Archytas clan: Difference between revisions

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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| 22 & 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

@@ Line 107: / Line 107: @@
 : ''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| 22 & 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
@@ Line 126: / Line 128: @@
 === 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
@@ Line 221: / Line 221: @@
 == Quasisuper ==
 {{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𝄫).
+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
@@ Line 240: / Line 241: @@
 === 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