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Escapade family: Difference between revisions

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[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~16875/16384 = 55.3052
* [[WE]]: ~2 = 1199.8082{{c}}, ~16875/16384 = 55.2840{{c}}
* [[POTE]]: ~2 = 1200.000, ~16875/16384 = 55.293
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16875/16384 = 55.2961{{c}}


{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}


[[Badness]] (Smith): 0.083778
[[Badness]] (Sintel): 1.965


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Line 42: Line 42:


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
Since (an ideally slightly flat) 4/3 is split in three by the interval of 3 generators, it makes sense to equate that interval to [[11/10]] by tempering out 4000/3993, and therefore the generator to {{nowrap|(11/10)/(16/15) {{=}} [[33/32]]}}; this does minimal damage to the temperament. This structure in 2.3.5.11 occurs in all extensions of escapade to include prime 7, and therefore will be considered the fount of all further extensions.
Since (an ideally slightly flat) 4/3 is split into 9 generators, it makes sense to equate the 3-generator interval to [[11/10]] by tempering out 4000/3993, and therefore the generator to {{nowrap|(11/10)/(16/15) {{=}} [[33/32]]}}; this does minimal damage to the temperament. This structure in 2.3.5.11 occurs in all extensions of escapade to include prime 7, and therefore will be considered the fount of all further extensions.


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11
Line 50: Line 50:
Mapping: {{Mapping| 1 2 2 3 | 0 -9 7 10 }}
Mapping: {{Mapping| 1 2 2 3 | 0 -9 7 10 }}


Optimal tuning (CTE): ~2 = 1200.0000, ~33/32 = 55.2760
Optimal tunings:
* WE: ~2 = 1199.7406{{c}}, ~33/32 = 55.2653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2770{{c}}


{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 87, 152, 369, 521e, 1194bcee, 1715bceeee }}
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e }}


Badness: 0.0107
Badness (Sintel): 0.335


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Line 77: Line 79:
Comma list: 496/495 ({{monzo| 4 -2 -1 -1 1 }}), 961/960 ({{monzo| -6 -1 -1 0 2 }}), 4000/3993 ({{monzo| 5 -1 3 -3 0 }})
Comma list: 496/495 ({{monzo| 4 -2 -1 -1 1 }}), 961/960 ({{monzo| -6 -1 -1 0 2 }}), 4000/3993 ({{monzo| 5 -1 3 -3 0 }})


Mapping: {{mapping| 1 2 2 3 5 | 0 -9 7 10 -1 }}
{{Mapping|legend=0| 1 2 2 3 5 | 0 -9 7 10 -1 }}


Optimal tuning (CTE): ~2 = 1200.000, ~32/31 = 55.276
Optimal tunings:
* WE: ~2 = 1199.8050{{c}}, ~32/31 = 55.2669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/31 = 55.2759{{c}}


{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }}
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e }}


Badness (Sintel): 0.251
Badness (Sintel): 0.251
Line 129: Line 133:
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9190{{c}}, ~28/27 = 55.1186{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 55.1242{{c}}


{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}


[[Badness]] (Smith): 0.088746
[[Badness]] (Sintel): 2.246


=== 11-limit ===
=== 11-limit ===
Line 140: Line 146:
Comma list: 245/243, 385/384, 4000/3993
Comma list: 245/243, 385/384, 4000/3993


Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}
{{Mapping|legend=0| 1 2 2 4 3 | 0 -9 7 -26 10 }}


Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.126
Optimal tunings:
* WE: ~2 = 1199.9480{{c}}, ~28/27 = 55.1241{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1271{{c}}


{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}
{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}


Badness (Smith): 0.035844
Badness (Sintel): 1.185


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Line 169: Line 177:
Comma list: 245/243, 352/351, 385/384, 625/624
Comma list: 245/243, 352/351, 385/384, 625/624


Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}
{{Mapping|legend=0| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}


Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.138
Optimal tunings:
* WE: ~2 = 1199.9926{{c}}, ~28/27 = 55.1378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1382{{c}}


{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}


Badness (Smith): 0.031366
Badness (Sintel): 1.296


== Alphaquarter ==
== Alphaquarter ==
Line 189: Line 199:
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16128/15625 = 55.243
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7349{{c}}, ~16128/15625 = 55.2306{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16128/15625 = 55.2405{{c}}


{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}


[[Badness]] (Smith): 0.116594
[[Badness]] (Sintel): 2.951


=== 11-limit ===
=== 11-limit ===
Line 200: Line 212:
Comma list: 3025/3024, 4000/3993, 5120/5103
Comma list: 3025/3024, 4000/3993, 5120/5103


Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}
{{Mapping|legend=0| 1 2 2 0 3 | 0 -9 7 61 10 }}


Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.243
Optimal tunings:
* WE: ~2 = 1199.7229{{c}}, ~33/32 = 55.2303{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2407{{c}}


{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}


Badness (Smith): 0.029638
Badness (Sintel): 0.980


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Line 229: Line 243:
Comma list: 352/351, 625/624, 847/845, 1575/1573
Comma list: 352/351, 625/624, 847/845, 1575/1573


Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
{{Mapping|legend=0| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}


Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.236
Optimal tunings:
* WE: ~2 = 1199.6491{{c}}, ~33/32 = 55.2200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2328{{c}}


{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}
{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}


Badness (Smith): 0.025344
Badness (Sintel): 1.047


== Escapist ==
== Escapist ==
Line 249: Line 265:
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.9926{{c}}, ~49/48 = 55.2809{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 55.3479{{c}}


{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}


[[Badness]] (Smith): 0.077950
[[Badness]] (Sintel): 1.973


=== 11-limit ===
=== 11-limit ===
Line 260: Line 278:
Comma list: 99/98, 176/175, 2560/2541
Comma list: 99/98, 176/175, 2560/2541


Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}
{{Mapping|legend=0| 1 2 2 3 3 | 0 -9 7 -4 10 }}


Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.354
Optimal tunings:
* WE: ~2 = 1199.0859{{c}}, ~33/32 = 55.3117{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.3574{{c}}


{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}


Badness (Smith): 0.036700
Badness (Sintel): 1.213


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Line 289: Line 309:
Comma list: 78/77, 99/98, 176/175, 507/500
Comma list: 78/77, 99/98, 176/175, 507/500


Mapping: {{mapping| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}
{{Mapping|legend=0| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}


Optimal tuning (POTE): ~2 = 1200.000, ~26/25 = 55.550
Optimal tunings:
* WE: ~2 = 1199.5949{{c}}, ~33/32 = 55.5317{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.5480{{c}}


{{Optimal ET sequence|legend=0| 21, 22, 43 }}
{{Optimal ET sequence|legend=0| 21, 22, 43 }}


Badness (Smith): 0.035261
Badness (Sintel): 1.457


= Weak extensions =
= Weak extensions =
Line 320: Line 342:
{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }}
{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }}


: mapping generators: ~343/243, ~16875/16384
: mapping generators: ~343/243, ~405/392


[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8762{{c}}, ~405/392 = 55.3089{{c}}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~405/392 = 55.3273{{c}}


{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}
{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}


[[Badness]] (Smith): 0.059241
[[Badness]] (Sintel): 1.499


=== 11-limit ===
=== 11-limit ===
Line 333: Line 357:
Comma list: 540/539, 4000/3993, 5632/5625
Comma list: 540/539, 4000/3993, 5632/5625


Mapping: {{mapping| 2 4 4 7 6 | 0 -9 7 -15 10 }}
{{Mapping|legend=0| 2 4 4 7 6 | 0 -9 7 -15 10 }}


Optimal tuning (POTE): ~99/70 = 600.000, ~33/32 = 55.304
Optimal tunings:
* WE: ~99/70 = 599.8383{{c}}, ~33/32 = 55.2895{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/32 = 55.3081{{c}}


{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282 }}
{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282 }}


Badness (Smith): 0.024619
Badness (Sintel): 0.814


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Line 362: Line 388:
Comma list: 540/539, 729/728, 1575/1573, 3584/3575
Comma list: 540/539, 729/728, 1575/1573, 3584/3575


Mapping: {{mapping| 2 4 4 7 6 11 | 0 -9 7 -15 10 -39 }}
{{Mapping|legend=0| 2 4 4 7 6 11 | 0 -9 7 -15 10 -39 }}


Optimal tuning (POTE): ~99/70 = 600.000, ~33/32 = 55.325
Optimal tunings:
* WE: ~99/70 = 599.8331{{c}}, ~33/32 = 55.3093{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/32 = 55.3295{{c}}


{{Optimal ET sequence|legend=0| 22f, 108f, 130, 282 }}
{{Optimal ET sequence|legend=0| 22f, 108f, 130, 282 }}


Badness (Smith): 0.022887
Badness (Sintel): 0.946


==== Septisuperquad ====
==== Septisuperquad ====
Line 377: Line 405:
Comma list: 351/350, 364/363, 540/539, 4096/4095
Comma list: 351/350, 364/363, 540/539, 4096/4095


Mapping: {{mapping| 2 4 4 7 6 5 | 0 -9 7 -15 10 26 }}
{{Mapping|legend=0| 2 4 4 7 6 5 | 0 -9 7 -15 10 26 }}


Optimal tuning (POTE): ~55/39 = 600.000, ~33/32 = 55.359
Optimal tunings:
* WE: ~55/39 = 599.9152{{c}}, ~33/32 = 55.3509{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~33/32 = 55.3584{{c}}


{{Optimal ET sequence|legend=0| 22, 108, 130 }}
{{Optimal ET sequence|legend=0| 22, 108, 130 }}


Badness (Smith): 0.033038
Badness (Sintel): 1.365


== Arch ==
== Arch ==
Line 397: Line 427:
: mapping generators: ~2, ~64/63
: mapping generators: ~2, ~64/63


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~64/63 = 27.668
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9246{{c}}, ~64/63 = 27.6662{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/63 = 27.6676{{c}}


{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}
{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}


[[Badness]] (Smith): 0.094345
[[Badness]] (Sintel): 2.388


=== 11-limit ===
=== 11-limit ===
Line 408: Line 440:
Comma list: 441/440, 3136/3125, 4000/3993
Comma list: 441/440, 3136/3125, 4000/3993


Mapping: {{mapping| 1 2 2 2 3 | 0 -18 14 35 20 }}
{{Mapping|legend=0| 1 2 2 2 3 | 0 -18 14 35 20 }}


Optimal tuning (POTE): ~2 = 1200.000, ~64/63 = 27.663
Optimal tunings:
* WE: ~2 = 1199.8347{{c}}, ~64/63 = 27.6590{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/63 = 27.6617{{c}}


{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 911cde }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 911cde }}


Badness (Smith): 0.036541
Badness (Sintel): 1.208


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Line 437: Line 471:
Comma list: 364/363, 441/440, 676/675, 3136/3125
Comma list: 364/363, 441/440, 676/675, 3136/3125


Mapping: {{mapping| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }}
{{Mapping|legend=0| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }}


Optimal tuning (POTE): ~2 = 1200.000, ~64/63 = 27.660
Optimal tunings:
* WE: ~2 = 1199.8733{{c}}, ~64/63 = 27.6569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/63 = 27.6594{{c}}


{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}


Badness (Smith): 0.019504
Badness (Sintel): 0.806


[[Category:Escapade family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Escapade_family"