72edo: Difference between revisions
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== Notations == | == Notations == | ||
===Sagittal notation=== | === Sagittal notation === | ||
This notation uses the same sagittal sequence as EDOs [[65edo#Sagittal notation|65-EDO]] and [[79edo#Sagittal notation|79]], and is a superset of the notations for EDOs [[36edo#Sagittal notation|36]], [[24edo#Sagittal notation|24]], [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], [[8edo#Sagittal notation|8]], and [[6edo#Sagittal notation|6]]. | This notation uses the same sagittal sequence as EDOs [[65edo#Sagittal notation|65-EDO]] and [[79edo#Sagittal notation|79]], and is a superset of the notations for EDOs [[36edo#Sagittal notation|36]], [[24edo#Sagittal notation|24]], [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], [[8edo#Sagittal notation|8]], and [[6edo#Sagittal notation|6]]. | ||
==== Evo flavor ==== | |||
<imagemap> | <imagemap> | ||
File:72-EDO_Evo_Sagittal.svg | File:72-EDO_Evo_Sagittal.svg | ||
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</imagemap> | </imagemap> | ||
====Revo flavor==== | ==== Revo flavor ==== | ||
<imagemap> | <imagemap> | ||
File:72-EDO_Revo_Sagittal.svg | File:72-EDO_Revo_Sagittal.svg | ||
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</imagemap> | </imagemap> | ||
====Evo-SZ flavor==== | ==== Evo-SZ flavor ==== | ||
<imagemap> | <imagemap> | ||
File:72-EDO_Evo-SZ_Sagittal.svg | File:72-EDO_Evo-SZ_Sagittal.svg | ||
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=== Ups and downs notation === | === Ups and downs notation === | ||
Using [[ | Using [[Helmholtz–Ellis]] accidentals, 72edo can also be notated using [[ups and downs notation]]: | ||
{{Sharpness-sharp6|72}} | {{Sharpness-sharp6|72}} | ||
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== JI approximation == | == JI approximation == | ||
[[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]] | [[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]] | ||
=== Z function === | === Z function === | ||
72edo is the ninth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[The Riemann Zeta Function and Tuning #The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72. | 72edo is the ninth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[The Riemann Zeta Function and Tuning #The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72. | ||
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{| class="commatable wikitable center-1 center-2 right-4" | {| class="commatable wikitable center-1 center-2 right-4" | ||
|- | |||
! [[Harmonic limit|Prime<br>limit]] | ! [[Harmonic limit|Prime<br>limit]] | ||
! [[Ratio]]<ref group="note">{{rd}}</ref> | ! [[Ratio]]<ref group="note">{{rd}}</ref> | ||
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== Zeta properties == | == Zeta properties == | ||
===Zeta peak index=== | === Zeta peak index === | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
!ZPI | ! colspan="3" | Tuning | ||
!Steps per octave | ! colspan="3" | Strength | ||
!Step size (cents) | ! colspan="2" | Closest EDO | ||
!Height | ! colspan="2" | Integer limit | ||
!Integral | |- | ||
! ZPI | |||
! Steps per octave | |||
! Step size (cents) | |||
! Height | |||
! Integral | |||
! Gap | ! Gap | ||
!EDO | ! EDO | ||
!Octave (cents) | ! Octave (cents) | ||
!Consistent | ! Consistent | ||
! Distinct | ! Distinct | ||
|- | |- | ||
|[[380zpi]] | | [[380zpi]] | ||
|71.9506065993786 | | 71.9506065993786 | ||
|16.6781081733140 | | 16.6781081733140 | ||
|9.157547 | | 9.157547 | ||
|1.625363 | | 1.625363 | ||
|19.964746 | | 19.964746 | ||
|72edo | | 72edo | ||
| 1200.82378847861 | | 1200.82378847861 | ||
|18 | | 18 | ||
|13 | | 13 | ||
|} | |} | ||
== Scales == | == Scales == | ||
* [[Smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]] | * [[Smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]] | ||
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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 | * [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon – 72 edo'']{{dead link}} | ||
* [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''] | ||