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Archytas clan: Difference between revisions

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{{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.  
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[[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 }}
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==== 7-limit extensions ====
==== 7-limit extensions ====
The second comma in the comma list defines which [[7-limit]] family member we are looking at:  
The second comma in the comma list defines which [[7-limit]] family member we are looking at:  
* [[#Schism|Schism]] adds 360/343, for an [[exotemperament]] well tuned around [[12edo]];  
* [[#Schism|Schism]] adds 360/343, for a tuning around [[12edo]];  
* Dominant adds [[36/35]], for a tuning flat of [[17edo|17c-edo]];  
* 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]];  
* [[#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]];  
* [[#Superpyth|Superpyth]] adds [[245/243]], for a tuning between 22edo and [[27edo]];  
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* ''[[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]]


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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 }}
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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 }}
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{{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 }}
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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 }}
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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 }}
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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 }}
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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 }}
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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 }}
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{{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 }}
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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 }}
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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 }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}
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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 }}
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== Ultrapyth ==
== Ultrapyth ==
{{Main| 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).
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).


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{{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 }}
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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 }}
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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 }}
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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 }}
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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 }}
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{{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 }}


Line 392: Line 386:


[[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 }}
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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 }}
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== 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
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{{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 }}
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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 }}
Line 465: Line 459:


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 }}
Line 480: Line 474:


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 }}
Line 495: Line 489:


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 }}
Line 510: Line 504:


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 }}
Line 525: Line 519:


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 }}
Line 533: Line 527:


== 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
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: 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 }}
Line 563: Line 558:


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 }}
Line 578: Line 573:


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 }}
Line 593: Line 588:


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 }}
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== 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 607: Line 604:
[[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 }}
Line 628: Line 623:
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 }}
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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 }}
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{{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
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{{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 }}
Line 685: Line 678:


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 }}
Line 700: Line 693:


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