@@ Line 1: / Line 1: @@
-The '''archytas clan''' (or '''archy family''') tempers out the [[64/63|Archytas' comma]], 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to [[81/80]] in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3's and 7's quite sharp, such as those of [[22edo]].
+{{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]].
+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| Superpyth }}
 [[Subgroup]]: 2.3.7
 [[Comma list]]: 64/63
-{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}
+{{Mapping|legend=3| 1 0 6 | 0 1 -2 }}
-: sval mapping generators: ~2, ~3
+{{Mapping|legend=5| 1 0 0 6 | 0 1 0 -2 }}
+: mapping generators: ~2, ~3
-{{Mapping|legend=3| 1 1 0 4 | 0 1 0 -2 }}
+[[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 }}
-: [[gencom]]: [2 3/2; 64/63]
+{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
-[[Optimal tuning]] ([[POTE]]): ~3/2 = 709.321
+[[Badness]] (Sintel): 0.159
-{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
 Scales: [[archy5]], [[archy7]], [[archy12]]
 === Overview to extensions ===
-Adding [[50/49]] to the list of commas gives pajara, [[36/35]] gives dominant, [[16/15]] gives mother, [[126/125]] gives augene, [[28/27]] gives blacksmith, [[245/243]] gives superpyth, [[250/243]] gives porcupine, [[686/675]] gives beatles, [[360/343]] gives schism, [[3125/3087]] gives passion, [[2430/2401]] gives quasisuper, and [[4375/4374]] gives modus.
+==== 7-limit extensions ====
+The second comma in the comma list defines which [[7-limit]] family member we are looking at:
-Discussed under their respective 5-limit families are:
+* [[#Schism|Schism]] adds [[360/343]], for a tuning around [[12edo]];
-* ''[[Mother]]'' → [[Father family #Mother|Father family]]
+* [[Meantone family #Dominant|Dominant]] adds [[36/35]], for a tuning between [[12edo]] and [[17edo|17c-edo]];
-* ''[[Dominant]]'' → [[Meantone family #Dominant|Meantone family]]
+* [[#Quasisuper|Quasisuper]] adds [[2430/2401]], for a tuning between 17c-edo and [[22edo]];
-* ''[[Augene]]'' → [[Augmented family #Augene|Augmented]]
+* [[#Superpyth|Superpyth]] adds [[245/243]], for a tuning between 22edo and [[27edo]];
-* [[Porcupine]] → [[Porcupine family #Septimal porcupine|Porcupine family]]
+* [[#Quasiultra|Quasiultra]] adds [[33614/32805]], for a tuning between 27edo and [[32edo]];
-* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]]
+* [[#Ultrapyth|Ultrapyth]] adds [[6860/6561]], for a tuning sharp of 32edo;
-* ''[[Blacksmith]]'' → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
+* Mother adds [[16/15]], for an exotemperament well tuned around [[5edo]].
-* ''[[Catalan]]'' → [[Kleismic family #Catalan|Kleismic family]]
-* ''[[Modus]]'' → [[Tetracot family #Modus|Tetracot family]]
-* ''[[Passion]]'' → [[Passion family #Septimal passion|Passion family]]
-* ''[[Immunized]]'' → [[Immunity family #Immunized|Immunity family]]
-The rest are considered below.
-=== Supra ===
-Subgroup: 2.3.7.11
-Comma list: 64/63, 99/98
-Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
-Gencom mapping: {{mapping| 1 1 0 4 7 | 0 1 0 -2 -6 }}
-: gencom: [2 3/2; 64/63 99/98]
-Optimal tuning (POTE): ~3/2 = 707.192
-{{Optimal ET sequence|legend=1| 5, 12, 17, 39d, 56d }}
-Scales: [[supra7]], [[supra12]]
-==== Supraphon ====
-Subgroup: 2.3.7.11.13
-Comma list: 64/63, 78/77, 99/98
-Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
-Gencom mapping: {{mapping| 1 1 0 4 7 9 | 0 1 0 -2 -6 -9 }}
-: gencom: [2 3/2; 64/63 78/77 99/98]
-Optimal tuning (POTE): ~3/2 = 706.137
-{{Optimal ET sequence|legend=1| 12f, 17 }}
-Scales: [[supra7]], [[supra12]]
-=== Suhajira ===
-Subgroup: 2.3.7.11
-Comma list: 64/63, 243/242
-Sval mapping: {{mapping| 1 1 4 2 | 0 2 -4 5 }}
-: sval mapping generators: ~2, ~11/9
-Gencom mapping: {{mapping| 1 1 0 4 2 | 0 1 0 -4 5 }}
-: gencom: [2 3/2; 64/63 99/98]
-Optimal tuning (POTE): ~11/9 = 353.958
-{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de }}
-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: {{mapping| 1 1 4 2 4 | 0 2 -4 5 -1 }}
-Gencom mapping: {{mapping| 1 1 0 4 2 4 | 0 1 0 -4 5 -1 }}
+These all use the same generators as archy.
-: gencom: [2 3/2; 64/63 78/77 99/98]
+[[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.
-Optimal tuning (POTE): ~11/9 = 353.775
+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]]
-{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de }}
+The rest are considered below.
-Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
+==== 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]].
 == Superpyth ==
 {{Main| Superpyth }}
+: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''
-In the 5-limit, superpyth tempers out 20480/19683 (sayo comma). This temperament has a fifth generator of ~3/2 = ~710¢ and 5/4 is equated to nine generators minus 5 octaves. A weak extension [[Jubilismic clan #Bipyth|bipyth]] (10cd&amp;22) tempers out the same 5-limit comma as the superpyth, but with a half-octave period and the jubilisma (50/49) rather than the Archytas comma tempered out.
+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.
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 20480/19683
-{{Mapping|legend=1| 1 0 -12 | 0 1 9 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 710.078
+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.
-{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
-[[Badness]]: 0.135141
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
@@ Line 131: / Line 79: @@
 {{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}
-{{Multival|legend=1| 1 9 -2 12 -6 -30 }}
+[[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 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 710.291
+{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
-{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49 }}
+[[Badness]] (Sintel): 0.818
-[[Badness]]: 0.032318
 === 11-limit ===
@@ Line 144: / Line 94: @@
 Comma list: 64/63, 100/99, 245/243
-Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}
+{{Mapping|legend=0| 1 0 -12 6 -22 | 0 1 9 -2 16 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 710.175
+Optimal tunings:
+* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}
-{{Optimal ET sequence|legend=1| 22, 27e, 49 }}
+{{Optimal ET sequence|legend=0| 22, 27e, 49 }}
-Badness: 0.024976
+Badness (Sintel): 0.826
 ==== 13-limit ====
@@ Line 157: / Line 109: @@
 Comma list: 64/63, 78/77, 91/90, 100/99
-Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
+{{Mapping|legend=0| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
+Optimal tunings:
+* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}
+{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
+Badness (Sintel): 1.02
+==== Uberpyth ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 64/63, 100/99, 144/143, 245/243
+{{Mapping|legend=0| 1 0 -12 6 -22 26 | 0 1 9 -2 16 -14 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 710.479
+Optimal tunings:
+* WE: ~2 = 1196.6666{{c}}, ~3/2 = 708.3602{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.2878{{c}}
-{{Optimal ET sequence|legend=1| 22, 27e, 49, 76bcde }}
+{{Optimal ET sequence|legend=0| 22f, 27e, 49f, 125bcddeeeff, 174bbcdddeeeeffff }}
-Badness: 0.024673
+Badness (Sintel): 1.11
 ==== Thomas ====
@@ Line 170: / Line 139: @@
 Comma list: 64/63, 100/99, 169/168, 245/243
-Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}
+{{Mapping|legend=0| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 355.036
+Optimal tunings:
+* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}
-{{Optimal ET sequence|legend=1| 17e, 27e, 44, 71d }}
+{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}
-Badness: 0.049183
+Badness (Sintel): 2.03
 === 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
 Comma list: 55/54, 64/63, 99/98
-Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}
+{{Mapping|legend=0| 1 0 -12 6 13 | 0 1 9 -2 -6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 5, 12c, 17, 22 }}
+{{Optimal ET sequence|legend=0| 5, 17, 22 }}
-Badness: 0.032768
+Badness (Sintel): 1.08
 ==== 13-limit ====
@@ Line 196: / Line 171: @@
 Comma list: 55/54, 64/63, 65/63, 99/98
-Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}
+{{Mapping|legend=0| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 17, 22, 83cdf }}
+{{Optimal ET sequence|legend=0| 5f, 17, 22 }}
-Badness: 0.036336
+Badness (Sintel): 1.50
 == Quasisuper ==
+{{Main|Quasisuper}}
+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
@@ Line 211: / Line 192: @@
 {{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}
-{{Multival|legend=1| 1 -13 -2 -23 -6 32 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 708.328
+: [[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 }}
 {{Optimal ET sequence|legend=1| 17c, 22, 61d }}
-[[Badness]]: 0.063794
+[[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: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}
+{{Mapping|legend=0| 1 0 23 6 13 | 0 1 -13 -2 -6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 17c, 22, 39d, 61d }}
+{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}
-Badness: 0.032203
+Badness (Sintel): 1.06
 ==== 13-limit ====
@@ Line 239: / Line 222: @@
 Comma list: 64/63, 78/77, 91/90, 121/120
-Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}
+{{Mapping|legend=0| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
+{{Optimal ET sequence|legend=0| 17c, 22, 39d }}
-Badness: 0.030219
+Badness (Sintel): 1.25
 === Quasisoup ===
@@ Line 252: / Line 237: @@
 Comma list: 55/54, 64/63, 2430/2401
-Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
+{{Mapping|legend=0| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 5ce, 17ce, 22 }}
+{{Optimal ET sequence|legend=0| 22 }}
-Badness: 0.083490
+Badness (Sintel): 2.76
 == Ultrapyth ==
-Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[The Biosphere #Oceanfront|oceanfront]] temperament, with mapping 5th harmonic to 14 fifths.
+{{Main| Ultrapyth }}
+: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Ultrapyth (5-limit)]].''
+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𝄪).
 [[Subgroup]]: 2.3.5.7
@@ Line 269: / Line 259: @@
 {{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
-{{Multival|legend=1| 1 14 -2 20 -6 -44 }}
+[[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 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 713.651
+{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
-{{Optimal ET sequence|legend=1| 5, 32, 37 }}
+[[Badness]] (Sintel): 2.74
-[[Badness]]: 0.108466
 === 11-limit ===
@@ Line 282: / Line 274: @@
 Comma list: 55/54, 64/63, 2401/2376
-Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}
+{{Mapping|legend=0| 1 0 -20 6 21 | 0 1 14 -2 -11 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 5, 32, 37 }}
+{{Optimal ET sequence|legend=0| 5, 32, 37 }}
-Badness: 0.068238
+Badness (Sintel): 2.26
 ==== 13-limit ====
@@ Line 295: / Line 289: @@
 Comma list: 55/54, 64/63, 91/90, 1573/1568
-Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
+{{Mapping|legend=0| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 5, 32, 37 }}
+{{Optimal ET sequence|legend=0| 5, 32, 37 }}
-Badness: 0.049172
+Badness (Sintel): 2.03
 === Ultramarine ===
@@ Line 308: / Line 304: @@
 Comma list: 64/63, 100/99, 3773/3645
-Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}
+{{Mapping|legend=0| 1 0 -20 6 -38 | 0 1 14 -2 26 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bce, 116bbce }}
+{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}
-Badness: 0.078068
+Badness (Sintel): 2.58
 ==== 13-limit ====
@@ Line 321: / Line 319: @@
 Comma list: 64/63, 91/90, 100/99, 847/845
-Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
+{{Mapping|legend=0| 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 }}
+Badness (Sintel): 1.89
+== 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𝄫♭).
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 713.811
+[[Comma list]]: 64/63, 33614/32805
-{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bcef, 116bbcef }}
+{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
-Badness: 0.045653
+[[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 ==
-{{see also|Schismatic family #Schism}}
+{{See also| Schismatic family #Schism }}
+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
@@ Line 338: / Line 359: @@
 {{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 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.556
+{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
-{{Optimal ET sequence|legend=1| 12, 41d, 53d }}
+[[Badness]] (Sintel): 1.43
-[[Badness]]: 0.056648
 === 11-limit ===
@@ Line 351: / Line 374: @@
 Comma list: 45/44, 64/63, 99/98
-Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}
+{{Mapping|legend=0| 1 0 15 6 13 | 0 1 -8 -2 -6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 12, 29de, 41de }}
+{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}
-Badness: 0.037482
+Badness (Sintel): 1.24
 == Beatles ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #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 {{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.
+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
@@ Line 368: / Line 397: @@
 {{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}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 355.904
+: [[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 }}
-{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
+{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
-[[Badness]]: 0.045872
+[[Badness]] (Sintel): 1.16
 ; 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 ===
@@ Line 384: / Line 415: @@
 Comma list: 64/63, 100/99, 686/675
-Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
+{{Mapping|legend=0| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 27e, 37, 64be, 91bcde }}
+{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}
-Badness: 0.045639
+Badness (Sintel): 1.51
 ==== 13-limit ====
@@ Line 397: / Line 430: @@
 Comma list: 64/63, 91/90, 100/99, 169/168
-Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
+{{Mapping|legend=0| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 27e, 37, 64be }}
+{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}
-Badness: 0.030161
+Badness (Sintel): 1.25
 === Ringo ===
@@ Line 410: / Line 445: @@
 Comma list: 56/55, 64/63, 540/539
-Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}
+{{Mapping|legend=0| 1 1 5 4 2 | 0 2 -9 -4 5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 10, 17c, 27e }}
+{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
-Badness: 0.032863
+Badness (Sintel): 1.09
 ==== 13-limit ====
@@ Line 423: / Line 460: @@
 Comma list: 56/55, 64/63, 78/77, 91/90
-Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
+{{Mapping|legend=0| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 10, 17c, 27e }}
+{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
-Badness: 0.022619
+Badness (Sintel): 0.935
 === Beetle ===
@@ Line 436: / Line 475: @@
 Comma list: 55/54, 64/63, 686/675
-Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15}}
+{{Mapping|legend=0| 1 1 5 4 -1 | 0 2 -9 -4 15 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 10, 27, 37 }}
+{{Optimal ET sequence|legend=0| 10, 27, 37 }}
-Badness: 0.058084
+Badness (Sintel): 1.92
 ==== 13-limit ====
@@ Line 449: / Line 490: @@
 Comma list: 55/54, 64/63, 91/90, 169/168
-Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
+{{Mapping|legend=0| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
+Optimal tunings:
+* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}
+{{Optimal ET sequence|legend=0| 10, 27, 37 }}
-Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 356.701
+Badness (Sintel): 1.40
-{{Optimal ET sequence|legend=1| 10, 27, 37 }}
+== Progress ==
+{{Distinguish| Progression }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
-Badness: 0.033971
+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.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 64/63, 392/375
+{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
+: mapping generators: ~2, ~10/7
+[[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 }}
+{{Optimal ET sequence|legend=1| 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|legend=0| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
+Optimal tunings:
+* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}
+{{Optimal ET sequence|legend=0| 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|legend=0| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
+Optimal tunings:
+* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}
+{{Optimal ET sequence|legend=0| 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|legend=0| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
+Optimal tunings:
+* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}
+{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}
+Badness (Sintel): 1.35
+== Kangaroo ==
+: ''For 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
+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.
+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]].
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 64/63, 6125/5832
+{{Mapping|legend=1| 1 0 -3 6 | 0 3 10 -6 }}
+: mapping generators: ~2, ~35/24
+[[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 }}
+{{Optimal ET sequence|legend=1| 15, 32, 47b }}
+[[Badness]] (Sintel): 2.91
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 55/54, 64/63, 245/242
+{{Mapping|legend=0| 1 0 -3 6 4 | 0 3 10 -6 -1 }}
+Optimal tunings:
+* WE: ~2 = 1197.8893{{c}}, ~16/11 = 637.1922{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 638.2605{{c}}
+{{Optimal ET sequence|legend=0| 15, 32, 47b }}
+Badness (Sintel): 1.65
 == Fervor ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Fervor]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 464: / Line 616: @@
 [[Comma list]]: 64/63, 9604/9375
-{{Mapping|legend=1| 1 4 -2 -2 | 0 -5 9 10 }}
+{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
+: mapping generators: ~2, ~10/7
-{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 577.776
+: [[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 }}
 {{Optimal ET sequence|legend=1| 2, 25, 27 }}
-[[Badness]]: 0.108455
+[[Badness]] (Sintel): 2.74
 === 11-limit ===
@@ Line 479: / Line 634: @@
 Comma list: 56/55, 64/63, 1350/1331
-Mapping: {{mapping| 1 4 -2 -2 3 | 0 -5 9 10 1 }}
+{{Mapping|legend=0| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 2, 25e, 27e }}
+{{Optimal ET sequence|legend=0| 2, 25e, 27e }}
-Badness: 0.052054
+Badness (Sintel): 1.72
 === 13-limit ===
@@ Line 492: / Line 649: @@
 Comma list: 56/55, 64/63, 78/77, 507/500
-Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}
+{{Mapping|legend=0| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 2f, 25ef, 27e }}
+{{Optimal ET sequence|legend=0| 2f, 27e }}
-Badness: 0.039705
+Badness (Sintel): 1.64
-== Progress ==
+== Sixix ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Progress]].''
+: ''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. 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]]).
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 64/63, 392/375
+[[Comma list]]: 64/63, 3125/2916
-{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
+{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
-{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 562.122
+[[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 }}
-{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 79bcc, 111bcc }}
+{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
-[[Badness]]: 0.066400
+[[Badness]] (Sintel): 4.02
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 56/55, 64/63, 77/75
+Comma list: 55/54, 64/63, 125/121
-Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
+{{Mapping|legend=0| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 47bc, 79bcce }}
+{{Optimal ET sequence|legend=0| 7, 25e, 32 }}
-Badness: 0.031036
+Badness (Sintel): 2.34
-==== 13-limit ====
+=== 13-limit ===
 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|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}}
+{{Optimal ET sequence|legend=0| 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|legend=0| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
+Optimal tunings:
+* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}
+{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
+Badness (Sintel): 2.00
+== Subgroup extensions ==
+=== Supra (2.3.7.11) ===
+Subgroup: 2.3.7.11
-Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
+Comma list: 64/63, 99/98
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.365
+{{Mapping|legend=2| 1 0 6 13 | 0 1 -2 -6 }}
-{{Optimal ET sequence|legend=1| 15, 17c, 32cf }}
+{{Mapping|legend=4| 1 0 0 6 13 | 0 1 0 -2 -6 }}
-Badness: 0.026214
+Optimal tunings:
+* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}
-==== Progressive ====
+{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 26/25, 56/55, 64/63, 77/75
+Badness (Sintel): 0.352
-Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
+Scales: [[supra7]], [[supra12]]
-Optimal tuning (POTE): ~2 = 1\1, ~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𝄫).
-{{Optimal ET sequence|legend=1| 15f, 17c, 32c, 49c }}
+Subgroup: 2.3.7.11.13
-Badness: 0.032721
+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 }}
-[[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|legend=1| 1 3 4 | 0 -5 -6 }}
+{{Optimal ET sequence|legend=0| 12f, 17 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 338.365
+Badness (Sintel): 0.498
-{{Optimal ET sequence|legend=1| 7, 25, 32 }}
+Scales: [[supra7]], [[supra12]]
-[[Badness]]: 0.153088
+=== 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.
-=== 7-limit ===
+[[Subgroup]]: 2.3.7.13/5
-Subgroup: 2.3.5.7
-[[Comma list]]: 64/63, 3125/2916
+[[Comma list]]: 64/63 ({{monzo| 6 -2 -1 0 }}), 91/90 ({{monzo| -1 -2 1 1 }})
-{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
+{{Mapping|legend=3| 1 0 6 -5 | 0 1 -2 4 }}
-{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}
+[[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}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 337.442
+{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 32, 59 }}
-{{Optimal ET sequence|legend=1| 7, 25, 32 }}
+[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
-[[Badness]]: 0.158903
+Scales: [[Oceanfront scales]]
 [[Category:Archytas clan| ]] <!-- main article -->
+[[Category:Archytas| ]] <!-- key article -->
 [[Category:Temperament clans]]
-[[Category:Rank 2]]
+[[Category:Catalogs of rank-2 temperaments]]

Archytas clan: Difference between revisions

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