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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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 revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-12-16 16:34:32 UTC</tt>.<br>
: The original revision id was <tt>570337903</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]


The //archytas clan// 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|81/80]] in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of [[22edo]]. Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, and 2430/2401 gives quasisuper. Pajara is discussed under [[diaschismic family]], dominant under [[meantone family]], augene under [[augmented family]] and porcupine under [[porcupine family]]. The rest are considered below.
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.  


=Mother=  
== Archy ==
Commas: 16/15, 21/20
{{Main| Superpyth }}


[[POTE tuning|POTE generator]]: 721.569
[[Subgroup]]: 2.3.7


Map: [&lt;1 0 4 6|, &lt;0 1 -1 -2|]
[[Comma list]]: 64/63
Wedgie: &lt;&lt;1 -1 -2 -4 -6 -2||
EDOs: 5, 148, 153
Badness: 0.0242


=Blacksmith=
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}
Commas: 28/27, 49/48


[[POTE tuning|POTE generator]]: ~5/4 = 392.767
: sval mapping generators: ~2, ~3


Map: [&lt;5 8 0 14|, &lt;0 0 1 0|]
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}
Wedgie: &lt;&lt;0 5 0 8 0 -14||
EDOs: 5, 10, 15, 40b, 55b
Badness: 0.0256


==11-limit==
: [[gencom]]: [2 3; 64/63]
Commas: 28/27, 49/48, 55/54


POTE generator: ~5/4 = 394.948
[[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 }}


Map: [&lt;5 8 0 14 29|, &lt;0 0 1 0 -1|]
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde
Badness: 0.0246


==13-limit==
[[Badness]] (Sintel): 0.159
Commas: 28/27, 40/39, 49/48, 55/54


POTE generator: ~5/4 = 391.0367
Scales: [[archy5]], [[archy7]], [[archy12]]


Map: [&lt;5 8 0 14 29 7|, &lt;0 0 1 0 -1 1|]
=== Overview to extensions ===
EDOs: 5, 10, 15, 25e, 40bef
==== 7-limit extensions ====
Badness: 0.0205
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]].  


==Farrier==
These all use the same generators as archy.
Commas: 28/27, 49/48, 77/75


POTE generator: ~5/4 = 398.070
[[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.  


Map: [&lt;5 8 0 14 -6|, &lt;0 0 1 0 2|]
Temperaments discussed elsewhere are:  
EDOs: 5e, 15
* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
Badness: 0.0292
* [[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]]


===13-limit===
Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.
Commas: 28/27, 40/39, 49/48, 66/65


POTE generator: ~5/4 = 396.812
==== 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]].


Map: [&lt;5 8 0 14 -6 7|, &lt;0 0 1 0 2 1|]
=== Supra ===
EDOs: 5e, 10e, 15
Subgroup: 2.3.7.11
Badness: 0.0223


==Ferrum==
Comma list: 64/63, 99/98
Commas: 28/27, 35/33, 49/48


POTE generator: ~5/4 = 374.763
Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}


Map: [&lt;5 8 0 14 6|, &lt;0 0 1 0 1|]
Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}
EDOs: 10
Badness: 0.0309


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


[[image:blacksmith10.jpg]]
Optimal tunings:  
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}


==Blackwood==
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}
//Blackwood (5-limit)//


Comma: 256/243
Badness (Sintel): 0.352


[[POTE tuning|POTE generator]]: 399.594
Scales: [[supra7]], [[supra12]]


Map: [&lt;5 8 0|, &lt;0 0 1|]
==== Supraphon ====
EDOs: 5, 10, 15
Subgroup: 2.3.7.11.13
Badness: 0.0638


=Superpyth=
Comma list: 64/63, 78/77, 99/98
[[Superpyth|Main article]]
Commas: 64/63, 245/243


[[POTE tuning|POTE generator]]: 710.291
Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}


Map: [&lt;1 0 -12 6|, &lt;0 1 9 -2|]
Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
Wedgie: &lt;&lt;1 9 -2 12 -6 -30||
EDOs: 5, 17, 22, 27, 49
Badness: 0.0323


==11-limit==
: gencom: [2 3; 64/63 78/77 99/98]
Commas: 64/63, 100/99, 245/243


[[POTE tuning|POTE generator]]: 710.175
Optimal tunings:  
* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}


Map: [&lt;1 0 -12 6 -22|, &lt;0 1 9 -2 16|]
{{Optimal ET sequence|legend=0| 12f, 17 }}
EDOs: 22, 49
Badness: 0.0250


==13-limit==
Badness (Sintel): 0.498
Commas: 64/63, 78/77, 91/90, 100/99


POTE generator: ~3/2 = 710.479
Scales: [[supra7]], [[supra12]]


Map: [&lt;1 0 -12 6 -22 -17|, &lt;0 1 9 -2 16 13|]
== Superpyth ==
EDOs: 22, 27e, 49, 76bcde
{{Main| Superpyth }}
Badness: 0.0247
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


==Suprapyth==
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.
Commas: 55/54, 64/63, 99/98


POTE generator: ~3/2 = 709.495
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -12 6 13|, &lt;0 1 9 -2 -6|]
[[Comma list]]: 64/63, 245/243
EDOs: 5, 7, 12, 17, 22
Badness: 0.0328


===13-limit===
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}
Commas: 55/54, 64/63, 65/63, 364/363


POTE generator: ~3/2 = 708.703
[[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 }}


Map: [&lt;1 0 -12 6 13 18|, &lt;0 1 9 -2 -6 -9|]
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
EDOs: 17, 22, 83cdf
Badness: 0.0363


=Beatles=
[[Badness]] (Sintel): 0.818
Comma: 524288/492075


POTE generator: ~512/405 = 355.930
=== 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).  


Map: [&lt;1 1 5|,&lt;0 2 -9|]
Subgroup: 2.3.5.7.11
EDOs: 10, 17c, 27, 64b, 91bc, 118bc
Badness: 0.3585


==7-limit==
Comma list: 64/63, 100/99, 245/243
Commas: 64/63, 686/675


[[POTE tuning|POTE generator]]: ~49/40 = 355.904
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}


Map: [&lt;1 1 5 4|,&lt;0 2 -9 -4|]
Optimal tunings:  
Wedgie: &lt;&lt;2 -9 -4 -19 -12 16||
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
EDOs: 10, 17c, 27, 64b, 91bcd, 118bcd
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}
Badness: 0.0459


Music: [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3|Beatles Improv]] by Herman Miller
{{Optimal ET sequence|legend=0| 22, 27e, 49 }}


==11-limit==
Badness (Sintel): 0.826
Commas: 64/63, 100/99, 686/675


POTE generator: ~49/40 = 356.140
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 5 4 10|,&lt;0 2 -9 -4 -22|]
Comma list: 64/63, 78/77, 91/90, 100/99
EDOs: 27e, 37, 64be, 91bcde
Badness: 0.0456


==13-limit==
Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
Commas: 64/63, 91/90, 100/99, 169/168


POTE generator: ~16/13 = 356.229
Optimal tunings:  
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}


Map: [&lt;1 1 5 4 10 4|,&lt;0 2 -9 -4 -22 -1|]
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
EDOs: 27e, 37, 64be
Badness: 0.0302


==Ringo==
Badness (Sintel): 1.02
Commas: 56/55, 64/63, 540/539


POTE generator: ~11/9 = 355.419
==== Thomas ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 5 4 2|,&lt;0 2 -9 -4 5|]
Comma list: 64/63, 100/99, 169/168, 245/243
EDOs: 10, 17c, 27e
Badness: 0.0329


===13-limit===
Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}
Commas: 56/55, 64/63, 78/77, 91/90


POTE generator: ~11/9 = 355.456
Optimal tunings:  
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


Map: [&lt;1 1 5 4 2 4|,&lt;0 2 -9 -4 5 -1|]
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}
EDOs: 10, 17c, 27e
Badness: 0.0226


=Schism=
Badness (Sintel): 2.03
Commas: 64/63, 360/343


[[POTE tuning|POTE generator]]: ~3/2 = 701.556
=== Suprapyth ===
Suprapyth finds the ~11/8 at the diminished fifth (C–Gb), and finds the ~13/8 at the diminished seventh (C–Bbb).  


Map: [&lt;1 0 15 6|, &lt;0 1 -8 -2|]
Subgroup: 2.3.5.7.11
Wedgie: &lt;&lt;1 -8 -2 -15 -6 18||
EDOs: 12, 41d, 53d
Badness: 0.0566


==11-limit==
Comma list: 55/54, 64/63, 99/98
Commas: 45/44, 64/63, 99/98


POTE generator ~3/2 = 702.136
Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}


Map: [&lt;1 0 15 6 13|, &lt;0 1 -8 -2 -6|]
Optimal tunings:  
EDOs: 12, 29de, 41de
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
Badness: 0.0375
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


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


==5-limit==
Badness (Sintel): 1.08
Comma: 262144/253125


POTE generator: ~16/15 = 98.670
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2|, &lt;0 -5 4|]
Comma list: 55/54, 64/63, 65/63, 99/98
EDOs: 11, 12, 49, 61, 73
Badness: 0.1686


===Passive===
Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}
Commas: 225/224, 256/245


POTE generator: ~16/15 = 98.809
Optimal tunings:  
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}


Map: [&lt;1 2 2 3|, &lt;0 -5 4 -2|]
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}
EDOs: 1, 11, 12, 49d
Badness: 0.0751


==7-limit==
Badness (Sintel): 1.50
Commas: 64/63, 3125/3087


[[POTE tuning|POTE generator]]: ~16/15 = 98.153
== Quasisuper ==
Quasisuper can be described as {{nowrap| 17c & 22 }}, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–Gbb).  


Map: [&lt;1 2 2 2|, &lt;0 -5 4 10|]
[[Subgroup]]: 2.3.5.7
Wedgie: &lt;&lt;5 -4 -10 -18 -30 -12||
Generators: 2, 16/15
EDOs: 12, 37, 49, 110bcd
Badness: 0.0623


==11-limit==
[[Comma list]]: 64/63, 2430/2401
Commas: 64/63, 100/99, 1375/1372


POTE generator: ~16/15 = 98.019
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}


Map: [&lt;1 2 2 2 2|, &lt;0 -5 4 10 18|]
[[Optimal tuning]]s:  
EDOs: 12, 37, 49
* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}
Badness: 0.0408
: [[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}
Commas: 64/63 100/99 196/195 275/273


POTE generator: ~16/15 = 97.910
[[Badness]] (Sintel): 1.61


Map: [&lt;1 2 2 2 2 2|, &lt;0 -5 4 10 18 21|]
=== Quasisupra ===
EDOs: 12f, 37, 49f
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).
Badness: 0.0309


=Fervor=
Subgroup: 2.3.5.7.11
Comma: 67108864/61509375


POTE generator: ~64/45 = 577.705
Comma list: 64/63, 99/98, 121/120


Map: [&lt;1 4 -2|, &lt;0 -5 9|]
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}
EDOs: 25, 27
Badness: 0.8526


==7-limit==
Optimal tunings:
Commas: 64/63, 9604/9375
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


POTE generator: ~7/5 = 577.777
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}


Map: [&lt;1 4 -2 -2|, &lt;0 -5 9 10|]
Badness (Sintel): 1.06
Wedgie: &lt;&lt;5 -9 -10 -26 -30 2||
EDOs: 25, 27
Badness: 0.1085


==11-limit==  
==== 13-limit ====
Commas: 56/55, 64/63, 1350/1331
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 577.850
Comma list: 64/63, 78/77, 91/90, 121/120


Map: [&lt;1 4 -2 -2 3|, &lt;0 -5 9 10 1|]
Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}
EDOs: 25e, 27e
Badness: 0.0521


==13-limit==
Optimal tunings:
Commas: 56/55, 64/63, 78/77, 507/500
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


POTE generator: ~7/5 = 578.060
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


Map: [&lt;1 4 -2 -2 3 -4|, &lt;0 -5 9 10 1 16|]
Badness (Sintel): 1.25
EDOs: 27e
Badness: 0.0397


=Quasisuper=  
=== Quasisoup ===
Commas: 64/63, 2430/2401
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: 708.328
Comma list: 55/54, 64/63, 2430/2401


Map: [&lt;1 0 23 6|, &lt;0 1 -13 -2|]
Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
Wedgie: &lt;&lt;1 -13 -2 -23 -2 -6 32||
EDOs: 22, 61
Badness: 0.0638


==Quasisupra==
Optimal tunings:
Commas: 64/63, 99/98, 121/120
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


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).
{{Optimal ET sequence|legend=0| 22 }}


POTE generator: ~3/2 = 708.205
Badness (Sintel): 2.76


Map: [&lt;1 2 -3 2 1|, &lt;0 -1 13 2 6|]
== Ultrapyth ==
EDOs: 17c, 22, 27c, 39d, 61d
{{Main| Ultrapyth }}
Badness: 0.0322


===13-limit===
Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[the Biosphere #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–Cx).
Commas: 64/63, 78/77, 91/90, 121/120


POTE generator: ~3/2 = 708.004
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 23 6 13 18|, &lt;0 1 -13 -2 -6 -9|]
[[Comma list]]: 64/63, 6860/6561
EDOs: 17c, 22, 39d, 61df, 100bcdf
Badness: 0.0302


==Quasisoup==
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
Commas: 55/54, 64/63, 2430/2401


POTE generator: ~3/2 = 709.021
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.2673{{c}}, ~3/2 = 712.0258{{c}}
: [[error map]]: {{val| -2.733 +7.338 -1.557 -3.808 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5430{{c}}
: error map: {{val| 0.000 +11.588 +3.288 +4.088 }}


Map: [&lt;1 0 23 6 -22|, &lt;0 1 -13 -2 16|]
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
EDOs: 22
Badness: 0.0835


=Progress=
[[Badness]] (Sintel): 2.74
Comma: 32768/30375


POTE generator: ~64/45 = 561.264
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 5|, &lt;0 3 -5|]
Comma list: 55/54, 64/63, 2401/2376
EDOs: 4, 13, 15, 32c, 47bc, 62bc
Badness: 0.2461


==7-limit==
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}
Commas: 64/63, 392/375


POTE generator: ~7/5 = 562.122
Optimal tunings:  
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


Map: [&lt;1 0 5 6|, &lt;0 3 -5 -6|]
{{Optimal ET sequence|legend=0| 5, 32, 37 }}
Wedgie: &lt;&lt;3 -5 -6 -15 -18 0||
EDOs: 13, 15, 32c, 79bcc, 111bcc
Badness: 0.0664


==11-limit==
Badness (Sintel): 2.26
Commas: 56/55, 64/63, 77/75


POTE generator: ~7/5 = 562.085
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 5 6 4|, &lt;0 3 -5 -6 -1|]
Comma list: 55/54, 64/63, 91/90, 1573/1568
EDOs: 13, 15, 32c, 47bc, 79bcce
Badness: 0.0310


==13-limit==
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
Commas: 56/55, 64/63, 66/65, 77/75


POTE generator: ~7/5 = 562.365
Optimal tunings:  
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


Map: [&lt;1 0 5 6 4 0|, &lt;0 3 -5 -6 -1 7|]
{{Optimal ET sequence|legend=0| 5, 32, 37 }}
EDOs: 15, 17c, 32cf
Badness: 0.0262


==Progressive==
Badness (Sintel): 2.03
Commas: 26/25, 56/55, 64/63, 77/75


POTE generator: ~7/5 = 563.239
=== Ultramarine ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 5 6 4 9|, &lt;0 3 -5 -6 -1 -10|]
Comma list: 64/63, 100/99, 3773/3645
EDOs: 15f, 17c, 32c, 49c
 
Badness: 0.0327</pre></div>
Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Archytas clan&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:80:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Mother"&gt;Mother&lt;/a&gt;&lt;/div&gt;
Optimal tunings:  
&lt;!-- ws:end:WikiTextTocRule:81 --&gt;&lt;!-- ws:start:WikiTextTocRule:82: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Blacksmith"&gt;Blacksmith&lt;/a&gt;&lt;/div&gt;
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
&lt;!-- ws:end:WikiTextTocRule:82 --&gt;&lt;!-- ws:start:WikiTextTocRule:83: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Blacksmith-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}
&lt;!-- ws:end:WikiTextTocRule:83 --&gt;&lt;!-- ws:start:WikiTextTocRule:84: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Blacksmith-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:84 --&gt;&lt;!-- ws:start:WikiTextTocRule:85: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Blacksmith-Farrier"&gt;Farrier&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}
&lt;!-- ws:end:WikiTextTocRule:85 --&gt;&lt;!-- ws:start:WikiTextTocRule:86: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Blacksmith-Farrier-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:86 --&gt;&lt;!-- ws:start:WikiTextTocRule:87: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Blacksmith-Ferrum"&gt;Ferrum&lt;/a&gt;&lt;/div&gt;
Badness (Sintel): 2.58
&lt;!-- ws:end:WikiTextTocRule:87 --&gt;&lt;!-- ws:start:WikiTextTocRule:88: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Blacksmith-Blackwood"&gt;Blackwood&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:88 --&gt;&lt;!-- ws:start:WikiTextTocRule:89: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Superpyth"&gt;Superpyth&lt;/a&gt;&lt;/div&gt;
==== 13-limit ====
&lt;!-- ws:end:WikiTextTocRule:89 --&gt;&lt;!-- ws:start:WikiTextTocRule:90: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Superpyth-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:end:WikiTextTocRule:90 --&gt;&lt;!-- ws:start:WikiTextTocRule:91: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Superpyth-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:91 --&gt;&lt;!-- ws:start:WikiTextTocRule:92: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Superpyth-Suprapyth"&gt;Suprapyth&lt;/a&gt;&lt;/div&gt;
Comma list: 64/63, 91/90, 100/99, 847/845
&lt;!-- ws:end:WikiTextTocRule:92 --&gt;&lt;!-- ws:start:WikiTextTocRule:93: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Superpyth-Suprapyth-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:93 --&gt;&lt;!-- ws:start:WikiTextTocRule:94: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Beatles"&gt;Beatles&lt;/a&gt;&lt;/div&gt;
Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
&lt;!-- ws:end:WikiTextTocRule:94 --&gt;&lt;!-- ws:start:WikiTextTocRule:95: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Beatles-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:95 --&gt;&lt;!-- ws:start:WikiTextTocRule:96: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Beatles-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
Optimal tunings:  
&lt;!-- ws:end:WikiTextTocRule:96 --&gt;&lt;!-- ws:start:WikiTextTocRule:97: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Beatles-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
&lt;!-- ws:end:WikiTextTocRule:97 --&gt;&lt;!-- ws:start:WikiTextTocRule:98: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Beatles-Ringo"&gt;Ringo&lt;/a&gt;&lt;/div&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}
&lt;!-- ws:end:WikiTextTocRule:98 --&gt;&lt;!-- ws:start:WikiTextTocRule:99: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Beatles-Ringo-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Schism"&gt;Schism&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}
&lt;!-- ws:end:WikiTextTocRule:100 --&gt;&lt;!-- ws:start:WikiTextTocRule:101: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Schism-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:101 --&gt;&lt;!-- ws:start:WikiTextTocRule:102: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Passion"&gt;Passion&lt;/a&gt;&lt;/div&gt;
Badness (Sintel): 1.89
&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Passion-5-limit"&gt;5-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Passion-5-limit-Passive"&gt;Passive&lt;/a&gt;&lt;/div&gt;
== Quasiultra ==
&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Passion-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
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).
&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Passion-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Passion-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Fervor"&gt;Fervor&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Fervor-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
[[Comma list]]: 64/63, 33614/32805
&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Fervor-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextTocRule:111: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Fervor-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
&lt;!-- ws:end:WikiTextTocRule:111 --&gt;&lt;!-- ws:start:WikiTextTocRule:112: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Quasisuper"&gt;Quasisuper&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:112 --&gt;&lt;!-- ws:start:WikiTextTocRule:113: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quasisuper-Quasisupra"&gt;Quasisupra&lt;/a&gt;&lt;/div&gt;
[[Optimal tuning]]s:
&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Quasisuper-Quasisupra-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
* [[WE]]: ~2 = 1196.9257{{c}}, ~3/2 = 709.6211{{c}}
&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quasisuper-Quasisoup"&gt;Quasisoup&lt;/a&gt;&lt;/div&gt;
: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}
&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Progress"&gt;Progress&lt;/a&gt;&lt;/div&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5429{{c}}
&lt;!-- ws:end:WikiTextTocRule:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Progress-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
: error map: {{val| 0.000 +9.588 +5.914 +8.088 }}
&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Progress-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:118 --&gt;&lt;!-- ws:start:WikiTextTocRule:119: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Progress-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
&lt;!-- ws:end:WikiTextTocRule:119 --&gt;&lt;!-- ws:start:WikiTextTocRule:120: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Progress-Progressive"&gt;Progressive&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:120 --&gt;&lt;!-- ws:start:WikiTextTocRule:121: --&gt;&lt;/div&gt;
[[Badness]] (Sintel): 3.34
&lt;!-- ws:end:WikiTextTocRule:121 --&gt;&lt;br /&gt;
 
The &lt;em&gt;archytas clan&lt;/em&gt; tempers out the &lt;a class="wiki_link" href="/64_63"&gt;Archytas comma&lt;/a&gt;, 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to &lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt; in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;. Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, and 2430/2401 gives quasisuper. Pajara is discussed under &lt;a class="wiki_link" href="/diaschismic%20family"&gt;diaschismic family&lt;/a&gt;, dominant under &lt;a class="wiki_link" href="/meantone%20family"&gt;meantone family&lt;/a&gt;, augene under &lt;a class="wiki_link" href="/augmented%20family"&gt;augmented family&lt;/a&gt; and porcupine under &lt;a class="wiki_link" href="/porcupine%20family"&gt;porcupine family&lt;/a&gt;. The rest are considered below.&lt;br /&gt;
== Schism ==
&lt;br /&gt;
{{See also| Schismatic family #Schism }}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Mother"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Mother&lt;/h1&gt;
 
Commas: 16/15, 21/20&lt;br /&gt;
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.
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 721.569&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
Map: [&amp;lt;1 0 4 6|, &amp;lt;0 1 -1 -2|]&lt;br /&gt;
[[Comma list]]: 64/63, 360/343
Wedgie: &amp;lt;&amp;lt;1 -1 -2 -4 -6 -2||&lt;br /&gt;
 
EDOs: 5, 148, 153&lt;br /&gt;
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}
Badness: 0.0242&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Blacksmith"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Blacksmith&lt;/h1&gt;
* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}
Commas: 28/27, 49/48&lt;br /&gt;
: [[error map]]: {{val| -2.640 -4.583 -4.896 +20.588 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7376{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~5/4 = 392.767&lt;br /&gt;
: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}
&lt;br /&gt;
 
Map: [&amp;lt;5 8 0 14|, &amp;lt;0 0 1 0|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
Wedgie: &amp;lt;&amp;lt;0 5 0 8 0 -14||&lt;br /&gt;
 
EDOs: 5, 10, 15, 40b, 55b&lt;br /&gt;
[[Badness]] (Sintel): 1.43
Badness: 0.0256&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Blacksmith-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11
Commas: 28/27, 49/48, 55/54&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 45/44, 64/63, 99/98
POTE generator: ~5/4 = 394.948&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}
Map: [&amp;lt;5 8 0 14 29|, &amp;lt;0 0 1 0 -1|]&lt;br /&gt;
 
EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde&lt;br /&gt;
Optimal tunings:  
Badness: 0.0246&lt;br /&gt;
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Blacksmith-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
 
Commas: 28/27, 40/39, 49/48, 55/54&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}
&lt;br /&gt;
 
POTE generator: ~5/4 = 391.0367&lt;br /&gt;
Badness (Sintel): 1.24
&lt;br /&gt;
 
Map: [&amp;lt;5 8 0 14 29 7|, &amp;lt;0 0 1 0 -1 1|]&lt;br /&gt;
== Beatles ==
EDOs: 5, 10, 15, 25e, 40bef&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''
Badness: 0.0205&lt;br /&gt;
 
&lt;br /&gt;
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.
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Blacksmith-Farrier"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Farrier&lt;/h2&gt;
 
Commas: 28/27, 49/48, 77/75&lt;br /&gt;
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.
&lt;br /&gt;
 
POTE generator: ~5/4 = 398.070&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
Map: [&amp;lt;5 8 0 14 -6|, &amp;lt;0 0 1 0 2|]&lt;br /&gt;
[[Comma list]]: 64/63, 686/675
EDOs: 5e, 15&lt;br /&gt;
 
Badness: 0.0292&lt;br /&gt;
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Blacksmith-Farrier-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h3&gt;
[[Optimal tuning]]s:  
Commas: 28/27, 40/39, 49/48, 66/65&lt;br /&gt;
* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -3.376 +4.475 +2.682 -1.940 }}
POTE generator: ~5/4 = 396.812&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +10.209 +8.949 +6.847 }}
Map: [&amp;lt;5 8 0 14 -6 7|, &amp;lt;0 0 1 0 2 1|]&lt;br /&gt;
 
EDOs: 5e, 10e, 15&lt;br /&gt;
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
Badness: 0.0223&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 1.16
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Blacksmith-Ferrum"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Ferrum&lt;/h2&gt;
 
Commas: 28/27, 35/33, 49/48&lt;br /&gt;
; Music
&lt;br /&gt;
* [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]]
POTE generator: ~5/4 = 374.763&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
Map: [&amp;lt;5 8 0 14 6|, &amp;lt;0 0 1 0 1|]&lt;br /&gt;
Subgroup: 2.3.5.7.11
EDOs: 10&lt;br /&gt;
 
Badness: 0.0309&lt;br /&gt;
Comma list: 64/63, 100/99, 686/675
&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
&lt;!-- ws:start:WikiTextLocalImageRule:122:&amp;lt;img src=&amp;quot;/file/view/blacksmith10.jpg/271383136/blacksmith10.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/blacksmith10.jpg/271383136/blacksmith10.jpg" alt="blacksmith10.jpg" title="blacksmith10.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:122 --&gt;&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Blacksmith-Blackwood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Blackwood&lt;/h2&gt;
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
&lt;em&gt;Blackwood (5-limit)&lt;/em&gt;&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}
&lt;br /&gt;
 
Comma: 256/243&lt;br /&gt;
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 399.594&lt;br /&gt;
Badness (Sintel): 1.51
&lt;br /&gt;
 
Map: [&amp;lt;5 8 0|, &amp;lt;0 0 1|]&lt;br /&gt;
==== 13-limit ====
EDOs: 5, 10, 15&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.0638&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 64/63, 91/90, 100/99, 169/168
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Superpyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Superpyth&lt;/h1&gt;
 
&lt;a class="wiki_link" href="/Superpyth"&gt;Main article&lt;/a&gt;&lt;br /&gt;
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
Commas: 64/63, 245/243&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 710.291&lt;br /&gt;
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}
Map: [&amp;lt;1 0 -12 6|, &amp;lt;0 1 9 -2|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 9 -2 12 -6 -30||&lt;br /&gt;
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}
EDOs: 5, 17, 22, 27, 49&lt;br /&gt;
 
Badness: 0.0323&lt;br /&gt;
Badness (Sintel): 1.25
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Superpyth-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
=== Ringo ===
Commas: 64/63, 100/99, 245/243&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 710.175&lt;br /&gt;
Comma list: 56/55, 64/63, 540/539
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -12 6 -22|, &amp;lt;0 1 9 -2 16|]&lt;br /&gt;
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}
EDOs: 22, 49&lt;br /&gt;
 
Badness: 0.0250&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Superpyth-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}
Commas: 64/63, 78/77, 91/90, 100/99&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
POTE generator: ~3/2 = 710.479&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.09
Map: [&amp;lt;1 0 -12 6 -22 -17|, &amp;lt;0 1 9 -2 16 13|]&lt;br /&gt;
 
EDOs: 22, 27e, 49, 76bcde&lt;br /&gt;
==== 13-limit ====
Badness: 0.0247&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Superpyth-Suprapyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Suprapyth&lt;/h2&gt;
Comma list: 56/55, 64/63, 78/77, 91/90
Commas: 55/54, 64/63, 99/98&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
POTE generator: ~3/2 = 709.495&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;1 0 -12 6 13|, &amp;lt;0 1 9 -2 -6|]&lt;br /&gt;
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
EDOs: 5, 7, 12, 17, 22&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}
Badness: 0.0328&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Superpyth-Suprapyth-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;13-limit&lt;/h3&gt;
 
Commas: 55/54, 64/63, 65/63, 364/363&lt;br /&gt;
Badness (Sintel): 0.935
&lt;br /&gt;
 
POTE generator: ~3/2 = 708.703&lt;br /&gt;
=== Beetle ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;1 0 -12 6 13 18|, &amp;lt;0 1 9 -2 -6 -9|]&lt;br /&gt;
 
EDOs: 17, 22, 83cdf&lt;br /&gt;
Comma list: 55/54, 64/63, 686/675
Badness: 0.0363&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="Beatles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Beatles&lt;/h1&gt;
 
Comma: 524288/492075&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
POTE generator: ~512/405 = 355.930&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 5|,&amp;lt;0 2 -9|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
EDOs: 10, 17c, 27, 64b, 91bc, 118bc&lt;br /&gt;
 
Badness: 0.3585&lt;br /&gt;
Badness (Sintel): 1.92
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Beatles-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;7-limit&lt;/h2&gt;
==== 13-limit ====
Commas: 64/63, 686/675&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~49/40 = 355.904&lt;br /&gt;
Comma list: 55/54, 64/63, 91/90, 169/168
&lt;br /&gt;
 
Map: [&amp;lt;1 1 5 4|,&amp;lt;0 2 -9 -4|]&lt;br /&gt;
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
Wedgie: &amp;lt;&amp;lt;2 -9 -4 -19 -12 16||&lt;br /&gt;
 
EDOs: 10, 17c, 27, 64b, 91bcd, 118bcd&lt;br /&gt;
Optimal tunings:  
Badness: 0.0459&lt;br /&gt;
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}
Music: &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3" rel="nofollow"&gt;Beatles Improv&lt;/a&gt; by Herman Miller&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Beatles-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;11-limit&lt;/h2&gt;
 
Commas: 64/63, 100/99, 686/675&lt;br /&gt;
Badness (Sintel): 1.40
&lt;br /&gt;
 
POTE generator: ~49/40 = 356.140&lt;br /&gt;
== Progress ==
&lt;br /&gt;
{{Distinguish| Progression }}
Map: [&amp;lt;1 1 5 4 10|,&amp;lt;0 2 -9 -4 -22|]&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
EDOs: 27e, 37, 64be, 91bcde&lt;br /&gt;
 
Badness: 0.0456&lt;br /&gt;
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.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Beatles-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;13-limit&lt;/h2&gt;
[[Subgroup]]: 2.3.5.7
Commas: 64/63, 91/90, 100/99, 169/168&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 64/63, 392/375
POTE generator: ~16/13 = 356.229&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
Map: [&amp;lt;1 1 5 4 10 4|,&amp;lt;0 2 -9 -4 -22 -1|]&lt;br /&gt;
 
EDOs: 27e, 37, 64be&lt;br /&gt;
: mapping generators: ~2, ~10/7
Badness: 0.0302&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Beatles-Ringo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Ringo&lt;/h2&gt;
* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}
Commas: 56/55, 64/63, 540/539&lt;br /&gt;
: [[error map]]: {{val| -4.862 +3.925 +12.908 -9.759 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 638.0791{{c}}
POTE generator: ~11/9 = 355.419&lt;br /&gt;
: error map: {{val| 0.000 +12.282 +23.291 +2.700 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 5 4 2|,&amp;lt;0 2 -9 -4 5|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
EDOs: 10, 17c, 27e&lt;br /&gt;
 
Badness: 0.0329&lt;br /&gt;
[[Badness]] (Sintel): 1.68
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc18"&gt;&lt;a name="Beatles-Ringo-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;13-limit&lt;/h3&gt;
=== 11-limit ===
Commas: 56/55, 64/63, 78/77, 91/90&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
POTE generator: ~11/9 = 355.456&lt;br /&gt;
Comma list: 56/55, 64/63, 77/75
&lt;br /&gt;
 
Map: [&amp;lt;1 1 5 4 2 4|,&amp;lt;0 2 -9 -4 5 -1|]&lt;br /&gt;
Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
EDOs: 10, 17c, 27e&lt;br /&gt;
 
Badness: 0.0226&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc19"&gt;&lt;a name="Schism"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Schism&lt;/h1&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}
Commas: 64/63, 360/343&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 701.556&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.03
Map: [&amp;lt;1 0 15 6|, &amp;lt;0 1 -8 -2|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 -8 -2 -15 -6 18||&lt;br /&gt;
==== 13-limit ====
EDOs: 12, 41d, 53d&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.0566&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 56/55, 64/63, 66/65, 77/75
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Schism-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;11-limit&lt;/h2&gt;
 
Commas: 45/44, 64/63, 99/98&lt;br /&gt;
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
&lt;br /&gt;
 
POTE generator ~3/2 = 702.136&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
Map: [&amp;lt;1 0 15 6 13|, &amp;lt;0 1 -8 -2 -6|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}
EDOs: 12, 29de, 41de&lt;br /&gt;
 
Badness: 0.0375&lt;br /&gt;
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc21"&gt;&lt;a name="Passion"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;Passion&lt;/h1&gt;
Badness (Sintel): 1.08
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Passion-5-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;5-limit&lt;/h2&gt;
==== Progressive ====
Comma: 262144/253125&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
POTE generator: ~16/15 = 98.670&lt;br /&gt;
Comma list: 26/25, 56/55, 64/63, 77/75
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2|, &amp;lt;0 -5 4|]&lt;br /&gt;
Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
EDOs: 11, 12, 49, 61, 73&lt;br /&gt;
 
Badness: 0.1686&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc23"&gt;&lt;a name="Passion-5-limit-Passive"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;Passive&lt;/h3&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}
Commas: 225/224, 256/245&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}
POTE generator: ~16/15 = 98.809&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.35
Map: [&amp;lt;1 2 2 3|, &amp;lt;0 -5 4 -2|]&lt;br /&gt;
 
EDOs: 1, 11, 12, 49d&lt;br /&gt;
== Fervor ==
Badness: 0.0751&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc24"&gt;&lt;a name="Passion-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;7-limit&lt;/h2&gt;
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.
Commas: 64/63, 3125/3087&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~16/15 = 98.153&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 64/63, 9604/9375
Map: [&amp;lt;1 2 2 2|, &amp;lt;0 -5 4 10|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;5 -4 -10 -18 -30 -12||&lt;br /&gt;
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
Generators: 2, 16/15&lt;br /&gt;
 
EDOs: 12, 37, 49, 110bcd&lt;br /&gt;
: mapping generators: ~2, ~10/7
Badness: 0.0623&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc25"&gt;&lt;a name="Passion-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;11-limit&lt;/h2&gt;
* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}
Commas: 64/63, 100/99, 1375/1372&lt;br /&gt;
: [[error map]]: {{val| -3.726 +3.230 +4.980 -1.550 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 622.3179{{c}}
POTE generator: ~16/15 = 98.019&lt;br /&gt;
: error map: {{val| 0.000 +9.634 +12.826 +7.996 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 2 2|, &amp;lt;0 -5 4 10 18|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 2, 25, 27 }}
EDOs: 12, 37, 49&lt;br /&gt;
 
Badness: 0.0408&lt;br /&gt;
[[Badness]] (Sintel): 2.74
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc26"&gt;&lt;a name="Passion-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;13-limit&lt;/h2&gt;
=== 11-limit ===
Commas: 64/63 100/99 196/195 275/273&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
POTE generator: ~16/15 = 97.910&lt;br /&gt;
Comma list: 56/55, 64/63, 1350/1331
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 2 2 2|, &amp;lt;0 -5 4 10 18 21|]&lt;br /&gt;
Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}
EDOs: 12f, 37, 49f&lt;br /&gt;
 
Badness: 0.0309&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc27"&gt;&lt;a name="Fervor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;Fervor&lt;/h1&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}
Comma: 67108864/61509375&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}
POTE generator: ~64/45 = 577.705&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.72
Map: [&amp;lt;1 4 -2|, &amp;lt;0 -5 9|]&lt;br /&gt;
 
EDOs: 25, 27&lt;br /&gt;
=== 13-limit ===
Badness: 0.8526&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc28"&gt;&lt;a name="Fervor-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;7-limit&lt;/h2&gt;
Comma list: 56/55, 64/63, 78/77, 507/500
Commas: 64/63, 9604/9375&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}
POTE generator: ~7/5 = 577.777&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;1 4 -2 -2|, &amp;lt;0 -5 9 10|]&lt;br /&gt;
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
Wedgie: &amp;lt;&amp;lt;5 -9 -10 -26 -30 2||&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}
EDOs: 25, 27&lt;br /&gt;
 
Badness: 0.1085&lt;br /&gt;
{{Optimal ET sequence|legend=0| 2f, 27e }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc29"&gt;&lt;a name="Fervor-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;11-limit&lt;/h2&gt;
Badness (Sintel): 1.64
Commas: 56/55, 64/63, 1350/1331&lt;br /&gt;
 
&lt;br /&gt;
== Sixix ==
POTE generator: ~7/5 = 577.850&lt;br /&gt;
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sixix (5-limit)]].''
&lt;br /&gt;
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}
Map: [&amp;lt;1 4 -2 -2 3|, &amp;lt;0 -5 9 10 1|]&lt;br /&gt;
 
EDOs: 25e, 27e&lt;br /&gt;
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.
Badness: 0.0521&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc30"&gt;&lt;a name="Fervor-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;13-limit&lt;/h2&gt;
 
Commas: 56/55, 64/63, 78/77, 507/500&lt;br /&gt;
[[Comma list]]: 64/63, 3125/2916
&lt;br /&gt;
 
POTE generator: ~7/5 = 578.060&lt;br /&gt;
{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 4 -2 -2 3 -4|, &amp;lt;0 -5 9 10 1 16|]&lt;br /&gt;
[[Optimal tuning]]s:  
EDOs: 27e&lt;br /&gt;
* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}
Badness: 0.0397&lt;br /&gt;
: [[error map]]: {{val| -1.097 +9.086 -13.503 +2.508 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 337.4588{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc31"&gt;&lt;a name="Quasisuper"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;Quasisuper&lt;/h1&gt;
: error map: {{val| 0.000 +10.751 -11.066 +5.762 }}
Commas: 64/63, 2430/2401&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 708.328&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 4.02
Map: [&amp;lt;1 0 23 6|, &amp;lt;0 1 -13 -2|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 -13 -2 -23 -2 -6 32||&lt;br /&gt;
=== 11-limit ===
EDOs: 22, 61&lt;br /&gt;
Subgroup: 2.3.5.7.11
Badness: 0.0638&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 55/54, 64/63, 125/121
&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc32"&gt;&lt;a name="Quasisuper-Quasisupra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;Quasisupra&lt;/h2&gt;
 
Commas: 64/63, 99/98, 121/120&lt;br /&gt;
Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
&lt;br /&gt;
 
Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament &lt;a class="wiki_link" href="/supra"&gt;supra&lt;/a&gt;, with the quasisuper mapping of 5 thrown in (rather than the superpyth mapping of 5, which results in suprapyth).&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
POTE generator: ~3/2 = 708.205&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 -3 2 1|, &amp;lt;0 -1 13 2 6|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}
EDOs: 17c, 22, 27c, 39d, 61d&lt;br /&gt;
 
Badness: 0.0322&lt;br /&gt;
Badness (Sintel): 2.34
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc33"&gt;&lt;a name="Quasisuper-Quasisupra-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;13-limit&lt;/h3&gt;
=== 13-limit ===
Commas: 64/63, 78/77, 91/90, 121/120&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
POTE generator: ~3/2 = 708.004&lt;br /&gt;
Comma list: 40/39, 55/54, 64/63, 125/121
&lt;br /&gt;
 
Map: [&amp;lt;1 0 23 6 13 18|, &amp;lt;0 1 -13 -2 -6 -9|]&lt;br /&gt;
Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
EDOs: 17c, 22, 39d, 61df, 100bcdf&lt;br /&gt;
 
Badness: 0.0302&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc34"&gt;&lt;a name="Quasisuper-Quasisoup"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;Quasisoup&lt;/h2&gt;
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}
Commas: 55/54, 64/63, 2430/2401&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
POTE generator: ~3/2 = 709.021&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.91
Map: [&amp;lt;1 0 23 6 -22|, &amp;lt;0 1 -13 -2 16|]&lt;br /&gt;
 
EDOs: 22&lt;br /&gt;
=== 17-limit ===
Badness: 0.0835&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc35"&gt;&lt;a name="Progress"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;Progress&lt;/h1&gt;
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121
Comma: 32768/30375&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
POTE generator: ~64/45 = 561.264&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;1 0 5|, &amp;lt;0 3 -5|]&lt;br /&gt;
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
EDOs: 4, 13, 15, 32c, 47bc, 62bc&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}
Badness: 0.2461&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc36"&gt;&lt;a name="Progress-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;7-limit&lt;/h2&gt;
 
Commas: 64/63, 392/375&lt;br /&gt;
Badness (Sintel): 2.00
&lt;br /&gt;
 
POTE generator: ~7/5 = 562.122&lt;br /&gt;
[[Category:Archytas clan| ]] <!-- main article -->
&lt;br /&gt;
[[Category:Temperament clans]]
Map: [&amp;lt;1 0 5 6|, &amp;lt;0 3 -5 -6|]&lt;br /&gt;
[[Category:Pages with mostly numerical content]]
Wedgie: &amp;lt;&amp;lt;3 -5 -6 -15 -18 0||&lt;br /&gt;
[[Category:Rank 2]]
EDOs: 13, 15, 32c, 79bcc, 111bcc&lt;br /&gt;
Badness: 0.0664&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc37"&gt;&lt;a name="Progress-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;11-limit&lt;/h2&gt;
Commas: 56/55, 64/63, 77/75&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 562.085&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 5 6 4|, &amp;lt;0 3 -5 -6 -1|]&lt;br /&gt;
EDOs: 13, 15, 32c, 47bc, 79bcce&lt;br /&gt;
Badness: 0.0310&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc38"&gt;&lt;a name="Progress-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt;13-limit&lt;/h2&gt;
Commas: 56/55, 64/63, 66/65, 77/75&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 562.365&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 5 6 4 0|, &amp;lt;0 3 -5 -6 -1 7|]&lt;br /&gt;
EDOs: 15, 17c, 32cf&lt;br /&gt;
Badness: 0.0262&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc39"&gt;&lt;a name="Progress-Progressive"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt;Progressive&lt;/h2&gt;
Commas: 26/25, 56/55, 64/63, 77/75&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 563.239&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 5 6 4 9|, &amp;lt;0 3 -5 -6 -1 -10|]&lt;br /&gt;
EDOs: 15f, 17c, 32c, 49c&lt;br /&gt;
Badness: 0.0327&lt;/body&gt;&lt;/html&gt;</pre></div>
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