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{{Technical data page}}
{{Technical data page}}
<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 & 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.


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.
== 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]].


= 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 18: 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"
|+ 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 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.
 
== 2.3.5.11 subgroup ==
Since (an ideally slightly flat) 4/3 is split in three by the interval of 3 generators, it makes sense to equate that interval to [[11/10]] by tempering out [[4000/3993]], and therefore the generator to {{nowrap|(11/10)/(16/15) {{=}} [[33/32]]}}; this does minimal damage to the temperament. This structure in 2.3.5.11 occurs in all extensions of escapade to include prime 7, and therefore will be considered the fount of all further extensions.


Subgroup: 2.3.5.11
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 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"
|+ 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
|-
| 11/8 || 551.318 || 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]]
| 5/4 || 386.932 || +0.618
|-
|-
| [[#Escaped|Escaped]] || -26 || ↑ 87
| 11/8 || 552.760 || +1.442
|}
|}
<nowiki />* Defined by intersection with other documented 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, 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 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"
|+ 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 =
{| 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"
|+ 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 150: 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.


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]]: 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"
|+ 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 198: 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"
|+ 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 244: 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 3 3 3 | 0 -9 7 -4 10 15 }}
 
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=0| 21, 22, 43 }}
 
Badness (Sintel): 1.457
 
= 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: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.236
: mapping generators: ~343/243, ~405/392


{{Optimal ET sequence|legend=1| 65d, 87, 152f, 239f }}
[[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}}


Badness: 0.025344
{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}


== Weak extensions ==
[[Badness]] (Sintel): 1.499


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


Line 263: 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"
|+ 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 || 702.070 || +0.115
| 3/2 || 702.070 || +0.115
|-
|-
| 5/4 || 386.314 || 387.279 || +0.965
| 5/4 || 387.279 || +0.965
|-
|-
| 7/4 || 968.826 || 970.117 || +1.291
| 7/4 || 970.117 || +1.291
|-
|-
| 11/8 || 551.318 || 553.255 || +1.937
| 11/8 || 553.255 || +1.937
|}
|}


Line 292: 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 307: 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=0| 22, 108, 130 }}
 
Badness (Sintel): 1.365
 
== 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]]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| 22, 108, 130 }}
{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}


Badness: 0.033038
[[Badness]] (Sintel): 2.388


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


Line 320: 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"
|+ 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 || 702.178 || +0.223
| 3/2 || 702.178 || +0.223
|-
|-
| 5/4 || 386.314 || 387.195 || +0.881
| 5/4 || 387.195 || +0.881
|-
|-
| 7/4 || 968.826 || 967.987 || -0.839
| 7/4 || 967.987 || -0.839
|-
|-
| 11/8 || 551.318 || 553.135 || +1.817
| 11/8 || 553.135 || +1.817
|}
|}


Line 349: 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]]
Retrieved from "https://en.xen.wiki/w/Escapade_family"