94edo: Difference between revisions
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Its step size is close to that of [[144/143]], which is consistently represented in this tuning system. | Its step size is close to that of [[144/143]], which is consistently represented in this tuning system. | ||
94edo can be thought of as two sets of [[47edo]] offset by one step of 94edo. It inherits from 47edo's good approximations of primes 5, 7, 13 and 17, while it dramatically improves on prime 3, as well as primes 11, 19 and 23 to a lesser degree. | 94edo can be thought of as two sets of [[47edo]] offset by one step of 94edo. It inherits from 47edo's good approximations of primes 5, 7, 13, and 17, while it dramatically improves on prime 3, as well as primes 11, 19, and 23 to a lesser degree. | ||
=== Significance of cassandra === | === Significance of cassandra === | ||
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94edo is an excellent edo for [[Carlos Beta]] scale, since the difference between 1 step of Carlos Beta and 5 steps of 94edo is only 0.00314534 cents. | 94edo is an excellent edo for [[Carlos Beta]] scale, since the difference between 1 step of Carlos Beta and 5 steps of 94edo is only 0.00314534 cents. | ||
===Prime harmonics=== | === Prime harmonics === | ||
{{Harmonics in equal|94|columns=11}} | {{Harmonics in equal|94|columns=11}} | ||
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{{See also | Table of 94edo intervals }} | {{See also | Table of 94edo intervals }} | ||
Assuming [[23-limit]] [[patent val]] {{val| 94 149 218 264 325 348 384 399 425 }}, here is a table of intervals as approximated by [[94edo]] steps, and their corresponding 13-limit well-ordered extended diatonic interval names. 'S/s' indicates alteration by the septimal comma, [[64/63]]; 'K/k' indicates alteration by the syntonic comma, [[81/80]]; 'U/u' by the undecimal quartertone, [[33/32]]; 'L/l' by pentacircle comma, [[896/891]]; 'O/o' by [[45/44]]; 'R/r' by the rastma, [[243/242]]; 'T/t' by the tridecimal quartertone, [[1053/1024]]; and finally, 'H/h', by [[40/39]]. Capital letters alter downward, lowercase alter upwards. Important 13-limit intervals approximated that are not associated with the extended diatonic interval names are added in brackets. Multiple alterations by 'K' down from augmented and major, or up from diminished and minor intervals are also added in brackets, along with their associated (5-limit) intervals. | Assuming [[23-limit]] [[patent val]] {{val| 94 149 218 264 325 348 384 399 425 }}, here is a table of intervals as approximated by [[94edo]] steps, and their corresponding 13-limit well-ordered extended diatonic interval names. 'S/s' indicates alteration by the septimal comma, [[64/63]]; 'K/k' indicates alteration by the syntonic comma, [[81/80]]; 'U/u' by the undecimal quartertone, [[33/32]]; 'L/l' by pentacircle comma, [[896/891]]; 'O/o' by [[45/44]]; 'R/r' by the rastma, [[243/242]]; 'T/t' by the tridecimal quartertone, [[1053/1024]]; and finally, 'H/h', by [[40/39]]. Capital letters alter downward, lowercase alter upwards. Important 13-limit intervals approximated that are not associated with the extended diatonic interval names are added in brackets. Multiple alterations by 'K' down from augmented and major, or up from diminished and minor intervals are also added in brackets, along with their associated (5-limit) intervals. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ style="font-size: 105%;" | 94edo [[SKULO interval names#WOFED interval names|WOFED interval names]] | |+ style="font-size: 105%;" | 94edo [[SKULO interval names#WOFED interval names|WOFED interval names]] | ||
| Line 800: | Line 801: | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
| Line 812: | Line 814: | ||
| {{monzo| 149 -94 }} | | {{monzo| 149 -94 }} | ||
| {{mapping| 94 149 }} | | {{mapping| 94 149 }} | ||
| | | −0.054 | ||
| 0.054 | | 0.054 | ||
| 0.43 | | 0.43 | ||
| Line 943: | Line 945: | ||
| [[Kleischismic]] | | [[Kleischismic]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for [[garibaldi temperament]], shows us how to extend it to the 23-limit. | Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for [[garibaldi temperament]], shows us how to extend it to the 23-limit. | ||
* 46&94 {{multival| 8 30 -18 -4 -28 8 -24 2 … }} | * {{nowrap|46 & 94}} {{multival| 8 30 -18 -4 -28 8 -24 2 … }} | ||
* 68&94 {{multival| 20 28 2 -10 24 20 34 52 … }} | * {{nowrap|68 & 94}} {{multival| 20 28 2 -10 24 20 34 52 … }} | ||
* 53&94 {{multival| 1 -8 -14 23 20 -46 -3 -35 … }} (one garibaldi) | * {{nowrap|53 & 94}} {{multival| 1 -8 -14 23 20 -46 -3 -35 … }} (one garibaldi) | ||
* 41&94 {{multival| 1 -8 -14 23 20 48 -3 -35 … }} (another garibaldi, only differing in the mappings of 17 and 23) | * {{nowrap|41 & 94}} {{multival| 1 -8 -14 23 20 48 -3 -35 … }} (another garibaldi, only differing in the mappings of 17 and 23) | ||
* 135&94 {{multival| 1 -8 -14 23 20 48 -3 59 … }} (another garibaldi) | * {{nowrap|135 & 94}} {{multival| 1 -8 -14 23 20 48 -3 59 … }} (another garibaldi) | ||
* 130&94 {{multival| 6 -48 10 -50 26 6 -18 -22 … }} (a pogo extension) | * {{nowrap|130 & 94}} {{multival| 6 -48 10 -50 26 6 -18 -22 … }} (a pogo extension) | ||
* 58&94 {{multival| 6 46 10 44 26 6 -18 -22 … }} (a supers extension) | * {{nowrap|58 & 94}} {{multival| 6 46 10 44 26 6 -18 -22 … }} (a supers extension) | ||
* 50&94 {{multival| 24 -4 40 -12 10 24 22 6 … }} | * {{nowrap|50 & 94}} {{multival| 24 -4 40 -12 10 24 22 6 … }} | ||
* 72&94 {{multival| 12 -2 20 -6 52 12 -36 -44 … }} (a gizzard extension) | * {{nowrap|72 & 94}} {{multival| 12 -2 20 -6 52 12 -36 -44 … }} (a gizzard extension) | ||
* 80&94 {{multival| 18 44 30 38 -16 18 40 28 … }} | * {{nowrap|80 & 94}} {{multival| 18 44 30 38 -16 18 40 28 … }} | ||
* 94 solo {{multival| 12 -2 20 -6 -42 12 -36 -44 … }} (a rank one temperament!) | * 94 solo {{multival| 12 -2 20 -6 -42 12 -36 -44 … }} (a rank one temperament!) | ||
Temperaments to which 94et can be detempered: | Temperaments to which 94et can be detempered: | ||
* [[Satin]] (94&311) {{multival| 3 70 -42 69 -34 50 85 83 … }} | * [[Satin]] ({{nowrap|94 & 311}}) {{multival| 3 70 -42 69 -34 50 85 83 … }} | ||
* 94&422 {{multival| 8 124 -18 90 -28 102 164 96 … }} | * {{nowrap|94 & 422}} {{multival| 8 124 -18 90 -28 102 164 96 … }} | ||
== Scales == | == Scales == | ||