Archytas clan: Difference between revisions

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== Ultrapyth ==

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).

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== Beatles ==

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

Beatles tempers out 686/675, which may also be characterized by saying it tempers out [[2401/2400]]. It may be described as the {{nowrap| 10 & 17c }} temperament. It splits the fifth into two neutral-third generators of 49/40~60/49; its [[ploidacot]] is dicot. 5/4 may be found at -9 generator steps, as a semidiminished fourth (C–Fd). 27edo is an obvious tuning, though 17c-edo and 37edo are among the possibilities.

Beatles extends easily to the no-11 13-limit, as the generator can be interpreted as ~16/13, tempering out 91/90, 169/168, and 196/195.

[[Subgroup]]: 2.3.5.7

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== Progress ==

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

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.

[[Subgroup]]: 2.3.5.7

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== Fervor ==

: ''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

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

: mapping generators: ~2, ~7/5

: mapping generators: ~2, ~10/7

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.000, ~7/5 = ~~577~~.~~353~~

* [[CTE]]: ~2 = 1200.000, ~10/7 = 622.647

: [[error map]]: {{val| 0.000 +11.278 +9.867 +4.709 }}

* [[POTE]]: ~2 = 1200.000, ~7/5 = ~~577~~.~~776~~

* [[POTE]]: ~2 = 1200.000, ~10/7 = 622.224

: error map: {{val| 0.000 +9.163 +13.673 +8.937 }}

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

Mapping: {{mapping| 1 4 ~~-2 -2 3~~ | 0 -5 9 10 1 }}

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~7/5 = ~~577~~.~~296~~

* CTE: ~2 = 1200.000, ~10/7 = 622.704

* POTE: ~2 = 1200.000, ~7/5 = ~~577~~.~~850~~

* POTE: ~2 = 1200.000, ~10/7 = 622.150

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

Mapping: {{mapping| 1 ~~4 -2 -2 3~~ -4 | 0 -5 9 10 1 16 }}

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~7/5 = ~~577~~.~~374~~

* CTE: ~2 = 1200.000, ~10/7 = 622.626

* POTE: ~2 = 1200.000, ~7/5 = ~~578~~.~~060~~

* POTE: ~2 = 1200.000, ~10/7 = 621.940

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Sixix 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.

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

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 == Ultrapyth ==
-{{main|Ultrapyth}}
+{{Main| Ultrapyth }}
 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).
@@ Line 412: / Line 412: @@
 == Beatles ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''
+Beatles tempers out 686/675, which may also be characterized by saying it tempers out [[2401/2400]]. It may be described as the {{nowrap| 10 & 17c }} temperament. It splits the fifth into two neutral-third generators of 49/40~60/49; its [[ploidacot]] is dicot. 5/4 may be found at -9 generator steps, as a semidiminished fourth (C–Fd). 27edo is an obvious tuning, though 17c-edo and 37edo are among the possibilities.
+Beatles extends easily to the no-11 13-limit, as the generator can be interpreted as ~16/13, tempering out 91/90, 169/168, and 196/195.
 [[Subgroup]]: 2.3.5.7
@@ Line 523: / Line 527: @@
 == Progress ==
+{{Distinguish| Progression }}
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 590: / Line 597: @@
 == Fervor ==
 : ''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
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 [[Comma list]]: 64/63, 9604/9375
-{{Mapping|legend=1| 1 4 -2 -2 | 0 -5 9 10 }}
+{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
-: mapping generators: ~2, ~7/5
+: mapping generators: ~2, ~10/7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~7/5 = 577.353
+* [[CTE]]: ~2 = 1200.000, ~10/7 = 622.647
 : [[error map]]: {{val| 0.000 +11.278 +9.867 +4.709 }}
-* [[POTE]]: ~2 = 1200.000, ~7/5 = 577.776
+* [[POTE]]: ~2 = 1200.000, ~10/7 = 622.224
 : error map: {{val| 0.000 +9.163 +13.673 +8.937 }}
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 Comma list: 56/55, 64/63, 1350/1331
-Mapping: {{mapping| 1 4 -2 -2 3 | 0 -5 9 10 1 }}
+Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~7/5 = 577.296
+* CTE: ~2 = 1200.000, ~10/7 = 622.704
-* POTE: ~2 = 1200.000, ~7/5 = 577.850
+* POTE: ~2 = 1200.000, ~10/7 = 622.150
 {{Optimal ET sequence|legend=0| 2, 25e, 27e }}
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 Comma list: 56/55, 64/63, 78/77, 507/500
-Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}
+Mapping: {{mapping| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~7/5 = 577.374
+* CTE: ~2 = 1200.000, ~10/7 = 622.626
-* POTE: ~2 = 1200.000, ~7/5 = 578.060
+* POTE: ~2 = 1200.000, ~10/7 = 621.940
 {{Optimal ET sequence|legend=0| 2f, 27e }}
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 {{See also| Dual-fifth temperaments #Dual-3 Sixix }}
-Sixix 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.
+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