Escapade family: Difference between revisions
Add back 7-limit data. Spell 1\1 etc. in decimal. Note the badness metric. Style |
m Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| " Tags: Mobile edit Mobile web edit |
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== 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]]. | |||
=== 5-limit === | === 5-limit === | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 20: | Line 22: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1199.8082{{c}}, ~16875/16384 = 55.2840{{c}} | ||
* [[ | * [[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]] ( | [[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" | ||
| Line 40: | Line 42: | ||
=== 2.3.5.11 subgroup === | === 2.3.5.11 subgroup === | ||
Since (an ideally slightly flat) 4/3 is split | 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. | ||
Subgroup: 2.3.5.11 | Subgroup: 2.3.5.11 | ||
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Mapping: {{Mapping| 1 2 2 3 | 0 -9 7 10 }} | Mapping: {{Mapping| 1 2 2 3 | 0 -9 7 10 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e }} | ||
Badness: 0. | 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" | ||
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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|legend=0| 1 2 2 3 5 | 0 -9 7 10 -1 }} | |||
Optimal | 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=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e | {{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e }} | ||
Badness ( | 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" | ||
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|} | |} | ||
= Strong extensions = | |||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|+ style="font-size: 105%;" | Map to strong full 11-limit extensions | |+ 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* | ! rowspan="1" | Extension !! rowspan="1" | Mapping of 7 !! rowspan="1" | Tuning range* | ||
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<nowiki/>* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions | <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]]'' | ||
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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]]. | ||
=== 7-limit === | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 126: | Line 133: | ||
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }} | {{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 }} | {{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 2.246 | ||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 245/243, 385/384, 4000/3993 | Comma list: 245/243, 385/384, 4000/3993 | ||
{{Mapping|legend=0| 1 2 2 4 3 | 0 -9 7 -26 10 }} | |||
Optimal | 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=0| 22, 65, 87, 196, 283 }} | {{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }} | ||
Badness ( | 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" | ||
| Line 168: | Line 177: | ||
Comma list: 245/243, 352/351, 385/384, 625/624 | Comma list: 245/243, 352/351, 385/384, 625/624 | ||
{{Mapping|legend=0| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }} | |||
Optimal | 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=0| 22, 65, 87, 283 }} | {{Optimal ET sequence|legend=0| 22, 65, 87, 283 }} | ||
Badness ( | Badness (Sintel): 1.296 | ||
== Alphaquarter == | |||
''[[#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 | 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 | ||
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{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }} | {{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 }} | {{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 2.951 | ||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 3025/3024, 4000/3993, 5120/5103 | Comma list: 3025/3024, 4000/3993, 5120/5103 | ||
{{Mapping|legend=0| 1 2 2 0 3 | 0 -9 7 61 10 }} | |||
Optimal | 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=0| 65d, 87, 152, 239, 391 }} | {{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }} | ||
Badness ( | 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" | ||
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Comma list: 352/351, 625/624, 847/845, 1575/1573 | Comma list: 352/351, 625/624, 847/845, 1575/1573 | ||
{{Mapping|legend=0| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }} | |||
Optimal | 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=0| 65d, 87, 152f, 239f }} | {{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }} | ||
Badness ( | Badness (Sintel): 1.047 | ||
== Escapist == | |||
''[[#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 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 | [[Subgroup]]: 2.3.5.7 | ||
| Line 248: | Line 265: | ||
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }} | {{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 }} | {{Optimal ET sequence|legend=1| 21, 22, 43, 65d }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 1.973 | ||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 99/98, 176/175, 2560/2541 | Comma list: 99/98, 176/175, 2560/2541 | ||
{{Mapping|legend=0| 1 2 2 3 3 | 0 -9 7 -4 10 }} | |||
Optimal | 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=0| 21, 22, 43, 65d }} | {{Optimal ET sequence|legend=0| 21, 22, 43, 65d }} | ||
Badness ( | 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" | ||
| Line 290: | Line 309: | ||
Comma list: 78/77, 99/98, 176/175, 507/500 | Comma list: 78/77, 99/98, 176/175, 507/500 | ||
{{Mapping|legend=0| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }} | |||
Optimal | 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 }} | {{Optimal ET sequence|legend=0| 21, 22, 43 }} | ||
Badness ( | Badness (Sintel): 1.457 | ||
= Weak extensions = | |||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|+ style="font-size: 105%;" | Map to weak extensions | |+ style="font-size: 105%;" | Map to weak extensions | ||
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|} | |} | ||
== Septisuperfourth == | |||
''[[#Weak extensions|Return to map]]'' | ''[[#Weak extensions|Return to map]]'' | ||
=== 7-limit === | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }} | {{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }} | ||
: mapping generators: ~343/243, ~ | : mapping generators: ~343/243, ~405/392 | ||
{{ | [[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}} | |||
{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }} | {{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 1.499 | ||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 540/539, 4000/3993, 5632/5625 | Comma list: 540/539, 4000/3993, 5632/5625 | ||
{{Mapping|legend=0| 2 4 4 7 6 | 0 -9 7 -15 10 }} | |||
Optimal | 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=0| 22, 86, 108, 130, 152, 282 }} | {{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282 }} | ||
Badness ( | 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" | ||
| Line 364: | Line 388: | ||
Comma list: 540/539, 729/728, 1575/1573, 3584/3575 | Comma list: 540/539, 729/728, 1575/1573, 3584/3575 | ||
{{Mapping|legend=0| 2 4 4 7 6 11 | 0 -9 7 -15 10 -39 }} | |||
Optimal | 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=0| 22f, 108f, 130, 282 }} | {{Optimal ET sequence|legend=0| 22f, 108f, 130, 282 }} | ||
Badness ( | Badness (Sintel): 0.946 | ||
==== Septisuperquad ==== | ==== Septisuperquad ==== | ||
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Comma list: 351/350, 364/363, 540/539, 4096/4095 | Comma list: 351/350, 364/363, 540/539, 4096/4095 | ||
{{Mapping|legend=0| 2 4 4 7 6 5 | 0 -9 7 -15 10 26 }} | |||
Optimal | 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 }} | {{Optimal ET sequence|legend=0| 22, 108, 130 }} | ||
Badness ( | Badness (Sintel): 1.365 | ||
== Arch == | |||
''[[#Weak extensions|Return to map]]'' | ''[[#Weak extensions|Return to map]]'' | ||
=== 7-limit === | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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: mapping generators: ~2, ~64/63 | : 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| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }} | {{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 2.388 | ||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 441/440, 3136/3125, 4000/3993 | Comma list: 441/440, 3136/3125, 4000/3993 | ||
{{Mapping|legend=0| 1 2 2 2 3 | 0 -18 14 35 20 }} | |||
Optimal | 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=0| 43, 87, 130, 217, 347e, 911cde }} | {{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 911cde }} | ||
Badness ( | 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" | ||
| Line 440: | Line 471: | ||
Comma list: 364/363, 441/440, 676/675, 3136/3125 | Comma list: 364/363, 441/440, 676/675, 3136/3125 | ||
{{Mapping|legend=0| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }} | |||
Optimal | 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=0| 43, 87, 130, 217, 347e, 564e }} | {{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }} | ||
Badness ( | Badness (Sintel): 0.806 | ||
[[Category:Escapade family| ]] <!-- main article --> | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||