Escapade family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,044 edits
mNo edit summary
Hkm (talk | contribs)
autopatrolled
960 edits
 
(17 intermediate revisions by 5 users not shown)
Line 2: Line 2:


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


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]].
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 &amp; 22)}}, tempering out [[225/224]] and mapping 7 to −4 generators; escaped {{nowrap|(22 &amp; 87)}}, tempering out [[245/243]] and mapping 7 to −26 generators; alphaquarter {{nowrap|(65d &amp; 87)}}, tempering out [[5120/5103]] and mapping 7 to 61 generators; septisuperfourth (aka biscapade) {{nowrap|(22 &amp; 86)}}, tempering out [[6144/6125]], splitting the octave in half and mapping 7 to −15 generators; and arch {{nowrap|(43 &amp; 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.


== Escapade (5-limit) ==
== 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


Line 19: Line 22:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~16875/16384 = 55.3052
* [[WE]]: ~2 = 1199.8082{{c}}, ~16875/16384 = 55.2840{{c}}
* [[POTE]]: ~2 = 1\1, ~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]]: 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 38: Line 41:
|}
|}


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


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


Optimal tuning (CTE): ~2 = 1\1, ~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 69: Line 71:
| 11/8 || 552.760 || +1.442
| 11/8 || 552.760 || +1.442
|}
|}
== Strong extensions ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong full 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


=== 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 {{nowrap|S31 {{=}} [[961/960]]}} so that the generator is equated to {{nowrap|[[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 ({{monzo|4 -2 -1 -1 1}}), 961/960 ({{monzo|-6 -1 -1 0 2}}), 4000/3993 ({{monzo|5 -1 3 -3 0}})
Comma list: 496/495 ({{monzo| 4 -2 -1 -1 1 }}), 961/960 ({{monzo| -6 -1 -1 0 2 }}), 4000/3993 ({{monzo| 5 -1 3 -3 0 }})


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


Optimal tuning (CTE): ~2 = 1\1, ~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=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 }}


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 115: Line 105:
|}
|}


=== Escapist ===
= 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
 
== Escaped ==
''[[#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 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]]s:
* [[WE]]: ~2 = 1199.9190{{c}}, ~28/27 = 55.1186{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 55.1242{{c}}
 
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
 
[[Badness]] (Sintel): 2.246


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


Badness: 0.036700
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 139: Line 163:
! CTE tuning !! Deviation from just
! CTE tuning !! Deviation from just
|-
|-
| 3/2 || 701.626 || -0.329
| 3/2 || 703.831 || +1.876
|-
|-
| 5/4 || 387.624 || +1.310
| 5/4 || 385.909 || -0.405
|-
|-
| 7/4 || 978.501 || +9.675
| 7/4 || 966.624 || -2.202
|-
|-
| 11/8 || 553.749 || +2.431
| 11/8 || 551.299 || -0.019
|}
|}


Line 151: Line 175:
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 3 3 3 | 0 -9 7 -4 10 15 }}
Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 55.550
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=1| 21, 22, 43 }}
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}


Badness: 0.035261
Badness (Sintel): 1.296


=== Escaped ===
== Alphaquarter ==
''[[#Strong extensions|Return to the map]]''
''[[#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 &amp; 27}} temperament (sensi extension).''
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


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]].
[[Comma list]]: 5120/5103, 29360128/29296875
 
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7349{{c}}, ~16128/15625 = 55.2306{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16128/15625 = 55.2405{{c}}
 
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
 
[[Badness]] (Sintel): 2.951


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


Badness: 0.035844
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 187: Line 229:
! CTE tuning !! Deviation from just
! CTE tuning !! Deviation from just
|-
|-
| 3/2 || 703.831 || +1.876
| 3/2 || 702.918 || +0.963
|-
|-
| 5/4 || 385.909 || -0.405
| 5/4 || 386.620 || +0.306
|-
|-
| 7/4 || 966.624 || -2.202
| 7/4 || 969.113 || +0.287
|-
|-
| 11/8 || 551.299 || -0.019
| 11/8 || 552.314 || +0.996
|}
|}


Line 199: Line 241:
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 4 3 2 | 0 -9 7 -26 10 37 }}
Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138
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=1| 22, 65, 87, 283 }}
{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}


Badness: 0.031366
Badness (Sintel): 1.047


=== Alphaquarter ===
== Escapist ==
''[[#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.
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
 
[[Comma list]]: 225/224, 12288/12005
 
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.9926{{c}}, ~49/48 = 55.2809{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 55.3479{{c}}


{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}
[[Badness]] (Sintel): 1.973
=== 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 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=1| 65d, 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}


Badness: 0.029638
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 233: Line 295:
! CTE tuning !! Deviation from just
! CTE tuning !! Deviation from just
|-
|-
| 3/2 || 702.918 || +0.963
| 3/2 || 701.626 || -0.329
|-
|-
| 5/4 || 386.620 || +0.306
| 5/4 || 387.624 || +1.310
|-
|-
| 7/4 || 969.113 || +0.287
| 7/4 || 978.501 || +9.675
|-
|-
| 11/8 || 552.314 || +0.996
| 11/8 || 553.749 || +2.431
|}
|}


Line 245: Line 307:
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 0 3 2 | 0 -9 7 61 10 37 }}
Mapping: {{mapping| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.236
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=1| 65d, 87, 152f, 239f }}
{{Optimal ET sequence|legend=0| 21, 22, 43 }}


Badness: 0.025344
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 268: Line 332:
|}
|}


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


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 6144/6125, 118098/117649
{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }}
: mapping generators: ~343/243, ~405/392
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8762{{c}}, ~405/392 = 55.3089{{c}}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~405/392 = 55.3273{{c}}
{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}
[[Badness]] (Sintel): 1.499
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 277: 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 = 1\2, ~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=1| 22, 86, 108, 130, 152, 282 }}
{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282 }}


Badness: 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 306: 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 = 1\2, ~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=1| 22f, 108f, 130, 282 }}
{{Optimal ET sequence|legend=0| 22f, 108f, 130, 282 }}


Badness: 0.022887
Badness (Sintel): 0.946


==== Septisuperquad ====
==== Septisuperquad ====
Line 321: 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 = 1\2, ~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=1| 22, 108, 130 }}
{{Optimal ET sequence|legend=0| 22, 108, 130 }}


Badness: 0.033038
Badness (Sintel): 1.365


=== Arch ===
== Arch ==
''[[#Weak extensions|Return to map]]''
''[[#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]]s:
* [[WE]]: ~2 = 1199.9246{{c}}, ~64/63 = 27.6662{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/63 = 27.6676{{c}}
{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}
[[Badness]] (Sintel): 2.388
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 336: 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 = 1\1, ~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=1| 43, 87, 130, 217, 347e, 911cde }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 911cde }}


Badness: 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 365: 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 = 1\1, ~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=1| 43, 87, 130, 217, 347e, 564e }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}


Badness: 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]]

Latest revision as of 23:18, 30 May 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

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:

  • WE: ~2 = 1199.8082 ¢, ~16875/16384 = 55.2840 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16875/16384 = 55.2961 ¢

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

Badness (Sintel): 1.965

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

Mapping: [⟨1 2 2 3], ⟨0 -9 7 10]]

Optimal tunings:

  • WE: ~2 = 1199.7406 ¢, ~33/32 = 55.2653 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.2770 ¢

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

Badness (Sintel): 0.335

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 tunings:

  • WE: ~2 = 1199.8050 ¢, ~32/31 = 55.2669 ¢
  • CWE: ~2 = 1200.0000 ¢, ~32/31 = 55.2759 ¢

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

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 tunings:

  • WE: ~2 = 1199.9190 ¢, ~28/27 = 55.1186 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 55.1242 ¢

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

Badness (Sintel): 2.246

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 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 55.1271 ¢

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

Badness (Sintel): 1.185

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 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): 1.296

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 tunings:

  • WE: ~2 = 1199.7349 ¢, ~16128/15625 = 55.2306 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16128/15625 = 55.2405 ¢

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

Badness (Sintel): 2.951

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 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.2407 ¢

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

Badness (Sintel): 0.980

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 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

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 tunings:

  • WE: ~2 = 1198.9926 ¢, ~49/48 = 55.2809 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 55.3479 ¢

Optimal ET sequence: 21, 22, 43, 65d

Badness (Sintel): 1.973

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 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.3574 ¢

Optimal ET sequence: 21, 22, 43, 65d

Badness (Sintel): 1.213

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 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

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, ~405/392

Optimal tunings:

  • WE: ~343/243 = 599.8762 ¢, ~405/392 = 55.3089 ¢
  • CWE: ~343/243 = 600.0000 ¢, ~405/392 = 55.3273 ¢

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

Badness (Sintel): 1.499

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 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/32 = 55.3081 ¢

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

Badness (Sintel): 0.814

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 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

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): 1.365

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 tunings:

  • WE: ~2 = 1199.9246 ¢, ~64/63 = 27.6662 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/63 = 27.6676 ¢

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

Badness (Sintel): 2.388

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 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/63 = 27.6617 ¢

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

Badness (Sintel): 1.208

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 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

Retrieved from "https://en.xen.wiki/index.php?title=Escapade_family&oldid=231365"