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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 is the ninth [[zeta integral edo]]. It is the second edo (after [[58edo|58]]) to be [[consistency|distinctly consistent]] in the [[11-odd-limit]], 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 error (about 4 cents), and the first edo to be consistent in the 12- and 13-[[odd prime sum limit|odd-prime-sum-limit]].  
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 (octave reduced) is so closely mapped on [[acoustic phi]] that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament.
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 ==
=== Ups and downs 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}}
[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp6}}
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}


=== Sagittal notation ===
=== Sagittal notation ===
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| 516.7
| 516.7
| 27/20
| 27/20
| [[Marvo]] / [[zarvo]]
| [[Gravity]] / [[marvo]] / [[zarvo]]
|-
|-
| 1
| 1
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| 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 lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
<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'']{{dead link}} in the [[lazysunday]] scale
* [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'']{{dead link}}
* [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'']{{dead link}}
* [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]]
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