Xenharmonic Wiki

Archytas clan: Difference between revisions

← Older edit
VisualWikitext
Unify to 3 decimal places
m Update
 
(46 intermediate revisions by 7 users not shown)
Line 1: Line 1:
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]].  
{{Technical data page}}
The '''archytas clan''' (or '''archy family''') [[tempering out|tempers out]] the [[64/63|Archytas' comma]], 64/63. This means a stack of two [[3/2]] fifths [[octave reduction|octave-reduced]] equals a whole tone of [[8/7]][[~]][[9/8]] tempered together; two of these tones or equivalently four stacked fifths octave-reduced equal a [[9/7]] major third. Note the similarity in function to [[81/80]] in meantone, where four stacked fifths octave-reduced equal a [[5/4]] major third. This leads to tunings with 3's and 7's quite sharp, such as those of [[22edo]], [[27edo]], or [[49edo]].  


This article focuses on rank-2 temperaments. See [[Archytas family]] for the [[rank-3 temperament]] resulting from tempering out 64/63 alone in the full 7-limit.  
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.  
Line 12: Line 13:
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}


: sval mapping generators: ~2, ~3
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}
 
: mapping generators: ~2, ~3
{{Mapping|legend=3| 1 1 0 4 | 0 1 0 -2 }}
 
: [[gencom]]: [2 3/2; 64/63]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3/2 = 709.595
* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}
* [[POTE]]: ~3/2 = 709.321
: [[error map]]: {{val| -3.045 +2.522 +3.952 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}
: error map: {{val| 0.000 +7.435 +12.394 }}


{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}


[[Badness]]: 0.00464
[[Badness]] (Sintel): 0.159


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


=== Overview to extensions ===
=== Overview to extensions ===
Adding [[245/243]] to the list of commas gives superpyth; [[2430/2401]] gives quasisuper; [[36/35]] gives dominant; 360/343 gives schism; 6860/6561 gives ultrapyth; 33614/32805 gives quasiultra; [[16/15]] gives mother. These all use the same generators as archy.  
==== 7-limit extensions ====
The second comma in the comma list defines which [[7-limit]] family member we are looking at:
* [[#Schism|Schism]] adds [[360/343]], for a tuning around [[12edo]];
* [[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]].  
 
These all use the same generators as archy.  


[[686/675]] gives beatles, splitting the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a semioctave period. [[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, splitting the twelfth in five. [[3125/3087]] gives passion, splitting the fourth in five. [[28/27]] gives blacksmith with a 1/5-octave period. Finally, [[15625/15552]] gives catalan, splitting the twelfth in six.  
[[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.  


Discussed under their respective 5-limit families are:  
Temperaments discussed elsewhere are:  
* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
* ''[[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]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* ''[[Dichotic]]'' (+25/24) → [[Dicot family #Dichotic|Dicot family]]
* ''[[Suhajira]]'' (+243/242) → [[Neutral clan #Suhajira|Neutral clan]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Augene]] (+126/125) → [[Augmented family #Augene|Augmented family]]
* [[Augene]] (+126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* ''[[Modus]]'' (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
* [[Modus]] (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
* ''[[Blacksmith]]'' (+28/27) → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]


The rest are considered below.
Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.
 
==== Subgroup extensions ====
Omitting prime 5, archy can be extended to the 2.3.7.11 subgroup by identifying 11/8 as a diminished fourth (C–G♭). This is called supra, given right below. Discussed elsewhere is [[suhajira]] of the [[rastmic clan #Suhajira|rastmic clan]].


=== Supra ===
=== Supra ===
Line 55: Line 68:
Comma list: 64/63, 99/98
Comma list: 64/63, 99/98


Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
Subgroup-val mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}


Gencom mapping: {{mapping| 1 1 0 4 7 | 0 1 0 -2 -6 }}
Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}
 
: gencom: [2 3/2; 64/63 99/98]


Optimal tunings:  
Optimal tunings:  
* CTE: ~3/2 = 708.456
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
* POTE: ~3/2 = 707.192
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}


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


Badness: 0.00933
Badness (Sintel): 0.352


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]


==== Supraphon ====
==== Supraphon ====
This extension maps [[13/11]] to the minor third (C–E♭), [[12/11]][[~]][[14/13]] to the augmented unison (C–C♯), and [[13/12]] to the diminished third (C–E𝄫).
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


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


Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
Subgroup-val mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}


Gencom mapping: {{mapping| 1 1 0 4 7 9 | 0 1 0 -2 -6 -9 }}
Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
 
: gencom: [2 3/2; 64/63 78/77 99/98]


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 707.344
* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
* POTE: ~2 = 1\1, ~3/2 = 706.137
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, 17 }}
{{Optimal ET sequence|legend=0| 12f, 17 }}


Badness: 0.0111
Badness (Sintel): 0.498


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]
Line 94: Line 105:
== Superpyth ==
== Superpyth ==
{{Main| Superpyth }}
{{Main| Superpyth }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


In the 5-limit, superpyth tempers out 20480/19683. This temperament has a fifth generator of ~3/2 = ~710¢ and ~5/4 is found at +9 generator steps, as an augmented second (C-D#). It also has a weak extension, [[Jubilismic clan #Bipyth|bipyth]] (10cd & 22), tempering out the same 5-limit comma as the superpyth, but with a half-octave period and the jubilisma (50/49) rather than the Archytas comma tempered out.
Superpyth, virtually the canonical extension, adds [[245/243]] and [[1728/1715]] to the comma list and can be described as {{nowrap| [[22edo|22]] [[&]] [[27edo|27]] }}. ~[[5/4]] is found at +9 generator steps, as an augmented second (C–D♯). In the [[11-limit]] it finds the ~[[11/8]] at +16 generator steps, as a double-augmented second (C–D𝄪). [[49edo]] remains an obvious tuning choice in either case.  
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 20480/19683


{{Mapping|legend=1| 1 0 -12 | 0 1 9 }}
Extending superpyth to the [[13-limit]] is more diffcult. Tridecimal superpyth finds the ~[[13/8]] at +13 generator steps, as a double-augmented fourth (C–F𝄪), for which 27edo can be recommended as a tuning since it is the only [[13-odd-limit]] [[diamond monotone]] tuning. The other extension, called uberpyth, is more flexible with its tunings, but unfortunately tends to tune the 13 very sharp.


[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 709.393
* [[POTE]]: ~2 = 1\1, ~3/2 = 710.078
{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
[[Badness]]: 0.135141
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 117: Line 116:


{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}
{{Multival|legend=1| 1 9 -2 12 -6 -30 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 709.591
* [[WE]]: ~2 = 1197.0549{{c}}, ~3/2 = 708.5478{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 710.291
: [[error map]]: {{val| -2.945 +3.648 -0.548 +2.298 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1193{{c}}
: error map: {{val| 0.000 +8.164 +4.760 +10.935 }}


{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}


[[Badness]]: 0.032318
[[Badness]] (Sintel): 0.818


=== 11-limit ===
=== 11-limit ===
The canonical extension to the 13-limit finds the ~11/8 at +16 generator steps, as a double augmented second (C-Dx) and finds the ~13/8 at +13 generator steps, as a double augmented fourth (C-Fx).
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 138: Line 135:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 709.514
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* POTE: ~2 = 1\1, ~3/2 = 710.175
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


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


Badness: 0.024976
Badness (Sintel): 0.826


==== 13-limit ====
==== 13-limit ====
Line 153: Line 150:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 709.836
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* POTE: ~2 = 1\1, ~3/2 = 710.479
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}
 
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
 
Badness (Sintel): 1.02
 
==== Uberpyth ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 64/63, 100/99, 144/143, 245/243
 
Mapping: {{mapping| 1 0 -12 6 -22 26 | 0 1 9 -2 16 -14 }}
 
Optimal tunings:
* WE: ~2 = 1196.6666{{c}}, ~3/2 = 708.3602{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.2878{{c}}


Optimal ET sequence: {{Optimal ET sequence| 22, 27e, 49, 76bcde }}
{{Optimal ET sequence|legend=0| 22f, 27e, 49f, 125bcddeeeff, 174bbcdddeeeeffff }}


Badness: 0.024673
Badness (Sintel): 1.11


==== Thomas ====
==== Thomas ====
Line 168: Line 180:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~16/13 = 354.759
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* POTE: ~2 = 1\1, ~16/13 = 355.036
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


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


Badness: 0.049183
Badness (Sintel): 2.03


=== Suprapyth ===
=== Suprapyth ===
Suprapyth finds the ~11/8 at the diminished fifth (C-Gb), and finds the ~13/8 at the diminished seventh (C-Bbb).  
Suprapyth finds the ~11/8 at the diminished fifth (C–G♭), and finds the ~13/8 at the diminished seventh (C–B𝄫).  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 185: Line 197:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 709.356
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* POTE: ~2 = 1\1, ~3/2 = 709.495
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


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


Badness: 0.032768
Badness (Sintel): 1.08


==== 13-limit ====
==== 13-limit ====
Line 200: Line 212:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 708.702
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* POTE: ~2 = 1\1, ~3/2 = 708.703
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}


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


Badness: 0.036336
Badness (Sintel): 1.50


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 215: Line 229:


{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}
{{Multival|legend=1| 1 -13 -2 -23 -6 32 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 708.769
* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 708.328
: [[error map]]: {{val| -3.017 +1.486 -0.435 +6.190 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 708.3716{{c}}
: error map: {{val| 0.000 +6.417 +4.855 +14.431 }}


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


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


=== Quasisupra ===
=== Quasisupra ===
Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament [[supra]], with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 236: Line 248:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 708.713
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* POTE: ~2 = 1\1, ~3/2 = 708.205
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


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


Badness: 0.032203
Badness (Sintel): 1.06


==== 13-limit ====
==== 13-limit ====
Line 251: Line 263:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 708.411
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* POTE: ~2 = 1\1, ~3/2 = 708.004
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


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


Badness: 0.030219
Badness (Sintel): 1.25


=== Quasisoup ===
=== Quasisoup ===
Line 266: Line 278:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 709.043
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* POTE: ~2 = 1\1, ~3/2 = 709.021
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


Optimal ET sequence: {{Optimal ET sequence| 22 }}
{{Optimal ET sequence|legend=0| 22 }}


Badness: 0.083490
Badness (Sintel): 2.76


== Ultrapyth ==
== Ultrapyth ==
Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[The Biosphere #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double augmented unison (C-Cx).
{{Main| Ultrapyth }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Ultrapyth (5-limit)]].''
 
Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[the Biosphere #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 281: Line 296:


{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
{{Multival|legend=1| 1 14 -2 20 -6 -44 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 713.218
* [[WE]]: ~2 = 1197.2673{{c}}, ~3/2 = 712.0258{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 713.651
: [[error map]]: {{val| -2.733 +7.338 -1.557 -3.808 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5430{{c}}
: error map: {{val| 0.000 +11.588 +3.288 +4.088 }}


{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}


[[Badness]]: 0.108466
[[Badness]] (Sintel): 2.74


=== 11-limit ===
=== 11-limit ===
Line 300: Line 315:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 713.282
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* POTE: ~2 = 1\1, ~3/2 = 713.395
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


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


Badness: 0.068238
Badness (Sintel): 2.26


==== 13-limit ====
==== 13-limit ====
Line 315: Line 330:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 713.309
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* POTE: ~2 = 1\1, ~3/2 = 713.500
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


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


Badness: 0.049172
Badness (Sintel): 2.03


=== Ultramarine ===
=== Ultramarine ===
Line 330: Line 345:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 713.395
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* POTE: ~2 = 1\1, ~3/2 = 713.791
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


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


Badness: 0.078068
Badness (Sintel): 2.58


==== 13-limit ====
==== 13-limit ====
Line 345: Line 360:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 713.708
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* POTE: ~2 = 1\1, ~3/2 = 713.811
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}


Optimal ET sequence: {{Optimal ET sequence| 5e, 32e, 37, 79bcef }}
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}


Badness: 0.045653
Badness (Sintel): 1.89


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 360: Line 375:


{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
{{Multival|legend=1| 1 -18 -2 -31 -6 46 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 711.838
* [[WE]]: ~2 = 1196.9257{{c}}, ~3/2 = 709.6211{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 711.5429
: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5429{{c}}
: error map: {{val| 0.000 +9.588 +5.914 +8.088 }}


{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}


[[Badness]]: 0.132
[[Badness]] (Sintel): 3.34


== Schism ==
== Schism ==
{{See also| Schismatic family #Schism }}
{{See also| Schismatic family #Schism }}


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 383: Line 398:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.270
* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 701.556
: [[error map]]: {{val| -2.640 -4.583 -4.896 +20.588 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7376{{c}}
{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}


{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}


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


=== 11-limit ===
=== 11-limit ===
Line 400: Line 415:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 703.383
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* POTE: ~2 = 1\1, ~3/2 = 702.136
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


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


Badness: 0.037482
Badness (Sintel): 1.24


== Beatles ==
== Beatles ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Beatles]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''
 
Beatles tempers out 686/675, which may also be characterized by saying it tempers out [[2401/2400]]. It may be described as the {{nowrap| 10 & 17c }} temperament. It splits the fifth into two neutral-third generators of 49/40~60/49; its [[ploidacot]] is dicot. 5/4 may be found at -9 generator steps, as a semidiminished fourth (C–Fd). 27edo is an obvious tuning, though 17c-edo and 37edo are among the possibilities.
 
Beatles extends easily to the no-11 13-limit, as the generator can be interpreted as ~16/13, tempering out 91/90, 169/168, and 196/195.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 415: Line 434:


{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~49/40 = 356.634
* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}
* [[POTE]]: ~2 = 1\1, ~49/40 = 355.904
: [[error map]]: {{val| -3.376 +4.475 +2.682 -1.940 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}
: error map: {{val| 0.000 +10.209 +8.949 +6.847 }}


{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}


[[Badness]]: 0.045872
[[Badness]] (Sintel): 1.16


; Music
; Music
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 ''Beatles Improv''] by [[Herman Miller]]
* [https://web.archive.org/web/20201127013829/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 ''Beatles Improv''] by [[Herman Miller]]


=== 11-limit ===
=== 11-limit ===
Line 437: Line 456:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~49/40 = 356.719
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* POTE: ~2 = 1\1, ~49/40 = 356.140
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


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


Badness: 0.045639
Badness (Sintel): 1.51


==== 13-limit ====
==== 13-limit ====
Line 452: Line 471:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~16/13 = 356.722
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* POTE: ~2 = 1\1, ~16/13 = 356.229
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


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


Badness: 0.030161
Badness (Sintel): 1.25


=== Ringo ===
=== Ringo ===
Line 467: Line 486:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 355.992
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* POTE: ~2 = 1\1, ~11/9 = 355.419
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


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


Badness: 0.032863
Badness (Sintel): 1.09


==== 13-limit ====
==== 13-limit ====
Line 482: Line 501:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 356.004
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* POTE: ~2 = 1\1, ~11/9 = 355.456
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}


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


Badness: 0.022619
Badness (Sintel): 0.935


=== Beetle ===
=== Beetle ===
Line 494: Line 513:
Comma list: 55/54, 64/63, 686/675
Comma list: 55/54, 64/63, 686/675


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~49/40 = 356.694
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* POTE: ~2 = 1\1, ~49/40 = 356.710
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


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


Badness: 0.058084
Badness (Sintel): 1.92


==== 13-limit ====
==== 13-limit ====
Line 512: Line 531:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~16/13 = 356.700
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* POTE: ~2 = 1\1, ~16/13 = 356.701
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


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


Badness: 0.033971
Badness (Sintel): 1.40


== Progress ==
== Progress ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #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
[[Subgroup]]: 2.3.5.7
Line 529: Line 551:


: mapping generators: ~2, ~10/7
: mapping generators: ~2, ~10/7
{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~10/7 = 638.782
* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}
* [[POTE]]: ~2 = 1\1, ~10/7 = 637.878
: [[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 }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


[[Badness]]: 0.066400
[[Badness]] (Sintel): 1.68


=== 11-limit ===
=== 11-limit ===
Line 548: Line 570:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/7 = 638.846
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
* POTE: ~2 = 1\1, ~10/7 = 637.915
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}


Optimal ET sequence: {{Optimal ET sequence| 2, 13, 15, 32c, 47bc }}
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}


Badness: 0.031036
Badness (Sintel): 1.03


==== 13-limit ====
==== 13-limit ====
Line 563: Line 585:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/7 = 637.871
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
* POTE: ~2 = 1\1, ~10/7 = 637.635
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}


Optimal ET sequence: {{Optimal ET sequence| 15, 17c, 32cf }}
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}


Badness: 0.026214
Badness (Sintel): 1.08


==== Progressive ====
==== Progressive ====
Line 578: Line 600:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/7 = 637.797
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
* POTE: ~2 = 1\1, ~10/7 = 636.761
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}


Optimal ET sequence: {{Optimal ET sequence| 2f, 15f, 17c }}
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}


Badness: 0.032721
Badness (Sintel): 1.35


== Fervor ==
== Fervor ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #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
[[Subgroup]]: 2.3.5.7
Line 592: Line 616:
[[Comma list]]: 64/63, 9604/9375
[[Comma list]]: 64/63, 9604/9375


{{Mapping|legend=1| 1 4 -2 -2 | 0 -5 9 10 }}
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~7/5 = 577.353
* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}
* [[POTE]]: ~2 = 1\1, ~7/5 = 577.776
: [[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 }}
{{Optimal ET sequence|legend=1| 2, 25, 27 }}


[[Badness]]: 0.108455
[[Badness]] (Sintel): 2.74


=== 11-limit ===
=== 11-limit ===
Line 611: Line 635:
Comma list: 56/55, 64/63, 1350/1331
Comma list: 56/55, 64/63, 1350/1331


Mapping: {{mapping| 1 4 -2 -2 3 | 0 -5 9 10 1 }}
Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/5 = 577.296
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* POTE: ~2 = 1\1, ~7/5 = 577.850
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}


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


Badness: 0.052054
Badness (Sintel): 1.72


=== 13-limit ===
=== 13-limit ===
Line 626: Line 650:
Comma list: 56/55, 64/63, 78/77, 507/500
Comma list: 56/55, 64/63, 78/77, 507/500


Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}
Mapping: {{mapping| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/5 = 577.374
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
* POTE: ~2 = 1\1, ~7/5 = 578.060
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}


Optimal ET sequence: {{Optimal ET sequence| 2f, 27e }}
{{Optimal ET sequence|legend=0| 2f, 27e }}


Badness: 0.039705
Badness (Sintel): 1.64


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


Sixix is related to the [[kleismic]] family in a way similar to the one between [[meantone]] and [[mavila]]. In both cases the generator is nominally a 6/5 and the complexity to generate major and minor chords is the same, but in sixix it is tuned extremely sharply, to the point where the 3rd and 5th harmonics are reached by going down instead of up, inverting the logic of chord construction.
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
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3125/2916
 
{{Mapping|legend=1| 1 3 4 | 0 -5 -6 }}
 
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~6/5 = 338.005
* [[POTE]]: ~2 = 1\1, ~6/5 = 338.365
 
{{Optimal ET sequence|legend=1| 7, 25, 32, 39c }}
 
[[Badness]]: 0.153088
 
=== 7-limit ===
Subgroup: 2.3.5.7


[[Comma list]]: 64/63, 3125/2916
[[Comma list]]: 64/63, 3125/2916


{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~6/5 = 337.519
* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}
* [[POTE]]: ~2 = 1\1, ~6/5 = 337.442
: [[error map]]: {{val| -1.097 +9.086 -13.503 +2.508 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 337.4588{{c}}
: error map: {{val| 0.000 +10.751 -11.066 +5.762 }}


{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}


[[Badness]]: 0.158903
[[Badness]] (Sintel): 4.02


=== 11-limit ===
=== 11-limit ===
Line 682: Line 690:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 337.749
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
* POTE: ~2 = 1\1, ~6/5 = 337.564
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32 }}
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}


Badness: 0.070799
Badness (Sintel): 2.34


=== 13-limit ===
=== 13-limit ===
Line 697: Line 705:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 337.793
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
* POTE: ~2 = 1\1, ~6/5 = 337.483
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


Badness: 0.046206
Badness (Sintel): 1.91


=== 17-limit ===
=== 17-limit ===
Line 712: Line 720:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 337.629
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
* POTE: ~2 = 1\1, ~6/5 = 337.513
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


Badness: 0.039224
Badness (Sintel): 2.00


[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Archytas_clan"