Xenharmonic Wiki

Archytas clan: Difference between revisions

← Older edit
VisualWikitext
mNo edit summary
m →Archy: schism isn't an exo
 
(10 intermediate revisions by 4 users not shown)
Line 20: Line 20:


[[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 }}
Line 34: Line 34:
==== 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]];  
Line 57: Line 57:
* ''[[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]]


Line 77: Line 77:


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 }}
Line 98: Line 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 }}
Line 118: Line 118:


{{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 }}
Line 141: Line 139:


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 }}
Line 156: Line 154:


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 }}
Line 171: Line 169:


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 }}
Line 188: Line 186:


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 }}
Line 203: Line 201:


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 }}
Line 218: Line 216:


{{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 }}
Line 241: Line 237:


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 }}
Line 256: Line 252:


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 }}
Line 271: Line 267:


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 }}
Line 279: Line 275:


== Ultrapyth ==
== Ultrapyth ==
{{main|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).
Line 288: Line 284:


{{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 }}
Line 309: Line 303:


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 }}
Line 324: Line 318:


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 }}
Line 339: Line 333:


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 }}
Line 354: Line 348:


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 }}
Line 369: Line 363:


{{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 394: 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 }}
Line 413: Line 403:


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 }}
Line 422: Line 412:
== 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 428: Line 422:


{{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 }}
Line 452: Line 444:


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 467: 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 482: 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 497: 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 512: 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 527: 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 535: 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
Line 544: Line 539:


: 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 565: 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 580: 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 595: 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 }}
Line 604: Line 597:
== 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 609: 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 630: 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 }}
Line 645: Line 638:
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 }}
Line 659: Line 652:
{{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 666: Line 659:


{{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 687: 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 702: 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 }}
Line 717: Line 708:


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 }}
Line 726: Line 717:
[[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"