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Escapade family: Difference between revisions

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<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.]]
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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 ==
== Escapade ==
=== 5-limit ===
=== 5-limit ===
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 21: Line 20:


[[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"
Line 49: Line 48:
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 }}
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| 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 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.
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 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 (Dirichlet): 0.251
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| 31/16 || 1144.724 || -0.311
| 31/16 || 1144.724 || -0.311
|}
|}
== 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


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


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


[[Subgroup]]: 2.3.5.7
[[Comma list]]: 245/243, 65625/65536
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}
{{Multival|legend=1| 9 -7 26 -32 16 80 }}
[[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


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


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


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


Badness: 0.035844
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"
Line 156: Line 170:
Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}
Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138
Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.138


{{Optimal ET sequence|legend=1| 22, 65, 87, 283 }}
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}


Badness: 0.031366
Badness (Smith): 0.031366


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


[[Subgroup]]: 2.3.5.7
[[Comma list]]: 5120/5103, 29360128/29296875
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}
{{Multival|legend=1| 9 -7 -61 -32 -122 -122 }}
[[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
Subgroup: 2.3.5.7.11


Line 173: Line 202:
Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}
Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}


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


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


Badness: 0.029638
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"
Line 202: Line 231:
Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.236
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.236
 
{{Optimal ET sequence|legend=1| 65d, 87, 152f, 239f }}


Badness: 0.025344
{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}


Badness (Smith): 0.025344


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


[[Subgroup]]: 2.3.5.7
[[Comma list]]: 225/224, 12288/12005
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}
{{Multival|legend=1| 9 -7 4 -32 -19 29 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}
[[Badness]] (Smith): 0.077950
==== 11-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 220: Line 263:
Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}
Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}


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


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


Badness: 0.036700
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"
Line 249: Line 292:
Mapping: {{mapping| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}
Mapping: {{mapping| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 55.550
Optimal tuning (POTE): ~2 = 1200.000, ~26/25 = 55.550


{{Optimal ET sequence|legend=1| 21, 22, 43 }}
{{Optimal ET sequence|legend=0| 21, 22, 43 }}


Badness: 0.035261
Badness (Smith): 0.035261


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


[[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, ~16875/16384
{{Multival|legend=1| 18 -14 30 -64 -3 109 }}
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320
{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}
[[Badness]] (Smith): 0.059241
==== 11-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 277: Line 337:
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"
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Line 306: Line 366:
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 321: Line 381:
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=1| 22, 108, 130 }}
{{Optimal ET sequence|legend=0| 22, 108, 130 }}


Badness: 0.033038
Badness (Smith): 0.033038


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


[[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
{{Multival|legend=1| 18 -14 -35 -64 -106 -42 }}
[[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 336: Line 413:
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"
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Line 365: Line 442:
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:Escapade family| ]] <!-- main article -->
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Escapade_family"