@@ 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 12: / Line 15: @@
 : sval mapping generators: ~2, ~3
-{{Mapping|legend=3| 1 1 0 4 | 0 1 0 -2 }}
+{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}
-: [[gencom]]: [2 3/2; 64/63]
+: [[gencom]]: [2 3; 64/63]
-[[Optimal tuning]] ([[POTE]]): ~3/2 = 709.321
+[[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 }}
 {{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]];
+* 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:
+[[686/675]] gives beatles, splitting the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a semioctave period. [[392/375]] gives progress, splitting the twelfth in three. [[250/243]] gives porcupine, splitting the fourth in three. [[126/125]] gives augene with a 1/3-octave period. [[4375/4374]] gives modus, splitting the fifth in four. [[3125/3024]] gives brightstone. [[9604/9375]] gives fervor. [[3125/2916]] gives sixix. [[3125/3087]] gives passion. Those split the generator in five in various ways. [[28/27]] gives blacksmith with a 1/5-octave period. Finally, [[15625/15552]] gives catalan, splitting the twelfth in six.
-* ''[[Mother]]'' → [[Father family #Mother|Father family]]
-* ''[[Dominant]]'' → [[Meantone family #Dominant|Meantone family]]
-* ''[[Augene]]'' → [[Augmented family #Augene|Augmented]]
-* [[Porcupine]] → [[Porcupine family #Septimal porcupine|Porcupine family]]
-* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]]
-* ''[[Blacksmith]]'' → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
-* ''[[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]]
-* ''[[Suhajira]]'' → [[Neutral clan #Suhajira|Neutral clan]]
-* ''[[Brightstone]]'' → [[Magic family #Brightstone|Magic family]]
-The rest are considered below.
+Temperaments discussed elsewhere are:
+* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
+* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
+* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
+* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
+* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
+* [[Augene]] (+126/125) → [[Augmented family #Augene|Augmented family]]
+* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
+* ''[[Modus]]'' (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
+* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
+* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
+* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
+* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
+Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.
+==== Subgroup extensions ====
+Omitting prime 5, archy can be extended to the 2.3.7.11 subgroup by identifying 11/8 as a diminished fourth (C–Gb). This is called supra, given right below. Discussed elsewhere is [[suhajira]] of the [[neutral clan #Suhajira|neutral clan]].
 === Supra ===
@@ Line 50: / Line 72: @@
 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 mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}
+: gencom: [2 3; 64/63 99/98]
-: gencom: [2 3/2; 64/63 99/98]
+Optimal tunings:
+* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}
-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]]
@@ Line 67: / Line 93: @@
 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 mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
+: gencom: [2 3; 64/63 78/77 99/98]
-: gencom: [2 3/2; 64/63 78/77 99/98]
+Optimal tunings:
+* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}
-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| 22 & 27 }}. ~5/4 is found at +9 generator steps, as an augmented second (C–D#). 49edo remains an obvious tuning choice.
-[[Subgroup]]: 2.3.5
-[[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
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
@@ Line 102: / Line 119: @@
 {{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 ===
-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).
+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 119: / Line 138: @@
 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 132: / Line 153: @@
 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
 ==== Thomas ====
@@ Line 145: / Line 168: @@
 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–Gb), and finds the ~13/8 at the diminished seventh (C–Bbb).
 Subgroup: 2.3.5.7.11
@@ Line 160: / Line 185: @@
 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 173: / Line 200: @@
 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).
+Quasisuper can be described as {{nowrap| 17c & 22 }}, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–Gbb).
 [[Subgroup]]: 2.3.5.7
@@ Line 188: / Line 217: @@
 {{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 ===
@@ Line 205: / Line 236: @@
 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 218: / Line 251: @@
 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 231: / Line 266: @@
 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 }}
+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).
 [[Subgroup]]: 2.3.5.7
@@ Line 246: / Line 285: @@
 {{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 261: / Line 302: @@
 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 274: / Line 317: @@
 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 287: / Line 332: @@
 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 300: / Line 347: @@
 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 }}
+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–Abbb).
+[[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| 5e, 32e, 37, 79bcef, 116bbcef }}
+{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
-Badness: 0.045653
+[[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–Fb) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53dd val) can be used.
 [[Subgroup]]: 2.3.5.7
@@ Line 317: / Line 385: @@
 {{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 332: / Line 402: @@
 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 347: / Line 423: @@
 {{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
-{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}
+: [[error map]]: {{val| -3.376 +4.475 +2.682 -1.940 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}
+: error map: {{val| 0.000 +10.209 +8.949 +6.847 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 355.904
+{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
-{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
+[[Badness]] (Sintel): 1.16
-[[Badness]]: 0.045872
 ; 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 365: / Line 443: @@
 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 378: / Line 458: @@
 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 391: / Line 473: @@
 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 404: / Line 488: @@
 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 415: / Line 501: @@
 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 430: / Line 518: @@
 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 4 -2 -2 3 | 0 -5 9 10 1 }}
+Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 577.850
+Optimal tunings:
+* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}
-{{Optimal ET sequence|legend=1| 2, 25e, 27e }}
+{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}
-Badness: 0.052054
+Badness (Sintel): 1.03
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 56/55, 64/63, 78/77, 507/500
+Comma list: 56/55, 64/63, 66/65, 77/75
+Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
-Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}
+Optimal tunings:
+* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 578.060
+{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
-{{Optimal ET sequence|legend=1| 2f, 25ef, 27e }}
+Badness (Sintel): 1.08
-Badness: 0.039705
+==== Progressive ====
+Subgroup: 2.3.5.7.11.13
-== Progress ==
+Comma list: 26/25, 56/55, 64/63, 77/75
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Progress]].''
-[[Subgroup]]: 2.3.5.7
+Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
-[[Comma list]]: 64/63, 392/375
+Optimal tunings:
+* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}
-{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
+{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}
-{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
+Badness (Sintel): 1.35
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 562.122
+== Fervor ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
-{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 79bcc, 111bcc }}
+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.
-[[Badness]]: 0.066400
+[[Subgroup]]: 2.3.5.7
-=== 11-limit ===
+[[Comma list]]: 64/63, 9604/9375
-Subgroup: 2.3.5.7.11
-Comma list: 56/55, 64/63, 77/75
+{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
-Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
+: mapping generators: ~2, ~10/7
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.085
+[[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, 47bc, 79bcce }}
+{{Optimal ET sequence|legend=1| 2, 25, 27 }}
-Badness: 0.031036
+[[Badness]] (Sintel): 2.74
-==== 13-limit ====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11
-Comma list: 56/55, 64/63, 66/65, 77/75
+Comma list: 56/55, 64/63, 1350/1331
-Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
+Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.365
+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| 15, 17c, 32cf }}
+{{Optimal ET sequence|legend=0| 2, 25e, 27e }}
-Badness: 0.026214
+Badness (Sintel): 1.72
-==== Progressive ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 26/25, 56/55, 64/63, 77/75
+Comma list: 56/55, 64/63, 78/77, 507/500
-Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
+Mapping: {{mapping| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 563.239
+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| 15f, 17c, 32c, 49c }}
+{{Optimal ET sequence|legend=0| 2f, 27e }}
-Badness: 0.032721
+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 }}
-[[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.
-[[Comma list]]: 3125/2916
+[[Subgroup]]: 2.3.5.7
-{{Mapping|legend=1| 1 3 4 | 0 -5 -6 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 338.365
-{{Optimal ET sequence|legend=1| 7, 25, 32 }}
-[[Badness]]: 0.153088
-=== 7-limit ===
-Subgroup: 2.3.5.7
 [[Comma list]]: 64/63, 3125/2916
@@ Line 557: / Line 660: @@
 {{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
-{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 337.442
+: [[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| 7, 25, 32 }}
+{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
-[[Badness]]: 0.158903
+[[Badness]] (Sintel): 4.02
 === 11-limit ===
@@ Line 572: / Line 677: @@
 Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 337.564
+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: {{Optimal ET sequence| 7, 25e, 32 }}
+{{Optimal ET sequence|legend=0| 7, 25e, 32 }}
-Badness: 0.070799
+Badness (Sintel): 2.34
 === 13-limit ===
@@ Line 585: / Line 692: @@
 Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 337.483
+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: {{Optimal ET sequence| 7, 25e, 32f }}
+{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
-Badness: 0.046206
+Badness (Sintel): 1.91
 === 17-limit ===
@@ Line 598: / Line 707: @@
 Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 337.513
+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: {{Optimal ET sequence| 7, 25e, 32f }}
+{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
-Badness: 0.039224
+Badness (Sintel): 2.00
 [[Category:Archytas clan| ]] <!-- main article -->
 [[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Rank 2]]

Archytas clan: Difference between revisions