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 ===
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 4294967296/4271484375 ({{monzo|32 -7 -9}})
[[Comma list]]: 4294967296/4271484375


{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}
{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}
: mapping generators: ~2, ~16875/16384
: mapping generators: ~2, ~16875/16384


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8082{{c}}, ~16875/16384 = 55.2840{{c}}
* [[WE]]: ~2 = 1199.8082{{c}}, ~16875/16384 = 55.2840{{c}}
: [[error map]]: {{val| -0.192 +0.105 +0.291 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16875/16384 = 55.2961{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16875/16384 = 55.2961{{c}}
: error map: {{val| 0.000 +0.380 +0.759 }}


{{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]] (Sintel): 1.965
[[Badness]] (Sintel): 1.965
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.253 || +0.298
|-
| 5/4 || 387.136 || +0.823
|}


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
Since (an ideally slightly flat) 4/3 is split into 9 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


Comma list: 4000/3993 ({{monzo|5 -1 3 -3}}), 5632/5625 ({{monzo|9 -2 -4 1}})
Comma list: 4000/3993, 5632/5625


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


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1199.7406{{c}}, ~33/32 = 55.2653{{c}}
* WE: ~2 = 1199.7406{{c}}, ~33/32 = 55.2653{{c}}
: error map: {{val| -0.259 +0.139 +0.024 +0.557 }}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2770{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2770{{c}}
: error map: {{val| 0.000 +0.552 +0.625 +1.452 }}


{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e }}
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e }}
Line 58: Line 46:
Badness (Sintel): 0.335
Badness (Sintel): 0.335


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
=== 2.3.5.11.31 subgroup ===
|+ style="font-size: 105%;" | Harmonics
One may note that the generator represents the square root of [[16/15]] and therefore it would be logical to also temper out [[961/960]] ({{S|31}}) so that the generator is equated to {{nowrap| [[32/31]][[~]][[31/30]] }} in addition to 33/32.
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.516 || +0.561
|-
| 5/4 || 386.932 || +0.618
|-
| 11/8 || 552.760 || +1.442
|}


=== 2.3.5.11.31 subgroup ===
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]].
One may note that the generator represents the square root of [[16/15]] and therefore it would be logical to also temper out {{nowrap| S31 {{=}} [[961/960]] }} so that the generator is equated to {{nowrap| [[32/31]] ~ [[31/30]] }} in addition to 33/32.


Subgroup: 2.3.5.11.31
Subgroup: 2.3.5.11.31


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, 961/960, 4000/3993


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


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1199.8050{{c}}, ~32/31 = 55.2669{{c}}
* WE: ~2 = 1199.8050{{c}}, ~32/31 = 55.2669{{c}}
: error map: {{val| -0.195 +0.253 +0.165 +0.766 -1.277 }}
* CWE: ~2 = 1200.0000{{c}}, ~32/31 = 55.2759{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/31 = 55.2759{{c}}
: error map: {{val| 0.000 +0.562 +0.617 +1.441 -0.311 }}


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


Badness (Sintel): 0.251
Badness (Sintel): 0.251
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.518 || +0.563
|-
| 5/4 || 386.931 || +0.617
|-
| 11/8 || 552.758 || +1.440
|-
| 31/16 || 1144.724 || -0.311
|}


= Strong extensions =
= Strong extensions =
Line 120: Line 82:


== Escaped ==
== Escaped ==
''[[#Strong extensions|Return to the map]]''
This temperament was also known as ''sensa'' in earlier materials because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with the {{nowrap| 19e & 27 }} temperament (sensi extension).''
 
This temperament was also known as "sensa" in earlier materials because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with the {{nowrap| 19e & 27 }} temperament (sensi extension).''


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 ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 135: Line 94:
[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9190{{c}}, ~28/27 = 55.1186{{c}}
* [[WE]]: ~2 = 1199.9190{{c}}, ~28/27 = 55.1186{{c}}
: [[error map]]: {{val| -0.081 +1.816 -0.646 -2.232 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 55.1242{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 55.1242{{c}}
: error map: {{val| 0.000 +1.927 -0.444 -2.056 }}


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


[[Badness]] (Sintel): 2.246
[[Badness]] (Sintel): 2.25


=== 11-limit ===
=== 11-limit ===
Line 146: Line 107:
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 tunings:
Optimal tunings:
* WE: ~2 = 1199.9480{{c}}, ~28/27 = 55.1241{{c}}
* WE: ~2 = 1199.9480{{c}}, ~28/27 = 55.1241{{c}}
: error map: {{val| -0.052 +1.824 -0.549 -2.261 -0.233 }}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1271{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1271{{c}}
: error map: {{val| 0.000 +1.901 -0.424 -2.131 -0.047 }}


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


Badness (Sintel): 1.185
Badness (Sintel): 1.18


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
=== 13-limit ===
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 703.831 || +1.876
|-
| 5/4 || 385.909 || -0.405
|-
| 7/4 || 966.624 || -2.202
|-
| 11/8 || 551.299 || -0.019
|}
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 tunings:
Optimal tunings:
Line 185: Line 132:
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}


Badness (Sintel): 1.296
Badness (Sintel): 1.30


== Alphaquarter ==
== Alphaquarter ==
''[[#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 prime 7; 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 ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 201: Line 145:
[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7349{{c}}, ~16128/15625 = 55.2306{{c}}
* [[WE]]: ~2 = 1199.7349{{c}}, ~16128/15625 = 55.2306{{c}}
: [[error map]]: {{val| -0.265 +0.439 -0.230 +0.242 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16128/15625 = 55.2405{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16128/15625 = 55.2405{{c}}
: error map: {{val| 0.000 +0.880 +0.370 +0.846 }}


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


[[Badness]] (Sintel): 2.951
[[Badness]] (Sintel): 2.95


=== 11-limit ===
=== 11-limit ===
Line 212: Line 158:
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 tunings:
Optimal tunings:
* WE: ~2 = 1199.7229{{c}}, ~33/32 = 55.2303{{c}}
* WE: ~2 = 1199.7229{{c}}, ~33/32 = 55.2303{{c}}
: error map: {{val| -0.277 +0.418 -0.256 +0.220 +0.153 }}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2407{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2407{{c}}
: error map: {{val| 0.000 +0.879 +0.371 +0.858 +1.089 }}


{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}
Line 222: Line 170:
Badness (Sintel): 0.980
Badness (Sintel): 0.980


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
=== 13-limit ===
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.918 || +0.963
|-
| 5/4 || 386.620 || +0.306
|-
| 7/4 || 969.113 || +0.287
|-
| 11/8 || 552.314 || +0.996
|}
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 tunings:
Optimal tunings:
Line 254: Line 186:


== Escapist ==
== Escapist ==
''[[#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 ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 267: Line 196:
[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.9926{{c}}, ~49/48 = 55.2809{{c}}
* [[WE]]: ~2 = 1198.9926{{c}}, ~49/48 = 55.2809{{c}}
: [[error map]]: {{val| -1.007 -1.498 -1.363 +7.028 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 55.3479{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 55.3479{{c}}
: error map: {{val| 0.000 -0.086 +1.122 +9.782 }}


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


[[Badness]] (Sintel): 1.973
[[Badness]] (Sintel): 1.97


=== 11-limit ===
=== 11-limit ===
Line 278: Line 209:
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 tunings:
Optimal tunings:
* WE: ~2 = 1199.0859{{c}}, ~33/32 = 55.3117{{c}}
* WE: ~2 = 1199.0859{{c}}, ~33/32 = 55.3117{{c}}
: error map: {{val| -0.914 -1.588 -0.960 +7.185 -0.944 }}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.3574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.3574{{c}}
: error map: {{val| 0.000 -0.172 +1.188 +9.745 +2.256 }}


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


Badness (Sintel): 1.213
Badness (Sintel): 1.21


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
=== 13-limit ===
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 701.626 || -0.329
|-
| 5/4 || 387.624 || +1.310
|-
| 7/4 || 978.501 || +9.675
|-
| 11/8 || 553.749 || +2.431
|}
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 tunings:
Optimal tunings:
Line 333: Line 250:


== Septisuperfourth ==
== Septisuperfourth ==
''[[#Weak extensions|Return to map]]''
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 341: Line 255:


{{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, ~405/392
: mapping generators: ~343/243, ~405/392


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8762{{c}}, ~405/392 = 55.3089{{c}}
* [[WE]]: ~343/243 = 599.8762{{c}}, ~405/392 = 55.3089{{c}}
: [[error map]]: {{val| -0.248 -0.230 +0.353 +0.674 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~405/392 = 55.3273{{c}}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~405/392 = 55.3273{{c}}
: error map: {{val| 0.000 +0.100 +0.977 +1.265 }}


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


[[Badness]] (Sintel): 1.499
[[Badness]] (Sintel): 1.50


=== 11-limit ===
=== 11-limit ===
Line 357: Line 272:
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 tunings:
Optimal tunings:
* WE: ~99/70 = 599.8383{{c}}, ~33/32 = 55.2895{{c}}
* WE: ~99/70 = 599.8383{{c}}, ~33/32 = 55.2895{{c}}
: error map: {{val| -0.323 -0.207 +0.066 +0.700 +0.606 }}
* CWE: ~99/70 = 600.0000{{c}}, ~33/32 = 55.3081{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/32 = 55.3081{{c}}
: error map: {{val| 0.000 +0.272 +0.843 +1.553 +1.763 }}


{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282 }}
{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282, 434de, 716dee, 1150cdddeee }}


Badness (Sintel): 0.814
Badness (Sintel): 0.814
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.070 || +0.115
|-
| 5/4 || 387.279 || +0.965
|-
| 7/4 || 970.117 || +1.291
|-
| 11/8 || 553.255 || +1.937
|}


==== 13-limit ====
==== 13-limit ====
Line 388: Line 289:
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 tunings:
Optimal tunings:
Line 394: Line 295:
* CWE: ~99/70 = 600.0000{{c}}, ~33/32 = 55.3295{{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, 976cddeeeff }}


Badness (Sintel): 0.946
Badness (Sintel): 0.946


==== Septisuperquad ====
==== Septisuperquad ====
This temperament is also known as "biscapade".
This temperament is also known as ''biscapade''.


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 405: Line 306:
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 tunings:
Optimal tunings:
Line 413: Line 314:
{{Optimal ET sequence|legend=0| 22, 108, 130 }}
{{Optimal ET sequence|legend=0| 22, 108, 130 }}


Badness (Sintel): 1.365
Badness (Sintel): 1.37


== Arch ==
== Arch ==
''[[#Weak extensions|Return to map]]''
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 424: Line 322:


{{Mapping|legend=1| 1 2 2 2 | 0 -18 14 35 }}
{{Mapping|legend=1| 1 2 2 2 | 0 -18 14 35 }}
: mapping generators: ~2, ~64/63
: mapping generators: ~2, ~64/63


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9246{{c}}, ~64/63 = 27.6662{{c}}
* [[WE]]: ~2 = 1199.9246{{c}}, ~64/63 = 27.6662{{c}}
: [[error map]]: {{val| -0.075 -0.097 +0.862 -0.661 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/63 = 27.6676{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/63 = 27.6676{{c}}
: error map: {{val| 0.000 +0.029 +1.032 -0.461 }}


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


[[Badness]] (Sintel): 2.388
[[Badness]] (Sintel): 2.39


=== 11-limit ===
=== 11-limit ===
Line 440: Line 339:
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 tunings:
Optimal tunings:
* WE: ~2 = 1199.8347{{c}}, ~64/63 = 27.6590{{c}}
* WE: ~2 = 1199.8347{{c}}, ~64/63 = 27.6590{{c}}
: error map: {{val| -0.165 -0.147 +0.581 -1.092 +1.366 }}
* CWE: ~2 = 1200.0000{{c}}, ~64/63 = 27.6617{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/63 = 27.6617{{c}}
: error map: {{val| 0.000 +0.134 +0.950 -0.667 1.916 }}


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


Badness (Sintel): 1.208
Badness (Sintel): 1.21


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
=== 13-limit ===
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.178 || +0.223
|-
| 5/4 || 387.195 || +0.881
|-
| 7/4 || 967.987 || -0.839
|-
| 11/8 || 553.135 || +1.817
|}
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 tunings:
Optimal tunings:
Line 481: Line 366:
Badness (Sintel): 0.806
Badness (Sintel): 0.806


[[Category:Escapade family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]

Latest revision as of 09:34, 4 October 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
Escapade.png
An image of the tuning spectrum of 2.3.5.11 escapade, in terms of the generator; Edo patent val tunings are marked with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled.

The escapade family of temperaments tempers out the escapade comma, [32 -7 -9⟩, of size 9.492 cents. The defining feature of this comma is splitting 5/3 into sixteen quartertones of which 5/4 makes up seven and 4/3 makes up nine; therefore 16/15 is two generator steps. It most naturally manifests as a 2.3.5.11-subgroup temperament, tempering out 4000/3993 and 5632/5625.

Extensions of escapade to incorporate prime 7 (and therefore the full 11-limit) include escapist (21 & 22), tempering out 225/224 and mapping 7 to −4 generators; escaped (22 & 87), tempering out 245/243 and mapping 7 to −26 generators; alphaquarter (65d & 87), tempering out 5120/5103 and mapping 7 to 61 generators; septisuperfourth (a.k.a. biscapade) (22 & 86), tempering out 6144/6125, splitting the octave in half and mapping 7 to −15 generators; and arch (43 & 87), tempering out 3136/3125 and splitting the generator into two 64/63 intervals; all are considered below.

Escapade

Subgroup: 2.3.5

Comma list: 4294967296/4271484375

Mapping: [⟨1 2 2], ⟨0 -9 7]]

mapping generators: ~2, ~16875/16384

Optimal tunings:

  • WE: ~2 = 1199.8082 ¢, ~16875/16384 = 55.2840 ¢
error map: ⟨-0.192 +0.105 +0.291]
  • CWE: ~2 = 1200.0000 ¢, ~16875/16384 = 55.2961 ¢
error map: ⟨0.000 +0.380 +0.759]

Optimal ET sequence: 21, 22, 43, 65, 152, 217, 586, 803

Badness (Sintel): 1.965

2.3.5.11 subgroup

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 (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

Comma list: 4000/3993, 5632/5625

Subgroup-val mapping: [⟨1 2 2 3], ⟨0 -9 7 10]]

Optimal tunings:

  • WE: ~2 = 1199.7406 ¢, ~33/32 = 55.2653 ¢
error map: ⟨-0.259 +0.139 +0.024 +0.557]
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.2770 ¢
error map: ⟨0.000 +0.552 +0.625 +1.452]

Optimal ET sequence: 21, 22, 43, 65, 87, 152, 369, 521e

Badness (Sintel): 0.335

2.3.5.11.31 subgroup

One may note that the generator represents the square root of 16/15 and therefore it would be logical to also temper out 961/960 (S31) so that the generator is equated to 32/31~31/30 in addition to 33/32.

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.

Subgroup: 2.3.5.11.31

Comma list: 496/495, 961/960, 4000/3993

Subgroup-val mapping: [⟨1 2 2 3 5], ⟨0 -9 7 10 -1]]

Optimal tunings:

  • WE: ~2 = 1199.8050 ¢, ~32/31 = 55.2669 ¢
error map: ⟨-0.195 +0.253 +0.165 +0.766 -1.277]
  • CWE: ~2 = 1200.0000 ¢, ~32/31 = 55.2759 ¢
error map: ⟨0.000 +0.562 +0.617 +1.441 -0.311]

Optimal ET sequence: 21, 22, 43, 65, 87, 152, 369, 521e, 673e

Badness (Sintel): 0.251

Strong extensions

Map to strong full 7- and 11-limit extensions
Extension Mapping of 7 Tuning range*
Escapist -4 ↓ 65
Alphaquarter +61 ↑ 65
↓ 87
Escaped -26 ↑ 87

* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions

Escaped

This temperament was also known as sensa in earlier materials because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. Not to be confused with the 19e & 27 temperament (sensi extension).

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.

Subgroup: 2.3.5.7

Comma list: 245/243, 65625/65536

Mapping: [⟨1 2 2 4], ⟨0 -9 7 -26]]

Optimal tunings:

  • WE: ~2 = 1199.9190 ¢, ~28/27 = 55.1186 ¢
error map: ⟨-0.081 +1.816 -0.646 -2.232]
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 55.1242 ¢
error map: ⟨0.000 +1.927 -0.444 -2.056]

Optimal ET sequence: 22, 65, 87, 196, 283

Badness (Sintel): 2.25

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 4000/3993

Mapping: [⟨1 2 2 4 3], ⟨0 -9 7 -26 10]]

Optimal tunings:

  • WE: ~2 = 1199.9480 ¢, ~28/27 = 55.1241 ¢
error map: ⟨-0.052 +1.824 -0.549 -2.261 -0.233]
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 55.1271 ¢
error map: ⟨0.000 +1.901 -0.424 -2.131 -0.047]

Optimal ET sequence: 22, 65, 87, 196, 283

Badness (Sintel): 1.18

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 245/243, 352/351, 385/384, 625/624

Mapping: [⟨1 2 2 4 3 2], ⟨0 -9 7 -26 10 37]]

Optimal tunings:

  • WE: ~2 = 1199.9926 ¢, ~28/27 = 55.1378 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 55.1382 ¢

Optimal ET sequence: 22, 65, 87, 283

Badness (Sintel): 1.30

Alphaquarter

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.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 29360128/29296875

Mapping: [⟨1 2 2 0], ⟨0 -9 7 61]]

Optimal tunings:

  • WE: ~2 = 1199.7349 ¢, ~16128/15625 = 55.2306 ¢
error map: ⟨-0.265 +0.439 -0.230 +0.242]
  • CWE: ~2 = 1200.0000 ¢, ~16128/15625 = 55.2405 ¢
error map: ⟨0.000 +0.880 +0.370 +0.846]

Optimal ET sequence: 65d, 87, 152, 239, 391

Badness (Sintel): 2.95

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 5120/5103

Mapping: [⟨1 2 2 0 3], ⟨0 -9 7 61 10]]

Optimal tunings:

  • WE: ~2 = 1199.7229 ¢, ~33/32 = 55.2303 ¢
error map: ⟨-0.277 +0.418 -0.256 +0.220 +0.153]
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.2407 ¢
error map: ⟨0.000 +0.879 +0.371 +0.858 +1.089]

Optimal ET sequence: 65d, 87, 152, 239, 391

Badness (Sintel): 0.980

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 625/624, 847/845, 1575/1573

Mapping: [⟨1 2 2 0 3 2], ⟨0 -9 7 61 10 37]]

Optimal tunings:

  • WE: ~2 = 1199.6491 ¢, ~33/32 = 55.2200 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.2328 ¢

Optimal ET sequence: 65d, 87, 152f, 239f

Badness (Sintel): 1.047

Escapist

This temperament makes the identification of the 4-generator interval, representing (16/15)2, 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.

Subgroup: 2.3.5.7

Comma list: 225/224, 12288/12005

Mapping: [⟨1 2 2 3], ⟨0 -9 7 -4]]

Optimal tunings:

  • WE: ~2 = 1198.9926 ¢, ~49/48 = 55.2809 ¢
error map: ⟨-1.007 -1.498 -1.363 +7.028]
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 55.3479 ¢
error map: ⟨0.000 -0.086 +1.122 +9.782]

Optimal ET sequence: 21, 22, 43, 65d

Badness (Sintel): 1.97

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 2560/2541

Mapping: [⟨1 2 2 3 3], ⟨0 -9 7 -4 10]]

Optimal tunings:

  • WE: ~2 = 1199.0859 ¢, ~33/32 = 55.3117 ¢
error map: ⟨-0.914 -1.588 -0.960 +7.185 -0.944]
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.3574 ¢
error map: ⟨0.000 -0.172 +1.188 +9.745 +2.256]

Optimal ET sequence: 21, 22, 43, 65d

Badness (Sintel): 1.21

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 99/98, 176/175, 507/500

Mapping: [⟨1 2 2 3 3 3], ⟨0 -9 7 -4 10 15]]

Optimal tunings:

  • WE: ~2 = 1199.5949 ¢, ~33/32 = 55.5317 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.5480 ¢

Optimal ET sequence: 21, 22, 43

Badness (Sintel): 1.457

Weak extensions

Map to weak extensions
Extensions Periods per octave Position of original generator
Number of generators Number of periods
Septisuperfourth period = 1/2 octave 1 generator + 0 periods
Arch period = octave 2 generators + 0 periods

Septisuperfourth

Subgroup: 2.3.5.7

Comma list: 6144/6125, 118098/117649

Mapping: [⟨2 4 4 7], ⟨0 -9 7 -15]]

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

Optimal tunings:

  • WE: ~343/243 = 599.8762 ¢, ~405/392 = 55.3089 ¢
error map: ⟨-0.248 -0.230 +0.353 +0.674]
  • CWE: ~343/243 = 600.0000 ¢, ~405/392 = 55.3273 ¢
error map: ⟨0.000 +0.100 +0.977 +1.265]

Optimal ET sequence: 22, 86, 108, 130, 152, 282

Badness (Sintel): 1.50

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 5632/5625

Mapping: [⟨2 4 4 7 6], ⟨0 -9 7 -15 10]]

Optimal tunings:

  • WE: ~99/70 = 599.8383 ¢, ~33/32 = 55.2895 ¢
error map: ⟨-0.323 -0.207 +0.066 +0.700 +0.606]
  • CWE: ~99/70 = 600.0000 ¢, ~33/32 = 55.3081 ¢
error map: ⟨0.000 +0.272 +0.843 +1.553 +1.763]

Optimal ET sequence: 22, 86, 108, 130, 152, 282, 434de, 716dee, 1150cdddeee

Badness (Sintel): 0.814

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573, 3584/3575

Mapping: [⟨2 4 4 7 6 11], ⟨0 -9 7 -15 10 -39]]

Optimal tunings:

  • WE: ~99/70 = 599.8331 ¢, ~33/32 = 55.3093 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/32 = 55.3295 ¢

Optimal ET sequence: 22f, 108f, 130, 282, 976cddeeeff

Badness (Sintel): 0.946

Septisuperquad

This temperament is also known as biscapade.

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 540/539, 4096/4095

Mapping: [⟨2 4 4 7 6 5], ⟨0 -9 7 -15 10 26]]

Optimal tunings:

  • WE: ~55/39 = 599.9152 ¢, ~33/32 = 55.3509 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~33/32 = 55.3584 ¢

Optimal ET sequence: 22, 108, 130

Badness (Sintel): 1.37

Arch

Subgroup: 2.3.5.7

Comma list: 3136/3125, 5250987/5242880

Mapping: [⟨1 2 2 2], ⟨0 -18 14 35]]

mapping generators: ~2, ~64/63

Optimal tunings:

  • WE: ~2 = 1199.9246 ¢, ~64/63 = 27.6662 ¢
error map: ⟨-0.075 -0.097 +0.862 -0.661]
  • CWE: ~2 = 1200.0000 ¢, ~64/63 = 27.6676 ¢
error map: ⟨0.000 +0.029 +1.032 -0.461]

Optimal ET sequence: 43, 87, 130, 217, 347

Badness (Sintel): 2.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4000/3993

Mapping: [⟨1 2 2 2 3], ⟨0 -18 14 35 20]]

Optimal tunings:

  • WE: ~2 = 1199.8347 ¢, ~64/63 = 27.6590 ¢
error map: ⟨-0.165 -0.147 +0.581 -1.092 +1.366]
  • CWE: ~2 = 1200.0000 ¢, ~64/63 = 27.6617 ¢
error map: ⟨0.000 +0.134 +0.950 -0.667 1.916]

Optimal ET sequence: 43, 87, 130, 217, 347e

Badness (Sintel): 1.21

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 3136/3125

Mapping: [⟨1 2 2 2 3 4], ⟨0 -18 14 35 20 -13]]

Optimal tunings:

  • WE: ~2 = 1199.8733 ¢, ~64/63 = 27.6569 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/63 = 27.6594 ¢

Optimal ET sequence: 43, 87, 130, 217, 347e, 564e

Badness (Sintel): 0.806

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