Archytas clan: Difference between revisions

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The '''archytas clan''' (or '''archy family''') tempers out the [[64/63|Archytas' comma]], 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to [[81/80]] in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 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 ==
== Archy ==
Line 12: Line 15:
: sval mapping generators: ~2, ~3
: 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 }}
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
[[Badness]] (Sintel): 0.159


Scales: [[archy5]], [[archy7]], [[archy12]]
Scales: [[archy5]], [[archy7]], [[archy12]]


=== Overview to extensions ===
=== 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 ===
=== Supra ===
Line 50: Line 72:
Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
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]]
Scales: [[supra7]], [[supra12]]
Line 67: Line 93:
Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
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]]
Scales: [[supra7]], [[supra12]]


== Superpyth ==
== Superpyth ==
{{Main| 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 & 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
[[Subgroup]]: 2.3.5.7


Line 102: Line 119:
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}
{{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 ===
=== 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
Subgroup: 2.3.5.7.11
Line 119: Line 138:
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}
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 ====
==== 13-limit ====
Line 132: Line 153:
Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
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 ====
==== Thomas ====
Line 145: Line 168:
Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}
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 ===
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
Subgroup: 2.3.5.7.11
Line 160: Line 185:
Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}
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 ====
==== 13-limit ====
Line 173: Line 200:
Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}
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 ==
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
[[Subgroup]]: 2.3.5.7
Line 188: Line 217:
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}
{{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 }}
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}


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


=== Quasisupra ===
=== Quasisupra ===
Line 205: Line 236:
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}
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 ====
==== 13-limit ====
Line 218: Line 251:
Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}
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 ===
=== Quasisoup ===
Line 231: Line 266:
Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
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 ==
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
[[Subgroup]]: 2.3.5.7
Line 246: Line 285:
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
{{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 ===
=== 11-limit ===
Line 261: Line 302:
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}
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 ====
==== 13-limit ====
Line 274: Line 317:
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
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 ===
=== Ultramarine ===
Line 287: Line 332:
Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}
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 ====
==== 13-limit ====
Line 300: Line 347:
Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
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).


{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bcef, 116bbcef }}
[[Subgroup]]: 2.3.5.7


Badness: 0.045653
[[Comma list]]: 64/63, 33614/32805


== Flutterpyth ==
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
Subgroup: 2.3.7.11.13.19


Comma list: 64/63, 209/208, 343/342, 364/363
[[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 }}


Mapping: {{mapping| 1 1 4 10 15 9 | 0 -1 -2 -11 -19 -8 }}
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}


Optimal tuning (CTE): ~3/2 = 713.459
[[Badness]] (Sintel): 3.34


== Schism ==
== 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-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
[[Subgroup]]: 2.3.5.7
Line 326: Line 385:
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}


{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}
: [[error map]]: {{val| -2.640 -4.583 -4.896 +20.588 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7376{{c}}
: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}


[[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 ===
=== 11-limit ===
Line 341: Line 402:
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}
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 ==
== 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
[[Subgroup]]: 2.3.5.7
Line 356: Line 423:
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}


{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}
[[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
; Music
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 ''Beatles Improv''] by [[Herman Miller]]
* [https://web.archive.org/web/20201127013829/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 ''Beatles Improv''] by [[Herman Miller]]


=== 11-limit ===
=== 11-limit ===
Line 374: Line 443:
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
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 ====
==== 13-limit ====
Line 387: Line 458:
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
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 ===
=== Ringo ===
Line 400: Line 473:
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}
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 ====
==== 13-limit ====
Line 413: Line 488:
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
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 ===
=== Beetle ===
Line 424: Line 501:
Comma list: 55/54, 64/63, 686/675
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 ====
==== 13-limit ====
Line 439: Line 518:
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
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=1| 10, 27, 37 }}
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Badness: 0.033971
Badness (Sintel): 1.40


== Fervor ==
== Progress ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Fervor]].''
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
 
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.


[[Subgroup]]: 2.3.5.7
[[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 ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 64/63, 1350/1331
Comma list: 56/55, 64/63, 77/75


Mapping: {{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
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
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 ==
== Sixix ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sixix (5-limit)]].''
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}
{{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
[[Comma list]]: 64/63, 3125/2916
Line 566: Line 660:
{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
{{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 ===
=== 11-limit ===
Line 581: Line 677:
Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
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 ===
=== 13-limit ===
Line 594: Line 692:
Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
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 ===
=== 17-limit ===
Line 607: Line 707:
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
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:Archytas clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 12:52, 22 July 2025

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

The archytas clan (or archy family) tempers out the Archytas' comma, 64/63. This means a stack of two 3/2 fifths octave-reduced equals a whole tone of 8/7~9/8 tempered together; two of these tones or equivalently four stacked fifths octave-reduced equal a 9/7 major third. Note the similarity in function to 81/80 in meantone, where four stacked fifths octave-reduced equal a 5/4 major third. This leads to tunings with 3's and 7's quite sharp, such as those of 22edo, 27edo, or 49edo.

This article focuses on rank-2 temperaments. See Archytas family for the rank-3 temperament resulting from tempering out 64/63 alone in the full 7-limit.

Archy

Main article: Superpyth

Subgroup: 2.3.7

Comma list: 64/63

Sval mapping: [⟨1 0 6], ⟨0 1 -2]]

sval mapping generators: ~2, ~3

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

gencom: [2 3; 64/63]

Optimal tunings:

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

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

Badness (Sintel): 0.159

Scales: archy5, archy7, archy12

Overview to extensions

7-limit extensions

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

These all use the same generators as archy.

686/675 gives beatles, splitting the fifth in two. 8748/8575 gives immunized, splitting the twelfth in two. 50/49 gives pajara with a semioctave period. 392/375 gives progress, splitting the twelfth in three. 250/243 gives porcupine, splitting the fourth in three. 126/125 gives augene with a 1/3-octave period. 4375/4374 gives modus, splitting the fifth in four. 3125/3024 gives brightstone. 9604/9375 gives fervor. 3125/2916 gives sixix. 3125/3087 gives passion. Those split the generator in five in various ways. 28/27 gives blacksmith with a 1/5-octave period. Finally, 15625/15552 gives catalan, splitting the twelfth in six.

Temperaments discussed elsewhere are:

Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.

Subgroup extensions

Omitting prime 5, archy can be extended to the 2.3.7.11 subgroup by identifying 11/8 as a diminished fourth (C–Gb). This is called supra, given right below. Discussed elsewhere is suhajira of the neutral clan.

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

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

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

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

Optimal tunings:

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

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

Badness (Sintel): 0.352

Scales: supra7, supra12

Supraphon

Subgroup: 2.3.7.11.13

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

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

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

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

Optimal tunings:

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

Optimal ET sequence: 12f, 17

Badness (Sintel): 0.498

Scales: supra7, supra12

Superpyth

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

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

Subgroup: 2.3.5.7

Comma list: 64/63, 245/243

Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]

Optimal tunings:

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

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

Badness (Sintel): 0.818

11-limit

The canonical extension to the 13-limit finds the ~11/8 at +16 generator steps, as a double-augmented second (C–Dx) and finds the ~13/8 at +13 generator steps, as a double-augmented fourth (C–Fx).

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 22, 27e, 49

Badness (Sintel): 0.826

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.02

Thomas

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 2.03

Suprapyth

Suprapyth finds the ~11/8 at the diminished fifth (C–Gb), and finds the ~13/8 at the diminished seventh (C–Bbb).

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 5, 17, 22

Badness (Sintel): 1.08

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 5f, 17, 22

Badness (Sintel): 1.50

Quasisuper

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

Subgroup: 2.3.5.7

Comma list: 64/63, 2430/2401

Mapping: [⟨1 0 23 6], ⟨0 1 -13 -2]]

Optimal tunings:

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

Optimal ET sequence: 17c, 22, 61d

Badness (Sintel): 1.61

Quasisupra

Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament supra, with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.06

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 17c, 22, 39d

Badness (Sintel): 1.25

Quasisoup

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 22

Badness (Sintel): 2.76

Ultrapyth

Main article: Ultrapyth

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

Subgroup: 2.3.5.7

Comma list: 64/63, 6860/6561

Mapping: [⟨1 0 -20 6], ⟨0 1 14 -2]]

Optimal tunings:

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

Optimal ET sequence: 5, 27c, 32, 37

Badness (Sintel): 2.74

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 5, 32, 37

Badness (Sintel): 2.26

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 5, 32, 37

Badness (Sintel): 2.03

Ultramarine

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 2.58

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.89

Quasiultra

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

Subgroup: 2.3.5.7

Comma list: 64/63, 33614/32805

Mapping: [⟨1 0 31 6], ⟨0 1 -18 -2]]

Optimal tunings:

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

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

Badness (Sintel): 3.34

Schism

See also: Schismatic family § Schism

Schism tempers out the schisma, mapping the ~5/4 to -8 fifths as a diminished fourth (C–Fb) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53dd val) can be used.

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]

Optimal tunings:

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

Optimal ET sequence: 5c, 7c, 12

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.24

Beatles

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

Beatles tempers out 686/675, which may also be characterized by saying it tempers out 2401/2400. It may be described as the 10 & 17c temperament. It splits the fifth into two neutral-third generators of 49/40~60/49; its ploidacot is dicot. 5/4 may be found at -9 generator steps, as a semidiminished fourth (C–Fd). 27edo is an obvious tuning, though 17c-edo and 37edo are among the possibilities.

Beatles extends easily to the no-11 13-limit, as the generator can be interpreted as ~16/13, tempering out 91/90, 169/168, and 196/195.

Subgroup: 2.3.5.7

Comma list: 64/63, 686/675

Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4]]

Optimal tunings:

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

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

Badness (Sintel): 1.16

Music

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.25

Ringo

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 5 4 2], ⟨0 2 -9 -4 5]]

Optimal tunings:

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

Optimal ET sequence: 10, 17c, 27e

Badness (Sintel): 1.09

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 10, 17c, 27e

Badness (Sintel): 0.935

Beetle

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 10, 27, 37

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 10, 27, 37

Badness (Sintel): 1.40

Progress

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

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

Subgroup: 2.3.5.7

Comma list: 64/63, 392/375

Mapping: [⟨1 0 5 6], ⟨0 3 -5 -6]]

mapping generators: ~2, ~10/7

Optimal tunings:

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

Optimal ET sequence: 2, 13, 15, 32c

Badness (Sintel): 1.68

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.03

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 15, 17c, 32cf

Badness (Sintel): 1.08

Progressive

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 2f, 15f, 17c

Badness (Sintel): 1.35

Fervor

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

Fervor tempers out 9704/9375 and may be described as 25 & 27. It splits the 6th harmonic into five generators of ~10/7; its ploidacot is beta-pentacot. 27edo is about as accurate as it can be tuned.

Subgroup: 2.3.5.7

Comma list: 64/63, 9604/9375

Mapping: [⟨1 -1 7 8], ⟨0 5 -9 -10]]

mapping generators: ~2, ~10/7

Optimal tunings:

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

Optimal ET sequence: 2, 25, 27

Badness (Sintel): 2.74

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 2, 25e, 27e

Badness (Sintel): 1.72

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 2f, 27e

Badness (Sintel): 1.64

Sixix

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

Sixix tempers out 3125/2916 and may be described as 25 & 32. It is related to the kleismic family in a way similar to the one between meantone and mavila. In both cases the generator is nominally a 6/5 and the complexity to generate major and minor chords is the same, but in sixix it is tuned extremely sharply, to the point where the 3rd and 5th harmonics are reached by going down instead of up, inverting the logic of chord construction. Its ploidacot is gamma-pentacot.

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/2916

Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]

Optimal tunings:

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

Optimal ET sequence: 7, 18d, 25, 32

Badness (Sintel): 4.02

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 7, 25e, 32

Badness (Sintel): 2.34

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 7, 25e, 32f

Badness (Sintel): 1.91

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

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

Optimal ET sequence: 7, 25e, 32f

Badness (Sintel): 2.00

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