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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.
{{Technical data page}}


Extensions of escapade 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 & 44), tempering out [[3136/3125]] and splitting the generator into two [[64/63]] intervals; all are considered below.
<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 ({{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 ==
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]].
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


[[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 15: 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"
|+ 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
|}


=== 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 (11/10)/(16/15) = [[33/32]]; this does minimal damage to the temperament.
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 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 }}


Badness: 0.0107
Badness: 0.0107
{| 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
|}


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


Mapping: {{Mapping| 1 2 2 3 5 | 0 -9 7 10 -1 }}
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }}


Optimal tuning (CTE): ~2 = 1\1, ~32/31 = 55.276
Badness (Sintel): 0.251


{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }}
{| 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 || 702.518 || +0.563
|-
| 5/4 || 386.931 || +0.617
|-
| 11/8 || 552.758 || +1.440
|-
| 31/16 || 1144.724 || -0.311
|}


Badness (Dirichlet): 0.251
= 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


== Escapist ==
== Escaped ==
This temperament makes the identification of the 4-generator interval, representing (16/15)<sup>2</sup>, with [[8/7]] by tempering out [[225/224]]; 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]].
''[[#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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 12288/12005
[[Comma list]]: 245/243, 65625/65536


{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}


{{Multival|legend=1| 9 -7 4 -32 -19 29 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 55.327
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
 
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}


[[Badness]]: 0.077950
[[Badness]] (Smith): 0.088746


=== 11-limit ===
=== 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 4 3 | 0 -9 7 -26 10 }}


Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}
Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.126


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.354
{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}


{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}
Badness (Smith): 0.035844


Badness: 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 || 703.831 || +1.876
|-
| 5/4 || 385.909 || -0.405
|-
| 7/4 || 966.624 || -2.202
|-
| 11/8 || 551.299 || -0.019
|}


=== 13-limit ===
==== 13-limit ====
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 tuning (POTE): ~2 = 1200.000, ~28/27 = 55.138


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


Badness: 0.035261
Badness (Smith): 0.031366


== Escaped ==
== Alphaquarter ==
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).''
''[[#Strong extensions|Return to the map]]''


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]].
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 65625/65536
[[Comma list]]: 5120/5103, 29360128/29296875


{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}


{{Multival|legend=1| 9 -7 26 -32 16 80 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16128/15625 = 55.243


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 55.122
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}


{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
[[Badness]] (Smith): 0.116594
 
[[Badness]]: 0.088746


=== 11-limit ===
=== 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 0 3 | 0 -9 7 61 10 }}


Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.243


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.126
{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}


{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
Badness (Smith): 0.029638


Badness: 0.035844
{| 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 ===
==== 13-limit ====
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 }}


Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.236


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


{{Optimal ET sequence|legend=1| 22, 65, 87, 283 }}
Badness (Smith): 0.025344


Badness: 0.031366
== Escapist ==
''[[#Strong extensions|Return to the map]]''


== Alphaquarter ==
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]].
Given the slightly sharp fifth 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.


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 29360128/29296875
[[Comma list]]: 225/224, 12288/12005


{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}


{{Multival|legend=1| 9 -7 -61 -32 -122 -122 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16128/15625 = 55.243
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}


{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
[[Badness]] (Smith): 0.077950
 
[[Badness]]: 0.116594


=== 11-limit ===
=== 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 3 3 | 0 -9 7 -4 10 }}


Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.354


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.243
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}


{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
Badness (Smith): 0.036700


Badness: 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 || 701.626 || -0.329
|-
| 5/4 || 387.624 || +1.310
|-
| 7/4 || 978.501 || +9.675
|-
| 11/8 || 553.749 || +2.431
|}


=== 13-limit ===
==== 13-limit ====
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 }}


Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
Optimal tuning (POTE): ~2 = 1200.000, ~26/25 = 55.550


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


{{Optimal ET sequence|legend=1| 65d, 87, 152f, 239f }}
Badness (Smith): 0.035261


Badness: 0.025344
= 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 ==
== Septisuperfourth ==
''[[#Weak extensions|Return to map]]''
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 191: Line 321:


: mapping generators: ~343/243, ~16875/16384
: 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 tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320
Line 198: Line 326:
{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}
{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}


[[Badness]]: 0.059241
[[Badness]] (Smith): 0.059241


=== 11-limit ===
=== 11-limit ===
Line 207: 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 (Smith): 0.024619


Badness: 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 220: 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 235: 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=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]]''
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 249: Line 396:


: mapping generators: ~2, ~64/63
: 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 tuning]] ([[POTE]]): ~2 = 1\1, ~64/63 = 27.668
Line 256: Line 401:
{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}
{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}


[[Badness]]: 0.094345
[[Badness]] (Smith): 0.094345


=== 11-limit ===
=== 11-limit ===
Line 265: 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=0| 43, 87, 130, 217, 347e, 911cde }}


{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347e, 911cde }}
Badness (Smith): 0.036541


Badness: 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 ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 278: 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]]
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