Archytas clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
m Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"
→Archy: + dichotic
 
(35 intermediate revisions by 6 users not shown)
Line 1: Line 1:
{{Technical data page}}
{{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]].  
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 13: 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|legend=3| 1 1 0 4 | 0 1 0 -2 }}
: mapping generators: ~2, ~3
 
: [[gencom]]: [2 3/2; 64/63]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.595
* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}
: [[error map]]: {{val| 0.000 +7.640 +11.984 }}
: [[error map]]: {{val| -3.045 +2.522 +3.952 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 709.321
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}
: error map: {{val| 0.000 +7.366 +12.532 }}
: 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]] (Smith): 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]].  


[[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.  
These all use the same generators as archy.
 
[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a 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.  


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
Line 40: Line 48:
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone 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]]
* ''[[Suhajira]]'' (+243/242) → [[Neutral clan #Suhajira|Neutral clan]]
* ''[[Dichotic]]'' (+25/24) → [[Dicot family #Dichotic|Dicot family]]
* ''[[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) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]


Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.  
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 58: Line 69:
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: ~2 = 1200.000, ~3/2 = 708.456
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 707.192
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}


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


Badness (Smith): 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 = 1200.000, ~3/2 = 707.344
* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 706.137
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}


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


Badness (Smith): 0.0111
Badness (Sintel): 0.498


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]
Line 99: Line 108:
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


Superpyth adds [[245/243]] and [[1728/1715]] to the comma list and can be described as 22 & 27. ~5/4 is found at +9 generator steps, as an augmented second (C–D#).  
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.
 
Extending superpyth to the [[13-limit]] is more diffcult. Tridecimal superpyth finds the ~[[13/8]] at +13 generator steps, as a double-augmented fourth (C–F𝄪), for which 27edo can be recommended as a tuning since it is the only [[13-odd-limit]] [[diamond monotone]] tuning. The other extension, called uberpyth, is more flexible with its tunings, but unfortunately tends to tune the 13 very sharp.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 106: Line 117:


{{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 = 1200.000, ~3/2 = 709.591
* [[WE]]: ~2 = 1197.0549{{c}}, ~3/2 = 708.5478{{c}}
: [[error map]]: {{val| 0.000 +7.636 +0.002 +11.993 }}
: [[error map]]: {{val| -2.945 +3.648 -0.548 +2.298 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 710.291
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1193{{c}}
: error map: {{val| 0.000 +8.336 +6.305 +10.592 }}
: 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]] (Smith): 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 129: Line 136:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 709.514
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 710.175
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


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


Badness (Smith): 0.024976
Badness (Sintel): 0.826


==== 13-limit ====
==== 13-limit ====
Line 144: Line 151:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 709.836
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 710.479
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}


{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}


Badness (Smith): 0.024673
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|legend=0| 22f, 27e, 49f, 125bcddeeeff, 174bbcdddeeeeffff }}
 
Badness (Sintel): 1.11


==== Thomas ====
==== Thomas ====
Line 159: Line 181:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~16/13 = 354.759
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* POTE: ~2 = 1200.000, ~16/13 = 355.036
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


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


Badness (Smith): 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 176: Line 198:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 709.356
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 709.495
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


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


Badness (Smith): 0.032768
Badness (Sintel): 1.08


==== 13-limit ====
==== 13-limit ====
Line 191: Line 213:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 708.702
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 708.703
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}


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


Badness (Smith): 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 206: Line 230:


{{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 = 1200.000, ~3/2 = 708.769
* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}
: [[error map]]: {{val| 0.000 +6.814 -0.310 +13.636 }}
: [[error map]]: {{val| -3.017 +1.486 -0.435 +6.190 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 708.238
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 708.3716{{c}}
: error map: {{val| 0.000 +6.283 +6.586 +14.697 }}
: 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]] (Smith): 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 229: Line 249:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 708.713
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 708.205
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


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


Badness (Smith): 0.032203
Badness (Sintel): 1.06


==== 13-limit ====
==== 13-limit ====
Line 244: Line 264:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 708.411
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 708.004
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


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


Badness (Smith): 0.030219
Badness (Sintel): 1.25


=== Quasisoup ===
=== Quasisoup ===
Line 259: Line 279:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 709.043
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 709.021
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


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


Badness (Smith): 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 274: Line 297:


{{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 = 1200.000, ~3/2 = 713.218
* [[WE]]: ~2 = 1197.2673{{c}}, ~3/2 = 712.0258{{c}}
: [[error map]]: {{val| 0.000 +11.263 -1.264 +4.738 }}
: [[error map]]: {{val| -2.733 +7.338 -1.557 -3.808 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 713.651
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5430{{c}}
: error map: {{val| 0.000 +11.696 +4.800 +3.872 }}
: 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]] (Smith): 0.108466
[[Badness]] (Sintel): 2.74


=== 11-limit ===
=== 11-limit ===
Line 295: Line 316:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 713.282
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 713.395
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


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


Badness (Smith): 0.068238
Badness (Sintel): 2.26


==== 13-limit ====
==== 13-limit ====
Line 310: Line 331:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 713.309
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 713.500
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


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


Badness (Smith): 0.049172
Badness (Sintel): 2.03


=== Ultramarine ===
=== Ultramarine ===
Line 325: Line 346:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 713.395
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 713.791
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


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


Badness (Smith): 0.078068
Badness (Sintel): 2.58


==== 13-limit ====
==== 13-limit ====
Line 340: Line 361:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 713.708
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 713.811
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}


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


Badness (Smith): 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 355: Line 376:


{{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 = 1200.000, ~3/2 = 711.838
* [[WE]]: ~2 = 1196.9257{{c}}, ~3/2 = 709.6211{{c}}
: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}
: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 711.543
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5429{{c}}
: error map: {{val| 0.000 +9.588 +5.914 +8.088 }}
: 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]] (Smith): 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 380: Line 399:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 702.270
* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}
: [[error map]]: {{val| 0.000 +0.315 -4.471 +26.635 }}
: [[error map]]: {{val| -2.640 -4.583 -4.896 +20.588 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 701.556
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7376{{c}}
: error map: {{val| 0.000 -0.399 +1.237 +28.062 }}
: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}
 
{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 399: Line 416:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 703.383
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 702.136
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


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


Badness (Smith): 0.037482
Badness (Sintel): 1.24


== Beatles ==
== Beatles ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit 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 414: Line 435:


{{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 = 1200.000, ~49/40 = 356.634
* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}
: [[error map]]: {{val| 0.000 +11.312 +3.984 +4.640 }}
: [[error map]]: {{val| -3.376 +4.475 +2.682 -1.940 }}
* [[POTE]]: ~2 = 1200.000, ~49/40 = 355.904
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}
: error map: {{val| 0.000 +9.853 +10.549 +7.558 }}
: 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]] (Smith): 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 438: Line 457:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~49/40 = 356.719
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* POTE: ~2 = 1200.000, ~49/40 = 356.140
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


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


Badness (Smith): 0.045639
Badness (Sintel): 1.51


==== 13-limit ====
==== 13-limit ====
Line 453: Line 472:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~16/13 = 356.722
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* POTE: ~2 = 1200.000, ~16/13 = 356.229
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


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


Badness (Smith): 0.030161
Badness (Sintel): 1.25


=== Ringo ===
=== Ringo ===
Line 468: Line 487:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 355.992
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* POTE: ~2 = 1200.000, ~11/9 = 355.419
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


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


Badness (Smith): 0.032863
Badness (Sintel): 1.09


==== 13-limit ====
==== 13-limit ====
Line 483: Line 502:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 356.004
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* POTE: ~2 = 1200.000, ~11/9 = 355.456
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}


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


Badness (Smith): 0.022619
Badness (Sintel): 0.935


=== Beetle ===
=== Beetle ===
Line 495: Line 514:
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 = 1200.000, ~49/40 = 356.694
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* POTE: ~2 = 1200.000, ~49/40 = 356.710
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


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


Badness (Smith): 0.058084
Badness (Sintel): 1.92


==== 13-limit ====
==== 13-limit ====
Line 513: Line 532:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~16/13 = 356.700
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* POTE: ~2 = 1200.000, ~16/13 = 356.701
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


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


Badness (Smith): 0.033971
Badness (Sintel): 1.40


== Progress ==
== Progress ==
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
: ''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 530: Line 552:


: 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 = 1200.000, ~10/7 = 638.782
* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}
: [[error map]]: {{val| 0.000 +14.391 +19.777 -1.518 }}
: [[error map]]: {{val| -4.862 +3.925 +12.908 -9.759 }}
* [[POTE]]: ~2 = 1200.000, ~10/7 = 637.878
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 638.0791{{c}}
: error map: {{val| 0.000 +11.679 +24.297 +3.907 }}
: 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]] (Smith): 0.066400
[[Badness]] (Sintel): 1.68


=== 11-limit ===
=== 11-limit ===
Line 551: Line 571:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 638.846
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 637.915
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}


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


Badness (Smith): 0.031036
Badness (Sintel): 1.03


==== 13-limit ====
==== 13-limit ====
Line 566: Line 586:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 637.871
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 637.635
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}


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


Badness (Smith): 0.026214
Badness (Sintel): 1.08


==== Progressive ====
==== Progressive ====
Line 581: Line 601:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 637.797
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 636.761
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}


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


Badness (Smith): 0.032721
Badness (Sintel): 1.35


== Fervor ==
== Fervor ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit 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 595: Line 617:
[[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 = 1200.000, ~7/5 = 577.353
* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}
: [[error map]]: {{val| 0.000 +11.278 +9.867 +4.709 }}
: [[error map]]: {{val| -3.726 +3.230 +4.980 -1.550 }}
* [[POTE]]: ~2 = 1200.000, ~7/5 = 577.776
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 622.3179{{c}}
: error map: {{val| 0.000 +9.163 +13.673 +8.937 }}
: 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]] (Smith): 0.108455
[[Badness]] (Sintel): 2.74


=== 11-limit ===
=== 11-limit ===
Line 616: Line 636:
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 = 1200.000, ~7/5 = 577.296
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* POTE: ~2 = 1200.000, ~7/5 = 577.850
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}


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


Badness (Smith): 0.052054
Badness (Sintel): 1.72


=== 13-limit ===
=== 13-limit ===
Line 631: Line 651:
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 = 1200.000, ~7/5 = 577.374
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
* POTE: ~2 = 1200.000, ~7/5 = 578.060
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}


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


Badness (Smith): 0.039705
Badness (Sintel): 1.64


== Sixix ==
== Sixix ==
Line 645: Line 665:
{{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]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 652: Line 672:


{{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 = 1200.000, ~6/5 = 337.519
* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}
: [[error map]]: {{val| 0.000 +10.449 -11.429 +6.366 }}
: [[error map]]: {{val| -1.097 +9.086 -13.503 +2.508 }}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 337.442
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 337.4588{{c}}
: error map: {{val| 0.000 +10.835 -10.965 +5.594 }}
: 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]] (Smith): 0.158903
[[Badness]] (Sintel): 4.02


=== 11-limit ===
=== 11-limit ===
Line 673: Line 691:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 337.749
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 337.564
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}


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


Badness (Smith): 0.070799
Badness (Sintel): 2.34


=== 13-limit ===
=== 13-limit ===
Line 688: Line 706:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 337.793
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 337.483
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}


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


Badness (Smith): 0.046206
Badness (Sintel): 1.91


=== 17-limit ===
=== 17-limit ===
Line 703: Line 721:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 337.629
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 337.513
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}


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


Badness (Smith): 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]]

Latest revision as of 10:06, 29 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The archytas clan (or archy family) tempers out the Archytas' comma, 64/63. This means a stack of two 3/2 fifths octave-reduced equals a whole tone of 8/7~9/8 tempered together; two of these tones or equivalently four stacked fifths octave-reduced equal a 9/7 major third. Note the similarity in function to 81/80 in meantone, where four stacked fifths octave-reduced equal a 5/4 major third. This leads to tunings with 3's and 7's quite sharp, such as those of 22edo, 27edo, or 49edo.

This article focuses on rank-2 temperaments. See Archytas family for the rank-3 temperament resulting from tempering out 64/63 alone in the full 7-limit.

Archy

Main article: Superpyth

Subgroup: 2.3.7

Comma list: 64/63

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

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

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1196.9552 ¢, ~3/2 = 707.5215 ¢
error map: ⟨-3.045 +2.522 +3.952]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.3901 ¢
error map: ⟨0.000 +7.435 +12.394]

Optimal ET sequence: 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd

Badness (Sintel): 0.159

Scales: archy5, archy7, archy12

Overview to extensions

7-limit extensions

The second comma in the comma list defines which 7-limit family member we are looking at:

These all use the same generators as archy.

25/24 gives dichotic. 686/675 gives beatles. Those split the fifth in two. 8748/8575 gives immunized, splitting the twelfth in two. 50/49 gives pajara with a 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.

Temperaments discussed elsewhere are:

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.

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

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

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

Optimal tunings:

  • WE: ~2 = 1197.2650 ¢, ~3/2 = 705.5803 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 707.4981 ¢

Optimal ET sequence: 5, 12, 17, 39d, 56d

Badness (Sintel): 0.352

Scales: supra7, supra12

Supraphon

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

Subgroup: 2.3.7.11.13

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

Subgroup-val mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]

Gencom mapping: [⟨1 0 0 6 13 18], ⟨0 1 0 -2 -6 -9]]

Optimal tunings:

  • WE: ~2 = 1197.1909 ¢, ~3/2 = 704.4836 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 706.4289 ¢

Optimal ET sequence: 12f, 17

Badness (Sintel): 0.498

Scales: supra7, supra12

Superpyth

Main article: Superpyth
For the 5-limit version, see Syntonic–diatonic equivalence continuum #Superpyth (5-limit).

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

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

Subgroup: 2.3.5.7

Comma list: 64/63, 245/243

Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]

Optimal tunings:

  • WE: ~2 = 1197.0549 ¢, ~3/2 = 708.5478 ¢
error map: ⟨-2.945 +3.648 -0.548 +2.298]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.1193 ¢
error map: ⟨0.000 +8.164 +4.760 +10.935]

Optimal ET sequence: 5, 17, 22, 27, 49, 174bbcddd

Badness (Sintel): 0.818

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1197.0673 ¢, ~3/2 = 708.4391 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.0129 ¢

Optimal ET sequence: 22, 27e, 49

Badness (Sintel): 0.826

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 100/99

Mapping: [⟨1 0 -12 6 -22 -17], ⟨0 1 9 -2 16 13]]

Optimal tunings:

  • WE: ~2 = 1197.3011 ¢, ~3/2 = 708.8813 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.3219 ¢

Optimal ET sequence: 22, 27e, 49, 76bcde

Badness (Sintel): 1.02

Uberpyth

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.11

Thomas

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 169/168, 245/243

Mapping: [⟨1 1 -3 4 -6 4], ⟨0 2 18 -4 32 -1]]

Optimal tunings:

  • WE: ~2 = 1197.4942 ¢, ~16/13 = 354.2950 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/13 = 354.9824 ¢

Optimal ET sequence: 27e, 44, 71d, 98bde

Badness (Sintel): 2.03

Suprapyth

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

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -12 6 13], ⟨0 1 9 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1198.6960 ¢, ~3/2 = 708.7235 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.4699 ¢

Optimal ET sequence: 5, 17, 22

Badness (Sintel): 1.08

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -12 6 13 18], ⟨0 1 9 -2 -6 -9]]

Optimal tunings:

  • WE: ~2 = 1199.9871 ¢, ~3/2 = 708.6952 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.7028 ¢

Optimal ET sequence: 5f, 17, 22

Badness (Sintel): 1.50

Quasisuper

Main article: Quasisuper

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

Subgroup: 2.3.5.7

Comma list: 64/63, 2430/2401

Mapping: [⟨1 0 23 6], ⟨0 1 -13 -2]]

Optimal tunings:

  • WE: ~2 = 1196.9830 ¢, ~3/2 = 706.4578 ¢
error map: ⟨-3.017 +1.486 -0.435 +6.190]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.3716 ¢
error map: ⟨0.000 +6.417 +4.855 +14.431]

Optimal ET sequence: 17c, 22, 61d

Badness (Sintel): 1.61

Quasisupra

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 23 6 13], ⟨0 1 -13 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1197.5675 ¢, ~3/2 = 706.7690 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.3200 ¢

Optimal ET sequence: 17c, 22, 39d, 61d

Badness (Sintel): 1.06

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 121/120

Mapping: [⟨1 0 23 6 13 18], ⟨0 1 -13 -2 -6 -9]]

Optimal tunings:

  • WE: ~2 = 1198.2543 ¢, ~3/2 = 706.9736 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.0936 ¢

Optimal ET sequence: 17c, 22, 39d

Badness (Sintel): 1.25

Quasisoup

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2430/2401

Mapping: [⟨1 0 23 6 -22], ⟨0 1 -13 -2 16]]

Optimal tunings:

  • WE: ~2 = 1198.8446 ¢, ~3/2 = 708.3388 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.0252 ¢

Optimal ET sequence: 22

Badness (Sintel): 2.76

Ultrapyth

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

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

Subgroup: 2.3.5.7

Comma list: 64/63, 6860/6561

Mapping: [⟨1 0 -20 6], ⟨0 1 14 -2]]

Optimal tunings:

  • WE: ~2 = 1197.2673 ¢, ~3/2 = 712.0258 ¢
error map: ⟨-2.733 +7.338 -1.557 -3.808]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.5430 ¢
error map: ⟨0.000 +11.588 +3.288 +4.088]

Optimal ET sequence: 5, 27c, 32, 37

Badness (Sintel): 2.74

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2401/2376

Mapping: [⟨1 0 -20 6 21], ⟨0 1 14 -2 -11]]

Optimal tunings:

  • WE: ~2 = 1198.0290 ¢, ~3/2 = 712.2235 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.3754 ¢

Optimal ET sequence: 5, 32, 37

Badness (Sintel): 2.26

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 1573/1568

Mapping: [⟨1 0 -20 6 21 -25], ⟨0 1 14 -2 -11 18]]

Optimal tunings:

  • WE: ~2 = 1198.1911 ¢, ~3/2 = 712.4243 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.4684 ¢

Optimal ET sequence: 5, 32, 37

Badness (Sintel): 2.03

Ultramarine

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 3773/3645

Mapping: [⟨1 0 -20 6 -38], ⟨0 1 14 -2 26]]

Optimal tunings:

  • WE: ~2 = 1197.2230 ¢, ~3/2 = 712.1393 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.6928 ¢

Optimal ET sequence: 5e, 32e, 37, 79bce

Badness (Sintel): 2.58

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 91/90, 100/99, 847/845

Mapping: [⟨1 0 -20 6 -38 -25], ⟨0 1 14 -2 26 18]]

Optimal tunings:

  • WE: ~2 = 1197.2739 ¢, ~3/2 = 712.1893 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.7079 ¢

Optimal ET sequence: 5e, 32e, 37, 79bcef

Badness (Sintel): 1.89

Quasiultra

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

Subgroup: 2.3.5.7

Comma list: 64/63, 33614/32805

Mapping: [⟨1 0 31 6], ⟨0 1 -18 -2]]

Optimal tunings:

  • WE: ~2 = 1196.9257 ¢, ~3/2 = 709.6211 ¢
error map: ⟨0.000 +9.883 +0.608 +7.499]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.5429 ¢
error map: ⟨0.000 +9.588 +5.914 +8.088]

Optimal ET sequence: 27, 86bd, 113bcd, 140bbcd

Badness (Sintel): 3.34

Schism

See also: Schismatic family § Schism

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

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]

Optimal tunings:

  • WE: ~2 = 1197.3598 ¢, ~3/2 = 700.0126 ¢
error map: ⟨-2.640 -4.583 -4.896 +20.588]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7376 ¢
error map: ⟨0.000 -0.217 -0.214 +27.699]

Optimal ET sequence: 5c, 7c, 12

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 64/63, 99/98

Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1196.1607 ¢, ~3/2 = 699.8897 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.4385 ¢

Optimal ET sequence: 5c, 7ce, 12, 29de

Badness (Sintel): 1.24

Beatles

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

Beatles tempers out 686/675, which may also be characterized by saying it tempers out 2401/2400. It may be described as the 10 & 17c temperament. It splits the fifth into two neutral-third generators of 49/40~60/49; its ploidacot is dicot. 5/4 may be found at -9 generator steps, as a semidiminished fourth (C–Fd). 27edo is an obvious tuning, though 17c-edo and 37edo are among the possibilities.

Beatles extends easily to the no-11 13-limit, as the generator can be interpreted as ~16/13, tempering out 91/90, 169/168, and 196/195.

Subgroup: 2.3.5.7

Comma list: 64/63, 686/675

Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4]]

Optimal tunings:

  • WE: ~2 = 1196.6244 ¢, ~49/40 = 354.9029 ¢
error map: ⟨-3.376 +4.475 +2.682 -1.940]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.0819 ¢
error map: ⟨0.000 +10.209 +8.949 +6.847]

Optimal ET sequence: 10, 17c, 27, 64b, 91bcd, 118bccd

Badness (Sintel): 1.16

Music

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 686/675

Mapping: [⟨1 1 5 4 10], ⟨0 2 -9 -4 -22]]

Optimal tunings:

  • WE: ~2 = 1196.7001 ¢, ~49/40 = 355.1606 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.2795 ¢

Optimal ET sequence: 10e, 17cee, 27e, 64be, 91bcdee

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 91/90, 100/99, 169/168

Mapping: [⟨1 1 5 4 10 4], ⟨0 2 -9 -4 -22 -1]]

Optimal tunings:

  • WE: ~2 = 1197.2504 ¢, ~16/13 = 355.4132 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/13 = 356.3273 ¢

Optimal ET sequence: 10e, 27e, 37, 64be

Badness (Sintel): 1.25

Ringo

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 540/539

Mapping: [⟨1 1 5 4 2], ⟨0 2 -9 -4 5]]

Optimal tunings:

  • WE: ~2 = 1195.4102 ¢, ~11/9 = 354.0597 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 355.5207 ¢

Optimal ET sequence: 10, 17c, 27e

Badness (Sintel): 1.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 91/90

Mapping: [⟨1 1 5 4 2 4], ⟨0 2 -9 -4 5 -1]]

Optimal tunings:

  • WE: ~2 = 1195.9943 ¢, ~11/9 = 354.2695 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 355.5398 ¢

Optimal ET sequence: 10, 17c, 27e

Badness (Sintel): 0.935

Beetle

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 686/675

Mapping: [⟨1 1 5 4 -1], ⟨0 2 -9 -4 15]]

Optimal tunings:

  • WE: ~2 = 1197.9660 ¢, ~49/40 = 356.1056 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.7075 ¢

Optimal ET sequence: 10, 27, 37

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 169/168

Mapping: [⟨1 1 5 4 -1 4], ⟨0 2 -9 -4 15 -1]]

Optimal tunings:

  • WE: ~2 = 1198.1741 ¢, ~16/13 = 356.1582 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/13 = 356.7008 ¢

Optimal ET sequence: 10, 27, 37

Badness (Sintel): 1.40

Progress

Not to be confused with Progression.
For the 5-limit version, see Miscellaneous 5-limit temperaments #Progress.

Progress tempers out 392/375 and may be described as 15 & 17c. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 32c-edo gives an obvious tuning.

Subgroup: 2.3.5.7

Comma list: 64/63, 392/375

Mapping: [⟨1 0 5 6], ⟨0 3 -5 -6]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1195.1377 ¢, ~10/7 = 635.2932 ¢
error map: ⟨-4.862 +3.925 +12.908 -9.759]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 638.0791 ¢
error map: ⟨0.000 +12.282 +23.291 +2.700]

Optimal ET sequence: 2, 13, 15, 32c

Badness (Sintel): 1.68

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 77/75

Mapping: [⟨1 0 5 6 4], ⟨0 3 -5 -6 -1]]

Optimal tunings:

  • WE: ~2 = 1195.4920 ¢, ~10/7 = 635.5183 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 638.0884 ¢

Optimal ET sequence: 2, 13, 15, 32c, 47bc

Badness (Sintel): 1.03

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 66/65, 77/75

Mapping: [⟨1 0 5 6 4 0], ⟨0 3 -5 -6 -1 7]]

Optimal tunings:

  • WE: ~2 = 1195.0786 ¢, ~10/7 = 635.0197 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 637.6691 ¢

Optimal ET sequence: 15, 17c, 32cf

Badness (Sintel): 1.08

Progressive

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 64/63, 77/75

Mapping: [⟨1 0 5 6 4 9], ⟨0 3 -5 -6 -1 -10]]

Optimal tunings:

  • WE: ~2 = 1196.0245 ¢, ~10/7 = 634.6516 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 636.9528 ¢

Optimal ET sequence: 2f, 15f, 17c

Badness (Sintel): 1.35

Fervor

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

Fervor tempers out 9704/9375 and may be described as 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: [⟨1 -1 7 8], ⟨0 5 -9 -10]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1196.2742 ¢, ~10/7 = 620.2918 ¢
error map: ⟨-3.726 +3.230 +4.980 -1.550]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.3179 ¢
error map: ⟨0.000 +9.634 +12.826 +7.996]

Optimal ET sequence: 2, 25, 27

Badness (Sintel): 2.74

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 1350/1331

Mapping: [⟨1 -1 7 8 4], ⟨0 5 -9 -10 -1]]

Optimal tunings:

  • WE: ~2 = 1195.4148 ¢, ~10/7 = 619.7729 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.2525 ¢

Optimal ET sequence: 2, 25e, 27e

Badness (Sintel): 1.72

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 507/500

Mapping: [⟨1 -1 7 8 4 12], ⟨0 5 -9 -10 -1 -16]]

Optimal tunings:

  • WE: ~2 = 1195.6284 ¢, ~10/7 = 619.6738 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.0631 ¢

Optimal ET sequence: 2f, 27e

Badness (Sintel): 1.64

Sixix

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Sixix (5-limit).
See also: Dual-fifth temperaments § Dual-3 Sixix

Sixix tempers out 3125/2916 and may be described as 25 & 32. It is related to the kleismic family in a way similar to the one between meantone and mavila. In both cases the generator is nominally a 6/5 and the complexity to generate major and minor chords is the same, but in sixix it is tuned extremely sharply, to the point where the 3rd and 5th harmonics are reached by going down instead of up, inverting the logic of chord construction. Its ploidacot is gamma-pentacot.

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/2916

Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]

Optimal tunings:

  • WE: ~2 = 1198.9028 ¢, ~6/5 = 337.1334 ¢
error map: ⟨-1.097 +9.086 -13.503 +2.508]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.4588 ¢
error map: ⟨0.000 +10.751 -11.066 +5.762]

Optimal ET sequence: 7, 18d, 25, 32

Badness (Sintel): 4.02

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 125/121

Mapping: [⟨1 3 4 0 6], ⟨0 -5 -6 10 -9]]

Optimal tunings:

  • WE: ~2 = 1198.5480 ¢, ~6/5 = 337.1557 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.6000 ¢

Optimal ET sequence: 7, 25e, 32

Badness (Sintel): 2.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 55/54, 64/63, 125/121

Mapping: [⟨1 3 4 0 6 4], ⟨0 -5 -6 10 -9 -1]]

Optimal tunings:

  • WE: ~2 = 1197.7111 ¢, ~6/5 = 336.8391 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.5336 ¢

Optimal ET sequence: 7, 25e, 32f

Badness (Sintel): 1.91

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 55/54, 64/63, 85/84, 125/121

Mapping: [⟨1 3 4 0 6 4 1], ⟨0 -5 -6 10 -9 -1 11]]

Optimal tunings:

  • WE: ~2 = 1197.7807 ¢, ~6/5 = 336.8884 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.5279 ¢

Optimal ET sequence: 7, 25e, 32f

Badness (Sintel): 2.00

Retrieved from "https://en.xen.wiki/index.php?title=Archytas_clan&oldid=231273"