Archytas clan: Difference between revisions

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==== 7-limit extensions ====

The second comma in the comma list defines which [[7-limit]] family member we are looking at:

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

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

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

* [[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]];

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

* [[#Ultrapyth|Ultrapyth]] adds 6860/6561, for a tuning sharp of 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 blackwood with a 1/5-octave period. Finally, [[15625/15552]] gives catalan, splitting the twelfth in six.

[[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:

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* [[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 #~~Augene~~|Augmented 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]]

* [[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]]

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

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

==== Subgroup extensions ====

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 [[~~neutral~~ clan #Suhajira|~~neutral~~ clan]].

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]].

=== Supra ===

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==== Supraphon ====

This extension maps [[12/11]][[~]][[14/13]] to ~~+7 generators~~ (C–C♯), and [[13/12]] to ~~-10 generators~~ (C–E𝄫)~~; [[17edo]] is the tuning equating all of them~~.

This extension maps [[13/11]] to the minor third (C–E♭), [[12/11]][[~]][[14/13]] to the augmented unison (C–C♯), and [[13/12]] to the diminished third (C–E𝄫).

Subgroup: 2.3.7.11.13

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: ''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| 22 & 27 }}. ~5/4 is found at +9 generator steps, as an augmented second (C–D♯). 49edo remains an obvious tuning choice.

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

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=== 11-limit ===

The canonical extension to the 13-limit finds the ~11/8 at +16 generator steps, as a double-augmented second (C–D𝄪) and finds the ~13/8 at +13 generator steps, as a double-augmented fourth (C–F𝄪).

Subgroup: 2.3.5.7.11

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== Quasisuper ==

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

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

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=== Quasisupra ===

Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament [[supra]], with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).

Subgroup: 2.3.5.7.11

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== Ultrapyth ==

: ''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 [[the Biosphere #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).

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 ==== 7-limit extensions ====
 The second comma in the comma list defines which [[7-limit]] family member we are looking at:
-* [[#Schism|Schism]] adds 360/343, for a tuning around [[12edo]];
+* [[#Schism|Schism]] adds [[360/343]], for a tuning around [[12edo]];
-* Dominant adds [[36/35]], for a tuning between [[12edo]] and [[17edo|17c-edo]];
+* [[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]];
+* [[#Quasiultra|Quasiultra]] adds [[33614/32805]], for a tuning between 27edo and [[32edo]];
-* [[#Ultrapyth|Ultrapyth]] adds 6860/6561, for a tuning sharp of 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 blackwood with a 1/5-octave period. Finally, [[15625/15552]] gives catalan, splitting the twelfth in six.
+[[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:
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 * [[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 #Augene|Augmented 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]]
+* [[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]]
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 * ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
 Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.
 ==== Subgroup extensions ====
-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 [[neutral clan #Suhajira|neutral clan]].
+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]].
 === Supra ===
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 ==== Supraphon ====
-This extension maps [[12/11]][[~]][[14/13]] to +7 generators (C–C♯), and [[13/12]] to -10 generators (C–E𝄫); [[17edo]] is the tuning equating all of them.
+This extension maps [[13/11]] to the minor third (C–E♭), [[12/11]][[~]][[14/13]] to the augmented unison (C–C♯), and [[13/12]] to the diminished third (C–E𝄫).
 Subgroup: 2.3.7.11.13
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 : ''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| 22 & 27 }}. ~5/4 is found at +9 generator steps, as an augmented second (C–D♯). 49edo remains an obvious tuning choice.
+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
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 === 11-limit ===
-The canonical extension to the 13-limit finds the ~11/8 at +16 generator steps, as a double-augmented second (C–D𝄪) and finds the ~13/8 at +13 generator steps, as a double-augmented fourth (C–F𝄪).
 Subgroup: 2.3.5.7.11
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 == Quasisuper ==
-Quasisuper can be described as {{nowrap| 17c & 22 }}, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–G𝄫).
+{{Main|Quasisuper}}
+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
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 === Quasisupra ===
-Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament [[supra]], with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).
 Subgroup: 2.3.5.7.11
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 == Ultrapyth ==
 {{Main| Ultrapyth }}
+: ''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 [[the Biosphere #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).