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


Map: [&lt;5 8 0 14|, &lt;0 0 1 0|]
: mapping generators: ~2, ~3
Wedgie: &lt;&lt;0 5 0 8 0 -14||
EDOs: 5, 10, 15, 40b, 55b
Badness: 0.0256


==11-limit==  
[[Optimal tuning]]s:
Commas: 28/27, 49/48, 55/54
* [[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 }}


POTE generator: ~5/4 = 394.948
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}


Map: [&lt;5 8 0 14 29|, &lt;0 0 1 0 -1|]
[[Badness]] (Sintel): 0.159
EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde
Badness: 0.0246


==13-limit==
Scales: [[archy5]], [[archy7]], [[archy12]]
Commas: 28/27, 40/39, 49/48, 55/54


POTE generator: ~5/4 = 391.0367
=== Overview to extensions ===
==== 7-limit extensions ====
The second comma in the comma list defines which [[7-limit]] family member we are looking at:  
* [[#Schism|Schism]] adds 360/343, for a tuning around [[12edo]];
* Dominant adds [[36/35]], for a tuning between [[12edo]] and [[17edo|17c-edo]];
* [[#Quasisuper|Quasisuper]] adds [[2430/2401]], for a tuning between 17c-edo and [[22edo]];
* [[#Superpyth|Superpyth]] adds [[245/243]], for a tuning between 22edo and [[27edo]];
* [[#Quasiultra|Quasiultra]] adds 33614/32805, for a tuning between 27edo and [[32edo]];
* [[#Ultrapyth|Ultrapyth]] adds 6860/6561, for a tuning sharp of 32edo;
* Mother adds [[16/15]], for an exotemperament well tuned around [[5edo]].  


Map: [&lt;5 8 0 14 29 7|, &lt;0 0 1 0 -1 1|]
These all use the same generators as archy.  
EDOs: 5, 10, 15, 25e, 40bef
Badness: 0.0205


==Farrier==
[[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.
Commas: 28/27, 49/48, 77/75


POTE generator: ~5/4 = 398.070
Temperaments discussed elsewhere are:  
* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Augene]] (+126/125) → [[Augmented family #Augene|Augmented family]]
* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* ''[[Modus]]'' (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]


Map: [&lt;5 8 0 14 -6|, &lt;0 0 1 0 2|]
Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.  
EDOs: 5e, 15
Badness: 0.0292


===13-limit===
==== Subgroup extensions ====
Commas: 28/27, 40/39, 49/48, 66/65
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]].


POTE generator: ~5/4 = 396.812
=== Supra ===
Subgroup: 2.3.7.11


Map: [&lt;5 8 0 14 -6 7|, &lt;0 0 1 0 2 1|]
Comma list: 64/63, 99/98
EDOs: 5e, 10e, 15
Badness: 0.0223


==Ferrum==
Subgroup-val mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
Commas: 28/27, 35/33, 49/48


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


Map: [&lt;5 8 0 14 6|, &lt;0 0 1 0 1|]
Optimal tunings:  
EDOs: 10
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
Badness: 0.0309
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}


{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}


[[image:blacksmith10.jpg]]
Badness (Sintel): 0.352


==Blackwood==
Scales: [[supra7]], [[supra12]]
//Blackwood (5-limit)//


Comma: 256/243
==== Supraphon ====
Subgroup: 2.3.7.11.13


[[POTE tuning|POTE generator]]: 399.594
Comma list: 64/63, 78/77, 99/98


Map: [&lt;5 8 0|, &lt;0 0 1|]
Subgroup-val mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
EDOs: 5, 10, 15
Badness: 0.0638


=Superpyth=
Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
[[Superpyth|Main article]]
Commas: 64/63, 245/243


[[POTE tuning|POTE generator]]: 710.291
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|, &lt;0 1 9 -2|]
{{Optimal ET sequence|legend=0| 12f, 17 }}
Wedgie: &lt;&lt;1 9 -2 12 -6 -30||
EDOs: 5, 17, 22, 27, 49
Badness: 0.0323


==11-limit==
Badness (Sintel): 0.498
Commas: 64/63, 100/99, 245/243


[[POTE tuning|POTE generator]]: 710.175
Scales: [[supra7]], [[supra12]]


Map: [&lt;1 0 -12 6 -22|, &lt;0 1 9 -2 16|]
== Superpyth ==
EDOs: 22, 49
{{Main| Superpyth }}
Badness: 0.0250
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


==13-limit==
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: 64/63, 78/77, 91/90, 100/99


POTE generator: ~3/2 = 710.479
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -12 6 -22 -17|, &lt;0 1 9 -2 16 13|]
[[Comma list]]: 64/63, 245/243
EDOs: 22, 27e, 49, 76bcde
Badness: 0.0247


==Suprapyth==
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}
Commas: 55/54, 64/63, 99/98


POTE generator: ~3/2 = 709.495
[[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|, &lt;0 1 9 -2 -6|]
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
EDOs: 5, 7, 12, 17, 22
Badness: 0.0328


===13-limit===
[[Badness]] (Sintel): 0.818
Commas: 55/54, 64/63, 65/63, 364/363


POTE generator: ~3/2 = 708.703
=== 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 0 -12 6 13 18|, &lt;0 1 9 -2 -6 -9|]
Subgroup: 2.3.5.7.11
EDOs: 17, 22, 83cdf
Badness: 0.0363


=Beatles=
Comma list: 64/63, 100/99, 245/243
Comma: 524288/492075


POTE generator: ~512/405 = 355.930
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}


Map: [&lt;1 1 5|,&lt;0 2 -9|]
Optimal tunings:  
EDOs: 10, 17c, 27, 64b, 91bc, 118bc
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
Badness: 0.3585
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 22, 27e, 49 }}
Commas: 64/63, 686/675


[[POTE tuning|POTE generator]]: ~49/40 = 355.904
Badness (Sintel): 0.826


Map: [&lt;1 1 5 4|,&lt;0 2 -9 -4|]
==== 13-limit ====
Wedgie: &lt;&lt;2 -9 -4 -19 -12 16||
Subgroup: 2.3.5.7.11.13
EDOs: 10, 17c, 27, 64b, 91bcd, 118bcd
Badness: 0.0459


Music: [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3|Beatles Improv]] by Herman Miller
Comma list: 64/63, 78/77, 91/90, 100/99


==11-limit==
Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
Commas: 64/63, 100/99, 686/675


POTE generator: ~49/40 = 356.140
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|,&lt;0 2 -9 -4 -22|]
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
EDOs: 27e, 37, 64be, 91bcde
Badness: 0.0456


==13-limit==
Badness (Sintel): 1.02
Commas: 64/63, 91/90, 100/99, 169/168


POTE generator: ~16/13 = 356.229
==== Thomas ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 5 4 10 4|,&lt;0 2 -9 -4 -22 -1|]
Comma list: 64/63, 100/99, 169/168, 245/243
EDOs: 27e, 37, 64be
Badness: 0.0302


==Ringo==
Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}
Commas: 56/55, 64/63, 540/539


POTE generator: ~11/9 = 355.419
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|,&lt;0 2 -9 -4 5|]
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}
EDOs: 10, 17c, 27e
Badness: 0.0329


===13-limit===
Badness (Sintel): 2.03
Commas: 56/55, 64/63, 78/77, 91/90


POTE generator: ~11/9 = 355.456
=== 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 1 5 4 2 4|,&lt;0 2 -9 -4 5 -1|]
Subgroup: 2.3.5.7.11
EDOs: 10, 17c, 27e
Badness: 0.0226


=Schism=
Comma list: 55/54, 64/63, 99/98
Commas: 64/63, 360/343


[[POTE tuning|POTE generator]]: ~3/2 = 701.556
Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}


Map: [&lt;1 0 15 6|, &lt;0 1 -8 -2|]
Optimal tunings:  
Wedgie: &lt;&lt;1 -8 -2 -15 -6 18||
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
EDOs: 12, 41d, 53d
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}
Badness: 0.0566


==11-limit==
{{Optimal ET sequence|legend=0| 5, 17, 22 }}
Commas: 45/44, 64/63, 99/98


POTE generator ~3/2 = 702.136
Badness (Sintel): 1.08


Map: [&lt;1 0 15 6 13|, &lt;0 1 -8 -2 -6|]
==== 13-limit ====
EDOs: 12, 29de, 41de
Subgroup: 2.3.5.7.11.13
Badness: 0.0375


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


==5-limit==
Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}
Comma: 262144/253125


POTE generator: ~16/15 = 98.670
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|, &lt;0 -5 4|]
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}
EDOs: 11, 12, 49, 61, 73
Badness: 0.1686


===Passive===
Badness (Sintel): 1.50
Commas: 225/224, 256/245


POTE generator: ~16/15 = 98.809
== 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 3|, &lt;0 -5 4 -2|]
[[Subgroup]]: 2.3.5.7
EDOs: 1, 11, 12, 49d
Badness: 0.0751


==7-limit==
[[Comma list]]: 64/63, 2430/2401
Commas: 64/63, 3125/3087


[[POTE tuning|POTE generator]]: ~16/15 = 98.153
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}


Map: [&lt;1 2 2 2|, &lt;0 -5 4 10|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;5 -4 -10 -18 -30 -12||
* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}
Generators: 2, 16/15
: [[error map]]: {{val| -3.017 +1.486 -0.435 +6.190 }}
EDOs: 12, 37, 49, 110bcd
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 708.3716{{c}}
Badness: 0.0623
: error map: {{val| 0.000 +6.417 +4.855 +14.431 }}


==11-limit==
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}
Commas: 64/63, 100/99, 1375/1372


POTE generator: ~16/15 = 98.019
[[Badness]] (Sintel): 1.61


Map: [&lt;1 2 2 2 2|, &lt;0 -5 4 10 18|]
=== Quasisupra ===
EDOs: 12, 37, 49
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: 0.0408


==13-limit==
Subgroup: 2.3.5.7.11
Commas: 64/63 100/99 196/195 275/273


POTE generator: ~16/15 = 97.910
Comma list: 64/63, 99/98, 121/120


Map: [&lt;1 2 2 2 2 2|, &lt;0 -5 4 10 18 21|]
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}
EDOs: 12f, 37, 49f
Badness: 0.0309


=Fervor=  
Optimal tunings:
Comma: 67108864/61509375
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


POTE generator: ~64/45 = 577.705
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}


Map: [&lt;1 4 -2|, &lt;0 -5 9|]
Badness (Sintel): 1.06
EDOs: 25, 27
Badness: 0.8526


==7-limit==  
==== 13-limit ====
Commas: 64/63, 9604/9375
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 577.777
Comma list: 64/63, 78/77, 91/90, 121/120


Map: [&lt;1 4 -2 -2|, &lt;0 -5 9 10|]
Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}
Wedgie: &lt;&lt;5 -9 -10 -26 -30 2||
EDOs: 25, 27
Badness: 0.1085


==11-limit==
Optimal tunings:
Commas: 56/55, 64/63, 1350/1331
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


POTE generator: ~7/5 = 577.850
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


Map: [&lt;1 4 -2 -2 3|, &lt;0 -5 9 10 1|]
Badness (Sintel): 1.25
EDOs: 25e, 27e
Badness: 0.0521


==13-limit==  
=== Quasisoup ===
Commas: 56/55, 64/63, 78/77, 507/500
Subgroup: 2.3.5.7.11


POTE generator: ~7/5 = 578.060
Comma list: 55/54, 64/63, 2430/2401


Map: [&lt;1 4 -2 -2 3 -4|, &lt;0 -5 9 10 1 16|]
Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
EDOs: 27e
Badness: 0.0397


=Quasisuper=  
Optimal tunings:
Commas: 64/63, 2430/2401
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


[[POTE tuning|POTE generator]]: 708.328
{{Optimal ET sequence|legend=0| 22 }}


Map: [&lt;1 0 23 6|, &lt;0 1 -13 -2|]
Badness (Sintel): 2.76
Wedgie: &lt;&lt;1 -13 -2 -23 -2 -6 32||
EDOs: 22, 61
Badness: 0.0638


==Quasisupra==  
== Ultrapyth ==
Commas: 64/63, 99/98, 121/120
{{Main| Ultrapyth }}


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).
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).


POTE generator: ~3/2 = 708.205
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 -3 2 1|, &lt;0 -1 13 2 6|]
[[Comma list]]: 64/63, 6860/6561
EDOs: 17c, 22, 27c, 39d, 61d
Badness: 0.0322


===13-limit===
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
Commas: 64/63, 78/77, 91/90, 121/120


POTE generator: ~3/2 = 708.004
[[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 13 18|, &lt;0 1 -13 -2 -6 -9|]
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
EDOs: 17c, 22, 39d, 61df, 100bcdf
Badness: 0.0302


==Quasisoup==
[[Badness]] (Sintel): 2.74
Commas: 55/54, 64/63, 2430/2401


POTE generator: ~3/2 = 709.021
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 23 6 -22|, &lt;0 1 -13 -2 16|]
Comma list: 55/54, 64/63, 2401/2376
EDOs: 22
Badness: 0.0835


=Progress=
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}
Comma: 32768/30375


POTE generator: ~64/45 = 561.264
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|, &lt;0 3 -5|]
{{Optimal ET sequence|legend=0| 5, 32, 37 }}
EDOs: 4, 13, 15, 32c, 47bc, 62bc
Badness: 0.2461


==7-limit==
Badness (Sintel): 2.26
Commas: 64/63, 392/375


POTE generator: ~7/5 = 562.122
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 5 6|, &lt;0 3 -5 -6|]
Comma list: 55/54, 64/63, 91/90, 1573/1568
Wedgie: &lt;&lt;3 -5 -6 -15 -18 0||
EDOs: 13, 15, 32c, 79bcc, 111bcc
Badness: 0.0664


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


POTE generator: ~7/5 = 562.085
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|, &lt;0 3 -5 -6 -1|]
{{Optimal ET sequence|legend=0| 5, 32, 37 }}
EDOs: 13, 15, 32c, 47bc, 79bcce
Badness: 0.0310


==13-limit==
Badness (Sintel): 2.03
Commas: 56/55, 64/63, 66/65, 77/75


POTE generator: ~7/5 = 562.365
=== Ultramarine ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 5 6 4 0|, &lt;0 3 -5 -6 -1 7|]
Comma list: 64/63, 100/99, 3773/3645
EDOs: 15, 17c, 32cf
Badness: 0.0262


==Progressive==
Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}
Commas: 26/25, 56/55, 64/63, 77/75


POTE generator: ~7/5 = 563.239
Optimal tunings:  
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


Map: [&lt;1 0 5 6 4 9|, &lt;0 3 -5 -6 -1 -10|]
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}
EDOs: 15f, 17c, 32c, 49c
 
Badness: 0.0327</pre></div>
Badness (Sintel): 2.58
<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;
==== 13-limit ====
&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;
Subgroup: 2.3.5.7.11.13
&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;
 
&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;
Comma list: 64/63, 91/90, 100/99, 847/845
&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;
 
&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;
Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
&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;
 
&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;
Optimal tunings:  
&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;
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
&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;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}
&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;
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}
&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;
Badness (Sintel): 1.89
&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;
== Quasiultra ==
&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;
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: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;
 
&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;
[[Subgroup]]: 2.3.5.7
&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;
 
&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;
[[Comma list]]: 64/63, 33614/32805
&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;
 
&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;
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
&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;
 
&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;
[[Optimal tuning]]s:
&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;
* [[WE]]: ~2 = 1196.9257{{c}}, ~3/2 = 709.6211{{c}}
&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;
: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}
&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;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5429{{c}}
&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;
: error map: {{val| 0.000 +9.588 +5.914 +8.088 }}
&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;
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
&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;
[[Badness]] (Sintel): 3.34
&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;
 
&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;
== Schism ==
&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;
{{See also| Schismatic family #Schism }}
&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;
 
&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;
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;!-- 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;
 
&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;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:end:WikiTextTocRule:120 --&gt;&lt;!-- ws:start:WikiTextTocRule:121: --&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:121 --&gt;&lt;br /&gt;
[[Comma list]]: 64/63, 360/343
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;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}
&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;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 721.569&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}}
Map: [&amp;lt;1 0 4 6|, &amp;lt;0 1 -1 -2|]&lt;br /&gt;
: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}
Wedgie: &amp;lt;&amp;lt;1 -1 -2 -4 -6 -2||&lt;br /&gt;
 
EDOs: 5, 148, 153&lt;br /&gt;
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
Badness: 0.0242&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 1.43
&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;
 
Commas: 28/27, 49/48&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~5/4 = 392.767&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 45/44, 64/63, 99/98
Map: [&amp;lt;5 8 0 14|, &amp;lt;0 0 1 0|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;0 5 0 8 0 -14||&lt;br /&gt;
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}
EDOs: 5, 10, 15, 40b, 55b&lt;br /&gt;
 
Badness: 0.0256&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
&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;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}
Commas: 28/27, 49/48, 55/54&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}
POTE generator: ~5/4 = 394.948&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.24
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;
== Beatles ==
Badness: 0.0246&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''
&lt;br /&gt;
 
&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;
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.
Commas: 28/27, 40/39, 49/48, 55/54&lt;br /&gt;
 
&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.
POTE generator: ~5/4 = 391.0367&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Map: [&amp;lt;5 8 0 14 29 7|, &amp;lt;0 0 1 0 -1 1|]&lt;br /&gt;
 
EDOs: 5, 10, 15, 25e, 40bef&lt;br /&gt;
[[Comma list]]: 64/63, 686/675
Badness: 0.0205&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
&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;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}
POTE generator: ~5/4 = 398.070&lt;br /&gt;
: [[error map]]: {{val| -3.376 +4.475 +2.682 -1.940 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}
Map: [&amp;lt;5 8 0 14 -6|, &amp;lt;0 0 1 0 2|]&lt;br /&gt;
: error map: {{val| 0.000 +10.209 +8.949 +6.847 }}
EDOs: 5e, 15&lt;br /&gt;
 
Badness: 0.0292&lt;br /&gt;
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
&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;
[[Badness]] (Sintel): 1.16
Commas: 28/27, 40/39, 49/48, 66/65&lt;br /&gt;
 
&lt;br /&gt;
; Music
POTE generator: ~5/4 = 396.812&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]]
&lt;br /&gt;
 
Map: [&amp;lt;5 8 0 14 -6 7|, &amp;lt;0 0 1 0 2 1|]&lt;br /&gt;
=== 11-limit ===
EDOs: 5e, 10e, 15&lt;br /&gt;
Subgroup: 2.3.5.7.11
Badness: 0.0223&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 64/63, 100/99, 686/675
&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;
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
&lt;br /&gt;
 
POTE generator: ~5/4 = 374.763&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
Map: [&amp;lt;5 8 0 14 6|, &amp;lt;0 0 1 0 1|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}
EDOs: 10&lt;br /&gt;
 
Badness: 0.0309&lt;br /&gt;
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}
&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.51
&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;
==== 13-limit ====
&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;
Subgroup: 2.3.5.7.11.13
&lt;em&gt;Blackwood (5-limit)&lt;/em&gt;&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 64/63, 91/90, 100/99, 169/168
Comma: 256/243&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 399.594&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;5 8 0|, &amp;lt;0 0 1|]&lt;br /&gt;
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
EDOs: 5, 10, 15&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}
Badness: 0.0638&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}
&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;
Badness (Sintel): 1.25
Commas: 64/63, 245/243&lt;br /&gt;
 
&lt;br /&gt;
=== Ringo ===
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 710.291&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -12 6|, &amp;lt;0 1 9 -2|]&lt;br /&gt;
Comma list: 56/55, 64/63, 540/539
Wedgie: &amp;lt;&amp;lt;1 9 -2 12 -6 -30||&lt;br /&gt;
 
EDOs: 5, 17, 22, 27, 49&lt;br /&gt;
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}
Badness: 0.0323&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&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;
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
Commas: 64/63, 100/99, 245/243&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 710.175&lt;br /&gt;
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -12 6 -22|, &amp;lt;0 1 9 -2 16|]&lt;br /&gt;
Badness (Sintel): 1.09
EDOs: 22, 49&lt;br /&gt;
 
Badness: 0.0250&lt;br /&gt;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&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;
 
Commas: 64/63, 78/77, 91/90, 100/99&lt;br /&gt;
Comma list: 56/55, 64/63, 78/77, 91/90
&lt;br /&gt;
 
POTE generator: ~3/2 = 710.479&lt;br /&gt;
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -12 6 -22 -17|, &amp;lt;0 1 9 -2 16 13|]&lt;br /&gt;
Optimal tunings:  
EDOs: 22, 27e, 49, 76bcde&lt;br /&gt;
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
Badness: 0.0247&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}
&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;
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
Commas: 55/54, 64/63, 99/98&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.935
POTE generator: ~3/2 = 709.495&lt;br /&gt;
 
&lt;br /&gt;
=== Beetle ===
Map: [&amp;lt;1 0 -12 6 13|, &amp;lt;0 1 9 -2 -6|]&lt;br /&gt;
Subgroup: 2.3.5.7.11
EDOs: 5, 7, 12, 17, 22&lt;br /&gt;
 
Badness: 0.0328&lt;br /&gt;
Comma list: 55/54, 64/63, 686/675
&lt;br /&gt;
 
&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;
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}
Commas: 55/54, 64/63, 65/63, 364/363&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~3/2 = 708.703&lt;br /&gt;
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}
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;
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
Badness: 0.0363&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.92
&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;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~512/405 = 355.930&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 55/54, 64/63, 91/90, 169/168
Map: [&amp;lt;1 1 5|,&amp;lt;0 2 -9|]&lt;br /&gt;
 
EDOs: 10, 17c, 27, 64b, 91bc, 118bc&lt;br /&gt;
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
Badness: 0.3585&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&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;
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
Commas: 64/63, 686/675&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~49/40 = 355.904&lt;br /&gt;
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 5 4|,&amp;lt;0 2 -9 -4|]&lt;br /&gt;
Badness (Sintel): 1.40
Wedgie: &amp;lt;&amp;lt;2 -9 -4 -19 -12 16||&lt;br /&gt;
 
EDOs: 10, 17c, 27, 64b, 91bcd, 118bcd&lt;br /&gt;
== Progress ==
Badness: 0.0459&lt;br /&gt;
{{Distinguish| Progression }}
&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
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;
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;!-- 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;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
POTE generator: ~49/40 = 356.140&lt;br /&gt;
[[Comma list]]: 64/63, 392/375
&lt;br /&gt;
 
Map: [&amp;lt;1 1 5 4 10|,&amp;lt;0 2 -9 -4 -22|]&lt;br /&gt;
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
EDOs: 27e, 37, 64be, 91bcde&lt;br /&gt;
 
Badness: 0.0456&lt;br /&gt;
: mapping generators: ~2, ~10/7
&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;
[[Optimal tuning]]s:  
Commas: 64/63, 91/90, 100/99, 169/168&lt;br /&gt;
* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -4.862 +3.925 +12.908 -9.759 }}
POTE generator: ~16/13 = 356.229&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 638.0791{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +12.282 +23.291 +2.700 }}
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;
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
Badness: 0.0302&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 1.68
&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;
 
Commas: 56/55, 64/63, 540/539&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~11/9 = 355.419&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 56/55, 64/63, 77/75
Map: [&amp;lt;1 1 5 4 2|,&amp;lt;0 2 -9 -4 5|]&lt;br /&gt;
 
EDOs: 10, 17c, 27e&lt;br /&gt;
Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
Badness: 0.0329&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&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;
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
Commas: 56/55, 64/63, 78/77, 91/90&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}
&lt;br /&gt;
 
POTE generator: ~11/9 = 355.456&lt;br /&gt;
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 5 4 2 4|,&amp;lt;0 2 -9 -4 5 -1|]&lt;br /&gt;
Badness (Sintel): 1.03
EDOs: 10, 17c, 27e&lt;br /&gt;
 
Badness: 0.0226&lt;br /&gt;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&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;
 
Commas: 64/63, 360/343&lt;br /&gt;
Comma list: 56/55, 64/63, 66/65, 77/75
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 701.556&lt;br /&gt;
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 15 6|, &amp;lt;0 1 -8 -2|]&lt;br /&gt;
Optimal tunings:  
Wedgie: &amp;lt;&amp;lt;1 -8 -2 -15 -6 18||&lt;br /&gt;
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
EDOs: 12, 41d, 53d&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}
Badness: 0.0566&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
&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;
Badness (Sintel): 1.08
&lt;br /&gt;
 
POTE generator ~3/2 = 702.136&lt;br /&gt;
==== Progressive ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Map: [&amp;lt;1 0 15 6 13|, &amp;lt;0 1 -8 -2 -6|]&lt;br /&gt;
 
EDOs: 12, 29de, 41de&lt;br /&gt;
Comma list: 26/25, 56/55, 64/63, 77/75
Badness: 0.0375&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
&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;
 
&lt;br /&gt;
Optimal tunings:  
&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;
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
Comma: 262144/253125&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}
&lt;br /&gt;
 
POTE generator: ~16/15 = 98.670&lt;br /&gt;
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2|, &amp;lt;0 -5 4|]&lt;br /&gt;
Badness (Sintel): 1.35
EDOs: 11, 12, 49, 61, 73&lt;br /&gt;
 
Badness: 0.1686&lt;br /&gt;
== Fervor ==
&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
&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;
 
Commas: 225/224, 256/245&lt;br /&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.
&lt;br /&gt;
 
POTE generator: ~16/15 = 98.809&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 3|, &amp;lt;0 -5 4 -2|]&lt;br /&gt;
[[Comma list]]: 64/63, 9604/9375
EDOs: 1, 11, 12, 49d&lt;br /&gt;
 
Badness: 0.0751&lt;br /&gt;
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
&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;
: mapping generators: ~2, ~10/7
Commas: 64/63, 3125/3087&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~16/15 = 98.153&lt;br /&gt;
* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -3.726 +3.230 +4.980 -1.550 }}
Map: [&amp;lt;1 2 2 2|, &amp;lt;0 -5 4 10|]&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 622.3179{{c}}
Wedgie: &amp;lt;&amp;lt;5 -4 -10 -18 -30 -12||&lt;br /&gt;
: error map: {{val| 0.000 +9.634 +12.826 +7.996 }}
Generators: 2, 16/15&lt;br /&gt;
 
EDOs: 12, 37, 49, 110bcd&lt;br /&gt;
{{Optimal ET sequence|legend=1| 2, 25, 27 }}
Badness: 0.0623&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 2.74
&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;
 
Commas: 64/63, 100/99, 1375/1372&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~16/15 = 98.019&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 56/55, 64/63, 1350/1331
Map: [&amp;lt;1 2 2 2 2|, &amp;lt;0 -5 4 10 18|]&lt;br /&gt;
 
EDOs: 12, 37, 49&lt;br /&gt;
Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}
Badness: 0.0408&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&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;
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
Commas: 64/63 100/99 196/195 275/273&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}
&lt;br /&gt;
 
POTE generator: ~16/15 = 97.910&lt;br /&gt;
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 2 2 2|, &amp;lt;0 -5 4 10 18 21|]&lt;br /&gt;
Badness (Sintel): 1.72
EDOs: 12f, 37, 49f&lt;br /&gt;
 
Badness: 0.0309&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&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;
 
Comma: 67108864/61509375&lt;br /&gt;
Comma list: 56/55, 64/63, 78/77, 507/500
&lt;br /&gt;
 
POTE generator: ~64/45 = 577.705&lt;br /&gt;
Mapping: {{mapping| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 4 -2|, &amp;lt;0 -5 9|]&lt;br /&gt;
Optimal tunings:  
EDOs: 25, 27&lt;br /&gt;
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
Badness: 0.8526&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}
&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;
{{Optimal ET sequence|legend=0| 2f, 27e }}
Commas: 64/63, 9604/9375&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.64
POTE generator: ~7/5 = 577.777&lt;br /&gt;
 
&lt;br /&gt;
== Sixix ==
Map: [&amp;lt;1 4 -2 -2|, &amp;lt;0 -5 9 10|]&lt;br /&gt;
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sixix (5-limit)]].''
Wedgie: &amp;lt;&amp;lt;5 -9 -10 -26 -30 2||&lt;br /&gt;
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}
EDOs: 25, 27&lt;br /&gt;
 
Badness: 0.1085&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.
&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;
[[Subgroup]]: 2.3.5.7
Commas: 56/55, 64/63, 1350/1331&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 64/63, 3125/2916
POTE generator: ~7/5 = 577.850&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
Map: [&amp;lt;1 4 -2 -2 3|, &amp;lt;0 -5 9 10 1|]&lt;br /&gt;
 
EDOs: 25e, 27e&lt;br /&gt;
[[Optimal tuning]]s:  
Badness: 0.0521&lt;br /&gt;
* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -1.097 +9.086 -13.503 +2.508 }}
&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;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 337.4588{{c}}
Commas: 56/55, 64/63, 78/77, 507/500&lt;br /&gt;
: error map: {{val| 0.000 +10.751 -11.066 +5.762 }}
&lt;br /&gt;
 
POTE generator: ~7/5 = 578.060&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 4 -2 -2 3 -4|, &amp;lt;0 -5 9 10 1 16|]&lt;br /&gt;
[[Badness]] (Sintel): 4.02
EDOs: 27e&lt;br /&gt;
 
Badness: 0.0397&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&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;
 
Commas: 64/63, 2430/2401&lt;br /&gt;
Comma list: 55/54, 64/63, 125/121
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 708.328&lt;br /&gt;
Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 23 6|, &amp;lt;0 1 -13 -2|]&lt;br /&gt;
Optimal tunings:  
Wedgie: &amp;lt;&amp;lt;1 -13 -2 -23 -2 -6 32||&lt;br /&gt;
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
EDOs: 22, 61&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}
Badness: 0.0638&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}
&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;
Badness (Sintel): 2.34
&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;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~3/2 = 708.205&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 40/39, 55/54, 64/63, 125/121
Map: [&amp;lt;1 2 -3 2 1|, &amp;lt;0 -1 13 2 6|]&lt;br /&gt;
 
EDOs: 17c, 22, 27c, 39d, 61d&lt;br /&gt;
Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
Badness: 0.0322&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&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;
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
Commas: 64/63, 78/77, 91/90, 121/120&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}
&lt;br /&gt;
 
POTE generator: ~3/2 = 708.004&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 23 6 13 18|, &amp;lt;0 1 -13 -2 -6 -9|]&lt;br /&gt;
Badness (Sintel): 1.91
EDOs: 17c, 22, 39d, 61df, 100bcdf&lt;br /&gt;
 
Badness: 0.0302&lt;br /&gt;
=== 17-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&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;
 
Commas: 55/54, 64/63, 2430/2401&lt;br /&gt;
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121
&lt;br /&gt;
 
POTE generator: ~3/2 = 709.021&lt;br /&gt;
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 23 6 -22|, &amp;lt;0 1 -13 -2 16|]&lt;br /&gt;
Optimal tunings:  
EDOs: 22&lt;br /&gt;
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
Badness: 0.0835&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}
&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;
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
Comma: 32768/30375&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 2.00
POTE generator: ~64/45 = 561.264&lt;br /&gt;
 
&lt;br /&gt;
[[Category:Archytas clan| ]] <!-- main article -->
Map: [&amp;lt;1 0 5|, &amp;lt;0 3 -5|]&lt;br /&gt;
[[Category:Temperament clans]]
EDOs: 4, 13, 15, 32c, 47bc, 62bc&lt;br /&gt;
[[Category:Pages with mostly numerical content]]
Badness: 0.2461&lt;br /&gt;
[[Category:Rank 2]]
&lt;br /&gt;
&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;
&lt;br /&gt;
POTE generator: ~7/5 = 562.122&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 5 6|, &amp;lt;0 3 -5 -6|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;3 -5 -6 -15 -18 0||&lt;br /&gt;
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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