Archytas clan: Difference between revisions

 

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

* [[#Schism|Schism]] adds 360/343, for a tuning around [[12edo]];  

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

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

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

[[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.  

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

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

* ''[[Modus]]'' (+4375/4374) → [[Tetracot family #Modus|Tetracot 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–Gb). This is called supra, given right below. Discussed elsewhere is [[suhajira]] of the [[neutral clan #Suhajira|neutral clan]].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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.  

Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}

Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}

Mapping: {{mapping| 1 0 -12 6 -22 26 | 0 1 9 -2 16 -14 }}

* WE: ~2 = 1196.667{{c}}, ~3/2 = 708.360{{c}}

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 710.288{{c}}

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

Badness (Sintel): 1.105

Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}

Suprapyth finds the ~11/8 at the diminished fifth (C–Gb), and finds the ~13/8 at the diminished seventh (C–Bbb).  

Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}

Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}

Quasisuper can be described as {{nowrap| 17c & 22 }}, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–Gbb).  

Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}

Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}

Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}

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).

Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}

Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}

Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}

Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}

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).  

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.  

Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}

Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}

Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}

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

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

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

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

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.  

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

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

Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}

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.  

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

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

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.  

Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}

Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}

Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}

[[Category:Rank 2]]