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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]].  
{{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:
== Archy ==
* ''[[Father]]'' → [[Father family #Mother]]
{{Main| Superpyth }}
* ''[[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.
[[Subgroup]]: 2.3.7


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


[[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]]
=== 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 ===
=== Supra ===
Line 41: Line 70:
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 }}


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


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]
Line 60: Line 91:
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 }}


Vals: {{Val list| 12f, 17 }}
Badness (Sintel): 0.498


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


=== Suhajira ===
== Superpyth ==
Subgroup: 2.3.7.11
{{Main| Superpyth }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


Comma list: 64/63, 243/242
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.


Sval mapping: [{{val| 1 1 4 2 }}, {{val| 0 2 -4 5 }}]
[[Subgroup]]: 2.3.5.7


Sval mapping generators: ~2, ~11/9
[[Comma list]]: 64/63, 245/243


Gencom mapping: [{{val| 1 1 0 4 2 }}, {{val| 0 1 0 -4 5 }}]
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}


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


POTE generator: ~11/9 = 353.958
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}


Vals: {{Val list| 7, 10, 17, 44e, 61de }}
[[Badness]] (Sintel): 0.818


Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
=== 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).


==== 2.3.7.11.13 ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.7.11.13


Comma list: 64/63, 78/77, 144/143
Comma list: 64/63, 100/99, 245/243


Sval mapping: [{{val| 1 1 4 2 4 }}, {{val| 0 2 -4 5 -1 }}]
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}


Sval mapping generators: ~2, ~11/9
Optimal tunings:
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


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


Gencom: [2 3/2; 64/63 78/77 99/98]
Badness (Sintel): 0.826


POTE generator: ~11/9 = 353.775
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 7, 10, 17, 44e, 61de }}
Comma list: 64/63, 78/77, 91/90, 100/99
 
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 }}]
 
{{Multival|legend=1| 1 9 -2 12 -6 -30 }}


[[POTE generator]]: ~3/2 = 710.291
Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}


{{Val list|legend=1| 5, 17, 22, 27, 49 }}
Optimal tunings:
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}


[[Badness]]: 0.032318
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}


=== 11-limit ===
Badness (Sintel): 1.02
Subgroup: 2.3.5.7.11


Comma list: 64/63, 100/99, 245/243
==== Thomas ====
 
Mapping: [{{val| 1 0 -12 6 -22 }}, {{val| 0 1 9 -2 16 }}]
 
POTE generator: ~3/2 = 710.175
 
Vals: {{Val list| 22, 27e, 49 }}
 
Badness: 0.024976
 
==== 13-limit ====
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}}


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


Badness: 0.024673
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


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 tunings:  
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


Vals: {{Val list| 5, 12c, 17, 22 }}
{{Optimal ET sequence|legend=0| 5, 17, 22 }}


Badness: 0.032768
Badness (Sintel): 1.08


==== 13-limit ====
==== 13-limit ====
Line 173: Line 198:
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}}


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


Badness: 0.036336
Badness (Sintel): 1.50


== Quasisuper ==
== Quasisuper ==
Subgroup: 2.3.5.7
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


[[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 }}


[[POTE generator]]: ~3/2 = 708.328
[[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 }}


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


[[Badness]]: 0.063794
[[Badness]] (Sintel): 1.61


=== Quasisupra ===
=== Quasisupra ===
Line 203: Line 234:
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 tunings:  
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


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


Badness: 0.032203
Badness (Sintel): 1.06


==== 13-limit ====
==== 13-limit ====
Line 216: Line 249:
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 tunings:  
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


Vals: {{Val list| 17c, 22, 39d, 61df, 100bcdf }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


Badness: 0.030219
Badness (Sintel): 1.25


=== Quasisoup ===
=== Quasisoup ===
Line 229: Line 264:
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}}


Vals: {{Val list| 5ce, 17ce, 22 }}
{{Optimal ET sequence|legend=0| 22 }}


Badness: 0.083490
Badness (Sintel): 2.76


== Ultrapyth ==
== Ultrapyth ==
=== 7-limit ===
{{Main| Ultrapyth }}
Subgroup: 2.3.5.7


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


[[Mapping]]: [{{val| 1 0 -20 6 }}, {{val| 0 1 14 -2 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 1 14 -2 20 -6 -44 }}
[[Comma list]]: 64/63, 6860/6561


[[POTE generator]]: ~3/2 = 713.651
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}


{{Val list|legend=1| 5, 32, 37 }}
[[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 }}


[[Badness]]: 0.108466
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}


==== 2.3.5.7.13 ====
[[Badness]] (Sintel): 2.74
Subgroup: 2.3.5.7.13
 
Comma list: 64/63, 91/90, 4394/4375
 
Mapping: [{{val| 1 0 -20 6 -25 }}, {{val| 0 1 14 -2 18 }}]
 
POTE generator: ~3/2 = 713.7453
 
Vals: {{Val list| 5, 32, 37, 79bc }}
 
Badness: 0.0547


=== 11-limit ===
=== 11-limit ===
Line 271: Line 300:
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 tunings:  
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


Vals: {{Val list| 5, 32, 37 }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Badness: 0.068238
Badness (Sintel): 2.26


==== 13-limit ====
==== 13-limit ====
Line 284: Line 315:
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 tunings:  
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


Vals: {{Val list| 5, 32, 37 }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Badness: 0.049172
Badness (Sintel): 2.03


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


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 tunings:  
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


Vals: {{Val list| 5e, 32e, 37, 79bce, 116bbce }}
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}


Badness: 0.078068
Badness (Sintel): 2.58


==== 13-limit ====
==== 13-limit ====
Line 310: Line 345:
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 }}
 
Optimal tunings:
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}


POTE generator: ~3/2 = 713.811
Badness (Sintel): 1.89


Vals: {{Val list| 5e, 32e, 37, 79bcef, 116bbcef }}
== 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).


Badness: 0.045653
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 64/63, 33614/32805
 
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
 
[[Badness]] (Sintel): 3.34


== Schism ==
== Schism ==
{{see also|Schismatic family #Schism}}
{{See also| Schismatic family #Schism }}


Subgroup: 2.3.5.7
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.
 
[[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 }}
[[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 }}


[[POTE generator]]: ~3/2 = 701.556
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}


{{Val list|legend=1| 12, 41d, 53d }}
[[Badness]] (Sintel): 1.43
 
[[Badness]]: 0.056648


=== 11-limit ===
=== 11-limit ===
Line 340: Line 400:
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 tunings:  
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


Vals: {{Val list| 12, 29de, 41de }}
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}


Badness: 0.037482
Badness (Sintel): 1.24


== Beatles ==
== Beatles ==
Subgroup: 2.3.5
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''


[[Comma]]: 524288/492075
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.


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


[[POTE generator]]: ~512/405 = 355.930
[[Subgroup]]: 2.3.5.7
 
{{Val list|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}
 
[[Badness]]: 0.358542
 
=== 7-limit ===
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 }}
[[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 }}


[[POTE generator]]: ~49/40 = 355.904
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}


{{Val list|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
[[Badness]] (Sintel): 1.16
 
[[Badness]]: 0.045872


; 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 ===
Line 384: Line 441:
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 tunings:  
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


Vals: {{Val list| 27e, 37, 64be, 91bcde }}
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}


Badness: 0.045639
Badness (Sintel): 1.51


==== 13-limit ====
==== 13-limit ====
Line 397: Line 456:
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 tunings:  
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


Vals: {{Val list| 27e, 37, 64be }}
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}


Badness: 0.030161
Badness (Sintel): 1.25


=== Ringo ===
=== Ringo ===
Line 410: Line 471:
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 tunings:  
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


Vals: {{Val list| 10, 17c, 27e }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


Badness: 0.032863
Badness (Sintel): 1.09


==== 13-limit ====
==== 13-limit ====
Line 423: Line 486:
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 tunings:  
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}


Vals: {{Val list| 10, 17c, 27e }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


Badness: 0.022619
Badness (Sintel): 0.935


=== Beetle ===
=== Beetle ===
Line 436: Line 501:
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 tunings:  
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


Vals: {{Val list| 10, 27, 37 }}
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Badness: 0.058084
Badness (Sintel): 1.92


==== 13-limit ====
==== 13-limit ====
Line 449: Line 516:
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 tunings:  
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


Vals: {{Val list| 10, 27, 37 }}
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Badness: 0.033971
Badness (Sintel): 1.40


== Fervor ==
== Progress ==
Subgroup: 2.3.5
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''


[[Comma]]: 67108864/61509375
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]]: [{{val| 1 4 -2 }}, {{val| 0 -5 9 }}]
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~64/45 = 577.705
[[Comma list]]: 64/63, 392/375
 
{{Val list|legend=1| 2, 25, 27 }}
 
[[Badness]]: 0.852612
 
=== 7-limit ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 64/63, 9604/9375


[[Mapping]]: [{{val| 1 4 -2 -2 }}, {{val| 0 -5 9 10 }}]
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}


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


[[POTE generator]]: ~7/5 = 577.776
[[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 }}


{{Val list|legend=1| 2, 25, 27 }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


[[Badness]]: 0.108455
[[Badness]] (Sintel): 1.68


=== 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, 77/75


Mapping: [{{val| 1 4 -2 -2 3 }}, {{val| 0 -5 9 10 1 }}]
Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}


POTE generator: ~7/5 = 577.850
Optimal tunings:  
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}


Vals: {{Val list| 2, 25e, 27e }}
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}


Badness: 0.052054
Badness (Sintel): 1.03


=== 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, 66/65, 77/75
 
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}


Mapping: [{{val| 1 4 -2 -2 3 -4 }}, {{val| 0 -5 9 10 1 16 }}]
Optimal tunings:  
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}


POTE generator: ~7/5 = 578.060
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}


Vals: {{Val list| 2f, 25ef, 27e }}
Badness (Sintel): 1.08


Badness: 0.039705
==== Progressive ====
Subgroup: 2.3.5.7.11.13


== Progress ==
Comma list: 26/25, 56/55, 64/63, 77/75
Subgroup: 2.3.5
 
Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}


[[Comma]]: 32768/30375
Optimal tunings:  
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}


[[Mapping]]: [{{val| 1 0 5 }}, {{val| 0 3 -5 }}]
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}


[[POTE generator]]: ~64/45 = 561.264
Badness (Sintel): 1.35


{{Val list|legend=1| 2, 13, 15, 32c, 47bc, 62bc }}
== Fervor ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''


Badness: 0.246073
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.  


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


[[Comma list]]: 64/63, 392/375
[[Comma list]]: 64/63, 9604/9375


[[Mapping]]: [{{val| 1 0 5 6 }}, {{val| 0 3 -5 -6 }}]
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}


{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
: mapping generators: ~2, ~10/7


[[POTE generator]]: ~7/5 = 562.122
[[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| 2, 13, 15, 32c, 79bcc, 111bcc }}
{{Optimal ET sequence|legend=1| 2, 25, 27 }}


[[Badness]]: 0.066400
[[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, 77/75
Comma list: 56/55, 64/63, 1350/1331


Mapping: [{{val| 1 0 5 6 4 }}, {{val| 0 3 -5 -6 -1 }}]
Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}


POTE generator: ~7/5 = 562.085
Optimal tunings:  
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}


Vals: {{Val list| 2, 13, 15, 32c, 47bc, 79bcce }}
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}


Badness: 0.031036
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, 66/65, 77/75
Comma list: 56/55, 64/63, 78/77, 507/500
 
Mapping: {{mapping| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}
 
Optimal tunings:
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}
 
{{Optimal ET sequence|legend=0| 2f, 27e }}
 
Badness (Sintel): 1.64
 
== Sixix ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sixix (5-limit)]].''
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}
 
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.


Mapping: [{{val| 1 0 5 6 4 0 }}, {{val| 0 3 -5 -6 -1 7 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~7/5 = 562.365
[[Comma list]]: 64/63, 3125/2916


Vals: {{Val list| 15, 17c, 32cf }}
{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}


Badness: 0.026214
[[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 }}


=== Progressive ===
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
Subgroup: 2.3.5.7.11.13


Comma list: 26/25, 56/55, 64/63, 77/75
[[Badness]] (Sintel): 4.02


Mapping: [{{val| 1 0 5 6 4 9 }}, {{val| 0 3 -5 -6 -1 -10 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~7/5 = 563.239
Comma list: 55/54, 64/63, 125/121


Vals: {{Val list| 15f, 17c, 32c, 49c }}
Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}


Badness: 0.032721
Optimal tunings:  
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}


== Sixix ==
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}
:''See also: [[Dual-fifth temperaments#Dual-3 Sixix]]''


Subgroup: 2.3.5
Badness (Sintel): 2.34


[[Comma]]: 3125/2916
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 1 3 4 }}, {{val| 0 -5 -6 }}]
Comma list: 40/39, 55/54, 64/63, 125/121


[[POTE generator]]: ~6/5 = 338.365
Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}


{{Val list|legend=1| 7, 25, 32 }}
Optimal tunings:
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}


[[Badness]]: 0.153088
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


=== 7-limit ===
Badness (Sintel): 1.91
Subgroup: 2.3.5.7


[[Comma list]]: 64/63, 3125/2916
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Mapping]]: [{{val| 1 3 4 0 }}, {{val| 0 -5 -6 10 }}]
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121


{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}


[[POTE generator]]: ~6/5 = 337.442
Optimal tunings:  
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}


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


[[Badness]]: 0.158903
Badness (Sintel): 2.00


[[Category:Regular temperament theory]]
[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Temperament clan]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 12:52, 22 July 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The archytas clan (or archy family) tempers out the Archytas' comma, 64/63. This means a stack of two 3/2 fifths 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.

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.

Archy

Main article: Superpyth

Subgroup: 2.3.7

Comma list: 64/63

Sval mapping: [⟨1 0 6], ⟨0 1 -2]]

sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 0 0 6], ⟨0 1 0 -2]]

gencom: [2 3; 64/63]

Optimal tunings:

  • WE: ~2 = 1196.9552 ¢, ~3/2 = 707.5215 ¢
error map: ⟨-3.045 +2.522 +3.952]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.3901 ¢
error map: ⟨0.000 +7.435 +12.394]

Optimal ET sequence: 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd

Badness (Sintel): 0.159

Scales: archy5, archy7, archy12

Overview to extensions

7-limit extensions

The second comma in the comma list defines which 7-limit family member we are looking at:

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:

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.

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]

Gencom mapping: [⟨1 0 0 6 13], ⟨0 1 0 -2 -6]]

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

Optimal tunings:

  • WE: ~2 = 1197.2650 ¢, ~3/2 = 705.5803 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 707.4981 ¢

Optimal ET sequence: 5, 12, 17, 39d, 56d

Badness (Sintel): 0.352

Scales: supra7, supra12

Supraphon

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 99/98

Sval mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]

Gencom mapping: [⟨1 0 0 6 13 18], ⟨0 1 0 -2 -6 -9]]

gencom: [2 3; 64/63 78/77 99/98]

Optimal tunings:

  • WE: ~2 = 1197.1909 ¢, ~3/2 = 704.4836 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 706.4289 ¢

Optimal ET sequence: 12f, 17

Badness (Sintel): 0.498

Scales: supra7, supra12

Superpyth

Main article: Superpyth
For the 5-limit version, see Syntonic–diatonic equivalence continuum #Superpyth (5-limit).

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

Subgroup: 2.3.5.7

Comma list: 64/63, 245/243

Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]

Optimal tunings:

  • WE: ~2 = 1197.0549 ¢, ~3/2 = 708.5478 ¢
error map: ⟨-2.945 +3.648 -0.548 +2.298]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.1193 ¢
error map: ⟨0.000 +8.164 +4.760 +10.935]

Optimal ET sequence: 5, 17, 22, 27, 49, 174bbcddd

Badness (Sintel): 0.818

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).

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -12 6 -22], ⟨0 1 9 -2 16]]

Optimal tunings:

  • WE: ~2 = 1197.0673 ¢, ~3/2 = 708.4391 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.0129 ¢

Optimal ET sequence: 22, 27e, 49

Badness (Sintel): 0.826

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 100/99

Mapping: [⟨1 0 -12 6 -22 -17], ⟨0 1 9 -2 16 13]]

Optimal tunings:

  • WE: ~2 = 1197.3011 ¢, ~3/2 = 708.8813 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.3219 ¢

Optimal ET sequence: 22, 27e, 49, 76bcde

Badness (Sintel): 1.02

Thomas

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 169/168, 245/243

Mapping: [⟨1 1 -3 4 -6 4], ⟨0 2 18 -4 32 -1]]

Optimal tunings:

  • WE: ~2 = 1197.4942 ¢, ~16/13 = 354.2950 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/13 = 354.9824 ¢

Optimal ET sequence: 27e, 44, 71d, 98bde

Badness (Sintel): 2.03

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

Comma list: 55/54, 64/63, 99/98

Mapping: [⟨1 0 -12 6 13], ⟨0 1 9 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1198.6960 ¢, ~3/2 = 708.7235 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.4699 ¢

Optimal ET sequence: 5, 17, 22

Badness (Sintel): 1.08

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 65/63, 99/98

Mapping: [⟨1 0 -12 6 13 18], ⟨0 1 9 -2 -6 -9]]

Optimal tunings:

  • WE: ~2 = 1199.9871 ¢, ~3/2 = 708.6952 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.7028 ¢

Optimal ET sequence: 5f, 17, 22

Badness (Sintel): 1.50

Quasisuper

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

Subgroup: 2.3.5.7

Comma list: 64/63, 2430/2401

Mapping: [⟨1 0 23 6], ⟨0 1 -13 -2]]

Optimal tunings:

  • WE: ~2 = 1196.9830 ¢, ~3/2 = 706.4578 ¢
error map: ⟨-3.017 +1.486 -0.435 +6.190]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.3716 ¢
error map: ⟨0.000 +6.417 +4.855 +14.431]

Optimal ET sequence: 17c, 22, 61d

Badness (Sintel): 1.61

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).

Subgroup: 2.3.5.7.11

Comma list: 64/63, 99/98, 121/120

Mapping: [⟨1 0 23 6 13], ⟨0 1 -13 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1197.5675 ¢, ~3/2 = 706.7690 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.3200 ¢

Optimal ET sequence: 17c, 22, 39d, 61d

Badness (Sintel): 1.06

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 121/120

Mapping: [⟨1 0 23 6 13 18], ⟨0 1 -13 -2 -6 -9]]

Optimal tunings:

  • WE: ~2 = 1198.2543 ¢, ~3/2 = 706.9736 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.0936 ¢

Optimal ET sequence: 17c, 22, 39d

Badness (Sintel): 1.25

Quasisoup

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2430/2401

Mapping: [⟨1 0 23 6 -22], ⟨0 1 -13 -2 16]]

Optimal tunings:

  • WE: ~2 = 1198.8446 ¢, ~3/2 = 708.3388 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.0252 ¢

Optimal ET sequence: 22

Badness (Sintel): 2.76

Ultrapyth

Main article: Ultrapyth

Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 oceanfront temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–Cx).

Subgroup: 2.3.5.7

Comma list: 64/63, 6860/6561

Mapping: [⟨1 0 -20 6], ⟨0 1 14 -2]]

Optimal tunings:

  • WE: ~2 = 1197.2673 ¢, ~3/2 = 712.0258 ¢
error map: ⟨-2.733 +7.338 -1.557 -3.808]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.5430 ¢
error map: ⟨0.000 +11.588 +3.288 +4.088]

Optimal ET sequence: 5, 27c, 32, 37

Badness (Sintel): 2.74

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2401/2376

Mapping: [⟨1 0 -20 6 21], ⟨0 1 14 -2 -11]]

Optimal tunings:

  • WE: ~2 = 1198.0290 ¢, ~3/2 = 712.2235 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.3754 ¢

Optimal ET sequence: 5, 32, 37

Badness (Sintel): 2.26

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 1573/1568

Mapping: [⟨1 0 -20 6 21 -25], ⟨0 1 14 -2 -11 18]]

Optimal tunings:

  • WE: ~2 = 1198.1911 ¢, ~3/2 = 712.4243 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.4684 ¢

Optimal ET sequence: 5, 32, 37

Badness (Sintel): 2.03

Ultramarine

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 3773/3645

Mapping: [⟨1 0 -20 6 -38], ⟨0 1 14 -2 26]]

Optimal tunings:

  • WE: ~2 = 1197.2230 ¢, ~3/2 = 712.1393 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.6928 ¢

Optimal ET sequence: 5e, 32e, 37, 79bce

Badness (Sintel): 2.58

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 91/90, 100/99, 847/845

Mapping: [⟨1 0 -20 6 -38 -25], ⟨0 1 14 -2 26 18]]

Optimal tunings:

  • WE: ~2 = 1197.2739 ¢, ~3/2 = 712.1893 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.7079 ¢

Optimal ET sequence: 5e, 32e, 37, 79bcef

Badness (Sintel): 1.89

Quasiultra

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

Subgroup: 2.3.5.7

Comma list: 64/63, 33614/32805

Mapping: [⟨1 0 31 6], ⟨0 1 -18 -2]]

Optimal tunings:

  • WE: ~2 = 1196.9257 ¢, ~3/2 = 709.6211 ¢
error map: ⟨0.000 +9.883 +0.608 +7.499]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.5429 ¢
error map: ⟨0.000 +9.588 +5.914 +8.088]

Optimal ET sequence: 27, 86bd, 113bcd, 140bbcd

Badness (Sintel): 3.34

Schism

See also: Schismatic family § Schism

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.

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]

Optimal tunings:

  • WE: ~2 = 1197.3598 ¢, ~3/2 = 700.0126 ¢
error map: ⟨-2.640 -4.583 -4.896 +20.588]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7376 ¢
error map: ⟨0.000 -0.217 -0.214 +27.699]

Optimal ET sequence: 5c, 7c, 12

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 64/63, 99/98

Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1196.1607 ¢, ~3/2 = 699.8897 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.4385 ¢

Optimal ET sequence: 5c, 7ce, 12, 29de

Badness (Sintel): 1.24

Beatles

For the 5-limit version, see Miscellaneous 5-limit temperaments #Beatles.

Beatles tempers out 686/675, which may also be characterized by saying it tempers out 2401/2400. It may be described as the 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.

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.

Subgroup: 2.3.5.7

Comma list: 64/63, 686/675

Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4]]

Optimal tunings:

  • WE: ~2 = 1196.6244 ¢, ~49/40 = 354.9029 ¢
error map: ⟨-3.376 +4.475 +2.682 -1.940]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.0819 ¢
error map: ⟨0.000 +10.209 +8.949 +6.847]

Optimal ET sequence: 10, 17c, 27, 64b, 91bcd, 118bccd

Badness (Sintel): 1.16

Music

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 686/675

Mapping: [⟨1 1 5 4 10], ⟨0 2 -9 -4 -22]]

Optimal tunings:

  • WE: ~2 = 1196.7001 ¢, ~49/40 = 355.1606 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.2795 ¢

Optimal ET sequence: 10e, 17cee, 27e, 64be, 91bcdee

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 91/90, 100/99, 169/168

Mapping: [⟨1 1 5 4 10 4], ⟨0 2 -9 -4 -22 -1]]

Optimal tunings:

  • WE: ~2 = 1197.2504 ¢, ~16/13 = 355.4132 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/13 = 356.3273 ¢

Optimal ET sequence: 10e, 27e, 37, 64be

Badness (Sintel): 1.25

Ringo

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 540/539

Mapping: [⟨1 1 5 4 2], ⟨0 2 -9 -4 5]]

Optimal tunings:

  • WE: ~2 = 1195.4102 ¢, ~11/9 = 354.0597 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 355.5207 ¢

Optimal ET sequence: 10, 17c, 27e

Badness (Sintel): 1.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 91/90

Mapping: [⟨1 1 5 4 2 4], ⟨0 2 -9 -4 5 -1]]

Optimal tunings:

  • WE: ~2 = 1195.9943 ¢, ~11/9 = 354.2695 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 355.5398 ¢

Optimal ET sequence: 10, 17c, 27e

Badness (Sintel): 0.935

Beetle

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 5 4 -1], ⟨0 2 -9 -4 15]]

Optimal tunings:

  • WE: ~2 = 1197.9660 ¢, ~49/40 = 356.1056 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.7075 ¢

Optimal ET sequence: 10, 27, 37

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 169/168

Mapping: [⟨1 1 5 4 -1 4], ⟨0 2 -9 -4 15 -1]]

Optimal tunings:

  • WE: ~2 = 1198.1741 ¢, ~16/13 = 356.1582 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/13 = 356.7008 ¢

Optimal ET sequence: 10, 27, 37

Badness (Sintel): 1.40

Progress

Not to be confused with Progression.
For the 5-limit version, see Miscellaneous 5-limit temperaments #Progress.

Progress tempers out 392/375 and may be described as 15 & 17c. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 32c-edo gives an obvious tuning.

Subgroup: 2.3.5.7

Comma list: 64/63, 392/375

Mapping: [⟨1 0 5 6], ⟨0 3 -5 -6]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1195.1377 ¢, ~10/7 = 635.2932 ¢
error map: ⟨-4.862 +3.925 +12.908 -9.759]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 638.0791 ¢
error map: ⟨0.000 +12.282 +23.291 +2.700]

Optimal ET sequence: 2, 13, 15, 32c

Badness (Sintel): 1.68

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 77/75

Mapping: [⟨1 0 5 6 4], ⟨0 3 -5 -6 -1]]

Optimal tunings:

  • WE: ~2 = 1195.4920 ¢, ~10/7 = 635.5183 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 638.0884 ¢

Optimal ET sequence: 2, 13, 15, 32c, 47bc

Badness (Sintel): 1.03

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 66/65, 77/75

Mapping: [⟨1 0 5 6 4 0], ⟨0 3 -5 -6 -1 7]]

Optimal tunings:

  • WE: ~2 = 1195.0786 ¢, ~10/7 = 635.0197 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 637.6691 ¢

Optimal ET sequence: 15, 17c, 32cf

Badness (Sintel): 1.08

Progressive

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 5 6 4 9], ⟨0 3 -5 -6 -1 -10]]

Optimal tunings:

  • WE: ~2 = 1196.0245 ¢, ~10/7 = 634.6516 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 636.9528 ¢

Optimal ET sequence: 2f, 15f, 17c

Badness (Sintel): 1.35

Fervor

For the 5-limit version, see Miscellaneous 5-limit temperaments #Fervor.

Fervor tempers out 9704/9375 and may be described as 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

Comma list: 64/63, 9604/9375

Mapping: [⟨1 -1 7 8], ⟨0 5 -9 -10]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1196.2742 ¢, ~10/7 = 620.2918 ¢
error map: ⟨-3.726 +3.230 +4.980 -1.550]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.3179 ¢
error map: ⟨0.000 +9.634 +12.826 +7.996]

Optimal ET sequence: 2, 25, 27

Badness (Sintel): 2.74

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 1350/1331

Mapping: [⟨1 -1 7 8 4], ⟨0 5 -9 -10 -1]]

Optimal tunings:

  • WE: ~2 = 1195.4148 ¢, ~10/7 = 619.7729 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.2525 ¢

Optimal ET sequence: 2, 25e, 27e

Badness (Sintel): 1.72

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 507/500

Mapping: [⟨1 -1 7 8 4 12], ⟨0 5 -9 -10 -1 -16]]

Optimal tunings:

  • WE: ~2 = 1195.6284 ¢, ~10/7 = 619.6738 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.0631 ¢

Optimal ET sequence: 2f, 27e

Badness (Sintel): 1.64

Sixix

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Sixix (5-limit).
See also: Dual-fifth temperaments § Dual-3 Sixix

Sixix tempers out 3125/2916 and may be described as 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.

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/2916

Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]

Optimal tunings:

  • WE: ~2 = 1198.9028 ¢, ~6/5 = 337.1334 ¢
error map: ⟨-1.097 +9.086 -13.503 +2.508]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.4588 ¢
error map: ⟨0.000 +10.751 -11.066 +5.762]

Optimal ET sequence: 7, 18d, 25, 32

Badness (Sintel): 4.02

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 125/121

Mapping: [⟨1 3 4 0 6], ⟨0 -5 -6 10 -9]]

Optimal tunings:

  • WE: ~2 = 1198.5480 ¢, ~6/5 = 337.1557 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.6000 ¢

Optimal ET sequence: 7, 25e, 32

Badness (Sintel): 2.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 55/54, 64/63, 125/121

Mapping: [⟨1 3 4 0 6 4], ⟨0 -5 -6 10 -9 -1]]

Optimal tunings:

  • WE: ~2 = 1197.7111 ¢, ~6/5 = 336.8391 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.5336 ¢

Optimal ET sequence: 7, 25e, 32f

Badness (Sintel): 1.91

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 3 4 0 6 4 1], ⟨0 -5 -6 10 -9 -1 11]]

Optimal tunings:

  • WE: ~2 = 1197.7807 ¢, ~6/5 = 336.8884 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.5279 ¢

Optimal ET sequence: 7, 25e, 32f

Badness (Sintel): 2.00

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