72edo: Difference between revisions
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== Theory == | == Theory == | ||
72edo approximates [[11-limit]] [[just intonation]] exceptionally well. It is [[consistent]] in the [[17-odd-limit]] and | 72edo approximates [[11-limit]] [[just intonation]] exceptionally well. It is the second edo (after [[58edo|58]]) to be [[consistent]] in the [[17-odd-limit]], and the second edo (also after 58) to be [[consistency|distinctly consistent]] in the [[11-odd-limit]], but it is the first edo to be [[consistency|consistent to distance 2]] in the 11-odd-limit, meaning every interval in the 11-odd-limit is approximated with less than 25% [[relative interval error|relative error]] (about 4 cents). It also has pretty good accuracy for the [[19-limit]], being almost consistent to the entire [[21-odd-limit]] with the only inconsistency occurring at [[19/13]] and its [[octave complement]]. It is the ninth [[zeta integral edo]]. | ||
The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33. | The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33. | ||
The 13th harmonic ( | The [[octave reduction|octave reduced]] [[13/1|13th harmonic]] is mapped on 50\72, an interval inherited from [[36edo]] (25\36) that is a very close approximation to [[acoustic phi]], and the [[17/1|17th]] and [[19/1|19th harmonics]] come from 12edo. | ||
72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]]. | 72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]]. | ||
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| 10 | | 10 | ||
| 166.7 | | 166.7 | ||
| [[11/10]] | | [[11/10]], [[21/19]] | ||
| {{UDnote|step=10}} | | {{UDnote|step=10}} | ||
|- | |- | ||
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| 20 | | 20 | ||
| 333.3 | | 333.3 | ||
| [[17/14]], [[39/32]], [[40/33]] | | [[17/14]], ''[[39/32]]'', [[40/33]] | ||
| {{UDnote|step=20}} | | {{UDnote|step=20}} | ||
|- | |- | ||
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| 52 | | 52 | ||
| 866.7 | | 866.7 | ||
| [[28/17]], [[33/20]], [[64/39]] | | [[28/17]], [[33/20]], ''[[64/39]]'' | ||
| {{UDnote|step=52}} | | {{UDnote|step=52}} | ||
|- | |- | ||
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| 62 | | 62 | ||
| 1033.3 | | 1033.3 | ||
| [[20/11]] | | [[20/11]], [[38/21]] | ||
| {{UDnote|step=62}} | | {{UDnote|step=62}} | ||
|- | |- | ||
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=== Proposed interval names and solfèges === | === Proposed interval names and solfèges === | ||
{| class="wikitable center-all right-2 left-4 left-7 mw-collapsible mw-collapsed" | {| class="wikitable center-all right-2 left-4 left-7 mw-collapsible mw-collapsed" | ||
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges | |+ style="font-size: 105%; white-space: nowrap;" | Table of proposed interval names and solfèges | ||
|- | |- | ||
! # | ! # | ||
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== Notation == | == Notation == | ||
=== | === Stein–Zimmermann–Gould notation === | ||
72edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc. | [[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows: | ||
{{Sharpness-sharp6-szg}} | |||
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals: | |||
{{Sharpness-sharp6-qt-szg}} | |||
=== Kite's ups and downs notation === | |||
72edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc. | |||
{{Ups and downs sharpness}} | {{Ups and downs sharpness}} | ||
Half-sharps and half-flats can be used to avoid triple arrows: | Half-sharps and half-flats can be used to avoid triple arrows: | ||
{{Ups and downs sharpness|72|true}} | {{Ups and downs sharpness|72|true}} | ||
=== Sagittal notation === | === Sagittal notation === | ||
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| 516.7 | | 516.7 | ||
| 27/20 | | 27/20 | ||
| [[ | | [[Gravity]] / [[marvo]] / [[zarvo]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 1,741: | Line 1,742: | ||
| 316.7<br>(50.0) | | 316.7<br>(50.0) | ||
| 6/5<br>(36/35) | | 6/5<br>(36/35) | ||
| [[Ennealimmal]] / ennealimnic | | [[Ennealimmal]] / ennealimnic / ennealiminal | ||
|- | |- | ||
| 9 | | 9 | ||
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| [[Gamelstearn]] | | [[Gamelstearn]] | ||
|} | |} | ||
<nowiki/>* [[Normal | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
== Octave stretch or compression == | == Octave stretch or compression == | ||
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; [[Jake Freivald]] | ; [[Jake Freivald]] | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday''] | * [https://web.archive.org/web/20201127014336/http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday''] in the [[lazysunday]] scale | ||
{{Wikipedia|In vain (Haas)}} | {{Wikipedia|In vain (Haas)}} | ||
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; [[Claudi Meneghin]] | ; [[Claudi Meneghin]] | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon – 72 edo''] | * [https://web.archive.org/web/20201127015744/http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon – 72 edo''] | ||
* [https://www.youtube.com/watch?v=zR0NDgh4944 ''The Miracle Canon'', 3-in-1 on a Ground] | * [https://www.youtube.com/watch?v=zR0NDgh4944 ''The Miracle Canon'', 3-in-1 on a Ground] | ||
* [https://www.youtube.com/watch?v=w6Bckog1eOM ''Sicilienne in Miracle''] | * [https://www.youtube.com/watch?v=w6Bckog1eOM ''Sicilienne in Miracle''] | ||
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; [[Prent Rodgers]] | ; [[Prent Rodgers]] | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 ''June Gloom #9''] | * [https://web.archive.org/web/20201127012907/http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 ''June Gloom #9''] | ||
; [[Gene Ward Smith]] | ; [[Gene Ward Smith]] | ||