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Escapade family: Difference between revisions

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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 &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 ==
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
* [[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"
|+ 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 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.
 
== 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 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 }}
Line 58: Line 59:
|+ 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
|-
| 3/2 || 701.955 || 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*
! CTE tuning !! Deviation from just
|-
|-
| [[#Escapist|Escapist]] || -4 || ↓ [[65edo|65]]
| 3/2 || 702.516 || +0.561
|-
|-
| [[#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 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 (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 from just
|-
| 3/2 || 702.518 || +0.563
|-
| 5/4 || 386.931 || +0.617
|-
| 11/8 || 552.758 || +1.440
|-
|-
! CTE tuning !! Deviation
| 31/16 || 1144.724 || -0.311
|}
 
= Strong extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong full 7- and 11-limit extensions
|-
|-
| 3/2 || 701.955 || 702.518 || +0.563
! rowspan="1" | Extension !! rowspan="1" | Mapping of 7 !! rowspan="1" | Tuning range*
|-
|-
| 5/4 || 386.314 || 386.931 || +0.617
| [[#Escapist|Escapist]] || -4 || ↓ [[65edo|65]]
|-
|-
| 11/8 || 551.318 || 552.758 || +1.440
| [[#Alphaquarter|Alphaquarter]] || +61 || ↑ 65 <br> ↓ [[87edo|87]]
|-
|-
| 31/16 || 1145.036 || 1144.724 || -0.311
| [[#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 ==
''[[#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]] ([[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


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


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


Badness: 0.036700
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"
|+ 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 167:
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 ==
''[[#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
 
[[Comma list]]: 5120/5103, 29360128/29296875


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]].
{{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
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 tuning (POTE): ~2 = 1200.000, ~33/32 = 55.243


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


Badness: 0.035844
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"
|+ 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 227:
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 tuning (POTE): ~2 = 1200.000, ~33/32 = 55.236


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


Badness: 0.031366
Badness (Smith): 0.025344


=== 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]] ([[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


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


{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}


Badness: 0.029638
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"
|+ 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 287:
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 tuning (POTE): ~2 = 1200.000, ~26/25 = 55.550


{{Optimal ET sequence|legend=1| 65d, 87, 152f, 239f }}
{{Optimal ET sequence|legend=0| 21, 22, 43 }}


Badness: 0.025344
Badness (Smith): 0.035261


== Weak extensions ==
= Weak extensions =
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong extensions
|+ style="font-size: 105%;" | Map to weak extensions
|-
|-
! rowspan="2" | Extensions !! rowspan="2" | Periods per octave !! colspan="2" | Position of original generator
! rowspan="2" | Extensions !! rowspan="2" | Periods per octave !! colspan="2" | Position of original generator
Line 262: Line 305:
! Number of generators !! Number of periods
! Number of generators !! Number of periods
|-
|-
| [[#Septisuperfourth|Septisuperfourth]] || 2 || 1 generator, || up 0 periods
| [[#Septisuperfourth|Septisuperfourth]] || period = 1/2 octave || 1 generator || + 0 periods
|-
|-
| [[#Arch|Arch]] || 1 || 2 generators, || up 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 }}


<nowiki />* Defined by intersection with other documented extensions
: mapping generators: ~343/243, ~16875/16384


=== Septisuperfourth ===
[[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 275: 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: 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"
|+ 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 304: 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 319: 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=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| 22, 108, 130 }}
{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}


Badness: 0.033038
[[Badness]] (Smith): 0.094345


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


Line 332: 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=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"
|+ 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 361: 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]]
Retrieved from "https://en.xen.wiki/w/Escapade_family"