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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>2012-04-14 00:20:12 UTC</tt>.<br>
: The original revision id was <tt>320372448</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|flat]]


The //archytas clan// tempers out the [[http://en.wikipedia.org/wiki/Septimal_comma|Archytas comma]], 64/63. 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 }}
: mapping generators: ~2, ~3


Map: [&lt;5 8 0 14|, &lt;0 0 1 0|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;0 5 0 8 0 -14||
* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}
EDOs: 5, 10, 15, 40b, 55b
: [[error map]]: {{val| -3.045 +2.522 +3.952 }}
Badness: 0.0256
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}
: error map: {{val| 0.000 +7.435 +12.394 }}


==11-limit==
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
Commas: 28/27, 49/48, 55/54


POTE generator: ~5/4 = 394.948
[[Badness]] (Sintel): 0.159


Map: [&lt;5 8 0 14 29|, &lt;0 0 1 0 -1|]
Scales: [[archy5]], [[archy7]], [[archy12]]
EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde
Badness: 0.0246


==13-limit==
=== Overview to extensions ===
Commas: 28/27, 40/39, 49/48, 55/54
==== 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]];
* [[Meantone family #Dominant|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]].


POTE generator: ~5/4 = 391.0367
These all use the same generators as archy.  


Map: [&lt;5 8 0 14 29 7|, &lt;0 0 1 0 -1 1|]
[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a 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 blackwood with a 1/5-octave period. Finally, [[15625/15552]] gives catalan, splitting the twelfth in six.  
EDOs: 5, 10, 15, 25e, 40bef
Badness: 0.0205


==Farrier==
Temperaments discussed elsewhere are:
Commas: 28/27, 49/48, 77/75
* ''[[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]]
* ''[[Dichotic]]'' (+25/24) → [[Dicot family #Dichotic|Dicot family]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Augene]] (+126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* [[Modus]] (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]


POTE generator: ~5/4 = 398.070
Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.


Map: [&lt;5 8 0 14 -6|, &lt;0 0 1 0 2|]
==== Subgroup extensions ====
EDOs: 5e, 15
Omitting prime 5, archy can be extended to the 2.3.7.11 subgroup by identifying 11/8 as a diminished fourth (C–G♭). This is called supra, given right below. Discussed elsewhere is [[suhajira]] of the [[rastmic clan #Suhajira|rastmic clan]].
Badness: 0.0292


==Ferrum==
=== Supra ===
Commas: 28/27, 35/33, 49/48
Subgroup: 2.3.7.11


POTE generator: ~5/4 = 374.763
Comma list: 64/63, 99/98


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


Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}


[[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
This extension maps [[13/11]] to the minor third (C–E♭), [[12/11]][[~]][[14/13]] to the augmented unison (C–C♯), and [[13/12]] to the diminished third (C–E𝄫).
Badness: 0.0638


=Superpyth=
Subgroup: 2.3.7.11.13
[[Superpyth|Main article]]
Commas: 64/63, 245/243


[[POTE tuning|POTE generator]]: 710.291
Comma list: 64/63, 78/77, 99/98


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


==11-limit==
Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
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


==Suprapyth==
Badness (Sintel): 0.498
Commas: 55/54, 64/63, 99/98


POTE generator: ~3/2 = 709.495
Scales: [[supra7]], [[supra12]]


Map: [&lt;1 0 -12 6 13|, &lt;0 1 9 -2 -6|]
== Superpyth ==
EDOs: 5, 7, 12, 17, 22
{{Main| Superpyth }}
Badness: 0.0328
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


=Beatles=
Superpyth, virtually the canonical extension, adds [[245/243]] and [[1728/1715]] to the comma list and can be described as {{nowrap| [[22edo|22]] [[&]] [[27edo|27]] }}. ~[[5/4]] is found at +9 generator steps, as an augmented second (C–D♯). In the [[11-limit]] it finds the ~[[11/8]] at +16 generator steps, as a double-augmented second (C–D𝄪). [[49edo]] remains an obvious tuning choice in either case.
Commas: 64/63, 686/675


[[POTE tuning|POTE generator]]: 355.904
Extending superpyth to the [[13-limit]] is more diffcult. Tridecimal superpyth finds the ~[[13/8]] at +13 generator steps, as a double-augmented fourth (C–F𝄪), for which 27edo can be recommended as a tuning since it is the only [[13-odd-limit]] [[diamond monotone]] tuning. The other extension, called uberpyth, is more flexible with its tunings, but unfortunately tends to tune the 13 very sharp.  


Map: [&lt;1 1 5 4|,&lt;0 2 -9 -4|]
[[Subgroup]]: 2.3.5.7
Wedgie: &lt;&lt;2 -9 -4 -19 -12 16||
EDOs: 10, 27, 118
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, 245/243


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


POTE generator: ~49/40 = 356.140
[[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 1 5 4 10|,&lt;0 2 -9 -4 -22|]
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
EDOs: 27e, 37, 64be, 91bcde
Badness: 0.0456


==Ringo==
[[Badness]] (Sintel): 0.818
Commas: 56/55, 64/63, 540/539


POTE generator: ~11/9 = 355.419
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


===13-limit===
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}
Commas: 56/55, 64/63, 78/77, 91/90


POTE generator: ~11/9 = 355.456
Optimal tunings:  
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


Map: [&lt;1 1 5 4 2 4|,&lt;0 2 -9 -4 5 -1|]
{{Optimal ET sequence|legend=0| 22, 27e, 49 }}
EDOs: 10, 17c, 27e
Badness: 0.0226


=Schism=
Badness (Sintel): 0.826
Commas: 64/63, 360/343


[[POTE tuning|POTE generator]]: ~3/2 = 701.556
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 15 6|, &lt;0 1 -8 -2|]
Comma list: 64/63, 78/77, 91/90, 100/99
Wedgie: &lt;&lt;1 -8 -2 -15 -6 18||
EDOs: 12, 41d, 53d
Badness: 0.0566


==11-limit==
Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
Commas: 45/44, 64/63, 99/98


POTE generator ~3/2 = 702.136
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 0 15 6 13|, &lt;0 1 -8 -2 -6|]
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
EDOs: 12, 29de, 41de
Badness: 0.0375


=Passion=
Badness (Sintel): 1.02
Commas: 64/63, 3125/3087


[[POTE tuning|POTE generator]]: 98.153
==== Uberpyth ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2 2|, &lt;0 -5 4 10|]
Comma list: 64/63, 100/99, 144/143, 245/243
Wedgie: &lt;&lt;5 -4 -10 -18 -30 -12||
Generators: 2, 16/15
EDOs: 12, 37, 49, 110, 379
Badness: 0.0623


=Quasisuper=
Mapping: {{mapping| 1 0 -12 6 -22 26 | 0 1 9 -2 16 -14 }}
Commas: 64/63, 2430/2401


[[POTE tuning|POTE generator]]: 708.328
Optimal tunings:  
* WE: ~2 = 1196.6666{{c}}, ~3/2 = 708.3602{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.2878{{c}}


Map: [&lt;1 0 23 6|, &lt;0 1 -13 -2|]
{{Optimal ET sequence|legend=0| 22f, 27e, 49f, 125bcddeeeff, 174bbcdddeeeeffff }}
Wedgie: &lt;&lt;1 -13 -2 -23 -2 -6 32||
EDOs: 22, 61
Badness: 0.0638


==Quasisupra==
Badness (Sintel): 1.11
Commas: 64/63, 99/98, 121/120


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).
==== Thomas ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 708.205
Comma list: 64/63, 100/99, 169/168, 245/243


Map: [&lt;1 2 -3 2 1|, &lt;0 -1 13 2 6|]
Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}
EDOs: 17c, 22, 27c, 39d, 61d
Badness: 0.0322


==Quasisoup==
Optimal tunings:
Commas: 55/54, 64/63, 2430/2401
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


POTE generator: ~3/2 = 709.021
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}


Map: [&lt;1 0 23 6 -22|, &lt;0 1 -13 -2 16|]
Badness (Sintel): 2.03
EDOs: 22
Badness: 0.0835


=Progress=
=== Suprapyth ===
Commas: 64/63, 392/375
Suprapyth finds the ~11/8 at the diminished fifth (C–G♭), and finds the ~13/8 at the diminished seventh (C–B𝄫).


POTE generator: ~7/5 = 562.122
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 5 6|, &lt;0 3 -5 -6|]
Comma list: 55/54, 64/63, 99/98
Wedgie: &lt;&lt;3 -5 -6 -15 -18 0||
EDOs: 13, 15, 32c, 79bcc, 111bcc
Badness: 0.0664


==11-limit==
Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}
Commas: 56/55, 64/63, 77/75


POTE generator: ~7/5 = 562.085
Optimal tunings:  
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


Map: [&lt;1 0 5 6 4|, &lt;0 3 -5 -6 -1|]
{{Optimal ET sequence|legend=0| 5, 17, 22 }}
EDOs: 13, 15, 32c, 47bc, 79bcce
Badness: 0.0310


==13-limit==
Badness (Sintel): 1.08
Commas: 56/55, 64/63, 66/65, 77/75


POTE generator: ~7/5 = 562.365
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 5 6 4 0|, &lt;0 3 -5 -6 -1 7|]
Comma list: 55/54, 64/63, 65/63, 99/98
EDOs: 15, 17c, 32cf
 
Badness: 0.0262
Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}
</pre></div>
 
<h4>Original HTML content:</h4>
Optimal tunings:  
<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:46:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;a href="#Mother"&gt;Mother&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt; | &lt;a href="#Blacksmith"&gt;Blacksmith&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt; | &lt;a href="#Superpyth"&gt;Superpyth&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt; | &lt;a href="#Beatles"&gt;Beatles&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt; | &lt;a href="#Schism"&gt;Schism&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt;&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt; | &lt;a href="#Passion"&gt;Passion&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt; | &lt;a href="#Quasisuper"&gt;Quasisuper&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt;&lt;!-- ws:end:WikiTextTocRule:65 --&gt;&lt;!-- ws:start:WikiTextTocRule:66: --&gt;&lt;!-- ws:end:WikiTextTocRule:66 --&gt;&lt;!-- ws:start:WikiTextTocRule:67: --&gt; | &lt;a href="#Progress"&gt;Progress&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:67 --&gt;&lt;!-- ws:start:WikiTextTocRule:68: --&gt;&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt;&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt;
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}
The &lt;em&gt;archytas clan&lt;/em&gt; tempers out the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow"&gt;Archytas comma&lt;/a&gt;, 64/63. 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;
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}
&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;
Badness (Sintel): 1.50
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 721.569&lt;br /&gt;
== Quasisuper ==
&lt;br /&gt;
{{Main|Quasisuper}}
Map: [&amp;lt;1 0 4 6|, &amp;lt;0 1 -1 -2|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 -1 -2 -4 -6 -2||&lt;br /&gt;
Quasisuper can be described as {{nowrap| 17c & 22 }}, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–G𝄫). The 11-limit version, quasisupra, can be viewed as an extension of the excellent 2.3.7.11-subgroup temperament [[supra]], with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).
EDOs: 5, 148, 153&lt;br /&gt;
 
Badness: 0.0242&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&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;
[[Comma list]]: 64/63, 2430/2401
Commas: 28/27, 49/48&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~5/4 = 392.767&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
Map: [&amp;lt;5 8 0 14|, &amp;lt;0 0 1 0|]&lt;br /&gt;
* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}
Wedgie: &amp;lt;&amp;lt;0 5 0 8 0 -14||&lt;br /&gt;
: [[error map]]: {{val| -3.017 +1.486 -0.435 +6.190 }}
EDOs: 5, 10, 15, 40b, 55b&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 708.3716{{c}}
Badness: 0.0256&lt;br /&gt;
: error map: {{val| 0.000 +6.417 +4.855 +14.431 }}
&lt;br /&gt;
 
&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;
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}
Commas: 28/27, 49/48, 55/54&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 1.61
POTE generator: ~5/4 = 394.948&lt;br /&gt;
 
&lt;br /&gt;
=== Quasisupra ===
Map: [&amp;lt;5 8 0 14 29|, &amp;lt;0 0 1 0 -1|]&lt;br /&gt;
Subgroup: 2.3.5.7.11
EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde&lt;br /&gt;
 
Badness: 0.0246&lt;br /&gt;
Comma list: 64/63, 99/98, 121/120
&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;
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}
Commas: 28/27, 40/39, 49/48, 55/54&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~5/4 = 391.0367&lt;br /&gt;
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}
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;
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}
Badness: 0.0205&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.06
&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;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~5/4 = 398.070&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 64/63, 78/77, 91/90, 121/120
Map: [&amp;lt;5 8 0 14 -6|, &amp;lt;0 0 1 0 2|]&lt;br /&gt;
 
EDOs: 5e, 15&lt;br /&gt;
Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}
Badness: 0.0292&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Blacksmith-Ferrum"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Ferrum&lt;/h2&gt;
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
Commas: 28/27, 35/33, 49/48&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}
&lt;br /&gt;
 
POTE generator: ~5/4 = 374.763&lt;br /&gt;
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}
&lt;br /&gt;
 
Map: [&amp;lt;5 8 0 14 6|, &amp;lt;0 0 1 0 1|]&lt;br /&gt;
Badness (Sintel): 1.25
EDOs: 10&lt;br /&gt;
 
Badness: 0.0309&lt;br /&gt;
=== Quasisoup ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextLocalImageRule:71:&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:71 --&gt;&lt;br /&gt;
Comma list: 55/54, 64/63, 2430/2401
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Blacksmith-Blackwood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Blackwood&lt;/h2&gt;
Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
&lt;em&gt;Blackwood (5-limit)&lt;/em&gt;&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Comma: 256/243&lt;br /&gt;
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 399.594&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 22 }}
Map: [&amp;lt;5 8 0|, &amp;lt;0 0 1|]&lt;br /&gt;
 
EDOs: 5, 10, 15&lt;br /&gt;
Badness (Sintel): 2.76
Badness: 0.0638&lt;br /&gt;
 
&lt;br /&gt;
== Ultrapyth ==
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Superpyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Superpyth&lt;/h1&gt;
{{Main| Ultrapyth }}
&lt;a class="wiki_link" href="/Superpyth"&gt;Main article&lt;/a&gt;&lt;br /&gt;
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Ultrapyth (5-limit)]].''
Commas: 64/63, 245/243&lt;br /&gt;
 
&lt;br /&gt;
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–C𝄪).
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 710.291&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
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;
[[Comma list]]: 64/63, 6860/6561
EDOs: 5, 17, 22, 27, 49&lt;br /&gt;
 
Badness: 0.0323&lt;br /&gt;
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Superpyth-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;11-limit&lt;/h2&gt;
[[Optimal tuning]]s:  
Commas: 64/63, 100/99, 245/243&lt;br /&gt;
* [[WE]]: ~2 = 1197.2673{{c}}, ~3/2 = 712.0258{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -2.733 +7.338 -1.557 -3.808 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 710.175&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5430{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +11.588 +3.288 +4.088 }}
Map: [&amp;lt;1 0 -12 6 -22|, &amp;lt;0 1 9 -2 16|]&lt;br /&gt;
 
EDOs: 22, 49&lt;br /&gt;
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
Badness: 0.0250&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 2.74
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Superpyth-Suprapyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Suprapyth&lt;/h2&gt;
 
Commas: 55/54, 64/63, 99/98&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~3/2 = 709.495&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 55/54, 64/63, 2401/2376
Map: [&amp;lt;1 0 -12 6 13|, &amp;lt;0 1 9 -2 -6|]&lt;br /&gt;
 
EDOs: 5, 7, 12, 17, 22&lt;br /&gt;
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}
Badness: 0.0328&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Beatles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Beatles&lt;/h1&gt;
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
Commas: 64/63, 686/675&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 355.904&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5, 32, 37 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 5 4|,&amp;lt;0 2 -9 -4|]&lt;br /&gt;
Badness (Sintel): 2.26
Wedgie: &amp;lt;&amp;lt;2 -9 -4 -19 -12 16||&lt;br /&gt;
 
EDOs: 10, 27, 118&lt;br /&gt;
==== 13-limit ====
Badness: 0.0459&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
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;
Comma list: 55/54, 64/63, 91/90, 1573/1568
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Beatles-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h2&gt;
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
Commas: 64/63, 100/99, 686/675&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~49/40 = 356.140&lt;br /&gt;
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}
Map: [&amp;lt;1 1 5 4 10|,&amp;lt;0 2 -9 -4 -22|]&lt;br /&gt;
 
EDOs: 27e, 37, 64be, 91bcde&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5, 32, 37 }}
Badness: 0.0456&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 2.03
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Beatles-Ringo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Ringo&lt;/h2&gt;
 
Commas: 56/55, 64/63, 540/539&lt;br /&gt;
=== Ultramarine ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~11/9 = 355.419&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 64/63, 100/99, 3773/3645
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 -20 6 -38 | 0 1 14 -2 26 }}
Badness: 0.0329&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Beatles-Ringo-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h3&gt;
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
Commas: 56/55, 64/63, 78/77, 91/90&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}
&lt;br /&gt;
 
POTE generator: ~11/9 = 355.456&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}
&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): 2.58
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:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Schism"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Schism&lt;/h1&gt;
 
Commas: 64/63, 360/343&lt;br /&gt;
Comma list: 64/63, 91/90, 100/99, 847/845
&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 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
&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 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
EDOs: 12, 41d, 53d&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}
Badness: 0.0566&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Schism-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;11-limit&lt;/h2&gt;
 
Commas: 45/44, 64/63, 99/98&lt;br /&gt;
Badness (Sintel): 1.89
&lt;br /&gt;
 
POTE generator ~3/2 = 702.136&lt;br /&gt;
== Quasiultra ==
&lt;br /&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–A𝄫♭).
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;
[[Subgroup]]: 2.3.5.7
Badness: 0.0375&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 64/63, 33614/32805
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc16"&gt;&lt;a name="Passion"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Passion&lt;/h1&gt;
 
Commas: 64/63, 3125/3087&lt;br /&gt;
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 98.153&lt;br /&gt;
[[Optimal tuning]]s:
&lt;br /&gt;
* [[WE]]: ~2 = 1196.9257{{c}}, ~3/2 = 709.6211{{c}}
Map: [&amp;lt;1 2 2 2|, &amp;lt;0 -5 4 10|]&lt;br /&gt;
: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}
Wedgie: &amp;lt;&amp;lt;5 -4 -10 -18 -30 -12||&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5429{{c}}
Generators: 2, 16/15&lt;br /&gt;
: error map: {{val| 0.000 +9.588 +5.914 +8.088 }}
EDOs: 12, 37, 49, 110, 379&lt;br /&gt;
 
Badness: 0.0623&lt;br /&gt;
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc17"&gt;&lt;a name="Quasisuper"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Quasisuper&lt;/h1&gt;
[[Badness]] (Sintel): 3.34
Commas: 64/63, 2430/2401&lt;br /&gt;
 
&lt;br /&gt;
== Schism ==
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 708.328&lt;br /&gt;
{{See also| Schismatic family #Schism }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 23 6|, &amp;lt;0 1 -13 -2|]&lt;br /&gt;
Schism tempers out the [[schisma]], mapping the ~5/4 to -8 fifths as a diminished fourth (C–F♭) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53dd val) can be used.
Wedgie: &amp;lt;&amp;lt;1 -13 -2 -23 -2 -6 32||&lt;br /&gt;
 
EDOs: 22, 61&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Badness: 0.0638&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 64/63, 360/343
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Quasisuper-Quasisupra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Quasisupra&lt;/h2&gt;
 
Commas: 64/63, 99/98, 121/120&lt;br /&gt;
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}
&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 tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}
POTE generator: ~3/2 = 708.205&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 2 -3 2 1|, &amp;lt;0 -1 13 2 6|]&lt;br /&gt;
: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}
EDOs: 17c, 22, 27c, 39d, 61d&lt;br /&gt;
 
Badness: 0.0322&lt;br /&gt;
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Quasisuper-Quasisoup"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Quasisoup&lt;/h2&gt;
[[Badness]] (Sintel): 1.43
Commas: 55/54, 64/63, 2430/2401&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
POTE generator: ~3/2 = 709.021&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
Map: [&amp;lt;1 0 23 6 -22|, &amp;lt;0 1 -13 -2 16|]&lt;br /&gt;
Comma list: 45/44, 64/63, 99/98
EDOs: 22&lt;br /&gt;
 
Badness: 0.0835&lt;br /&gt;
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc20"&gt;&lt;a name="Progress"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;Progress&lt;/h1&gt;
Optimal tunings:
Commas: 64/63, 392/375&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}}
POTE generator: ~7/5 = 562.122&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}
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;
Badness (Sintel): 1.24
EDOs: 13, 15, 32c, 79bcc, 111bcc&lt;br /&gt;
 
Badness: 0.0664&lt;br /&gt;
== Beatles ==
&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Progress-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;11-limit&lt;/h2&gt;
 
Commas: 56/55, 64/63, 77/75&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;br /&gt;
 
POTE generator: ~7/5 = 562.085&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;
 
Map: [&amp;lt;1 0 5 6 4|, &amp;lt;0 3 -5 -6 -1|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
EDOs: 13, 15, 32c, 47bc, 79bcce&lt;br /&gt;
 
Badness: 0.0310&lt;br /&gt;
[[Comma list]]: 64/63, 686/675
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Progress-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;13-limit&lt;/h2&gt;
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
Commas: 56/55, 64/63, 66/65, 77/75&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
POTE generator: ~7/5 = 562.365&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 }}
Map: [&amp;lt;1 0 5 6 4 0|, &amp;lt;0 3 -5 -6 -1 7|]&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}
EDOs: 15, 17c, 32cf&lt;br /&gt;
: error map: {{val| 0.000 +10.209 +8.949 +6.847 }}
Badness: 0.0262&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
 
[[Badness]] (Sintel): 1.16
 
; Music
* [https://web.archive.org/web/20201127013829/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 ''Beatles Improv''] by [[Herman Miller]]
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 64/63, 100/99, 686/675
 
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
 
Optimal tunings:
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}
 
Badness (Sintel): 1.51
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 64/63, 91/90, 100/99, 169/168
 
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
 
Optimal tunings:
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}
 
Badness (Sintel): 1.25
 
=== Ringo ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 64/63, 540/539
 
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}
 
Optimal tunings:
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}
 
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
 
Badness (Sintel): 1.09
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 64/63, 78/77, 91/90
 
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
 
Optimal tunings:  
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}
 
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
 
Badness (Sintel): 0.935
 
=== Beetle ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 64/63, 686/675
 
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}
 
Optimal tunings:
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}
 
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
 
Badness (Sintel): 1.92
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 64/63, 91/90, 169/168
 
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
 
Optimal tunings:
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}
 
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
 
Badness (Sintel): 1.40
 
== Progress ==
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
 
Progress tempers out 392/375 and may be described as {{nowrap| 15 & 17c }}. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 32c-edo gives an obvious tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 64/63, 392/375
 
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
 
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}
: [[error map]]: {{val| -4.862 +3.925 +12.908 -9.759 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 638.0791{{c}}
: error map: {{val| 0.000 +12.282 +23.291 +2.700 }}
 
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
 
[[Badness]] (Sintel): 1.68
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 64/63, 77/75
 
Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
 
Optimal tunings:  
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}
 
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}
 
Badness (Sintel): 1.03
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 64/63, 66/65, 77/75
 
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
 
Optimal tunings:
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}
 
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
 
Badness (Sintel): 1.08
 
==== Progressive ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 26/25, 56/55, 64/63, 77/75
 
Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
 
Optimal tunings:  
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}
 
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}
 
Badness (Sintel): 1.35
 
== Fervor ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
 
Fervor tempers out 9704/9375 and may be described as {{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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 64/63, 9604/9375
 
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
 
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}
: [[error map]]: {{val| -3.726 +3.230 +4.980 -1.550 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 622.3179{{c}}
: error map: {{val| 0.000 +9.634 +12.826 +7.996 }}
 
{{Optimal ET sequence|legend=1| 2, 25, 27 }}
 
[[Badness]] (Sintel): 2.74
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 64/63, 1350/1331
 
Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}
 
Optimal tunings:  
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}
 
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}
 
Badness (Sintel): 1.72
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 64/63, 78/77, 507/500
 
Mapping: {{mapping| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}
 
Optimal tunings:
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}
 
{{Optimal ET sequence|legend=0| 2f, 27e }}
 
Badness (Sintel): 1.64
 
== Sixix ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sixix (5-limit)]].''
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}
 
Sixix tempers out 3125/2916 and may be described as {{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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 64/63, 3125/2916
 
{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}
: [[error map]]: {{val| -1.097 +9.086 -13.503 +2.508 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 337.4588{{c}}
: error map: {{val| 0.000 +10.751 -11.066 +5.762 }}
 
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
 
[[Badness]] (Sintel): 4.02
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 64/63, 125/121
 
Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
 
Optimal tunings:  
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}
 
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}
 
Badness (Sintel): 2.34
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 55/54, 64/63, 125/121
 
Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
 
Optimal tunings:  
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}
 
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
 
Badness (Sintel): 1.91
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121
 
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
 
Optimal tunings:
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}
 
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
 
Badness (Sintel): 2.00
 
[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Archytas_clan"