45edo: Difference between revisions
Dave Keenan (talk | contribs) →Sagittal notation: Changed "b" ET anchor to "#Second-best fifth notation". |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
45edo effectively has two 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]] | 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 tempers out the [[quartisma]] and provides an excellent tuning for the 2. | 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]]. | |||
Otherwise, it can be treated as a 2.5/3.7/3 subgroup system (borrowing 5/3 from [[15edo]] and 7/3 from [[9edo]]) and is a good tuning for [[gariberttet]], defined by tempering out [[3125/3087]] in this subgroup, approximating 2/5-comma gariberttet. | |||
=== Odd harmonics === | === Odd harmonics === | ||
{{ | {{Harmonics in equal|45}} | ||
=== Octave stretch === | === Octave stretch === | ||
45edo's approximations of 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 are all improved by [[equal tuning|13ed11/9]], a [[Octave stretch|stretched-octave]] version of 45edo. The trade-off is a slightly worse 2/1. | |||
The tuning [[126ed7]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1 and 13/1, at the cost of a slightly worse 2/1. | The tuning [[126ed7]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1 and 13/1, at the cost of a slightly worse 2/1. | ||
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== Notation == | == Notation == | ||
=== | === Ups and Downs notation === | ||
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp. | |||
{{sharpness-sharp2a}} | |||
=== Quarter-tone notation === | |||
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used. | |||
{{sharpness-sharp2}} | |||
=== 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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Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation. | Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation. | ||
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. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
=== Commas === | === Commas === | ||
This is a partial list of the [[commas]] that | This is a partial list of the [[commas]] that 45et [[tempering out|tempers out]] with its patent [[val]], {{val| 45 71 104 126 143 156 167 }}. | ||
{| class="commatable wikitable center-1 center-2 right-4 center-5" | {| class="commatable wikitable center-1 center-2 right-4 center-5" | ||
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| [[Quartisma]] | | [[Quartisma]] | ||
|} | |} | ||
<references group="note" /> | |||
== Instruments == | == Instruments == | ||
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== Music == | == Music == | ||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/shorts/33tKBiWZvXM ''(short clip) Fantasy in 45edo''] (2025) | |||
; [[JUMBLE]] | ; [[JUMBLE]] | ||
* [https://www.youtube.com/watch?v=tbc_OxHp-ec ''Fishbowl''] (2023) | * [https://www.youtube.com/watch?v=tbc_OxHp-ec ''Fishbowl''] (2023) | ||
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* [https://www.youtube.com/watch?v=Kd4t_iKiKMA ''Fallen Angel''] (2024) | * [https://www.youtube.com/watch?v=Kd4t_iKiKMA ''Fallen Angel''] (2024) | ||
* [https://www.youtube.com/watch?v=DPztb8W6ykY ''Solar Guardian''] (2024) | * [https://www.youtube.com/watch?v=DPztb8W6ykY ''Solar Guardian''] (2024) | ||
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number --> | [[Category:Equal divisions of the octave|##]] <!-- 2-digit number --> | ||