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Biscapade is the same as septisuperfourth
 
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The '''escapade family''' tempers out the [[escapade comma]], {{monzo|32 -7 -9}}, of size 9.492 [[cent]]s.
{{Technical data page}}
 
<div style="float: right;">
[[File:Escapade.png|alt=Escapade.png|thumb|600x560px|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.]]
</div>
 
The '''escapade family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[escapade comma]], {{monzo| 32 -7 -9 }}, of size 9.492 [[cent]]s. 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|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 ({{nowrap| 21 & 22 }}), tempering out [[225/224]] and mapping 7 to −4 generators; escaped ({{nowrap| 22 & 87 }}), tempering out [[245/243]] and mapping 7 to −26 generators; alphaquarter ({{nowrap| 65d & 87 }}), tempering out [[5120/5103]] and mapping 7 to 61 generators; septisuperfourth (a.k.a. biscapade) ({{nowrap| 22 & 86 }}), tempering out [[6144/6125]], splitting the octave in half and mapping 7 to −15 generators; and arch ({{nowrap| 43 & 87 }}), tempering out [[3136/3125]] and splitting the generator into two [[64/63]] intervals; all are considered below.


== Escapade ==
== Escapade ==
Subgroup: 2.3.5
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
 
[[Comma list]]: 4294967296/4271484375 ({{monzo|32 -7 -9}})
 
{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}
 
: mapping generators: ~2, ~16875/16384
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~16875/16384 = 55.3052
* [[POTE]]: ~2 = 1200.000, ~16875/16384 = 55.293
 
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}
 
[[Badness]] (Smith): 0.083778
 
{| 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 ===
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.
 
Subgroup: 2.3.5.11
 
Comma list: 4000/3993 ({{monzo|5 -1 3 -3}}), 5632/5625 ({{monzo|9 -2 -4 1}})
 
Mapping: {{Mapping| 1 2 2 3 | 0 -9 7 10 }}
 
Optimal tuning (CTE): ~2 = 1200.0000, ~33/32 = 55.2760
 
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 87, 152, 369, 521e, 1194bcee, 1715bceeee }}
 
Badness: 0.0107
 
{| 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 || 702.516 || +0.561
|-
| 5/4 || 386.932 || +0.618
|-
| 11/8 || 552.760 || +1.442
|}
 
=== 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 {{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
 
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 }}
 
Optimal tuning (CTE): ~2 = 1200.000, ~32/31 = 55.276


[[Comma list]]: 4294967296/4271484375
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }}


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -9 7 }}]
Badness (Sintel): 0.251


[[POTE generator]]: ~16875/16384 = 55.293
{| 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
|}


{{Val list|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}
= Strong extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong full 7- and 11-limit extensions
|-
! rowspan="1" | Extension !! rowspan="1" | Mapping of 7 !! rowspan="1" | Tuning range*
|-
| [[#Escapist|Escapist]] || -4 || ↓ [[65edo|65]]
|-
| [[#Alphaquarter|Alphaquarter]] || +61 || ↑ 65 <br> ↓ [[87edo|87]]
|-
| [[#Escaped|Escaped]] || -26 || ↑ 87
|}
<nowiki/>* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions


[[Badness]]: 0.083778
== Escaped ==
''[[#Strong extensions|Return to the map]]''


=== Extensions ===
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).''
Extensions of escapade include (septimal) escapade, escaped and biscapade, considered below, as well as these temperaments discussed elsewhere:
* ''[[arch]]'', {3136/3125, 5250987/5242880} → [[Hemimean clan #Arch]]
* ''[[alphaquarter]]'', {5120/5103, 29360128/29296875} → [[Hemifamity temperaments #Alphaquarter]]


== Septimal escapade ==
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]]: 225/224, 12288/12005
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 2 2 3}}, {{val|0 -9 7 -4}}]
[[Comma list]]: 245/243, 65625/65536


{{Multival|legend=1|9 -7 4 -32 -19 29}}
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}


[[POTE generator]]: ~49/48 = 55.327
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122


{{Val list|legend=1| 21, 22, 43, 65d }}
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}


[[Badness]]: 0.077950
[[Badness]] (Smith): 0.088746


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 2560/2541
Comma list: 245/243, 385/384, 4000/3993
 
Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}


Mapping: [{{val|1 2 2 3 3}}, {{val|0 -9 7 -4 10}}]
Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.126


POTE generator: ~33/32 = 55.354
{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}


Vals: {{Val list| 21, 22, 43, 65d }}
Badness (Smith): 0.035844


Badness: 0.036700
{| 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 || 703.831 || +1.876
|-
| 5/4 || 385.909 || -0.405
|-
| 7/4 || 966.624 || -2.202
|-
| 11/8 || 551.299 || -0.019
|}


=== 13-limit ===
==== 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: 245/243, 352/351, 385/384, 625/624
 
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 ET sequence|legend=0| 22, 65, 87, 283 }}
 
Badness (Smith): 0.031366
 
== 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.
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 29360128/29296875
 
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16128/15625 = 55.243
 
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
 
[[Badness]] (Smith): 0.116594
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4000/3993, 5120/5103
 
Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.243
 
{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}
 
Badness (Smith): 0.029638
 
{| 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 || 702.918 || +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


Mapping: [{{val|1 2 2 3 3 3}}, {{val|0 -9 7 -4 10 15}}]
Comma list: 352/351, 625/624, 847/845, 1575/1573


POTE generator: ~26/25 = 55.550
Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}


Vals: {{Val list| 21, 22, 43 }}
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.236


Badness: 0.035261
{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}


== Escaped ==
Badness (Smith): 0.025344
{{See also|Sensamagic clan #Escaped}}


This temperament is also called as "sensa" because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with 19e&amp;27 temperament (sensi extension).''
== Escapist ==
''[[#Strong extensions|Return to the map]]''


Subgroup: 2.3.5.7
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]].


[[Comma list]]: 245/243, 65625/65536
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 2 4 }}, {{val| 0 -9 7 -26 }}]
[[Comma list]]: 225/224, 12288/12005


{{Multival|legend=1| 9 -7 26 -32 16 80 }}
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}


[[POTE generator]]: ~28/27 = 55.122
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327


{{Val list|legend=1| 22, 65, 87, 196, 283 }}
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}


[[Badness]]: 0.088746
[[Badness]] (Smith): 0.077950


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 245/243, 385/384, 4000/3993
Comma list: 99/98, 176/175, 2560/2541
 
Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}


Mapping: [{{val| 1 2 2 4 3 }}, {{val| 0 -9 7 -26 10 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.354


POTE generator: ~28/27 = 55.126
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}


Vals: {{Val list| 22, 65, 87, 196, 283 }}
Badness (Smith): 0.036700


Badness: 0.035844
{| 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 || 701.626 || -0.329
|-
| 5/4 || 387.624 || +1.310
|-
| 7/4 || 978.501 || +9.675
|-
| 11/8 || 553.749 || +2.431
|}


=== 13-limit ===
==== 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: 78/77, 99/98, 176/175, 507/500
 
Mapping: {{mapping| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~26/25 = 55.550


Mapping: [{{val| 1 2 2 4 3 2 }}, {{val| 0 -9 7 -26 10 37 }}]
{{Optimal ET sequence|legend=0| 21, 22, 43 }}


POTE generator: ~28/27 = 55.138
Badness (Smith): 0.035261


Vals: {{Val list| 22, 65, 87, 283 }}
= Weak extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to weak extensions
|-
! rowspan="2" | Extensions !! rowspan="2" | Periods per octave !! colspan="2" | Position of original generator
|-
! Number of generators !! Number of periods
|-
| [[#Septisuperfourth|Septisuperfourth]] || period = 1/2 octave || 1 generator || + 0 periods
|-
| [[#Arch|Arch]] || period = octave || 2 generators || + 0 periods
|}


Badness: 0.031366
== Septisuperfourth ==
''[[#Weak extensions|Return to map]]''


== Septisuperfourth aka biscapade ==
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 118098/117649
[[Comma list]]: 6144/6125, 118098/117649


[[Mapping]]: [{{val| 2 4 4 7 }}, {{val| 0 -9 7 -15 }}]
{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }}


{{Multival|legend=1| 18 -14 30 -64 -3 109 }}
: mapping generators: ~343/243, ~16875/16384


[[POTE generator]]: ~16875/16384 = 55.320
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320


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


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


=== 11-limit ===
=== 11-limit ===
Line 125: Line 333:
Comma list: 540/539, 4000/3993, 5632/5625
Comma list: 540/539, 4000/3993, 5632/5625


Mapping: [{{val| 2 4 4 7 6 }}, {{val| 0 -9 7 -15 10 }}]
Mapping: {{mapping| 2 4 4 7 6 | 0 -9 7 -15 10 }}


POTE generator: ~33/32 = 55.304
Optimal tuning (POTE): ~99/70 = 600.000, ~33/32 = 55.304


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


Badness: 0.024619
Badness (Smith): 0.024619


==== 13-limit (septisuperfourth) ====
{| 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 ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 2 4 4 7 6 11 }}, {{val| 0 -9 7 -15 10 -39 }}]
Mapping: {{mapping| 2 4 4 7 6 11 | 0 -9 7 -15 10 -39 }}


POTE generator: ~48/35 = 544.675
Optimal tuning (POTE): ~99/70 = 600.000, ~33/32 = 55.325


Vals: {{Val list| 22f, 108f, 130, 282 }}
{{Optimal ET sequence|legend=0| 22f, 108f, 130, 282 }}


Badness: 0.022887
Badness (Smith): 0.022887
 
==== Septisuperquad ====
This temperament is also known as "biscapade".


==== 13-limit (biscapade aka septisuperquad) ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val|2 4 4 7 6 5}}, {{val|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 ET sequence|legend=0| 22, 108, 130 }}
 
Badness (Smith): 0.033038
 
== Arch ==
''[[#Weak extensions|Return to map]]''
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3136/3125, 5250987/5242880
 
{{Mapping|legend=1| 1 2 2 2 | 0 -18 14 35 }}
 
: mapping generators: ~2, ~64/63
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~64/63 = 27.668
 
{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}
 
[[Badness]] (Smith): 0.094345
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3136/3125, 4000/3993
 
Mapping: {{mapping| 1 2 2 2 3 | 0 -18 14 35 20 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~64/63 = 27.663
 
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 911cde }}
 
Badness (Smith): 0.036541
 
{| 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 || 702.178 || +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
 
Comma list: 364/363, 441/440, 676/675, 3136/3125
 
Mapping: {{mapping| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }}


POTE generator: ~33/32 = 55.359
Optimal tuning (POTE): ~2 = 1200.000, ~64/63 = 27.660


Vals: {{Val list| 22, 108, 130 }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}


Badness: 0.033038
Badness (Smith): 0.019504


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:41, 24 June 2025

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

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

Comma list: 4294967296/4271484375 ([32 -7 -9⟩)

Mapping: [⟨1 2 2], ⟨0 -9 7]]

mapping generators: ~2, ~16875/16384

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~16875/16384 = 55.3052
  • POTE: ~2 = 1200.000, ~16875/16384 = 55.293

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

Badness (Smith): 0.083778

Harmonics
Prime harmonic Tunings
CTE tuning Deviation from just
3/2 702.253 +0.298
5/4 387.136 +0.823

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 (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 ([5 -1 3 -3⟩), 5632/5625 ([9 -2 -4 1⟩)

Mapping: [⟨1 2 2 3], ⟨0 -9 7 10]]

Optimal tuning (CTE): ~2 = 1200.0000, ~33/32 = 55.2760

Optimal ET sequence: 21, 22, 43, 65, 87, 152, 369, 521e, 1194bcee, 1715bceeee

Badness: 0.0107

Harmonics
Prime harmonic 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

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

Subgroup: 2.3.5.11.31

Comma list: 496/495 ([4 -2 -1 -1 1⟩), 961/960 ([-6 -1 -1 0 2⟩), 4000/3993 ([5 -1 3 -3 0⟩)

Mapping: [⟨1 2 2 3 5], ⟨0 -9 7 10 -1]]

Optimal tuning (CTE): ~2 = 1200.000, ~32/31 = 55.276

Optimal ET sequence: 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee

Badness (Sintel): 0.251

Harmonics
Prime harmonic 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

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

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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.

7-limit

Subgroup: 2.3.5.7

Comma list: 245/243, 65625/65536

Mapping: [⟨1 2 2 4], ⟨0 -9 7 -26]]

Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.122

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

Badness (Smith): 0.088746

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 tuning (POTE): ~2 = 1200.000, ~28/27 = 55.126

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

Badness (Smith): 0.035844

Harmonics
Prime harmonic 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

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.138

Optimal ET sequence: 22, 65, 87, 283

Badness (Smith): 0.031366

Alphaquarter

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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

Comma list: 5120/5103, 29360128/29296875

Mapping: [⟨1 2 2 0], ⟨0 -9 7 61]]

Optimal tuning (POTE): ~2 = 1200.000, ~16128/15625 = 55.243

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

Badness (Smith): 0.116594

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 tuning (POTE): ~2 = 1200.000, ~33/32 = 55.243

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

Badness (Smith): 0.029638

Harmonics
Prime harmonic 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

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.236

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

Badness (Smith): 0.025344

Escapist

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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.

7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 12288/12005

Mapping: [⟨1 2 2 3], ⟨0 -9 7 -4]]

Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 55.327

Optimal ET sequence: 21, 22, 43, 65d

Badness (Smith): 0.077950

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 tuning (POTE): ~2 = 1200.000, ~33/32 = 55.354

Optimal ET sequence: 21, 22, 43, 65d

Badness (Smith): 0.036700

Harmonics
Prime harmonic 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

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~26/25 = 55.550

Optimal ET sequence: 21, 22, 43

Badness (Smith): 0.035261

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

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7-limit

Subgroup: 2.3.5.7

Comma list: 6144/6125, 118098/117649

Mapping: [⟨2 4 4 7], ⟨0 -9 7 -15]]

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

Optimal tuning (POTE): ~343/243 = 1\2, ~16875/16384 = 55.320

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

Badness (Smith): 0.059241

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 tuning (POTE): ~99/70 = 600.000, ~33/32 = 55.304

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

Badness (Smith): 0.024619

Harmonics
Prime harmonic 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

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 tuning (POTE): ~99/70 = 600.000, ~33/32 = 55.325

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

Badness (Smith): 0.022887

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 tuning (POTE): ~55/39 = 600.000, ~33/32 = 55.359

Optimal ET sequence: 22, 108, 130

Badness (Smith): 0.033038

Arch

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7-limit

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 tuning (POTE): ~2 = 1\1, ~64/63 = 27.668

Optimal ET sequence: 43, 87, 130, 217, 347, 824c, 1171c, 1518cd

Badness (Smith): 0.094345

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 tuning (POTE): ~2 = 1200.000, ~64/63 = 27.663

Optimal ET sequence: 43, 87, 130, 217, 347e, 911cde

Badness (Smith): 0.036541

Harmonics
Prime harmonic 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

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~64/63 = 27.660

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

Badness (Smith): 0.019504

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