@@ 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
-[[Sval]] [[mapping]]: [{{val| 1 0 6 }}, {{val| 0 1 -2 }}]
+{{Mapping|legend=3| 1 0 6 | 0 1 -2 }}
-Sval mapping generators: ~2, ~3
-Gencom mapping: [{{val| 1 1 0 4 }}, {{val| 0 1 0 -2 }}]
+{{Mapping|legend=5| 1 0 0 6 | 0 1 0 -2 }}
+: mapping generators: ~2, ~3
-[[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]]
-== Supra ==
+=== Overview to extensions ===
+==== 7-limit extensions ====
-Subgroup: 2.3.7.11
+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]];
+* [[Meantone family #Dominant|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]].
-Comma list: 64/63, 99/98
+These all use the same generators as archy.
-Sval mapping: [{{val| 1 0 6 13 }}, {{val| 0 1 -2 -6 }}]
+[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a semi-octave period. [[392/375]] gives progress. [[6125/5832]] gives kangaroo. Those split 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. [[15625/15552]] gives catalan, splitting the twelfth in six. Finally, [[28/27]] gives blackwood and [[17496/16807]] gives [[gothic]], reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.
-Sval mapping generators: ~2, ~3
+Temperaments discussed elsewhere are:
+* ''[[Phlegyas]]'' (+10/9) → [[Very low accuracy temperaments #Phlegyas|Very low accuracy temperaments]]
+* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
+* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
+* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
+* ''[[Dichotic]]'' (+25/24) → [[Dicot family #Dichotic|Dicot family]]
+* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
+* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
+* [[Augene]] (+126/125) → [[Augmented family #Septimal augmented (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]]
+* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
+* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
+* ''[[Gothic]]'' (+17496/16807) → [[17th-octave temperaments #Gothic|17th-octave temperaments]]
-Gencom mapping: [{{val| 1 1 0 4 7 }}, {{val| 0 1 0 -2 -6 }}]
+The rest are considered below.
-Gencom: [2 3/2; 64/63 99/98]
+==== 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–G♭). This is called supra, given in [[#Subgroup extensions_2|#Subgroup extensions]]. Discussed elsewhere is [[suhajira]] of the [[rastmic clan #Suhajira|rastmic clan]].
-POTE generator: ~3/2 = 707.192
+== Superpyth ==
+{{Main| Superpyth }}
+: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''
-{{Val list|legend=1| 5, 12, 17, 39d, 56d }}
+Superpyth, virtually the canonical extension, adds [[245/243]] and [[1728/1715]] to the comma list and can be described as {{nowrap| [[22edo|22]] [[&]] [[27edo|27]] }}. ~[[5/4]] is found at +9 generator steps, as an augmented second (C–D♯). In the [[11-limit]] it finds the ~[[11/8]] at +16 generator steps, as a double-augmented second (C–D𝄪). [[49edo]] remains an obvious tuning choice in either case.
-Scales: [[supra7]], [[supra12]]
+Extending superpyth to the [[13-limit]] is more diffcult. Tridecimal superpyth finds the ~[[13/8]] at +13 generator steps, as a double-augmented fourth (C–F𝄪), for which 27edo can be recommended as a tuning since it is the only [[13-odd-limit]] [[diamond monotone]] tuning. The other extension, called uberpyth, is more flexible with its tunings, but unfortunately tends to tune the 13 very sharp.
-=== Supraphon ===
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.7.11.13
+[[Comma list]]: 64/63, 245/243
-Comma list: 64/63, 78/77, 99/98
+{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}
-Sval mapping: [{{val| 1 0 6 13 18 }}, {{val| 0 1 -2 -6 -9 }}]
+[[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 }}
-Sval mapping generators: ~2, ~3
+{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
-Gencom mapping: [{{val| 1 1 0 4 7 9 }}, {{val| 0 1 0 -2 -6 -9 }}]
+[[Badness]] (Sintel): 0.818
-Gencom: [2 3/2; 64/63 78/77 99/98]
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-POTE generator: ~3/2 = 706.137
+Comma list: 64/63, 100/99, 245/243
-{{Val list|legend=1| 12f, 17 }}
+{{Mapping|legend=0| 1 0 -12 6 -22 | 0 1 9 -2 16 }}
-Scales: [[supra7]], [[supra12]]
+Optimal tunings:
+* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}
-== Suhajira ==
+{{Optimal ET sequence|legend=0| 22, 27e, 49 }}
-Subgroup: 2.3.7.11
+Badness (Sintel): 0.826
-Comma list: 64/63, 243/242
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Sval mapping: [{{val| 1 1 4 2 }}, {{val| 0 2 -4 5 }}]
+Comma list: 64/63, 78/77, 91/90, 100/99
-Sval mapping generators: ~2, ~3
+{{Mapping|legend=0| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
-Gencom mapping: [{{val| 1 1 0 4 2 }}, {{val| 0 1 0 -4 5 }}]
+Optimal tunings:
+* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}
-Gencom: [2 3/2; 64/63 99/98]
+{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
-POTE generator: ~11/9 = 353.958
+Badness (Sintel): 1.02
-{{Val list|legend=1| 7, 10, 17, 44e, 61de }}
+==== Uberpyth ====
+Subgroup: 2.3.5.7.11.13
-Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
-=== 2.3.7.11.13 ===
-Subgroup: 2.3.7.11.13
-Comma list: 64/63, 78/77, 144/143
-Sval mapping: [{{val| 1 1 4 2 4 }}, {{val| 0 2 -4 5 -1 }}]
-Sval mapping generators: ~2, ~3
-Gencom mapping: [{{val| 1 1 0 4 2 4 }}, {{val| 0 1 0 -4 5 -1 }}]
-Gencom: [2 3/2; 64/63 78/77 99/98]
-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 }}]
-{{Multival|legend=1| 1 9 -2 12 -6 -30 }}
-[[POTE generator]]: ~3/2 = 710.291
-{{Val list|legend=1| 5, 17, 22, 27, 49 }}
-[[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 }}]
+Comma list: 64/63, 100/99, 144/143, 245/243
-POTE generator: ~3/2 = 710.175
+{{Mapping|legend=0| 1 0 -12 6 -22 26 | 0 1 9 -2 16 -14 }}
-{{Val list|legend=1| 22, 27e, 49 }}
+Optimal tunings:
+* WE: ~2 = 1196.6666{{c}}, ~3/2 = 708.3602{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.2878{{c}}
-Badness: 0.0250
+{{Optimal ET sequence|legend=0| 22f, 27e, 49f, 125bcddeeeff, 174bbcdddeeeeffff }}
-=== 13-limit ===
+Badness (Sintel): 1.11
+==== Thomas ====
 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|legend=0| 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–G♭), and finds the ~13/8 at the diminished seventh (C–B𝄫).
 Subgroup: 2.3.5.7.11
@@ Line 156: / Line 156: @@
 Comma list: 55/54, 64/63, 99/98
-Mapping: [{{val| 1 0 -12 6 13 }}, {{val| 0 1 9 -2 -6 }}]
+{{Mapping|legend=0| 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
 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|legend=0| 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 ==
+{{Main|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–G𝄫). The 11-limit version, quasisupra, can be viewed as an extension of the excellent 2.3.7.11-subgroup 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
 [[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 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 ===
 Subgroup: 2.3.5.7.11
 Comma list: 64/63, 99/98, 121/120
-Mapping: [{{val| 1 2 -3 2 1 }}, {{val| 0 -1 13 2 6 }}]
+{{Mapping|legend=0| 1 0 23 6 13 | 0 1 -13 -2 -6 }}
-POTE generator: ~3/2 = 708.205
-{{Val list|legend=1| 17c, 22, 39d, 61d }}
+Optimal tunings:
+* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}
-Badness: 0.0322
+{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}
-=== 13-limit ===
+Badness (Sintel): 1.06
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
 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|legend=0| 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
 Comma list: 55/54, 64/63, 2430/2401
-Mapping: [{{val| 1 0 23 6 -22 }}, {{val| 0 1 -13 -2 16 }}]
+{{Mapping|legend=0| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
-POTE generator: ~3/2 = 709.021
-{{Val list|legend=1| 22 }}
+Optimal tunings:
+* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}
-Badness: 0.0835
+{{Optimal ET sequence|legend=0| 22 }}
-= Schism =
+Badness (Sintel): 2.76
-{{see also|Schismatic family #Schism}}
-Subgroup: 2.3.5.7
+== Ultrapyth ==
+{{Main| Ultrapyth }}
+: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Ultrapyth (5-limit)]].''
-[[Comma list]]: 64/63, 360/343
+Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[#Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).
-[[Mapping]]: [{{val| 1 0 15 6 }}, {{val| 0 1 -8 -2 }}]
+[[Subgroup]]: 2.3.5.7
-{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
+[[Comma list]]: 64/63, 6860/6561
-[[POTE generator]]: ~3/2 = 701.556
+{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
-{{Val list|legend=1| 12, 41d, 53d }}
+[[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]]: 0.0566
+{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
-== 11-limit ==
+[[Badness]] (Sintel): 2.74
+=== 11-limit ===
 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|legend=0| 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]]
+Comma list: 55/54, 64/63, 91/90, 1573/1568
-== 5-limit (blackwood) ==
+{{Mapping|legend=0| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
-Subgroup: 2.3.5
+Optimal tunings:
+* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}
-Comma list: 256/243
+{{Optimal ET sequence|legend=0| 5, 32, 37 }}
-Mapping: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
+Badness (Sintel): 2.03
-Mapping generators: ~9/8, ~5
+=== Ultramarine ===
+Subgroup: 2.3.5.7.11
-POTE generator: ~5/4 = 399.594
+Comma list: 64/63, 100/99, 3773/3645
-{{Val list|legend=1| 5, 10, 15 }}
+{{Mapping|legend=0| 1 0 -20 6 -38 | 0 1 14 -2 26 }}
-Badness: 0.0638
+Optimal tunings:
+* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}
-== 7-limit ==
+{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}
-Subgroup: 2.3.5.7
+Badness (Sintel): 2.58
-Comma list: 28/27, 49/48
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Mapping: [{{val| 5 8 0 14 }}, {{val| 0 0 1 0 }}]
+Comma list: 64/63, 91/90, 100/99, 847/845
-Mapping generators: ~7/6, ~5
+{{Mapping|legend=0| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
-{{Multival|legend=1| 0 5 0 8 0 -14 }}
+Optimal tunings:
+* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}
-POTE generator: ~5/4 = 392.767
+{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}
-{{Val list|legend=1| 5, 10, 15, 40b, 55b }}
+Badness (Sintel): 1.89
-Badness: 0.0256
+== 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–A𝄫♭).
-== 11-limit ==
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11
+[[Comma list]]: 64/63, 33614/32805
-Comma list: 28/27, 49/48, 55/54
+{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
-Mapping: [{{val| 5 8 0 14 29 }}, {{val| 0 0 1 0 -1 }}]
+[[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 }}
-POTE generator: ~5/4 = 394.948
+{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
-{{Val list|legend=1| 5, 10, 15, 40be, 55be, 70bde, 85bcde}}
+[[Badness]] (Sintel): 3.34
-Badness: 0.0246
+== Schism ==
+{{See also| Schismatic family #Schism }}
-=== 13-limit ===
+Schism tempers out the [[schisma]], mapping the ~5/4 to -8 fifths as a diminished fourth (C–F♭) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53dd val) can be used.
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7
-Comma list: 28/27, 40/39, 49/48, 55/54
+[[Comma list]]: 64/63, 360/343
-Mapping: [{{val| 5 8 0 14 29 7 }}, {{val| 0 0 1 0 -1 1 }}]
-POTE generator: ~5/4 = 391.0367
+{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}
-{{Val list|legend=1| 5, 10, 15, 25e, 40bef}}
+[[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: 0.0205
+{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
-== Farrier ==
+[[Badness]] (Sintel): 1.43
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 28/27, 49/48, 77/75
+Comma list: 45/44, 64/63, 99/98
-Mapping: [{{val| 5 8 0 14 -6 }}, {{val| 0 0 1 0 2 }}]
+{{Mapping|legend=0| 1 0 15 6 13 | 0 1 -8 -2 -6 }}
-POTE generator: ~5/4 = 398.070
+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, 10e, 15 }}
+{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}
-Badness: 0.0292
+Badness (Sintel): 1.24
-=== 13-limit ===
+== Beatles ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''
-Subgroup: 2.3.5.7.11.13
+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: 28/27, 40/39, 49/48, 66/65
+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| 5 8 0 14 -6 7 }}, {{val| 0 0 1 0 2 1 }}]
+[[Subgroup]]: 2.3.5.7
-POTE generator: ~5/4 = 396.812
+[[Comma list]]: 64/63, 686/675
-{{Val list|legend=1| 5e, 10e, 15 }}
+{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
-Badness: 0.0223
+[[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 }}
-== Ferrum ==
+{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
-Subgroup: 2.3.5.7.11
+[[Badness]] (Sintel): 1.16
-Comma list: 28/27, 35/33, 49/48
+; Music
+* [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]]
-Mapping: [{{val| 5 8 0 14 6 }}, {{val| 0 0 1 0 1 }}]
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-POTE generator: ~5/4 = 374.763
+Comma list: 64/63, 100/99, 686/675
-{{Val list|legend=1| 5e, 10 }}
+{{Mapping|legend=0| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
-Badness: 0.0309
+Optimal tunings:
+* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}
-= Beatles =
+{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}
-== 5-limit ==
-Subgroup: 2.3.5
+Badness (Sintel): 1.51
-Comma list: 524288/492075
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Mapping: [{{val| 1 1 5 }}, {{val| 0 2 -9 }}]
+Comma list: 64/63, 91/90, 100/99, 169/168
-POTE generator: ~512/405 = 355.930
-{{Val list|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}
-Badness: 0.3585
-== 7-limit ==
-Subgroup: 2.3.5.7
-Comma list: 64/63, 686/675
+{{Mapping|legend=0| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
-Mapping: [{{val| 1 1 5 4 }}, {{val| 0 2 -9 -4 }}]
+Optimal tunings:
+* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}
-{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}
+{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}
-POTE generator: ~49/40 = 355.904
+Badness (Sintel): 1.25
-{{Val list|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
-Badness: 0.0459
-; Music
-* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 Beatles Improv] by Herman Miller
-== 11-limit ==
+=== Ringo ===
 Subgroup: 2.3.5.7.11
-Comma list: 64/63, 100/99, 686/675
+Comma list: 56/55, 64/63, 540/539
-Mapping: [{{val| 1 1 5 4 10 }}, {{val| 0 2 -9 -4 -22 }}]
-POTE generator: ~49/40 = 356.140
+{{Mapping|legend=0| 1 1 5 4 2 | 0 2 -9 -4 5 }}
-{{Val list|legend=1| 27e, 37, 64be, 91bcde }}
+Optimal tunings:
+* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}
-Badness: 0.0456
+{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
-=== 13-limit ===
+Badness (Sintel): 1.09
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 64/63, 91/90, 100/99, 169/168
+Comma list: 56/55, 64/63, 78/77, 91/90
-Mapping: [{{val| 1 1 5 4 10 4 }}, {{val| 0 2 -9 -4 -22 -1 }}]
+{{Mapping|legend=0| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
-POTE generator: ~16/13 = 356.229
+Optimal tunings:
+* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}
-{{Val list|legend=1| 27e, 37, 64be }}
+{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
-Badness: 0.0302
+Badness (Sintel): 0.935
-== Ringo ==
+=== Beetle ===
+Subgroup: 2.3.5.7.11
-Subgroup: 2.3.5.7.11
+Comma list: 55/54, 64/63, 686/675
-Comma list: 56/55, 64/63, 540/539
+{{Mapping|legend=0| 1 1 5 4 -1 | 0 2 -9 -4 15 }}
-Mapping: [{{val| 1 1 5 4 2 }}, {{val| 0 2 -9 -4 5 }}]
+Optimal tunings:
+* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}
-POTE generator: ~11/9 = 355.419
+{{Optimal ET sequence|legend=0| 10, 27, 37 }}
-{{Val list|legend=1| 10, 17c, 27e }}
+Badness (Sintel): 1.92
-Badness: 0.0329
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== 13-limit ===
+Comma list: 55/54, 64/63, 91/90, 169/168
-Subgroup: 2.3.5.7.11.13
+{{Mapping|legend=0| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
-Comma list: 56/55, 64/63, 78/77, 91/90
+Optimal tunings:
+* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}
-Mapping: [{{val| 1 1 5 4 2 4 }}, {{val| 0 2 -9 -4 5 -1 }}]
+{{Optimal ET sequence|legend=0| 10, 27, 37 }}
-POTE generator: ~11/9 = 355.456
+Badness (Sintel): 1.40
-{{Val list|legend=1| 10, 17c, 27e }}
+== Progress ==
+{{Distinguish| Progression }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
-Badness: 0.0226
+Progress tempers out [[392/375]] and may be described as {{nowrap| 15 & 17c }}. It splits the [[3/1|perfect twelfth]] into three generators of [[~]][[10/7]]; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]], and the 13-limit extension further equates it to [[13/9]]. [[32edo|32c-edo]] gives an obvious tuning.
-= Passion =
+[[Subgroup]]: 2.3.5.7
-== 5-limit ==
-Subgroup: 2.3.5
+[[Comma list]]: 64/63, 392/375
-Comma list: 262144/253125
+{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
+: mapping generators: ~2, ~10/7
-Mapping: [{{val| 1 2 2 }}, {{val| 0 -5 4 }}]
+[[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 }}
-Mapping generators: 2, 16/15
+{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
-POTE generator: ~16/15 = 98.670
+[[Badness]] (Sintel): 1.68
-{{Val list|legend=1| 12, 49, 61, 73 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness: 0.1686
+Comma list: 56/55, 64/63, 77/75
-== 7-limit ==
+{{Mapping|legend=0| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
-Subgroup: 2.3.5.7
+Optimal tunings:
+* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}
-Comma list: 64/63, 3125/3087
+{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}
-Mapping: [{{val| 1 2 2 2 }}, {{val| 0 -5 4 10 }}]
+Badness (Sintel): 1.03
-Mapping generators: 2, 16/15
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-{{Multival|legend=1| 5 -4 -10 -18 -30 -12 }}
+Comma list: 56/55, 64/63, 66/65, 77/75
-POTE generator: ~16/15 = 98.153
+{{Mapping|legend=0| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
-{{Val list|legend=1| 12, 37, 49, 110bcd }}
+Optimal tunings:
+* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}
-Badness: 0.0623
+{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
-== 11-limit ==
+Badness (Sintel): 1.08
-Subgroup: 2.3.5.7.11
+==== Progressive ====
+Subgroup: 2.3.5.7.11.13
-Comma list: 64/63, 100/99, 1375/1372
+Comma list: 26/25, 56/55, 64/63, 77/75
-Mapping: [{{val| 1 2 2 2 2 }}, {{val| 0 -5 4 10 18 }}]
+{{Mapping|legend=0| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
-POTE generator: ~16/15 = 98.019
+Optimal tunings:
+* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}
-{{Val list|legend=1| 12, 37, 49 }}
+{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}
-Badness: 0.0408
+Badness (Sintel): 1.35
-== 13-limit ==
+== Kangaroo ==
+: ''For 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
-Subgroup: 2.3.5.7.11.13
+Kangaroo splits the [[3/1|perfect twelfth]] into three [[35/24]][[~]][[40/27]] generators; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]]. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat.
-Comma list: 64/63, 100/99, 196/195, 275/273
+Originally the canonical extension of 5-limit kangaroo was [[trienstonic clan #Parakangaroo|parakangaroo]], but it was decanonized in 2024 in favor of this extension because parakangaroo is an [[exotemperament]] that only supports the [[patent val]] of [[15edo]], while this extension is a bit more practical since it is at least low-medium accuracy and supports the [[patent val]]s of 15edo, [[17edo]] and [[32edo]].
-Mapping: [{{val| 1 2 2 2 2 2 }}, {{val| 0 -5 4 10 18 21 }}]
+[[Subgroup]]: 2.3.5.7
-POTE generator: ~16/15 = 97.910
+[[Comma list]]: 64/63, 6125/5832
-{{Val list|legend=1| 12f, 25f, 37, 49f }}
+{{Mapping|legend=1| 1 0 -3 6 | 0 3 10 -6 }}
+: mapping generators: ~2, ~35/24
-Badness: 0.0309
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1198.1767{{c}}, ~35/24 = 637.2914{{c}}
+: [[error map]]: {{val| -1.823 +9.919 -7.930 -3.514 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 638.1977{{c}}
+: error map: {{val| 0.000 +12.638 -4.336 +1.988 }}
-= Fervor =
+{{Optimal ET sequence|legend=1| 15, 32, 47b }}
-== 5-limit ==
-Subgroup: 2.3.5
+[[Badness]] (Sintel): 2.91
-Comma list: 67108864/61509375
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: [{{val| 1 4 -2 }}, {{val| 0 -5 9 }}]
+Comma list: 55/54, 64/63, 245/242
-POTE generator: ~64/45 = 577.705
+{{Mapping|legend=0| 1 0 -3 6 4 | 0 3 10 -6 -1 }}
-{{Val list|legend=1| 25, 27 }}
+Optimal tunings:
+* WE: ~2 = 1197.8893{{c}}, ~16/11 = 637.1922{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 638.2605{{c}}
-Badness: 0.8526
+{{Optimal ET sequence|legend=0| 15, 32, 47b }}
-== 7-limit ==
+Badness (Sintel): 1.65
-Subgroup: 2.3.5.7
+== Fervor ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
-Comma list: 64/63, 9604/9375
+Fervor tempers out [[9604/9375]] and may be described as {{nowrap| 25 & 27 }}. It splits the [[6/1|6th harmonic]] into five generators of [[~]][[10/7]]; its ploidacot is beta-pentacot. The 11-limit extension simply equates the generator to [[16/11]]. [[27edo]] is about as accurate as it can be tuned.
-Mapping: [{{val| 1 4 -2 -2 }}, {{val| 0 -5 9 10 }}]
+[[Subgroup]]: 2.3.5.7
-{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}
+[[Comma list]]: 64/63, 9604/9375
-POTE generator: ~7/5 = 577.777
+{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
+: mapping generators: ~2, ~10/7
-{{Val list|legend=1| 25, 27 }}
+[[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 }}
-Badness: 0.1085
+{{Optimal ET sequence|legend=1| 2, 25, 27 }}
-== 11-limit ==
+[[Badness]] (Sintel): 2.74
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
 Comma list: 56/55, 64/63, 1350/1331
-Mapping: [{{val| 1 4 -2 -2 3 }}, {{val| 0 -5 9 10 1 }}]
+{{Mapping|legend=0| 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
 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|legend=0| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}
-POTE generator: ~7/5 = 578.060
+Optimal tunings:
+* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}
-{{Val list|legend=1| 27e }}
+{{Optimal ET sequence|legend=0| 2f, 27e }}
-Badness: 0.0397
+Badness (Sintel): 1.64
-= Progress =
+== Sixix ==
-== 5-limit ==
+: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sixix (5-limit)]].''
+{{See also| Dual-fifth temperaments #Dual-3 Sixix }}
-Subgroup: 2.3.5
+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. An important val not in the [[optimal ET sequence]] is 39c, that is, [[39edo]] using the second-best approximation of [[5/4]] (the 369-cent one inherited from [[13edo]]). It is also supported by [[57edo|57b-edo]], which uses the sharper of [[57edo]]'s two diatonic fifths.
-Comma list: 32768/30375
+[[Subgroup]]: 2.3.5.7
-Mapping: [{{val| 1 0 5 }}, {{val| 0 3 -5 }}]
+[[Comma list]]: 64/63, 3125/2916
-POTE generator: ~64/45 = 561.264
+{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
-{{Val list|legend=1| 15, 32c, 47bc, 62bc }}
+[[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 }}
-Badness: 0.2461
+{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
-== 7-limit ==
+[[Badness]] (Sintel): 4.02
-Subgroup: 2.3.5.7
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Comma list: 64/63, 392/375
+Comma list: 55/54, 64/63, 125/121
-Mapping: [{{val| 1 0 5 6 }}, {{val| 0 3 -5 -6 }}]
+{{Mapping|legend=0| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
-{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
+Optimal tunings:
+* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}
-POTE generator: ~7/5 = 562.122
+{{Optimal ET sequence|legend=0| 7, 25e, 32 }}
-{{Val list|legend=1| 15, 32c, 79bcc, 111bcc }}
+Badness (Sintel): 2.34
-Badness: 0.0664
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-== 11-limit ==
+Comma list: 40/39, 55/54, 64/63, 125/121
-Subgroup: 2.3.5.7.11
+{{Mapping|legend=0| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
+Optimal tunings:
+* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}
-Comma list: 56/55, 64/63, 77/75
+{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
-Mapping: [{{val| 1 0 5 6 4 }}, {{val| 0 3 -5 -6 -1 }}]
+Badness (Sintel): 1.91
-POTE generator: ~7/5 = 562.085
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-{{Val list|legend=1| 15, 32c, 47bc, 79bcce }}
+Comma list: 40/39, 55/54, 64/63, 85/84, 125/121
-Badness: 0.0310
+{{Mapping|legend=0| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
-=== 13-limit ===
+Optimal tunings:
+* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}
-Subgroup: 2.3.5.7.11.13
+{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
-Comma list: 56/55, 64/63, 66/65, 77/75
+Badness (Sintel): 2.00
-Mapping: [{{val| 1 0 5 6 4 0 }}, {{val| 0 3 -5 -6 -1 7 }}]
+== Subgroup extensions ==
+=== Supra (2.3.7.11) ===
+Subgroup: 2.3.7.11
-POTE generator: ~7/5 = 562.365
+Comma list: 64/63, 99/98
-{{Val list|legend=1| 15, 17c, 32cf }}
+{{Mapping|legend=2| 1 0 6 13 | 0 1 -2 -6 }}
-Badness: 0.0262
+{{Mapping|legend=4| 1 0 0 6 13 | 0 1 0 -2 -6 }}
-=== Progressive ===
+Optimal tunings:
+* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}
-Subgroup: 2.3.5.7.11.13
+{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}
-Comma list: 26/25, 56/55, 64/63, 77/75
+Badness (Sintel): 0.352
-Mapping: [{{val| 1 0 5 6 4 9 }}, {{val| 0 3 -5 -6 -1 -10 }}]
+Scales: [[supra7]], [[supra12]]
-POTE generator: ~7/5 = 563.239
+==== Supraphon ====
+This extension maps [[13/11]] to the minor third (C–E♭), [[12/11]][[~]][[14/13]] to the augmented unison (C–C♯), and [[13/12]] to the diminished third (C–E𝄫).
-{{Val list|legend=1| 15f, 17c, 32c, 49c }}
+Subgroup: 2.3.7.11.13
-Badness: 0.0327
+Comma list: 64/63, 78/77, 99/98
-= Sixix =
+{{Mapping|legend=2| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
-:''See also: [[Dual-fifth temperaments#Dual-3 Sixix]]''
-== 5-limit ==
-Subgroup: 2.3.5
+{{Mapping|legend=4| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
-Comma list: 3125/2916
+Optimal tunings:
+* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}
-Mapping: [{{val| 1 3 4 }}, {{val| 0 -5 -6 }}]
+{{Optimal ET sequence|legend=0| 12f, 17 }}
-POTE generator: ~6/5 = 338.365
+Badness (Sintel): 0.498
+Scales: [[supra7]], [[supra12]]
-{{Val list|legend=1| 7, 25, 32 }}
+=== Oceanfront (2.3.7.13/5) ===
+This is the [[restriction]] of [[#Ultrapyth|ultrapyth]] to the 2.3.7.13/5 subgroup, mapping [[13/10]] to the major third.
-Badness: 0.1531
+[[Subgroup]]: 2.3.7.13/5
-== 7-limit ==
+[[Comma list]]: 64/63 ({{monzo| 6 -2 -1 0 }}), 91/90 ({{monzo| -1 -2 1 1 }})
-Subgroup: 2.3.5.7
+{{Mapping|legend=3| 1 0 6 -5 | 0 1 -2 4 }}
-Comma list: 64/63, 3125/2916
+[[Optimal tuning]]s:
+* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1197.2656{{c}}, ~3/2 = 710.1993{{c}}
+* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.7413{{c}}
-Mapping: [{{val| 1 3 4 0 }}, {{val| 0 -5 -6 10 }}]
+{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 32, 59 }}
-POTE generator: ~6/5 = 337.4419
+[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
-{{Val list|legend=1| 7, 25, 32 }}
+Scales: [[Oceanfront scales]]
-[[Category:Regular temperament theory]]
 [[Category:Archytas clan| ]] <!-- main article -->
-[[Category:Temperament clan]]
+[[Category:Archytas| ]] <!-- key article -->
-[[Category:Rank 2]]
+[[Category:Temperament clans]]
+[[Category:Catalogs of rank-2 temperaments]]

Archytas clan: Difference between revisions

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