Escapade family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}


The '''escapade family''' 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]] temperament, tempering out [[4000/3993]] and [[5632/5625]].
<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 (21 & 22), tempering out [[225/224]] and mapping 7 to -4 generators; escaped (87 & 22), tempering out [[245/243]] and mapping 7 to -26 generators; alphaquarter (65d & 87), tempering out [[5120/5103]] and mapping 7 to 61 generators; septisuperfourth (aka 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.
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.


<div style="float:right;">
== Escapade ==
[[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.]]
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]].
</div>


= Escapade (5-limit) =
=== 5-limit ===
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


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


{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}
{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}
Line 19: Line 22:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~16875/16384 = 55.3052
* [[CTE]]: ~2 = 1200.0000, ~16875/16384 = 55.3052
* [[POTE]]: ~2 = 1\1, ~16875/16384 = 55.293
* [[POTE]]: ~2 = 1200.000, ~16875/16384 = 55.293


{{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]]: 0.083778
[[Badness]] (Smith): 0.083778


{| 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"
|+ style="font-size: 105%;" | Harmonics
|+ style="font-size: 105%;" | Harmonics
|-
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
|-
! CTE tuning !! Deviation
! CTE tuning !! Deviation from just
|-
|-
| 3/2 || 701.955 || 702.253 || +0.298
| 3/2 || 702.253 || +0.298
|-
|-
| 5/4 || 386.314 || 387.136 || +0.823
| 5/4 || 387.136 || +0.823
|}
|}


= Escapade =
=== 2.3.5.11 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]].
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.
 
== 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
Subgroup: 2.3.5.11


Comma list: 4000/3993, 5632/5625
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 }}
Mapping: {{Mapping| 1 2 2 3 | 0 -9 7 10 }}


Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 55.2760
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 }}
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 87, 152, 369, 521e, 1194bcee, 1715bceeee }}
Line 59: Line 59:
|+ style="font-size: 105%;" | Harmonics
|+ style="font-size: 105%;" | Harmonics
|-
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
|-
! CTE tuning !! Deviation
! CTE tuning !! Deviation from just
|-
|-
| 3/2 || 701.955 || 702.516 || +0.561
| 3/2 || 702.516 || +0.561
|-
|-
| 5/4 || 386.314 || 386.932 || +0.618
| 5/4 || 386.932 || +0.618
|-
|-
| 11/8 || 551.318 || 552.760 || +1.442
| 11/8 || 552.760 || +1.442
|}
|}
== Strong extensions ==


=== 2.3.5.11.31 subgroup ===
=== 2.3.5.11.31 subgroup ===
One may also note that the generator represents the square root of [[16/15]] and therefore it would be logical to temper out S31 = [[961/960]] so that the generator is equated to [[32/31]]~[[31/30]] in addition to 33/32.
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, 961/960, 4000/3993
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: {{mapping| 1 2 2 3 5 | 0 -9 7 10 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~32/31 = 55.276
Optimal tuning (CTE): ~2 = 1200.000, ~32/31 = 55.276


{{Optimal ET sequence|legend=1| 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, 1194bcee, 1867bceeee }}


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"
|+ style="font-size: 105%;" | Harmonics
|+ style="font-size: 105%;" | Harmonics
|-
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
|-
! CTE tuning !! Deviation
! CTE tuning !! Deviation from just
|-
|-
| 3/2 || 701.955 || 702.518 || +0.563
| 3/2 || 702.518 || +0.563
|-
|-
| 5/4 || 386.314 || 386.931 || +0.617
| 5/4 || 386.931 || +0.617
|-
|-
| 11/8 || 551.318 || 552.758 || +1.440
| 11/8 || 552.758 || +1.440
|-
|-
| 31/16 || 1145.036 || 1144.724 || -0.311
| 31/16 || 1144.724 || -0.311
|}
|}


=== Escapist ===
= Strong extensions =
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]].
{| 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
 
== 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).''
 
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|legend=1| 1 2 2 4 | 0 -9 7 -26 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122
 
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
 
[[Badness]] (Smith): 0.088746


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


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.354
Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.126


{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}
{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}


Badness: 0.036700
Badness (Smith): 0.035844


{| 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"
|+ style="font-size: 105%;" | Harmonics
|+ style="font-size: 105%;" | Harmonics
|-
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
|-
! CTE tuning !! Deviation
! CTE tuning !! Deviation from just
|-
|-
| 3/2 || 701.955 || 701.626 || -0.329
| 3/2 || 703.831 || +1.876
|-
|-
| 5/4 || 386.314 || 387.624 || +1.310
| 5/4 || 385.909 || -0.405
|-
|-
| 7/4 || 968.826 || 978.501 || +9.675
| 7/4 || 966.624 || -2.202
|-
|-
| 11/8 || 551.318 || 553.749 || +2.431
| 11/8 || 551.299 || -0.019
|}
|}


Line 137: Line 167:
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]]''


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


Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 55.550
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


{{Optimal ET sequence|legend=1| 21, 22, 43 }}
[[Comma list]]: 5120/5103, 29360128/29296875


Badness: 0.035261
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}


=== Escaped ===
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16128/15625 = 55.243
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 19e &amp; 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]].
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
 
[[Badness]] (Smith): 0.116594


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


Comma list: 245/243, 385/384, 4000/3993
Comma list: 3025/3024, 4000/3993, 5120/5103


Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}
Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.126
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.243


{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}


Badness: 0.035844
Badness (Smith): 0.029638


{| 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"
|+ style="font-size: 105%;" | Harmonics
|+ style="font-size: 105%;" | Harmonics
|-
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
|-
! CTE tuning !! Deviation
! CTE tuning !! Deviation from just
|-
|-
| 3/2 || 701.955 || 703.831 || +1.876
| 3/2 || 702.918 || +0.963
|-
|-
| 5/4 || 386.314 || 385.909 || -0.405
| 5/4 || 386.620 || +0.306
|-
|-
| 7/4 || 968.826 || 966.624 || -2.202
| 7/4 || 969.113 || +0.287
|-
|-
| 11/8 || 551.318 || 551.299 || -0.019
| 11/8 || 552.314 || +0.996
|}
|}


Line 183: Line 227:
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: 352/351, 625/624, 847/845, 1575/1573
 
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 ET sequence|legend=0| 65d, 87, 152f, 239f }}
 
Badness (Smith): 0.025344
 
== 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]].
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}
[[Comma list]]: 225/224, 12288/12005


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}


{{Optimal ET sequence|legend=1| 22, 65, 87, 283 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327


Badness: 0.031366
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}


=== Alphaquarter ===
[[Badness]] (Smith): 0.077950
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.


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


Comma list: 3025/3024, 4000/3993, 5120/5103
Comma list: 99/98, 176/175, 2560/2541


Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}
Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.243
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.354


{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}


Badness: 0.029638
Badness (Smith): 0.036700


{| 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"
|+ style="font-size: 105%;" | Harmonics
|+ style="font-size: 105%;" | Harmonics
|-
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
|-
! CTE tuning !! Deviation
! CTE tuning !! Deviation from just
|-
|-
| 3/2 || 701.955 || 702.918 || +0.963
| 3/2 || 701.626 || -0.329
|-
|-
| 5/4 || 386.314 || 386.620 || +0.306
| 5/4 || 387.624 || +1.310
|-
|-
| 7/4 || 968.826 || 969.113 || +0.287
| 7/4 || 978.501 || +9.675
|-
|-
| 11/8 || 551.318 || 552.314 || +0.996
| 11/8 || 553.749 || +2.431
|}
|}


Line 227: Line 287:
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: 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
 
{{Optimal ET sequence|legend=0| 21, 22, 43 }}
 
Badness (Smith): 0.035261
 
= 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
|}
 
== Septisuperfourth ==
''[[#Weak extensions|Return to map]]''
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
[[Comma list]]: 6144/6125, 118098/117649


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.236
{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }}


{{Optimal ET sequence|legend=1| 65d, 87, 152f, 239f }}
: mapping generators: ~343/243, ~16875/16384


Badness: 0.025344
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320


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


=== Septisuperfourth ===
[[Badness]] (Smith): 0.059241
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 246: Line 335:
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 = 1\2, ~33/32 = 55.304
Optimal tuning (POTE): ~99/70 = 600.000, ~33/32 = 55.304


{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}
{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282 }}


Badness: 0.024619
Badness (Smith): 0.024619
 
{| 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 259: Line 364:
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 = 1\2, ~33/32 = 55.325
Optimal tuning (POTE): ~99/70 = 600.000, ~33/32 = 55.325


{{Optimal ET sequence|legend=1| 22f, 108f, 130, 282 }}
{{Optimal ET sequence|legend=0| 22f, 108f, 130, 282 }}


Badness: 0.022887
Badness (Smith): 0.022887


==== Septisuperquad ====
==== Septisuperquad ====
Line 274: Line 379:
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 = 1\2, ~33/32 = 55.359
Optimal tuning (POTE): ~55/39 = 600.000, ~33/32 = 55.359
 
{{Optimal ET sequence|legend=0| 22, 108, 130 }}


{{Optimal ET sequence|legend=1| 22, 108, 130 }}
Badness (Smith): 0.033038


Badness: 0.033038
== Arch ==
''[[#Weak extensions|Return to map]]''


=== Arch ===
=== 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
Subgroup: 2.3.5.7.11


Line 287: Line 410:
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 = 1\1, ~64/63 = 27.663
Optimal tuning (POTE): ~2 = 1200.000, ~64/63 = 27.663


{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347e, 911cde }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 911cde }}


Badness: 0.036541
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 ====
==== 13-limit ====
Line 300: Line 439:
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 = 1\1, ~64/63 = 27.660
Optimal tuning (POTE): ~2 = 1200.000, ~64/63 = 27.660


{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347e, 564e }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}


Badness: 0.019504
Badness (Smith): 0.019504


[[Category:Temperament families]]
[[Category:Temperament families]]
[[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

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 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

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: [⟨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

Return to the map

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

Return to map

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

Return to map

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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