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 }}

: [[gencom]]: [2 3; 64/63]

* [[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}}

: error map: {{val| 0.000 +7.435 +12.394 }}

[[Badness]] (Sintel): 0.159

* 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.  

[[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.

Temperaments discussed elsewhere are:

 

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–Gb). This is called supra, given right below. Discussed elsewhere is [[suhajira]] of the [[neutral clan #Suhajira|neutral clan]].

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

: gencom: [2 3; 64/63 99/98]

* 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 }}

Badness (Sintel): 0.352

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

: gencom: [2 3; 64/63 78/77 99/98]

* 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 }}

Badness (Sintel): 0.498

Superpyth, virtually the canonical extension, adds [[245/243]] and [[1728/1715]] to the comma list and can be described as {{nowrap| 22 & 27 }}. ~5/4 is found at +9 generator steps, as an augmented second (C–D#). 49edo remains an obvious tuning choice.  

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

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

[[Badness]] (Sintel): 0.818

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).  

* 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

* 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 }}

Badness (Sintel): 1.02

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

* 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–Gb), and finds the ~13/8 at the diminished seventh (C–Bbb).  

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

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

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

Badness (Sintel): 1.08

* 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–Gbb).  

* [[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

* 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

* 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

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

* 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–Cx).

* [[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}}

[[Badness]] (Sintel): 2.74

* 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

* 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 }}

Badness (Sintel): 2.03

* 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

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

* 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–Abbb).  

* [[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–Fb) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53dd val) can be used.  

* [[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 }}

[[Badness]] (Sintel): 1.43

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

* 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 }}

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

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

[[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]]

* 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

* 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

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

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

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

Badness (Sintel): 1.09

* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}

* 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 }}

* 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

* 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

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

 

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

: [[error map]]: {{val| -4.862 +3.925 +12.908 -9.759 }}

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

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

[[Badness]] (Sintel): 1.68

* 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

* 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

* 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]].''

 

{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}

: mapping generators: ~2, ~10/7

* [[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 }}

[[Badness]] (Sintel): 2.74

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

* 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

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

* 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}}

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

[[Badness]] (Sintel): 4.02

* 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

* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}

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

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

Badness (Sintel): 1.91

* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}

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

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

Badness (Sintel): 2.00