@@ 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 ==
@@ Line 10: / Line 13: @@
 {{Mapping|legend=2| 1 0 6 | 0 1 -2 }}
-: sval mapping generators: ~2, ~3
+{{Mapping|legend=3| 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
-{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
+[[Badness]] (Sintel): 0.159
 Scales: [[archy5]], [[archy7]], [[archy12]]
 === Overview to extensions ===
-Adding [[245/243]] to the list of commas gives superpyth; [[2430/2401]] gives quasisuper; [[36/35]] gives dominant; [[360/343]] gives schism; [[16/15]] gives mother. These all use the same generators as archy.
+==== 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]];
+* [[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]].
-[[50/49]] gives pajara with a semioctave period. [[126/125]] gives augene with a 1/3-octave period. [[28/27]] gives blacksmith with a 1/5-octave period. [[686/675]] gives beatles, splitting the fifth in two. [[250/243]] gives porcupine, splitting the fourth in three. [[4375/4374]] gives modus, splitting the fifth in four. [[3125/3087]] gives passion, splitting the fourth in five.
+These all use the same generators as archy.
-Discussed under their respective 5-limit families are:
+[[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 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. [[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.
-* ''[[Mother]]'' → [[Father family #Mother|Father family]]
-* ''[[Dominant]]'' → [[Meantone family #Dominant|Meantone family]]
+Temperaments discussed elsewhere are:
-* ''[[Augene]]'' → [[Augmented family #Augene|Augmented]]
+* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
-* [[Porcupine]] → [[Porcupine family #Septimal porcupine|Porcupine family]]
+* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
-* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]]
+* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
-* ''[[Blacksmith]]'' → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
+* ''[[Dichotic]]'' (+25/24) → [[Dicot family #Dichotic|Dicot family]]
-* ''[[Catalan]]'' → [[Kleismic family #Catalan|Kleismic family]]
+* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
-* ''[[Modus]]'' → [[Tetracot family #Modus|Tetracot family]]
+* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
-* ''[[Passion]]'' → [[Passion family #Septimal passion|Passion family]]
+* [[Augene]] (+126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
-* ''[[Immunized]]'' → [[Immunity family #Immunized|Immunity family]]
+* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
-* ''[[Suhajira]]'' -> [[Neutral clan #Suhajira|Neutral clan]]
+* [[Modus]] (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
-The rest are considered below.
+* ''[[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]]
+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–G♭). This is called supra, given right below. Discussed elsewhere is [[suhajira]] of the [[rastmic clan #Suhajira|rastmic clan]].
 === Supra ===
@@ Line 46: / Line 69: @@
 Comma list: 64/63, 99/98
-Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
+Subgroup-val mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
-Gencom mapping: {{mapping| 1 1 0 4 7 | 0 1 0 -2 -6 }}
+Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}
-: 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}}
-Optimal tuning (POTE): ~3/2 = 707.192
+{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}
-{{Optimal ET sequence|legend=1| 5, 12, 17, 39d, 56d }}
+Badness (Sintel): 0.352
 Scales: [[supra7]], [[supra12]]
 ==== 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𝄫).
 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 }}
+Subgroup-val 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 mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
-: 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}}
-Optimal tuning (POTE): ~3/2 = 706.137
+{{Optimal ET sequence|legend=0| 12f, 17 }}
-{{Optimal ET sequence|legend=1| 12f, 17 }}
+Badness (Sintel): 0.498
 Scales: [[supra7]], [[supra12]]
 == 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. This temperament has a fifth generator of ~3/2 = ~710¢ and ~5/4 is found at +9 generator steps, as an augmented second (C-D#). It also has a weak extension, [[Jubilismic clan #Bipyth|bipyth]] (10cd &amp; 22), tempering 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
-{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
-[[Badness]]: 0.135141
+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.
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
@@ Line 100: / Line 118: @@
 {{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}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 710.291
+: [[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 ET sequence|legend=1| 5, 17, 22, 27, 49 }}
+{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
-[[Badness]]: 0.032318
+[[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
@@ Line 117: / Line 135: @@
 Mapping: {{mapping| 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 130: / Line 150: @@
 Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 710.479
+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=1| 22, 27e, 49, 76bcde }}
+{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
-Badness: 0.024673
+Badness (Sintel): 1.02
+==== Uberpyth ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 64/63, 100/99, 144/143, 245/243
+Mapping: {{mapping| 1 0 -12 6 -22 26 | 0 1 9 -2 16 -14 }}
+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=0| 22f, 27e, 49f, 125bcddeeeff, 174bbcdddeeeeffff }}
+Badness (Sintel): 1.11
 ==== Thomas ====
@@ Line 143: / Line 180: @@
 Mapping: {{mapping| 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-Gb), and finds the ~13/8 at the diminished seventh (C-Bbb).
+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 158: / Line 197: @@
 Mapping: {{mapping| 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 171: / Line 212: @@
 Mapping: {{mapping| 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 ==
-Quasisuper can be described as 17c & 22, with the ~5/4 mapped to -13 generator steps, as a double diminished fifth (C-Gbb).
+{{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 186: / Line 231: @@
 {{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
@@ Line 203: / Line 248: @@
 Mapping: {{mapping| 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 216: / Line 263: @@
 Mapping: {{mapping| 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 229: / Line 278: @@
 Mapping: {{mapping| 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, mapping the ~5/4 to +14 fifths as a double augmented unison (C-Cx).
+{{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 [[the Biosphere #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).
 [[Subgroup]]: 2.3.5.7
@@ Line 244: / Line 298: @@
 {{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}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 713.651
+: [[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 ET sequence|legend=1| 5, 32, 37 }}
+{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
-[[Badness]]: 0.108466
+[[Badness]] (Sintel): 2.74
 === 11-limit ===
@@ Line 259: / Line 315: @@
 Mapping: {{mapping| 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 272: / Line 330: @@
 Mapping: {{mapping| 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 285: / Line 345: @@
 Mapping: {{mapping| 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 298: / Line 360: @@
 Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 713.811
+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 }}
-{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bcef, 116bbcef }}
+Badness (Sintel): 1.89
-Badness: 0.045653
+== 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
+[[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 ==
 {{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.
+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 315: / Line 398: @@
 {{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}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.556
+: [[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 ET sequence|legend=1| 12, 41d, 53d }}
+{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
-[[Badness]]: 0.056648
+[[Badness]] (Sintel): 1.43
 === 11-limit ===
@@ Line 330: / Line 415: @@
 Mapping: {{mapping| 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 345: / Line 436: @@
 {{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 363: / Line 456: @@
 Mapping: {{mapping| 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 376: / Line 471: @@
 Mapping: {{mapping| 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 389: / Line 486: @@
 Mapping: {{mapping| 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 402: / Line 501: @@
 Mapping: {{mapping| 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 413: / Line 514: @@
 Comma list: 55/54, 64/63, 686/675
-Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15}}
+Mapping: {{mapping| 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 428: / Line 531: @@
 Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
+{{Optimal ET sequence|legend=0| 10, 27, 37 }}
-{{Optimal ET sequence|legend=1| 10, 27, 37 }}
+Badness (Sintel): 1.40
-Badness: 0.033971
+== Progress ==
+{{Distinguish| Progression }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
-== Fervor ==
+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.
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Fervor]].''
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 64/63, 9604/9375
+[[Comma list]]: 64/63, 392/375
-{{Mapping|legend=1| 1 4 -2 -2 | 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
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}
-{{Optimal ET sequence|legend=1| 2, 25, 27 }}
+{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
-[[Badness]]: 0.108455
+[[Badness]] (Sintel): 1.68
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 56/55, 64/63, 1350/1331
+Comma list: 56/55, 64/63, 77/75
+Mapping: {{mapping| 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
-Mapping: {{mapping| 1 4 -2 -2 3 | 0 -5 9 10 1 }}
+Comma list: 56/55, 64/63, 66/65, 77/75
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 577.850
+Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
-{{Optimal ET sequence|legend=1| 2, 25e, 27e }}
+Optimal tunings:
+* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}
-Badness: 0.052054
+{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
-=== 13-limit ===
+Badness (Sintel): 1.08
+==== Progressive ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 56/55, 64/63, 78/77, 507/500
+Comma list: 26/25, 56/55, 64/63, 77/75
+Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
-Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}
+Optimal tunings:
+* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 578.060
+{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}
-{{Optimal ET sequence|legend=1| 2f, 25ef, 27e }}
+Badness (Sintel): 1.35
-Badness: 0.039705
+== Fervor ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
-== Progress ==
+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.
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Progress]].''
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 64/63, 392/375
+[[Comma list]]: 64/63, 9604/9375
-{{Mapping|legend=1| 1 0 5 6 | 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
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}
-{{Optimal ET sequence|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 ===
 Subgroup: 2.3.5.7.11
-Comma list: 56/55, 64/63, 77/75
+Comma list: 56/55, 64/63, 1350/1331
-Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
+Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 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
-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: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.365
+[[Comma list]]: 64/63, 3125/2916
-{{Optimal ET sequence|legend=1| 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: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 563.239
+Comma list: 55/54, 64/63, 125/121
-{{Optimal ET sequence|legend=1| 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 list]]: 3125/2916
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=1| 1 3 4 | 0 -5 -6 }}
+Comma list: 40/39, 55/54, 64/63, 125/121
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 338.365
+Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
-{{Optimal ET sequence|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|legend=1| 1 3 4 0 | 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 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 7, 25, 32 }}
+{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
-[[Badness]]: 0.158903
+Badness (Sintel): 2.00
 [[Category:Archytas clan| ]] <!-- main article -->
 [[Category:Temperament clans]]
-[[Category:Rank 2]]
+[[Category:Catalogs of rank-2 temperaments]]

Archytas clan: Difference between revisions