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.  

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

: mapping generators: ~2, ~3

[[Optimal tuning]]s:  

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

: [[error map]]: {{val| -3.045 +2.522 +3.952 }}

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

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

* [[Meantone family #Dominant|Dominant]] adds [[36/35]], for a tuning between [[12edo]] and [[17edo|17c-edo]];  

* [[#Quasisuper|Quasisuper]] adds [[2430/2401]], for a tuning between 17c-edo and [[22edo]];  

* [[#Superpyth|Superpyth]] adds [[245/243]], for a tuning between 22edo and [[27edo]];  

* [[#Quasiultra|Quasiultra]] adds [[33614/32805]], for a tuning between 27edo and [[32edo]];  

* [[#Ultrapyth|Ultrapyth]] adds [[6860/6561]], for a tuning sharp of 32edo;  

* Mother adds [[16/15]], for an exotemperament well tuned around [[5edo]].  

 

These all use the same generators as archy.  

[[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:  

* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]

* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]

* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]

* ''[[Dichotic]]'' (+25/24) → [[Dicot family #Dichotic|Dicot family]]

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

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

* [[Augene]] (+126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]

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

* [[Modus]] (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]

* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]

* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]

* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]

* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]

Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.

 

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]].

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

Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}

Optimal tunings:  

* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}

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

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

Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}

Optimal tunings:  

* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}

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

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.  

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

: [[error map]]: {{val| -2.945 +3.648 -0.548 +2.298 }}

: 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

Optimal tunings:

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

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

Badness (Sintel): 0.826

Optimal tunings:

 

 

Badness (Sintel): 1.02

 

 

 

 

* 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

Optimal tunings:

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

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

Badness (Sintel): 2.03

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

Optimal tunings:

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

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

Badness (Sintel): 1.08

Optimal tunings:  

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

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

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

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).

* [[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}}

[[Badness]] (Sintel): 1.61

Optimal tunings:

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

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

Badness (Sintel): 1.06

Optimal tunings:

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

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

Badness (Sintel): 1.25

Optimal tunings:

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

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

Badness (Sintel): 2.76

 

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𝄪).

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

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

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

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

[[Badness]] (Sintel): 2.74

Optimal tunings:

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

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

Badness (Sintel): 2.26

Optimal tunings:

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

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

Badness (Sintel): 2.03

Optimal tunings:

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

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

Badness (Sintel): 2.58

Optimal tunings:

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

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

Badness (Sintel): 1.89

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𝄫♭).  

* [[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}}

[[Badness]] (Sintel): 3.34

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.  

[[Optimal tuning]]s:

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

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

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

[[Badness]] (Sintel): 1.43

Optimal tunings:

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

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

Badness (Sintel): 1.24

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''

 

 

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

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

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

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

[[Badness]] (Sintel): 1.16

* [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]]

Optimal tunings:

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

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

Badness (Sintel): 1.51

Optimal tunings:

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

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

Badness (Sintel): 1.25

Optimal tunings:

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

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

Badness (Sintel): 1.09

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}

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

Badness (Sintel): 0.935

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

Optimal tunings:

* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}

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

Badness (Sintel): 1.92

Optimal tunings:

* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}

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

Badness (Sintel): 1.40

== Progress ==

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''

 

[[Comma list]]: 64/63, 392/375

{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}

: mapping generators: ~2, ~10/7

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 638.0791{{c}}

: error map: {{val| 0.000 +12.282 +23.291 +2.700 }}

{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}

[[Badness]] (Sintel): 1.68

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

Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}

Optimal tunings:

* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}

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

Badness (Sintel): 1.03

==== 13-limit ====

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

Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}

Optimal tunings:

* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}

{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}

Optimal tunings:  

* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}

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

Badness (Sintel): 1.35

: ''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

[[Comma list]]: 64/63, 9604/9375

: mapping generators: ~2, ~10/7

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}

: [[error map]]: {{val| -3.726 +3.230 +4.980 -1.550 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 622.3179{{c}}

: error map: {{val| 0.000 +9.634 +12.826 +7.996 }}

{{Optimal ET sequence|legend=1| 2, 25, 27 }}

[[Badness]] (Sintel): 2.74

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}

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

Badness (Sintel): 1.72

=== 13-limit ===

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

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

Optimal tunings:

* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}

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

Badness (Sintel): 1.64

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

* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}

: [[error map]]: {{val| -1.097 +9.086 -13.503 +2.508 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 337.4588{{c}}

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

[[Badness]] (Sintel): 4.02

Optimal tunings:

* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}

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

Badness (Sintel): 2.34

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}

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

Badness (Sintel): 1.91

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}

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

Badness (Sintel): 2.00