~7/5 = 563.239
 
{{Val list|legend=1| 15f, 17c, 32c, 49c }}
 
Badness: 0.0327
 
= Sixix =
:''See also: [[Dual-fifth temperaments#Dual-3 Sixix]]''
== 5-limit ==
 
Subgroup: 2.3.5
 
Comma list: 3125/2916


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


POTE generator: ~6/5 = 338.365
Optimal tunings:  
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}


{{Val list|legend=1| 7, 25, 32 }}
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


Badness: 0.1531
Badness (Sintel): 1.91


== 7-limit ==
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121


Comma list: 64/63, 3125/2916
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}


Mapping: [{{val| 1 3 4 0 }}, {{val| 0 -5 -6 10 }}]
Optimal tunings:  
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}


POTE generator: ~6/5 = 337.4419
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


{{Val list|legend=1| 7, 25, 32 }}
Badness (Sintel): 2.00


[[Category:Regular temperament theory]]
[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Temperament clan]]
[[Category:Temperament clans]]
[[Category:Archytas]]
[[Category:Pages with mostly numerical content]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Archytas_clan"