Xenharmonic Wiki

Escapade family: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled
4,516 edits
No edit summary
 
(87 intermediate revisions by 9 users not shown)
Line 1: Line 1:
The '''escapade family''' tempers out the [[escapade_comma|escapade comma]], |32 -7 -9>, of size 9.492 [[cent]]s. It includes the [[Hemifamity temperaments|alphaquarter temperament]] and the [[Hemimean clan|arch temperament]].
{{Technical data page}}


=Escapade=
<div style="float: right;">
Comma: 4294967296/4271484375
[[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>


[[POTE_tuning|POTE generator]]: ~16875/16384 = 55.293
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]].


Map: [&lt;1 2 2|, &lt;0 -9 7|]
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.


EDOs: [[21edo|21]], [[22edo|22]], [[43edo|43]], [[65edo|65]], 152, 217, 586, 803
== 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]].


Badness: 0.0838
=== 5-limit ===
[[Subgroup]]: 2.3.5


==7-limit==
[[Comma list]]: 4294967296/4271484375 ({{monzo|32 -7 -9}})
Commas: 225/224, 12288/12005


POTE generator: ~49/48 = 55.327
{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}


Map: [&lt;1 2 2 3|, &lt;0 -9 7 -4|]
: mapping generators: ~2, ~16875/16384


Wedgie: &lt;&lt;9 -7 4 -32 -19 29||
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~16875/16384 = 55.3052
* [[POTE]]: ~2 = 1200.000, ~16875/16384 = 55.293


EDOs: 21, 22, 43, 65d
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}


Badness: 0.0779
[[Badness]] (Smith): 0.083778


==11-limit==
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Commas: 99/98, 176/175, 2560/2541
|+ 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
|}


POTE generator: ~49/48 = 55.354
=== 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.


Map: [&lt;1 2 2 3 3|, &lt;0 -9 7 -4 10|]
Subgroup: 2.3.5.11


EDOs: 21, 22, 43, 65d
Comma list: 4000/3993 ({{monzo|5 -1 3 -3}}), 5632/5625 ({{monzo|9 -2 -4 1}})


Badness: 0.0367
Mapping: {{Mapping| 1 2 2 3 | 0 -9 7 10 }}


==13-limit==
Optimal tuning (CTE): ~2 = 1200.0000, ~33/32 = 55.2760
Commas: 78/77, 99/98, 176/175, 507/500


POTE generator: ~26/25 = 55.550
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 87, 152, 369, 521e, 1194bcee, 1715bceeee }}


Map: [&lt;1 2 2 3 3 3|, &lt;0 -9 7 -4 10 15|]
Badness: 0.0107


EDOs: 21, 22, 43
{| 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
|}


Badness: 0.0353
=== 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.


=Escaped=
Subgroup: 2.3.5.11.31
Commas: 245/243, 65625/65536


[[POTE_tuning|POTE generator]]: ~28/27 = 55.122
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 }})


Map: [&lt;1 2 2 4|, &lt;0 -9 7 -26|]
Mapping: {{mapping| 1 2 2 3 5 | 0 -9 7 10 -1 }}


Wedgie: &lt;&lt;9 -7 26 -32 16 80||
Optimal tuning (CTE): ~2 = 1200.000, ~32/31 = 55.276


EDOs: 22, 65, [[87edo|87]], 196, 283
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }}


Badness: 0.0887
Badness (Sintel): 0.251


==11-limit==
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Commas: 245/243, 385/384, 3388/3375
|+ 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
|}


POTE generator: ~28/27 = 55.126
= 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


Map: [&lt;1 2 2 4 3|, &lt;0 -9 7 -26 10|]
== Escaped ==
''[[#Strong extensions|Return to the map]]''


EDOs: 22, 65, 87, 196, 283
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).''


Badness: 0.0358
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]].


==13-limit==
=== 7-limit ===
Commas: 245/243, 352/351, 364/363, 625/624
[[Subgroup]]: 2.3.5.7


POTE generator: ~28/27 = 55.138
[[Comma list]]: 245/243, 65625/65536


Map: [&lt;1 2 2 4 3 2|, &lt;0 -9 7 -26 10 37|]
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}


EDOs: 22, 65, 87
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122


Badness: 0.0314
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}


=Biscapade=
[[Badness]] (Smith): 0.088746
Commas: 6144/6125, 118098/117649


[[POTE_tuning|POTE generator]]: ~16875/16384 = 55.230
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 4 4 7|, &lt;0 -9 7 -15|]
Comma list: 245/243, 385/384, 4000/3993


Wedgie: &lt;&lt;18 -14 30 -64 -3 109||
Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}


EDOs: 22, [[86edo|86]], 108, 130, 152, 282
Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.126


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


==11-limit==
Badness (Smith): 0.035844
Commas: 540/539, 4000/3993, 5632/5625


POTE generator: ~33/32 = 55.304
{| 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
|}


Map: [&lt;2 4 4 7 6|, &lt;0 -9 7 -15 10|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 22, 86, 108, 130, 152, 282
Comma list: 245/243, 352/351, 385/384, 625/624


Badness:  
Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}


==13-limit==
Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.138
Commas: 351/350, 364/363, 540/539, 4096/4095


POTE generator: ~33/32 =55.359
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}


Map: [&lt;2 4 4 7 6 5|, &lt;0 -9 7 -15 10 26|]
Badness (Smith): 0.031366


EDOs: 22, 108, 130
== Alphaquarter ==
''[[#Strong extensions|Return to the map]]''


Badness:
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.


[[Category:escapade]]
=== 7-limit ===
[[Category:family]]
[[Subgroup]]: 2.3.5.7
[[Category:list]]
 
[[Category:overview]]
[[Comma list]]: 5120/5103, 29360128/29296875
[[Category:theory]]
 
{{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
 
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
 
[[Comma list]]: 225/224, 12288/12005
 
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}
 
[[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
 
Comma list: 99/98, 176/175, 2560/2541
 
Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.354
 
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}
 
Badness (Smith): 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 || 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: {{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
 
[[Comma list]]: 6144/6125, 118098/117649
 
{{Mapping|legend=1| 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|legend=1| 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: {{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|legend=0| 22, 86, 108, 130, 152, 282 }}
 
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 ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1575/1573, 3584/3575
 
Mapping: {{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|legend=0| 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: {{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 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~64/63 = 27.660
 
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}
 
Badness (Smith): 0.019504
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Escapade_family"