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Archytas clan: Difference between revisions

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→Archy: + rough tuning ranges of these extensions
Switch to Sintel's badness, per community consensus. Misc. fixes
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: sval mapping generators: ~2, ~3
: sval mapping generators: ~2, ~3


{{Mapping|legend=3| 1 1 0 4 | 0 1 0 -2 }}
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}


: [[gencom]]: [2 3/2; 64/63]
: [[gencom]]: [2 3; 64/63]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 27: Line 27:
{{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]] (Smith): 0.00464
[[Badness]] (Sintel): 0.159


Scales: [[archy5]], [[archy7]], [[archy12]]
Scales: [[archy5]], [[archy7]], [[archy12]]
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Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}


Gencom mapping: {{mapping| 1 1 0 4 7 | 0 1 0 -2 -6 }}
Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}


: gencom: [2 3/2; 64/63 99/98]
: gencom: [2 3; 64/63 99/98]


Optimal tunings:  
Optimal tunings:  
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{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}


Badness (Smith): 0.00933
Badness (Sintel): 0.352


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]
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Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}


Gencom mapping: {{mapping| 1 1 0 4 7 9 | 0 1 0 -2 -6 -9 }}
Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}


: gencom: [2 3/2; 64/63 78/77 99/98]
: gencom: [2 3; 64/63 78/77 99/98]


Optimal tunings:  
Optimal tunings:  
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{{Optimal ET sequence|legend=0| 12f, 17 }}
{{Optimal ET sequence|legend=0| 12f, 17 }}


Badness (Smith): 0.0111
Badness (Sintel): 0.498


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]
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: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


Superpyth adds [[245/243]] and [[1728/1715]] to the comma list and can be described as 22 & 27. ~5/4 is found at +9 generator steps, as an augmented second (C–D#).  
Superpyth 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#).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}


[[Badness]] (Smith): 0.032318
[[Badness]] (Sintel): 0.818


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


Badness (Smith): 0.024976
Badness (Sintel): 0.826


==== 13-limit ====
==== 13-limit ====
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{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}


Badness (Smith): 0.024673
Badness (Sintel): 1.02


==== Thomas ====
==== Thomas ====
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{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}


Badness (Smith): 0.049183
Badness (Sintel): 2.03


=== Suprapyth ===
=== Suprapyth ===
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{{Optimal ET sequence|legend=0| 5, 17, 22 }}
{{Optimal ET sequence|legend=0| 5, 17, 22 }}


Badness (Smith): 0.032768
Badness (Sintel): 1.08


==== 13-limit ====
==== 13-limit ====
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{{Optimal ET sequence|legend=0| 5f, 17, 22 }}
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}


Badness (Smith): 0.036336
Badness (Sintel): 1.50


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


[[Badness]] (Smith): 0.063794
[[Badness]] (Sintel): 1.61


=== Quasisupra ===
=== Quasisupra ===
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{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}


Badness (Smith): 0.032203
Badness (Sintel): 1.06


==== 13-limit ====
==== 13-limit ====
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* POTE: ~2 = 1200.000, ~3/2 = 708.004
* POTE: ~2 = 1200.000, ~3/2 = 708.004


{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61df, 100bcdf }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


Badness (Smith): 0.030219
Badness (Sintel): 1.25


=== Quasisoup ===
=== Quasisoup ===
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{{Optimal ET sequence|legend=0| 22 }}
{{Optimal ET sequence|legend=0| 22 }}


Badness (Smith): 0.083490
Badness (Sintel): 2.76


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


[[Badness]] (Smith): 0.108466
[[Badness]] (Sintel): 2.74


=== 11-limit ===
=== 11-limit ===
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{{Optimal ET sequence|legend=0| 5, 32, 37 }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Badness (Smith): 0.068238
Badness (Sintel): 2.26


==== 13-limit ====
==== 13-limit ====
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{{Optimal ET sequence|legend=0| 5, 32, 37 }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Badness (Smith): 0.049172
Badness (Sintel): 2.03


=== Ultramarine ===
=== Ultramarine ===
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{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}


Badness (Smith): 0.078068
Badness (Sintel): 2.58


==== 13-limit ====
==== 13-limit ====
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{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}


Badness (Smith): 0.045653
Badness (Sintel): 1.89


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


[[Badness]] (Smith): 0.132
[[Badness]] (Sintel): 3.34


== 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 (53dd val) can be used.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}


[[Badness]] (Smith): 0.056648
[[Badness]] (Sintel): 1.43


=== 11-limit ===
=== 11-limit ===
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{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}


Badness (Smith): 0.037482
Badness (Sintel): 1.24


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


[[Badness]] (Smith): 0.045872
[[Badness]] (Sintel): 1.16


; Music
; Music
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 ''Beatles Improv''] by [[Herman Miller]]
* [https://web.archive.org/web/20201127013829/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 ''Beatles Improv''] by [[Herman Miller]]


=== 11-limit ===
=== 11-limit ===
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{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}


Badness (Smith): 0.045639
Badness (Sintel): 1.51


==== 13-limit ====
==== 13-limit ====
Line 471: Line 471:
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}


Badness (Smith): 0.030161
Badness (Sintel): 1.25


=== Ringo ===
=== Ringo ===
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{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


Badness (Smith): 0.032863
Badness (Sintel): 1.09


==== 13-limit ====
==== 13-limit ====
Line 501: Line 501:
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


Badness (Smith): 0.022619
Badness (Sintel): 0.935


=== Beetle ===
=== Beetle ===
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Comma list: 55/54, 64/63, 686/675
Comma list: 55/54, 64/63, 686/675


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


Optimal tunings:  
Optimal tunings:  
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{{Optimal ET sequence|legend=0| 10, 27, 37 }}
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Badness (Smith): 0.058084
Badness (Sintel): 1.92


==== 13-limit ====
==== 13-limit ====
Line 531: Line 531:
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Badness (Smith): 0.033971
Badness (Sintel): 1.40


== Progress ==
== Progress ==
Line 554: Line 554:
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


[[Badness]] (Smith): 0.066400
[[Badness]] (Sintel): 1.68


=== 11-limit ===
=== 11-limit ===
Line 569: Line 569:
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}


Badness (Smith): 0.031036
Badness (Sintel): 1.03


==== 13-limit ====
==== 13-limit ====
Line 584: Line 584:
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}


Badness (Smith): 0.026214
Badness (Sintel): 1.08


==== Progressive ====
==== Progressive ====
Line 599: Line 599:
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}


Badness (Smith): 0.032721
Badness (Sintel): 1.35


== Fervor ==
== Fervor ==
Line 622: Line 622:
{{Optimal ET sequence|legend=1| 2, 25, 27 }}
{{Optimal ET sequence|legend=1| 2, 25, 27 }}


[[Badness]] (Smith): 0.108455
[[Badness]] (Sintel): 2.74


=== 11-limit ===
=== 11-limit ===
Line 637: Line 637:
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}


Badness (Smith): 0.052054
Badness (Sintel): 1.72


=== 13-limit ===
=== 13-limit ===
Line 652: Line 652:
{{Optimal ET sequence|legend=0| 2f, 27e }}
{{Optimal ET sequence|legend=0| 2f, 27e }}


Badness (Smith): 0.039705
Badness (Sintel): 1.64


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


[[Badness]] (Smith): 0.158903
[[Badness]] (Sintel): 4.02


=== 11-limit ===
=== 11-limit ===
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{{Optimal ET sequence|legend=0| 7, 25e, 32 }}
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}


Badness (Smith): 0.070799
Badness (Sintel): 2.34


=== 13-limit ===
=== 13-limit ===
Line 706: Line 706:
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


Badness (Smith): 0.046206
Badness (Sintel): 1.91


=== 17-limit ===
=== 17-limit ===
Line 721: Line 721:
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


Badness (Smith): 0.039224
Badness (Sintel): 2.00


[[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"