Archytas clan: Difference between revisions

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.

{{Main| Superpyth }}

[[Subgroup]]: 2.3.7

[[Comma list]]: 64/63

{{Mapping|legend=3| 1 0 6 | 0 1 -2 }}

{{Mapping|legend=5| 1 0 0 6 | 0 1 0 -2 }}

: mapping generators: ~2, ~3

* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}

[[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:

* [[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]];

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 semi-octave period. [[392/375]] gives progress. [[6125/5832]] gives kangaroo. Those split 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. [[15625/15552]] gives catalan, splitting the twelfth in six. Finally, [[28/27]] gives blackwood and [[17496/16807]] gives [[gothic]], reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.  

* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]

* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]

* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]

The rest are considered below.  

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 in [[#Subgroup extensions_2|#Subgroup extensions]]. Discussed elsewhere is [[suhajira]] of the [[rastmic clan #Suhajira|rastmic clan]].

== 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 {{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

[[Comma list]]: 64/63, 245/243

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

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1197.0549{{c}}, ~3/2 = 708.5478{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1193{{c}}

: error map: {{val| 0.000 +8.164 +4.760 +10.935 }}

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

[[Badness]] (Sintel): 0.818

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

{{Mapping|legend=0| 1 0 -12 6 -22 | 0 1 9 -2 16 }}

Optimal tunings:  

* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}

{{Optimal ET sequence|legend=0| 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|legend=0| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}

 

Optimal tunings:  

* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}

 

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

 

 

==== Uberpyth ====

Subgroup: 2.3.5.7.11.13

 

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

 

{{Mapping|legend=0| 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 }}

 

 

Subgroup: 2.3.5.7.11.13

 

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

 

{{Mapping|legend=0| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}

 

Optimal tunings:  

* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}

 

 

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|legend=0| 1 0 -12 6 13 | 0 1 9 -2 -6 }}

 

Optimal tunings:  

* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}

 

 

Badness (Sintel): 1.08

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

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

 

{{Mapping|legend=0| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}

 

Optimal tunings:  

* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}

 

 

Badness (Sintel): 1.50

 

 

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

 

[[Comma list]]: 64/63, 2430/2401

 

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

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}

: [[error map]]: {{val| -3.017 +1.486 -0.435 +6.190 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 708.3716{{c}}

: error map: {{val| 0.000 +6.417 +4.855 +14.431 }}

 

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

 

 

=== Quasisupra ===

 

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

 

 

* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}

 

{{Optimal ET sequence|legend=0| 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|legend=0| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}

 

* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}

 

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

 

Badness (Sintel): 1.25

 

Subgroup: 2.3.5.7.11

 

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

 

{{Mapping|legend=0| 1 0 23 6 -22 | 0 1 -13 -2 16 }}

 

* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}

 

 

Badness (Sintel): 2.76

 

: ''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|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|legend=1| 1 0 -20 6 | 0 1 14 -2 }}

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1197.2673{{c}}, ~3/2 = 712.0258{{c}}

: [[error map]]: {{val| -2.733 +7.338 -1.557 -3.808 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5430{{c}}

: error map: {{val| 0.000 +11.588 +3.288 +4.088 }}

 

 

 

=== 11-limit ===

 

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

 

{{Mapping|legend=0| 1 0 -20 6 21 | 0 1 14 -2 -11 }}

 

Optimal tunings:  

* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}

 

{{Optimal ET sequence|legend=0| 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|legend=0| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}

 

Optimal tunings:  

* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}

 

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

 

 

=== Ultramarine ===

Subgroup: 2.3.5.7.11

 

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

 

{{Mapping|legend=0| 1 0 -20 6 -38 | 0 1 14 -2 26 }}

 

* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}

 

{{Optimal ET sequence|legend=0| 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|legend=0| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}

 

* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}

 

 

Badness (Sintel): 1.89

 

== Quasiultra ==

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

 

[[Comma list]]: 64/63, 33614/32805

 

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

 

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1196.9257{{c}}, ~3/2 = 709.6211{{c}}

: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5429{{c}}

: error map: {{val| 0.000 +9.588 +5.914 +8.088 }}

 

 

 

== 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.

 

 

[[Comma list]]: 64/63, 360/343

 

{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}

 

* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}

: [[error map]]: {{val| -2.640 -4.583 -4.896 +20.588 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7376{{c}}

: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}

 

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

 

[[Badness]] (Sintel): 1.43

 

Subgroup: 2.3.5.7.11

 

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

 

{{Mapping|legend=0| 1 0 15 6 13 | 0 1 -8 -2 -6 }}

 

Optimal tunings:  

* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}

 

 

Badness (Sintel): 1.24

 

: ''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

 

[[Comma list]]: 64/63, 686/675

 

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

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}

: [[error map]]: {{val| -3.376 +4.475 +2.682 -1.940 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}

: error map: {{val| 0.000 +10.209 +8.949 +6.847 }}

 

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

 

 

; Music

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

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

 

{{Mapping|legend=0| 1 1 5 4 10 | 0 2 -9 -4 -22 }}

 

* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}

 

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

 

Badness (Sintel): 1.51

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

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

 

{{Mapping|legend=0| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}

 

Optimal tunings:  

* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}

 

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

 

Badness (Sintel): 1.25

 

Subgroup: 2.3.5.7.11

 

 

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

 

* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}

 

 

Badness (Sintel): 1.09

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Archytas| ]] <!-- key article -->