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Archytas clan: Difference between revisions

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m Spell out 1\1 in decimal, note the badness metric being used, and misc. cleanup
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3/2 = 709.595
* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.595
* [[POTE]]: ~3/2 = 709.321
* [[POTE]]: ~2 = 1200.000, ~3/2 = 709.321


{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}


[[Badness]]: 0.00464
[[Badness]] (Smith): 0.00464


Scales: [[archy5]], [[archy7]], [[archy12]]
Scales: [[archy5]], [[archy7]], [[archy12]]
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Discussed under their respective 5-limit families are:  
Discussed under their respective 5-limit families are:  
* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* ''[[Suhajira]]'' (+243/242) → [[Neutral clan #Suhajira|Neutral clan]]
* ''[[Suhajira]]'' (+243/242) → [[Neutral clan #Suhajira|Neutral clan]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
Line 62: Line 62:


Optimal tunings:  
Optimal tunings:  
* CTE: ~3/2 = 708.456
* CTE: ~2 = 1200.000, ~3/2 = 708.456
* POTE: ~3/2 = 707.192
* POTE: ~2 = 1200.000, ~3/2 = 707.192


Optimal ET sequence: {{Optimal ET sequence| 5, 12, 17, 39d, 56d }}
Optimal ET sequence: {{optimal ET sequence| 5, 12, 17, 39d, 56d }}


Badness: 0.00933
Badness (Smith): 0.00933


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 707.344
* CTE: ~2 = 1200.000, ~3/2 = 707.344
* POTE: ~2 = 1\1, ~3/2 = 706.137
* POTE: ~2 = 1200.000, ~3/2 = 706.137


Optimal ET sequence: {{Optimal ET sequence| 12f, 17 }}
Optimal ET sequence: {{optimal ET sequence| 12f, 17 }}


Badness: 0.0111
Badness (Smith): 0.0111


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]
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{{Main| Superpyth }}
{{Main| Superpyth }}


In the 5-limit, superpyth tempers out 20480/19683. This temperament has a fifth generator of ~3/2 = ~710¢ and ~5/4 is found at +9 generator steps, as an augmented second (C-D#). It also has a weak extension, [[Jubilismic clan #Bipyth|bipyth]] (10cd & 22), tempering out the same 5-limit comma as the superpyth, but with a half-octave period and the jubilisma (50/49) rather than the Archytas comma tempered out.
In the 5-limit, superpyth tempers out 20480/19683. This temperament has a fifth generator of ~3/2 = ~710¢ and ~5/4 is found at +9 generator steps, as an augmented second (C–D#). It also has a weak extension, [[Jubilismic clan #Bipyth|bipyth]] (10cd & 22), tempering out the same 5-limit comma as the superpyth, but with a half-octave period and the jubilisma (50/49) rather than the Archytas comma tempered out.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 709.393
* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.393
* [[POTE]]: ~2 = 1\1, ~3/2 = 710.078
* [[POTE]]: ~2 = 1200.000, ~3/2 = 710.078


{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}


[[Badness]]: 0.135141
[[Badness]] (Smith): 0.135141


=== 7-limit ===
=== 7-limit ===
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 709.591
* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.591
* [[POTE]]: ~2 = 1\1, ~3/2 = 710.291
* [[POTE]]: ~2 = 1200.000, ~3/2 = 710.291


{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}


[[Badness]]: 0.032318
[[Badness]] (Smith): 0.032318


=== 11-limit ===
=== 11-limit ===
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).  
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).  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 709.514
* CTE: ~2 = 1200.000, ~3/2 = 709.514
* POTE: ~2 = 1\1, ~3/2 = 710.175
* POTE: ~2 = 1200.000, ~3/2 = 710.175


Optimal ET sequence: {{Optimal ET sequence| 22, 27e, 49 }}
Optimal ET sequence: {{optimal ET sequence| 22, 27e, 49 }}


Badness: 0.024976
Badness (Smith): 0.024976


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 709.836
* CTE: ~2 = 1200.000, ~3/2 = 709.836
* POTE: ~2 = 1\1, ~3/2 = 710.479
* POTE: ~2 = 1200.000, ~3/2 = 710.479


Optimal ET sequence: {{Optimal ET sequence| 22, 27e, 49, 76bcde }}
Optimal ET sequence: {{optimal ET sequence| 22, 27e, 49, 76bcde }}


Badness: 0.024673
Badness (Smith): 0.024673


==== Thomas ====
==== Thomas ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~16/13 = 354.759
* CTE: ~2 = 1200.000, ~16/13 = 354.759
* POTE: ~2 = 1\1, ~16/13 = 355.036
* POTE: ~2 = 1200.000, ~16/13 = 355.036


Optimal ET sequence: {{Optimal ET sequence| 27e, 44, 71d, 98bde }}
Optimal ET sequence: {{optimal ET sequence| 27e, 44, 71d, 98bde }}


Badness: 0.049183
Badness (Smith): 0.049183


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 709.356
* CTE: ~2 = 1200.000, ~3/2 = 709.356
* POTE: ~2 = 1\1, ~3/2 = 709.495
* POTE: ~2 = 1200.000, ~3/2 = 709.495


Optimal ET sequence: {{Optimal ET sequence| 5, 17, 22 }}
Optimal ET sequence: {{optimal ET sequence| 5, 17, 22 }}


Badness: 0.032768
Badness (Smith): 0.032768


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 708.702
* CTE: ~2 = 1200.000, ~3/2 = 708.702
* POTE: ~2 = 1\1, ~3/2 = 708.703
* POTE: ~2 = 1200.000, ~3/2 = 708.703


Optimal ET sequence: {{Optimal ET sequence| 5f, 17, 22 }}
Optimal ET sequence: {{optimal ET sequence| 5f, 17, 22 }}


Badness: 0.036336
Badness (Smith): 0.036336


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 708.769
* [[CTE]]: ~2 = 1200.000, ~3/2 = 708.769
* [[POTE]]: ~2 = 1\1, ~3/2 = 708.328
* [[POTE]]: ~2 = 1200.000, ~3/2 = 708.328


{{Optimal ET sequence|legend=1| 17c, 22, 61d }}
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}


[[Badness]]: 0.063794
[[Badness]] (Smith): 0.063794


=== Quasisupra ===
=== Quasisupra ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 708.713
* CTE: ~2 = 1200.000, ~3/2 = 708.713
* POTE: ~2 = 1\1, ~3/2 = 708.205
* POTE: ~2 = 1200.000, ~3/2 = 708.205


Optimal ET sequence: {{Optimal ET sequence| 17c, 22, 39d, 61d }}
Optimal ET sequence: {{optimal ET sequence| 17c, 22, 39d, 61d }}


Badness: 0.032203
Badness (Smith): 0.032203


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 708.411
* CTE: ~2 = 1200.000, ~3/2 = 708.411
* POTE: ~2 = 1\1, ~3/2 = 708.004
* POTE: ~2 = 1200.000, ~3/2 = 708.004


Optimal ET sequence: {{Optimal ET sequence| 17c, 22, 39d, 61df, 100bcdf }}
Optimal ET sequence: {{optimal ET sequence| 17c, 22, 39d, 61df, 100bcdf }}


Badness: 0.030219
Badness (Smith): 0.030219


=== Quasisoup ===
=== Quasisoup ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 709.043
* CTE: ~2 = 1200.000, ~3/2 = 709.043
* POTE: ~2 = 1\1, ~3/2 = 709.021
* POTE: ~2 = 1200.000, ~3/2 = 709.021


Optimal ET sequence: {{Optimal ET sequence| 22 }}
Optimal ET sequence: {{optimal ET sequence| 22 }}


Badness: 0.083490
Badness (Smith): 0.083490


== Ultrapyth ==
== 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).
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).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 713.218
* [[CTE]]: ~2 = 1200.000, ~3/2 = 713.218
* [[POTE]]: ~2 = 1\1, ~3/2 = 713.651
* [[POTE]]: ~2 = 1200.000, ~3/2 = 713.651


{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}


[[Badness]]: 0.108466
[[Badness]] (Smith): 0.108466


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 713.282
* CTE: ~2 = 1200.000, ~3/2 = 713.282
* POTE: ~2 = 1\1, ~3/2 = 713.395
* POTE: ~2 = 1200.000, ~3/2 = 713.395


Optimal ET sequence: {{Optimal ET sequence| 5, 32, 37 }}
Optimal ET sequence: {{optimal ET sequence| 5, 32, 37 }}


Badness: 0.068238
Badness (Smith): 0.068238


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 713.309
* CTE: ~2 = 1200.000, ~3/2 = 713.309
* POTE: ~2 = 1\1, ~3/2 = 713.500
* POTE: ~2 = 1200.000, ~3/2 = 713.500


Optimal ET sequence: {{Optimal ET sequence| 5, 32, 37 }}
Optimal ET sequence: {{optimal ET sequence| 5, 32, 37 }}


Badness: 0.049172
Badness (Smith): 0.049172


=== Ultramarine ===
=== Ultramarine ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 713.395
* CTE: ~2 = 1200.000, ~3/2 = 713.395
* POTE: ~2 = 1\1, ~3/2 = 713.791
* POTE: ~2 = 1200.000, ~3/2 = 713.791


Optimal ET sequence: {{Optimal ET sequence| 5e, 32e, 37, 79bce }}
Optimal ET sequence: {{optimal ET sequence| 5e, 32e, 37, 79bce }}


Badness: 0.078068
Badness (Smith): 0.078068


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 713.708
* CTE: ~2 = 1200.000, ~3/2 = 713.708
* POTE: ~2 = 1\1, ~3/2 = 713.811
* POTE: ~2 = 1200.000, ~3/2 = 713.811


Optimal ET sequence: {{Optimal ET sequence| 5e, 32e, 37, 79bcef }}
Optimal ET sequence: {{optimal ET sequence| 5e, 32e, 37, 79bcef }}


Badness: 0.045653
Badness (Smith): 0.045653


== Quasiultra ==
== Quasiultra ==
Quasiultra is to ultrapyth what quasisuper is to superpyth. It is the 27 & 32 temperament, mapping the ~5/4 to -18 fifths as a double diminished sixth (C-Abbb).  
Quasiultra is to ultrapyth what quasisuper is to superpyth. It is the 27 & 32 temperament, mapping the ~5/4 to -18 fifths as a double diminished sixth (C–Abbb).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 711.838
* [[CTE]]: ~2 = 1200.000, ~3/2 = 711.838
* [[CWE]]: ~2 = 1\1, ~3/2 = 711.5429
* [[CWE]]: ~2 = 1200.000, ~3/2 = 711.5429


{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}


[[Badness]]: 0.132
[[Badness]] (Smith): 0.132


== Schism ==
== Schism ==
{{See also| Schismatic family #Schism }}
{{See also| Schismatic family #Schism }}


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 (53d val) can be used.  
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 (53d val) can be used.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.270
* [[CTE]]: ~2 = 1200.000, ~3/2 = 702.270
* [[POTE]]: ~2 = 1\1, ~3/2 = 701.556
* [[POTE]]: ~2 = 1200.000, ~3/2 = 701.556


{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
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{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}


[[Badness]]: 0.056648
[[Badness]] (Smith): 0.056648


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 703.383
* CTE: ~2 = 1200.000, ~3/2 = 703.383
* POTE: ~2 = 1\1, ~3/2 = 702.136
* POTE: ~2 = 1200.000, ~3/2 = 702.136


Optimal ET sequence: {{Optimal ET sequence| 5c, 7ce, 12, 29de }}
Optimal ET sequence: {{optimal ET sequence| 5c, 7ce, 12, 29de }}


Badness: 0.037482
Badness (Smith): 0.037482


== Beatles ==
== Beatles ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~49/40 = 356.634
* [[CTE]]: ~2 = 1200.000, ~49/40 = 356.634
* [[POTE]]: ~2 = 1\1, ~49/40 = 355.904
* [[POTE]]: ~2 = 1200.000, ~49/40 = 355.904


{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}


[[Badness]]: 0.045872
[[Badness]] (Smith): 0.045872


; Music
; Music
Line 437: Line 437:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~49/40 = 356.719
* CTE: ~2 = 1200.000, ~49/40 = 356.719
* POTE: ~2 = 1\1, ~49/40 = 356.140
* POTE: ~2 = 1200.000, ~49/40 = 356.140


Optimal ET sequence: {{Optimal ET sequence| 10e, 17cee, 27e, 64be, 91bcdee }}
Optimal ET sequence: {{optimal ET sequence| 10e, 17cee, 27e, 64be, 91bcdee }}


Badness: 0.045639
Badness (Smith): 0.045639


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~16/13 = 356.722
* CTE: ~2 = 1200.000, ~16/13 = 356.722
* POTE: ~2 = 1\1, ~16/13 = 356.229
* POTE: ~2 = 1200.000, ~16/13 = 356.229


Optimal ET sequence: {{Optimal ET sequence| 10e, 27e, 37, 64be }}
Optimal ET sequence: {{optimal ET sequence| 10e, 27e, 37, 64be }}


Badness: 0.030161
Badness (Smith): 0.030161


=== Ringo ===
=== Ringo ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 355.992
* CTE: ~2 = 1200.000, ~11/9 = 355.992
* POTE: ~2 = 1\1, ~11/9 = 355.419
* POTE: ~2 = 1200.000, ~11/9 = 355.419


Optimal ET sequence: {{Optimal ET sequence| 10, 17c, 27e }}
Optimal ET sequence: {{optimal ET sequence| 10, 17c, 27e }}


Badness: 0.032863
Badness (Smith): 0.032863


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 356.004
* CTE: ~2 = 1200.000, ~11/9 = 356.004
* POTE: ~2 = 1\1, ~11/9 = 355.456
* POTE: ~2 = 1200.000, ~11/9 = 355.456


Optimal ET sequence: {{Optimal ET sequence| 10, 17c, 27e }}
Optimal ET sequence: {{optimal ET sequence| 10, 17c, 27e }}


Badness: 0.022619
Badness (Smith): 0.022619


=== Beetle ===
=== Beetle ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~49/40 = 356.694
* CTE: ~2 = 1200.000, ~49/40 = 356.694
* POTE: ~2 = 1\1, ~49/40 = 356.710
* POTE: ~2 = 1200.000, ~49/40 = 356.710


Optimal ET sequence: {{Optimal ET sequence| 10, 27, 37 }}
Optimal ET sequence: {{optimal ET sequence| 10, 27, 37 }}


Badness: 0.058084
Badness (Smith): 0.058084


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~16/13 = 356.700
* CTE: ~2 = 1200.000, ~16/13 = 356.700
* POTE: ~2 = 1\1, ~16/13 = 356.701
* POTE: ~2 = 1200.000, ~16/13 = 356.701


Optimal ET sequence: {{Optimal ET sequence| 10, 27, 37 }}
Optimal ET sequence: {{optimal ET sequence| 10, 27, 37 }}


Badness: 0.033971
Badness (Smith): 0.033971


== Progress ==
== Progress ==
Line 533: Line 533:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~10/7 = 638.782
* [[CTE]]: ~2 = 1200.000, ~10/7 = 638.782
* [[POTE]]: ~2 = 1\1, ~10/7 = 637.878
* [[POTE]]: ~2 = 1200.000, ~10/7 = 637.878


{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


[[Badness]]: 0.066400
[[Badness]] (Smith): 0.066400


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/7 = 638.846
* CTE: ~2 = 1200.000, ~10/7 = 638.846
* POTE: ~2 = 1\1, ~10/7 = 637.915
* POTE: ~2 = 1200.000, ~10/7 = 637.915


Optimal ET sequence: {{Optimal ET sequence| 2, 13, 15, 32c, 47bc }}
Optimal ET sequence: {{optimal ET sequence| 2, 13, 15, 32c, 47bc }}


Badness: 0.031036
Badness (Smith): 0.031036


==== 13-limit ====
==== 13-limit ====
Line 563: Line 563:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/7 = 637.871
* CTE: ~2 = 1200.000, ~10/7 = 637.871
* POTE: ~2 = 1\1, ~10/7 = 637.635
* POTE: ~2 = 1200.000, ~10/7 = 637.635


Optimal ET sequence: {{Optimal ET sequence| 15, 17c, 32cf }}
Optimal ET sequence: {{optimal ET sequence| 15, 17c, 32cf }}


Badness: 0.026214
Badness (Smith): 0.026214


==== Progressive ====
==== Progressive ====
Line 578: Line 578:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/7 = 637.797
* CTE: ~2 = 1200.000, ~10/7 = 637.797
* POTE: ~2 = 1\1, ~10/7 = 636.761
* POTE: ~2 = 1200.000, ~10/7 = 636.761


Optimal ET sequence: {{Optimal ET sequence| 2f, 15f, 17c }}
Optimal ET sequence: {{optimal ET sequence| 2f, 15f, 17c }}


Badness: 0.032721
Badness (Smith): 0.032721


== Fervor ==
== Fervor ==
Line 599: Line 599:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~7/5 = 577.353
* [[CTE]]: ~2 = 1200.000, ~7/5 = 577.353
* [[POTE]]: ~2 = 1\1, ~7/5 = 577.776
* [[POTE]]: ~2 = 1200.000, ~7/5 = 577.776


{{Optimal ET sequence|legend=1| 2, 25, 27 }}
{{Optimal ET sequence|legend=1| 2, 25, 27 }}


[[Badness]]: 0.108455
[[Badness]] (Smith): 0.108455


=== 11-limit ===
=== 11-limit ===
Line 614: Line 614:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/5 = 577.296
* CTE: ~2 = 1200.000, ~7/5 = 577.296
* POTE: ~2 = 1\1, ~7/5 = 577.850
* POTE: ~2 = 1200.000, ~7/5 = 577.850


Optimal ET sequence: {{Optimal ET sequence| 2, 25e, 27e }}
Optimal ET sequence: {{optimal ET sequence| 2, 25e, 27e }}


Badness: 0.052054
Badness (Smith): 0.052054


=== 13-limit ===
=== 13-limit ===
Line 629: Line 629:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/5 = 577.374
* CTE: ~2 = 1200.000, ~7/5 = 577.374
* POTE: ~2 = 1\1, ~7/5 = 578.060
* POTE: ~2 = 1200.000, ~7/5 = 578.060


Optimal ET sequence: {{Optimal ET sequence| 2f, 27e }}
Optimal ET sequence: {{optimal ET sequence| 2f, 27e }}


Badness: 0.039705
Badness (Smith): 0.039705


== Sixix ==
== Sixix ==
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}
{{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 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.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 650: Line 650:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~6/5 = 338.005
* [[CTE]]: ~2 = 1200.000, ~6/5 = 338.005
* [[POTE]]: ~2 = 1\1, ~6/5 = 338.365
* [[POTE]]: ~2 = 1200.000, ~6/5 = 338.365


{{Optimal ET sequence|legend=1| 7, 25, 32, 39c }}
{{Optimal ET sequence|legend=1| 7, 25, 32, 39c }}


[[Badness]]: 0.153088
[[Badness]] (Smith): 0.153088


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 3125/2916
[[Comma list]]: 64/63, 3125/2916
Line 667: Line 667:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~6/5 = 337.519
* [[CTE]]: ~2 = 1200.000, ~6/5 = 337.519
* [[POTE]]: ~2 = 1\1, ~6/5 = 337.442
* [[POTE]]: ~2 = 1200.000, ~6/5 = 337.442


{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}


[[Badness]]: 0.158903
[[Badness]] (Smith): 0.158903


=== 11-limit ===
=== 11-limit ===
Line 682: Line 682:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 337.749
* CTE: ~2 = 1200.000, ~6/5 = 337.749
* POTE: ~2 = 1\1, ~6/5 = 337.564
* POTE: ~2 = 1200.000, ~6/5 = 337.564


Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32 }}
Optimal ET sequence: {{optimal ET sequence| 7, 25e, 32 }}


Badness: 0.070799
Badness (Smith): 0.070799


=== 13-limit ===
=== 13-limit ===
Line 697: Line 697:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 337.793
* CTE: ~2 = 1200.000, ~6/5 = 337.793
* POTE: ~2 = 1\1, ~6/5 = 337.483
* POTE: ~2 = 1200.000, ~6/5 = 337.483


Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
Optimal ET sequence: {{optimal ET sequence| 7, 25e, 32f }}


Badness: 0.046206
Badness (Smith): 0.046206


=== 17-limit ===
=== 17-limit ===
Line 712: Line 712:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 337.629
* CTE: ~2 = 1200.000, ~6/5 = 337.629
* POTE: ~2 = 1\1, ~6/5 = 337.513
* POTE: ~2 = 1200.000, ~6/5 = 337.513


Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
Optimal ET sequence: {{optimal ET sequence| 7, 25e, 32f }}


Badness: 0.039224
Badness (Smith): 0.039224


[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Archytas_clan"