45edo: Difference between revisions
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{{EDO intro|45}} | {{EDO intro|45}} | ||
== Theory == | == Theory == | ||
45edo effectively has two approximate major thirds, each almost equally far from [[just]], but as the flat one is slightly closer, it qualifies as a [[meantone]] temperament, forming a good approximation to [[2/5-comma meantone]]. It is a flat-tending system in the [[7-limit]], with 3, 5 and 7 all flat, but the 11 is sharp. | 45edo effectively has two approximate major thirds, each almost equally far from [[just]], but as the flat one is slightly closer, it qualifies as a [[meantone]] temperament, forming a good approximation to [[2/5-comma meantone]]. It is a flat-tending system in the [[7-limit]], with 3, 5, and 7 all flat, but the 11 is sharp. | ||
It provides the [[optimal patent val]] for [[flattone]] temperament, 7-limit rank-3 [[avicennmic]] temperament [[tempering out]] [[525/512]], the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank-4 temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is also the unique equal temperament tuning | It provides the [[optimal patent val]] for [[flattone]] temperament, 7-limit rank-3 [[avicennmic]] temperament [[tempering out]] [[525/512]], the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank-4 temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is also the unique equal temperament tuning whose patent val tempers out both the syntonic comma and the [[ennealimma]]. | ||
45edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.7/3.33 subgroup [[The Quartercache #Direct quartismic|direct quartismic]] temperament, in which it approximates the [[33/32]] quartertone with 2 steps and [[7/6]] with 10 steps. A bit more broadly, it maps the 2.17.25.27.33.63.65 subgroup to great precision; this is the part of the [[17-limit]] shared with [[270edo]]. | 45edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.7/3.33 subgroup [[The Quartercache #Direct quartismic|direct quartismic]] temperament, in which it approximates the [[33/32]] quartertone with 2 steps and [[7/6]] with 10 steps. A bit more broadly, it maps the 2.17.25.27.33.63.65 subgroup to great precision; this is the part of the [[17-limit]] shared with [[270edo]]. | ||
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== Notation == | == Notation == | ||
=== Sagittal notation === | |||
===Sagittal notation=== | |||
This notation uses the same sagittal sequence as EDOs [[52edo#Sagittal notation|52]] and [[59edo#Second-best fifth notation|59b]]. | This notation uses the same sagittal sequence as EDOs [[52edo#Sagittal notation|52]] and [[59edo#Second-best fifth notation|59b]]. | ||
==== Evo flavor ==== | |||
<imagemap> | <imagemap> | ||
File:45-EDO_Evo_Sagittal.svg | File:45-EDO_Evo_Sagittal.svg | ||
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</imagemap> | </imagemap> | ||
====Revo flavor==== | ==== Revo flavor ==== | ||
<imagemap> | <imagemap> | ||
File:45-EDO_Revo_Sagittal.svg | File:45-EDO_Revo_Sagittal.svg | ||
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</imagemap> | </imagemap> | ||
====Evo-SZ flavor==== | ==== Evo-SZ flavor ==== | ||
<imagemap> | <imagemap> | ||
File:45-EDO_Evo-SZ_Sagittal.svg | File:45-EDO_Evo-SZ_Sagittal.svg | ||
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In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO. | In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO. | ||
=== Quarter-tone notation === | |||
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used: | |||
{{sharpness-sharp2}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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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 | |- | ||
!Gap | ! ZPI | ||
!EDO | ! Steps per octave | ||
!Octave (cents) | ! Step size (cents) | ||
!Consistent | ! Height | ||
!Distinct | ! Integral | ||
! Gap | |||
! EDO | |||
! Octave (cents) | |||
! Consistent | |||
! Distinct | |||
|- | |- | ||
|[[207zpi]] | | [[207zpi]] | ||
|44.8397488079316 | | 44.8397488079316 | ||
|26.7619697233392 | | 26.7619697233392 | ||
|5.252141 | | 5.252141 | ||
|0.837163 | | 0.837163 | ||
|14.719415 | | 14.719415 | ||
|45edo | | 45edo | ||
|1204.28863755026 | | 1204.28863755026 | ||
|7 | | 7 | ||
|7 | | 7 | ||
|} | |} | ||
== Instruments == | == Instruments == | ||
'''Lumatone''' | '''Lumatone''' | ||