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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 8: Line 11:
[[Comma list]]: 64/63
[[Comma list]]: 64/63


{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}
{{Mapping|legend=3| 1 0 6 | 0 1 -2 }}


: sval mapping generators: ~2, ~3
{{Mapping|legend=5| 1 0 0 6 | 0 1 0 -2 }}
: mapping generators: ~2, ~3


{{Mapping|legend=3| 1 1 0 4 | 0 1 0 -2 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}
: [[error map]]: {{val| -3.045 +2.522 +3.952 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}
: error map: {{val| 0.000 +7.435 +12.394 }}


: [[gencom]]: [2 3/2; 64/63]
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 709.321
[[Badness]] (Sintel): 0.159
 
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}


Scales: [[archy5]], [[archy7]], [[archy12]]
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:
[[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.
* [[#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]];
Discussed under their respective 5-limit families are:
* [[#Quasisuper|Quasisuper]] adds [[2430/2401]], for a tuning between 17c-edo and [[22edo]];
* ''[[Mother]]'' → [[Father family #Mother|Father family]]
* [[#Superpyth|Superpyth]] adds [[245/243]], for a tuning between 22edo and [[27edo]];
* ''[[Dominant]]'' → [[Meantone family #Dominant|Meantone family]]
* [[#Quasiultra|Quasiultra]] adds [[33614/32805]], for a tuning between 27edo and [[32edo]];
* ''[[Augene]]'' → [[Augmented family #Augene|Augmented]]
* [[#Ultrapyth|Ultrapyth]] adds [[6860/6561]], for a tuning sharp of 32edo;  
* [[Porcupine]] → [[Porcupine family #Septimal porcupine|Porcupine family]]
* Mother adds [[16/15]], for an exotemperament well tuned around [[5edo]].
* [[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.
 
=== Supra ===
Subgroup: 2.3.7.11
 
Comma list: 64/63, 99/98
 
Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
 
Gencom mapping: {{mapping| 1 1 0 4 7 | 0 1 0 -2 -6 }}
 
: gencom: [2 3/2; 64/63 99/98]
 
Optimal tuning (POTE): ~3/2 = 707.192
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 39d, 56d }}
 
Scales: [[supra7]], [[supra12]]


==== Supraphon ====
These all use the same generators as archy.  
Subgroup: 2.3.7.11.13


Comma list: 64/63, 78/77, 99/98
[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a semi-octave period. [[392/375]] gives progress. [[6125/5832]] gives kangaroo. Those split the twelfth in three. [[250/243]] gives porcupine, splitting the fourth in three. [[126/125]] gives augene with a 1/3-octave period. [[4375/4374]] gives modus, splitting the fifth in four. [[3125/3024]] gives brightstone. [[9604/9375]] gives fervor. [[3125/2916]] gives sixix. [[3125/3087]] gives passion. Those split the generator in five in various ways. [[15625/15552]] gives catalan, splitting the twelfth in six. Finally, [[28/27]] gives blackwood and [[17496/16807]] gives [[gothic]], reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.


Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
Temperaments discussed elsewhere are:  
* ''[[Phlegyas]]'' (+10/9) → [[Very low accuracy temperaments #Phlegyas|Very low accuracy temperaments]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* ''[[Dichotic]]'' (+25/24) → [[Dicot family #Dichotic|Dicot family]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Augene]] (+126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* [[Modus]] (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
* ''[[Gothic]]'' (+17496/16807) → [[17th-octave temperaments #Gothic|17th-octave temperaments]]


Gencom mapping: {{mapping| 1 1 0 4 7 9 | 0 1 0 -2 -6 -9 }}
The rest are considered below.  
 
: gencom: [2 3/2; 64/63 78/77 99/98]
 
Optimal tuning (POTE): ~3/2 = 706.137
 
{{Optimal ET sequence|legend=1| 12f, 17 }}
 
Scales: [[supra7]], [[supra12]]


==== Subgroup extensions ====
Omitting prime 5, archy can be extended to the [[2.3.7.11 subgroup]] by identifying 11/8 as a diminished fourth (C–G♭). This is called supra, given in [[#Subgroup extensions_2|#Subgroup extensions]]. Discussed elsewhere is [[suhajira]] of the [[rastmic clan #Suhajira|rastmic clan]].


== Superpyth ==
== 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| [[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 }}
Extending superpyth to the [[13-limit]] is more diffcult. Tridecimal superpyth finds the ~[[13/8]] at +13 generator steps, as a double-augmented fourth (C–F𝄪), for which 27edo can be recommended as a tuning since it is the only [[13-odd-limit]] [[diamond monotone]] tuning. The other extension, called uberpyth, is more flexible with its tunings, but unfortunately tends to tune the 13 very sharp.


[[Optimal 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 79:
{{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).
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}
{{Mapping|legend=0| 1 0 -12 6 -22 | 0 1 9 -2 16 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 710.175
Optimal tunings:
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


{{Optimal ET sequence|legend=1| 22, 27e, 49 }}
{{Optimal ET sequence|legend=0| 22, 27e, 49 }}


Badness: 0.024976
Badness (Sintel): 0.826


==== 13-limit ====
==== 13-limit ====
Line 130: Line 109:
Comma list: 64/63, 78/77, 91/90, 100/99
Comma list: 64/63, 78/77, 91/90, 100/99


Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
{{Mapping|legend=0| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
 
Optimal tunings:
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}
 
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
 
Badness (Sintel): 1.02
 
==== Uberpyth ====
Subgroup: 2.3.5.7.11.13


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 710.479
Comma list: 64/63, 100/99, 144/143, 245/243


{{Optimal ET sequence|legend=1| 22, 27e, 49, 76bcde }}
{{Mapping|legend=0| 1 0 -12 6 -22 26 | 0 1 9 -2 16 -14 }}


Badness: 0.024673
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 ====
==== Thomas ====
Line 143: Line 139:
Comma list: 64/63, 100/99, 169/168, 245/243
Comma list: 64/63, 100/99, 169/168, 245/243


Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}
{{Mapping|legend=0| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 355.036
Optimal tunings:
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


{{Optimal ET sequence|legend=1| 17e, 27e, 44, 71d }}
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}


Badness: 0.049183
Badness (Sintel): 2.03


=== Suprapyth ===
=== Suprapyth ===
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
Subgroup: 2.3.5.7.11
Line 158: Line 156:
Comma list: 55/54, 64/63, 99/98
Comma list: 55/54, 64/63, 99/98


Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}
{{Mapping|legend=0| 1 0 -12 6 13 | 0 1 9 -2 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 709.495
Optimal tunings:
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


{{Optimal ET sequence|legend=1| 5, 12c, 17, 22 }}
{{Optimal ET sequence|legend=0| 5, 17, 22 }}


Badness: 0.032768
Badness (Sintel): 1.08


==== 13-limit ====
==== 13-limit ====
Line 171: Line 171:
Comma list: 55/54, 64/63, 65/63, 99/98
Comma list: 55/54, 64/63, 65/63, 99/98


Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}
{{Mapping|legend=0| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 708.703
Optimal tunings:  
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}


{{Optimal ET sequence|legend=1| 17, 22, 83cdf }}
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}


Badness: 0.036336
Badness (Sintel): 1.50


== Quasisuper ==
== 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
[[Subgroup]]: 2.3.5.7
Line 188: Line 192:
{{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 ===
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
Subgroup: 2.3.5.7.11


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


Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}
{{Mapping|legend=0| 1 0 23 6 13 | 0 1 -13 -2 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 708.205
Optimal tunings:
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


{{Optimal ET sequence|legend=1| 17c, 22, 39d, 61d }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}


Badness: 0.032203
Badness (Sintel): 1.06


==== 13-limit ====
==== 13-limit ====
Line 216: Line 222:
Comma list: 64/63, 78/77, 91/90, 121/120
Comma list: 64/63, 78/77, 91/90, 121/120


Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}
{{Mapping|legend=0| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 708.004
Optimal tunings:
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


{{Optimal ET sequence|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


Badness: 0.030219
Badness (Sintel): 1.25


=== Quasisoup ===
=== Quasisoup ===
Line 229: Line 237:
Comma list: 55/54, 64/63, 2430/2401
Comma list: 55/54, 64/63, 2430/2401


Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
{{Mapping|legend=0| 1 0 23 6 -22 | 0 1 -13 -2 16 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 709.021
Optimal tunings:
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


{{Optimal ET sequence|legend=1| 5ce, 17ce, 22 }}
{{Optimal ET sequence|legend=0| 22 }}


Badness: 0.083490
Badness (Sintel): 2.76


== Ultrapyth ==
== Ultrapyth ==
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 [[#Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 246: Line 259:
{{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 259: Line 274:
Comma list: 55/54, 64/63, 2401/2376
Comma list: 55/54, 64/63, 2401/2376


Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}
{{Mapping|legend=0| 1 0 -20 6 21 | 0 1 14 -2 -11 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 713.395
Optimal tunings:
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


{{Optimal ET sequence|legend=1| 5, 32, 37 }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Badness: 0.068238
Badness (Sintel): 2.26


==== 13-limit ====
==== 13-limit ====
Line 272: Line 289:
Comma list: 55/54, 64/63, 91/90, 1573/1568
Comma list: 55/54, 64/63, 91/90, 1573/1568


Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
{{Mapping|legend=0| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 713.500
Optimal tunings:
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


{{Optimal ET sequence|legend=1| 5, 32, 37 }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Badness: 0.049172
Badness (Sintel): 2.03


=== Ultramarine ===
=== Ultramarine ===
Line 285: Line 304:
Comma list: 64/63, 100/99, 3773/3645
Comma list: 64/63, 100/99, 3773/3645


Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}
{{Mapping|legend=0| 1 0 -20 6 -38 | 0 1 14 -2 26 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 713.791
Optimal tunings:
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bce, 116bbce }}
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}


Badness: 0.078068
Badness (Sintel): 2.58


==== 13-limit ====
==== 13-limit ====
Line 298: Line 319:
Comma list: 64/63, 91/90, 100/99, 847/845
Comma list: 64/63, 91/90, 100/99, 847/845


Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
{{Mapping|legend=0| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
 
Optimal tunings:
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 713.811
{{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 ==
== 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–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
[[Subgroup]]: 2.3.5.7
Line 317: Line 359:
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.556
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}
{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
: [[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, 29d, 41d, 53d }}
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}


[[Badness]]: 0.056648
[[Badness]] (Sintel): 1.43


=== 11-limit ===
=== 11-limit ===
Line 330: Line 374:
Comma list: 45/44, 64/63, 99/98
Comma list: 45/44, 64/63, 99/98


Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}
{{Mapping|legend=0| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.136
Optimal tunings:
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


{{Optimal ET sequence|legend=1| 12, 29de, 41de }}
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}


Badness: 0.037482
Badness (Sintel): 1.24


== Beatles ==
== 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 347: Line 397:
{{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}}
: [[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
; 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 363: Line 415:
Comma list: 64/63, 100/99, 686/675
Comma list: 64/63, 100/99, 686/675


Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
{{Mapping|legend=0| 1 1 5 4 10 | 0 2 -9 -4 -22 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 356.140
Optimal tunings:
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


{{Optimal ET sequence|legend=1| 27e, 37, 64be, 91bcde }}
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}


Badness: 0.045639
Badness (Sintel): 1.51


==== 13-limit ====
==== 13-limit ====
Line 376: Line 430:
Comma list: 64/63, 91/90, 100/99, 169/168
Comma list: 64/63, 91/90, 100/99, 169/168


Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
{{Mapping|legend=0| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 356.229
Optimal tunings:
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


{{Optimal ET sequence|legend=1| 27e, 37, 64be }}
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}


Badness: 0.030161
Badness (Sintel): 1.25


=== Ringo ===
=== Ringo ===
Line 389: Line 445:
Comma list: 56/55, 64/63, 540/539
Comma list: 56/55, 64/63, 540/539


Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}
{{Mapping|legend=0| 1 1 5 4 2 | 0 2 -9 -4 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 355.419
Optimal tunings:
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


{{Optimal ET sequence|legend=1| 10, 17c, 27e }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


Badness: 0.032863
Badness (Sintel): 1.09


==== 13-limit ====
==== 13-limit ====
Line 402: Line 460:
Comma list: 56/55, 64/63, 78/77, 91/90
Comma list: 56/55, 64/63, 78/77, 91/90


Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
{{Mapping|legend=0| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 355.456
Optimal tunings:
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}


{{Optimal ET sequence|legend=1| 10, 17c, 27e }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


Badness: 0.022619
Badness (Sintel): 0.935


=== Beetle ===
=== Beetle ===
Line 415: Line 475:
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|legend=0| 1 1 5 4 -1 | 0 2 -9 -4 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 356.710
Optimal tunings:
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


{{Optimal ET sequence|legend=1| 10, 27, 37 }}
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Badness: 0.058084
Badness (Sintel): 1.92


==== 13-limit ====
==== 13-limit ====
Line 428: Line 490:
Comma list: 55/54, 64/63, 91/90, 169/168
Comma list: 55/54, 64/63, 91/90, 169/168


Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
{{Mapping|legend=0| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}


Optimal 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 [[3/1|perfect twelfth]] into three generators of [[~]][[10/7]]; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]], and the 13-limit extension further equates it to [[13/9]]. [[32edo|32c-edo]] gives an obvious tuning.


[[Subgroup]]: 2.3.5.7
[[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 }}
 
: mapping generators: ~2, ~10/7
{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}


[[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|legend=0| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
 
Optimal tunings:
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}
 
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}
 
Badness (Sintel): 1.03
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 64/63, 66/65, 77/75


Mapping: {{mapping| 1 4 -2 -2 3 | 0 -5 9 10 1 }}
{{Mapping|legend=0| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 577.850
Optimal tunings:
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}


{{Optimal ET sequence|legend=1| 2, 25e, 27e }}
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}


Badness: 0.052054
Badness (Sintel): 1.08


=== 13-limit ===
==== Progressive ====
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: 26/25, 56/55, 64/63, 77/75
 
{{Mapping|legend=0| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
 
Optimal tunings:
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}


Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 578.060
Badness (Sintel): 1.35


{{Optimal ET sequence|legend=1| 2f, 25ef, 27e }}
== Kangaroo ==
: ''For 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''


Badness: 0.039705
Kangaroo splits the [[3/1|perfect twelfth]] into three [[35/24]][[~]][[40/27]] generators; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]]. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat.


== Progress ==
Originally the canonical extension of 5-limit kangaroo was [[trienstonic clan #Parakangaroo|parakangaroo]], but it was decanonized in 2024 in favor of this extension because parakangaroo is an [[exotemperament]] that only supports the [[patent val]] of [[15edo]], while this extension is a bit more practical since it is at least low-medium accuracy and supports the [[patent val]]s of 15edo, [[17edo]] and [[32edo]].
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Progress]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 392/375
[[Comma list]]: 64/63, 6125/5832


{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
{{Mapping|legend=1| 1 0 -3 6 | 0 3 10 -6 }}
: mapping generators: ~2, ~35/24


{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.1767{{c}}, ~35/24 = 637.2914{{c}}
: [[error map]]: {{val| -1.823 +9.919 -7.930 -3.514 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 638.1977{{c}}
: error map: {{val| 0.000 +12.638 -4.336 +1.988 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 562.122
{{Optimal ET sequence|legend=1| 15, 32, 47b }}


{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 79bcc, 111bcc }}
[[Badness]] (Sintel): 2.91
 
[[Badness]]: 0.066400


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 64/63, 77/75
Comma list: 55/54, 64/63, 245/242
 
{{Mapping|legend=0| 1 0 -3 6 4 | 0 3 10 -6 -1 }}


Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
Optimal tunings:  
* WE: ~2 = 1197.8893{{c}}, ~16/11 = 637.1922{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 638.2605{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.085
{{Optimal ET sequence|legend=0| 15, 32, 47b }}


{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 47bc, 79bcce }}
Badness (Sintel): 1.65


Badness: 0.031036
== Fervor ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''


==== 13-limit ====
Fervor tempers out [[9604/9375]] and may be described as {{nowrap| 25 & 27 }}. It splits the [[6/1|6th harmonic]] into five generators of [[~]][[10/7]]; its ploidacot is beta-pentacot. The 11-limit extension simply equates the generator to [[16/11]]. [[27edo]] is about as accurate as it can be tuned.  
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 64/63, 66/65, 77/75
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
[[Comma list]]: 64/63, 9604/9375


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.365
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
: mapping generators: ~2, ~10/7


{{Optimal ET sequence|legend=1| 15, 17c, 32cf }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}
: [[error map]]: {{val| -3.726 +3.230 +4.980 -1.550 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 622.3179{{c}}
: error map: {{val| 0.000 +9.634 +12.826 +7.996 }}


Badness: 0.026214
{{Optimal ET sequence|legend=1| 2, 25, 27 }}


==== Progressive ====
[[Badness]] (Sintel): 2.74
Subgroup: 2.3.5.7.11.13


Comma list: 26/25, 56/55, 64/63, 77/75
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
Comma list: 56/55, 64/63, 1350/1331


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 563.239
{{Mapping|legend=0| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}


{{Optimal ET sequence|legend=1| 15f, 17c, 32c, 49c }}
Optimal tunings:
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}


Badness: 0.032721
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}


== Sixix ==
Badness (Sintel): 1.72
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}


Sixix 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.
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Subgroup]]: 2.3.5
Comma list: 56/55, 64/63, 78/77, 507/500


[[Comma list]]: 3125/2916
{{Mapping|legend=0| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}


{{Mapping|legend=1| 1 3 4 | 0 -5 -6 }}
Optimal tunings:
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 338.365
{{Optimal ET sequence|legend=0| 2f, 27e }}


{{Optimal ET sequence|legend=1| 7, 25, 32 }}
Badness (Sintel): 1.64


[[Badness]]: 0.153088
== Sixix ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sixix (5-limit)]].''
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}


=== 7-limit ===
Sixix tempers out 3125/2916 and may be described as {{nowrap| 25 & 32 }}. It is related to the [[kleismic family]] in a way similar to the one between [[meantone]] and [[mavila]]. In both cases the generator is nominally a 6/5 and the complexity to generate major and minor chords is the same, but in sixix it is tuned extremely sharply, to the point where the 3rd and 5th harmonics are reached by going down instead of up, inverting the logic of chord construction. Its ploidacot is gamma-pentacot. An important val not in the [[optimal ET sequence]] is 39c, that is, [[39edo]] using the second-best approximation of [[5/4]] (the 369-cent one inherited from [[13edo]]).
Subgroup: 2.3.5.7
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 3125/2916
[[Comma list]]: 64/63, 3125/2916
Line 559: Line 671:
{{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}}
: [[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 tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 337.442
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}


{{Optimal ET sequence|legend=1| 7, 25, 32 }}
[[Badness]] (Sintel): 4.02
 
[[Badness]]: 0.158903


=== 11-limit ===
=== 11-limit ===
Line 572: Line 686:
Comma list: 55/54, 64/63, 125/121
Comma list: 55/54, 64/63, 125/121


Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
{{Mapping|legend=0| 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 585: Line 701:
Comma list: 40/39, 55/54, 64/63, 125/121
Comma list: 40/39, 55/54, 64/63, 125/121


Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
{{Mapping|legend=0| 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 598: Line 716:
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121


Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
{{Mapping|legend=0| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
 
Optimal tunings:
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}
 
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
 
Badness (Sintel): 2.00
 
== Subgroup extensions ==
=== Supra (2.3.7.11) ===
Subgroup: 2.3.7.11
 
Comma list: 64/63, 99/98
 
{{Mapping|legend=2| 1 0 6 13 | 0 1 -2 -6 }}
 
{{Mapping|legend=4| 1 0 0 6 13 | 0 1 0 -2 -6 }}
 
Optimal tunings:
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}
 
{{Optimal ET sequence|legend=0| 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
 
{{Mapping|legend=2| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
 
{{Mapping|legend=4| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
 
Optimal tunings:
* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 17 }}
 
Badness (Sintel): 0.498
 
Scales: [[supra7]], [[supra12]]
 
=== Oceanfront (2.3.7.13/5) ===
This is the [[restriction]] of [[#Ultrapyth|ultrapyth]] to the 2.3.7.13/5 subgroup, mapping [[13/10]] to the major third.
 
[[Subgroup]]: 2.3.7.13/5
 
[[Comma list]]: 64/63 ({{monzo| 6 -2 -1 0 }}), 91/90 ({{monzo| -1 -2 1 1 }})
 
{{Mapping|legend=3| 1 0 6 -5 | 0 1 -2 4 }}
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1197.2656{{c}}, ~3/2 = 710.1993{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.7413{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 337.513
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 32, 59 }}


Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents


Badness: 0.039224
Scales: [[Oceanfront scales]]


[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Archytas| ]] <!-- key article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]

Revision as of 21:13, 5 November 2026

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

Subgroup-val mapping: [⟨1 0 6], ⟨0 1 -2]]

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

mapping generators: ~2, ~3

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.

25/24 gives dichotic. 686/675 gives beatles. Those split the fifth in two. 8748/8575 gives immunized, splitting the twelfth in two. 50/49 gives pajara with a semi-octave period. 392/375 gives progress. 6125/5832 gives kangaroo. Those split the twelfth in three. 250/243 gives porcupine, splitting the fourth in three. 126/125 gives augene with a 1/3-octave period. 4375/4374 gives modus, splitting the fifth in four. 3125/3024 gives brightstone. 9604/9375 gives fervor. 3125/2916 gives sixix. 3125/3087 gives passion. Those split the generator in five in various ways. 15625/15552 gives catalan, splitting the twelfth in six. Finally, 28/27 gives blackwood and 17496/16807 gives gothic, reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.

Temperaments discussed elsewhere are:

The rest are considered below.

Subgroup extensions

Omitting prime 5, archy can be extended to the 2.3.7.11 subgroup by identifying 11/8 as a diminished fourth (C–G♭). This is called supra, given in #Subgroup extensions. Discussed elsewhere is suhajira of the rastmic clan.

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♯). 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.

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.

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

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

Uberpyth

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 144/143, 245/243

Mapping: [⟨1 0 -12 6 -22 26], ⟨0 1 9 -2 16 -14]]

Optimal tunings:

  • WE: ~2 = 1196.6666 ¢, ~3/2 = 708.3602 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.2878 ¢

Optimal ET sequence: 22f, 27e, 49f, 125bcddeeeff, 174bbcdddeeeeffff

Badness (Sintel): 1.11

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–G♭), and finds the ~13/8 at the diminished seventh (C–B𝄫).

Subgroup: 2.3.5.7.11

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

Mapping: [⟨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

Main article: Quasisuper

Quasisuper can be described as 17c & 22, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–G𝄫). The 11-limit version, quasisupra, can be viewed as an extension of the excellent 2.3.7.11-subgroup temperament supra, with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).

Subgroup: 2.3.5.7

Comma list: 64/63, 2430/2401

Mapping: [⟨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

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
For the 5-limit version, see Syntonic–diatonic equivalence continuum #Ultrapyth (5-limit).

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

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–A𝄫♭).

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–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

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. The 11-limit extension simply equates the generator to 16/11, and the 13-limit extension further equates it to 13/9. 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

Kangaroo

For 5-limit version, see Miscellaneous 5-limit temperaments #Kangaroo.

Kangaroo splits the perfect twelfth into three 35/24~40/27 generators; its ploidacot is alpha-tricot. The 11-limit extension simply equates the generator to 16/11. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~5/4 will be too flat.

Originally the canonical extension of 5-limit kangaroo was parakangaroo, but it was decanonized in 2024 in favor of this extension because parakangaroo is an exotemperament that only supports the patent val of 15edo, while this extension is a bit more practical since it is at least low-medium accuracy and supports the patent vals of 15edo, 17edo and 32edo.

Subgroup: 2.3.5.7

Comma list: 64/63, 6125/5832

Mapping: [⟨1 0 -3 6], ⟨0 3 10 -6]]

mapping generators: ~2, ~35/24

Optimal tunings:

  • WE: ~2 = 1198.1767 ¢, ~35/24 = 637.2914 ¢
error map: ⟨-1.823 +9.919 -7.930 -3.514]
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 638.1977 ¢
error map: ⟨0.000 +12.638 -4.336 +1.988]

Optimal ET sequence: 15, 32, 47b

Badness (Sintel): 2.91

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 245/242

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

Optimal tunings:

  • WE: ~2 = 1197.8893 ¢, ~16/11 = 637.1922 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 638.2605 ¢

Optimal ET sequence: 15, 32, 47b

Badness (Sintel): 1.65

Fervor

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

Fervor tempers out 9604/9375 and may be described as 25 & 27. It splits the 6th harmonic into five generators of ~10/7; its ploidacot is beta-pentacot. The 11-limit extension simply equates the generator to 16/11. 27edo is about as accurate as it can be tuned.

Subgroup: 2.3.5.7

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. An important val not in the optimal ET sequence is 39c, that is, 39edo using the second-best approximation of 5/4 (the 369-cent one inherited from 13edo).

Subgroup: 2.3.5.7

Comma list: 64/63, 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

Subgroup extensions

Supra (2.3.7.11)

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

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

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

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

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

Subgroup-val 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]]

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

Oceanfront (2.3.7.13/5)

This is the restriction of ultrapyth to the 2.3.7.13/5 subgroup, mapping 13/10 to the major third.

Subgroup: 2.3.7.13/5

Comma list: 64/63 ([6 -2 -1 0⟩), 91/90 ([-1 -2 1 1⟩)

Subgroup-val mapping: [⟨1 0 6 -5], ⟨0 1 -2 4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1197.2656 ¢, ~3/2 = 710.1993 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.7413 ¢

Optimal ET sequence: 5, 17, 22, 27, 32, 59

RMS error: 2.063 cents

Scales: Oceanfront scales

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