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Escapade family: Difference between revisions

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== Escapade ==
== Escapade ==
For intervals along the chain of generators in the 2.3.5.11.31 subgroup temperament, out to 22 generators up, see the third column of [[16ed5/3#Intervals]].
=== 5-limit ===
=== 5-limit ===
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 20: Line 22:


[[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 40: 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. For intervals along the chain of generators in the 2.3.5.11.31 subgroup temperament, out to 22 generators up, see the third column of [[16ed5/3#Intervals]].
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 48: 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 81:
Mapping: {{mapping| 1 2 2 3 5 | 0 -9 7 10 -1 }}
Mapping: {{mapping| 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 (Dirichlet): 0.251
Badness (Sintel): 0.251


{| 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 99: Line 105:
|}
|}


== Strong extensions ==
= Strong extensions =
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong full 7- and 11-limit extensions
|+ style="font-size: 105%;" | Map to strong full 7- and 11-limit extensions
Line 113: Line 119:
<nowiki/>* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions
<nowiki/>* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions


=== Escaped ===
== Escaped ==
''[[#Strong extensions|Return to the map]]''
''[[#Strong extensions|Return to the map]]''


Line 120: Line 126:
Here, [[245/243]] is tempered out so that [[9/7]] is equated to the square root of 5/3 (at 8 generators) present in the temperament. This works best where 5/3 is slightly flat, therefore on the end of the spectrum approaching [[22edo]].
Here, [[245/243]] is tempered out so that [[9/7]] is equated to the square root of 5/3 (at 8 generators) present in the temperament. This works best where 5/3 is slightly flat, therefore on the end of the spectrum approaching [[22edo]].


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


Line 127: Line 133:
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}


{{Multival|legend=1| 9 -7 26 -32 16 80 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.9190{{c}}, ~28/27 = 55.1186{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122
* [[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 ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 142: Line 148:
Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}
Mapping: {{mapping| 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 171: Line 179:
Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}
Mapping: {{mapping| 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 ==
''[[#Strong extensions|Return to the map]]''
''[[#Strong extensions|Return to the map]]''


Given the slightly sharp ~[[3/2]] in ideal tunings of escapade (between [[65edo]] and [[87edo]]), it does very little damage to temper out [[5120/5103]] to extend it to the [[7-limit]]; the cost is that the harmonic 7 is exceedingly complex, located all the way at 61 generators up.
Given the slightly sharp ~[[3/2]] in ideal tunings of escapade (between [[65edo]] and [[87edo]]), it does very little damage to temper out [[5120/5103]] to extend it to prime 7; the cost is that the harmonic 7 is exceedingly complex, located all the way at 61 generators up.


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


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 }}


{{Multival|legend=1| 9 -7 -61 -32 -122 -122 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.7349{{c}}, ~16128/15625 = 55.2306{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16128/15625 = 55.243
* [[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 ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 204: Line 214:
Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}
Mapping: {{mapping| 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 233: Line 245:
Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
Mapping: {{mapping| 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 ==
''[[#Strong extensions|Return to the map]]''
''[[#Strong extensions|Return to the map]]''


This temperament makes the identification of the 4-generator interval, representing (16/15)<sup>2</sup>, with [[8/7]] by tempering out [[225/224]] (along with [[12288/12005]]); however, this is somewhat inaccurate as the ~16/15 in escapade is slightly flat, while for a good marvel tuning it needs to be tempered sharpward to equate it with [[15/14]].
This temperament makes the identification of the 4-generator interval, representing (16/15)<sup>2</sup>, with [[8/7]] by tempering out [[225/224]] (along with [[12288/12005]]); however, this is somewhat inaccurate as the ~16/15 in escapade is slightly flat, while for a good marvel tuning it needs to be tempered sharpward to equate it with [[15/14]].


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


Line 251: Line 265:
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}


{{Multival|legend=1| 9 -7 4 -32 -19 29 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1198.9926{{c}}, ~49/48 = 55.2809{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327
* [[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 ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 266: Line 280:
Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}
Mapping: {{mapping| 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 295: Line 311:
Mapping: {{mapping| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}
Mapping: {{mapping| 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 =
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to weak extensions
|+ style="font-size: 105%;" | Map to weak extensions
Line 314: Line 332:
|}
|}


=== Septisuperfourth ===
== Septisuperfourth ==
''[[#Weak extensions|Return to map]]''
''[[#Weak extensions|Return to map]]''


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


Line 324: 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


{{Multival|legend=1| 18 -14 30 -64 -3 109 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~343/243 = 599.8762{{c}}, ~405/392 = 55.3089{{c}}
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320
* [[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 ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 341: Line 359:
Mapping: {{mapping| 2 4 4 7 6 | 0 -9 7 -15 10 }}
Mapping: {{mapping| 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 370: Line 390:
Mapping: {{mapping| 2 4 4 7 6 11 | 0 -9 7 -15 10 -39 }}
Mapping: {{mapping| 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 385: Line 407:
Mapping: {{mapping| 2 4 4 7 6 5 | 0 -9 7 -15 10 26 }}
Mapping: {{mapping| 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 ==
''[[#Weak extensions|Return to map]]''
''[[#Weak extensions|Return to map]]''


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


Line 403: Line 427:
: mapping generators: ~2, ~64/63
: mapping generators: ~2, ~64/63


{{Multival|legend=1| 18 -14 -35 -64 -106 -42 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.9246{{c}}, ~64/63 = 27.6662{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~64/63 = 27.668
* [[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 ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 418: Line 442:
Mapping: {{mapping| 1 2 2 2 3 | 0 -18 14 35 20 }}
Mapping: {{mapping| 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 447: Line 473:
Mapping: {{mapping| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }}
Mapping: {{mapping| 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:Temperament families]]
[[Category:Temperament families]]
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Escapade_family"