45edo: Difference between revisions

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== 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]]. It is the [[optimal patent val]] for [[flattone]] temperament, the [[7-limit]] 525/512 planar [[Avicennmic_temperaments|avicennmic]] temperament, the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank four temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. 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.

~~45edo tempers out~~ the [[~~quartisma~~]] ~~and provides an excellent tuning~~ for the ~~2.33~~/~~32.7/6 subgroup~~ [[~~The Quartercache#Direct quartismic|direct quartismic~~]] temperament, ~~in which it approximates 33~~/~~32 quartertone with 2 steps~~ and 7/~~6 with 10 steps~~. It is also the unique equal temperament tuning ~~that~~ tempers out both the syntonic comma and the [[ennealimma]].

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]].

~~=== Odd harmonics ===~~

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]].

~~{{harmonics~~ in ~~equal|45}}~~

~~=== Octave stretch ===~~

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.

~~45edo’s approximations of 3/1~~, 5/1, 7/~~1, 11~~/~~1, 13~~/1 and ~~17/1 are all improved by~~ [[~~Gallery of arithmetic pitch sequences#APS of farabs|APS3.21farab~~]], 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.~~

=== Odd harmonics ===

~~There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used for this same purpose: 207zpi, 208zpi and 209zpi. The main Zeta peak index page details all three tunings.~~

== Intervals ==

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| D

|}

== Notation ==

=== Ups and Downs notation ===

Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.

=== Quarter-tone notation ===

Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used.

=== Sagittal notation ===

This notation uses the same sagittal sequence as EDOs [[52edo#Sagittal notation|52]] and [[59edo#Second-best fifth notation|59b]].

==== Evo flavor ====

File:45-EDO_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 140 106 [[36/35]]

rect 140 80 300 106 [[1053/1024]]

default [[File:45-EDO_Evo_Sagittal.svg]]

</imagemap>

==== Revo flavor ====

File:45-EDO_Revo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 140 106 [[36/35]]

rect 140 80 300 106 [[1053/1024]]

default [[File:45-EDO_Revo_Sagittal.svg]]

</imagemap>

==== Evo-SZ flavor ====

File:45-EDO_Evo-SZ_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 511 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 140 106 [[36/35]]

rect 140 80 300 106 [[1053/1024]]

default [[File:45-EDO_Evo-SZ_Sagittal.svg]]

</imagemap>

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 ==

=== Commas ===

This is a partial list of the [[commas]] that ~~45edo~~ [[tempers out]] with its patent [[val]], {{val| 45 71 104 126 143 156 167 }}.

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"

Line 534:

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| [[Quartisma]]

|}

== Octave stretch and compression ==

45edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by an [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed6|116ed6]]. The trade-off is a slightly worse 2/1.

The tuning [[equal tuning|183ed17]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1, but at the cost of a noticeably worse 2/1 than 116ed6.

What follows is a comparison of compressed- and stretched-octave 45edo tunings.

; [[zpi|209zpi]]

* Step size: 26.550{{c}}, octave size: 1194.8{{c}}

Compressing the octave of 45edo by around 5{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 11.1{{c}}. The tuning 209zpi does this.

{{Harmonics in cet|26.550|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 209zpi}}

{{Harmonics in cet|26.550|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 209zpi (continued)}}

; 45edo

* Step size: 26.667{{c}}, octave size: 1200.0{{c}}

Pure-octaves 45edo approximates all harmonics up to 16 within 13.0{{c}}.

{{Harmonics in equal|45|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45edo}}

{{Harmonics in equal|45|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45edo (continued)}}

; [[WE|45et, 13-limit WE tuning]]

* Step size: 26.695{{c}}, octave size: 1201.3{{c}}

Stretching the octave of 45edo by around 1{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|26.695|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning}}

{{Harmonics in cet|26.695|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning (continued)}}

; [[ed12|161ed12]]

* Step size: Octave size: 1202.4{{c}}

Stretching the octave of 45edo by around 2.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 12.2{{c}}. The tuning 161ed12 does this.

{{Harmonics in equal|161|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 161ed12}}

{{Harmonics in equal|161|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 161ed12 (continued)}}

; [[ed6|116ed6]]

* Step size: Octave size: 1203.3{{c}}

Stretching the octave of 45edo by around 3{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 116ed6 does this. So does [[ed7|126ed7]] whose octave is identical within 0.1{{c}}.

{{Harmonics in equal|116|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 116ed6}}

{{Harmonics in equal|116|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 116ed6 (continued)}}

; [[WE|45et, 7-limit WE tuning]]

* Step size: 26.745{{c}}, octave size: 1203.5{{c}}

Stretching the octave of 45edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.6{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.

{{Harmonics in cet|26.745|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning}}

{{Harmonics in cet|26.745|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning (continued)}}

; [[zpi|207zpi]]

* Step size: 26.762{{c}}, octave size: 1204.3{{c}}

Stretching the octave of 45edo by around 4{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 207zpi does this.

{{Harmonics in cet|26.762|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 207zpi}}

{{Harmonics in cet|26.762|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 207zpi (continued)}}

; [[71edt]]

* Step size: 26.788{{c}}, octave size: 1205.5{{c}}

Stretching the octave of 45edo by around 5.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 71edt does this. So do the tunings [[ed5|104ed5]] and [[equal tuning|155ed11]] whose octave is identical within 0.3{{c}}.

{{Harmonics in equal|71|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 71edt}}

{{Harmonics in equal|71|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 71edt (continued)}}

; [[equal tuning|183ed17]]

* Octave size: 1206.1{{c}}

Stretching the octave of 45edo by around 6{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 13.3{{c}}. The tuning 183ed17 does this.

{{Harmonics in equal|183|17|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 183ed17}}

{{Harmonics in equal|183|17|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 183ed17 (continued)}}

== Instruments ==

Line 541:

Line 650:

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/33tKBiWZvXM ''(short clip) Fantasy in 45edo''] (2025)

; [[JUMBLE]]

* [https://www.youtube.com/watch?v=tbc_OxHp-ec ''Fishbowl''] (2023)

Line 546:

Line 658:

* [https://www.youtube.com/watch?v=Kd4t_iKiKMA ''Fallen Angel''] (2024)

* [https://www.youtube.com/watch?v=DPztb8W6ykY ''Solar Guardian''] (2024)

~~== Notes ==~~

~~<references group="note" />~~

[[Category:Equal divisions of the octave|##]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|45}}
+{{ED intro}}
 == Theory ==
-edo 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]]. It is the [[optimal patent val]] for [[flattone]] temperament, the [[7-limit]] 525/512 planar [[Avicennmic_temperaments|avicennmic]] temperament, the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank four temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is a flat-tending system in the 7-limit, with 3, 5 and 7 all flat, but the 11 is sharp.
+edo 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.
-edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.33/32.7/6 subgroup [[The Quartercache#Direct quartismic|direct quartismic]] temperament, in which it approximates 33/32 quartertone with 2 steps and 7/6 with 10 steps. It is also the unique equal temperament tuning that tempers out both the syntonic comma and the [[ennealimma]].
+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]].
-=== Odd harmonics ===
+edo 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]].
-{{harmonics in equal|45}}
-=== Octave stretch ===
+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.
-edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 are all improved by [[Gallery of arithmetic pitch sequences#APS of farabs|APS3.21farab]], 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.
+=== Odd harmonics ===
+{{Harmonics in equal|45}}
-There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used for this same purpose: 207zpi, 208zpi and 209zpi. The main Zeta peak index page details all three tunings.
 == Intervals ==
@@ Line 443: / Line 440: @@
 | D
 |}
+== 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]].
+==== Evo flavor ====
+<imagemap>
+File:45-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 140 106 [[36/35]]
+rect 140 80 300 106 [[1053/1024]]
+default [[File:45-EDO_Evo_Sagittal.svg]]
+</imagemap>
+==== Revo flavor ====
+<imagemap>
+File:45-EDO_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 140 106 [[36/35]]
+rect 140 80 300 106 [[1053/1024]]
+default [[File:45-EDO_Revo_Sagittal.svg]]
+</imagemap>
+==== Evo-SZ flavor ====
+<imagemap>
+File:45-EDO_Evo-SZ_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 511 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 140 106 [[36/35]]
+rect 140 80 300 106 [[1053/1024]]
+default [[File:45-EDO_Evo-SZ_Sagittal.svg]]
+</imagemap>
+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 ==
 === Commas ===
-This is a partial list of the [[commas]] that 45edo [[tempers out]] with its patent [[val]], {{val| 45 71 104 126 143 156 167 }}.
+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"
@@ Line 534: / Line 581: @@
 | [[Quartisma]]
 |}
+<references group="note" />
+== Octave stretch and compression ==
+edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by an [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed6|116ed6]]. The trade-off is a slightly worse 2/1.
+The tuning [[equal tuning|183ed17]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1, but at the cost of a noticeably worse 2/1 than 116ed6.
+What follows is a comparison of compressed- and stretched-octave 45edo tunings.
+; [[zpi|209zpi]]
+* Step size: 26.550{{c}}, octave size: 1194.8{{c}}
+Compressing the octave of 45edo by around 5{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 11.1{{c}}. The tuning 209zpi does this.
+{{Harmonics in cet|26.550|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 209zpi}}
+{{Harmonics in cet|26.550|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 209zpi (continued)}}
+; 45edo
+* Step size: 26.667{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 45edo approximates all harmonics up to 16 within 13.0{{c}}.
+{{Harmonics in equal|45|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45edo}}
+{{Harmonics in equal|45|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45edo (continued)}}
+; [[WE|45et, 13-limit WE tuning]]
+* Step size: 26.695{{c}}, octave size: 1201.3{{c}}
+Stretching the octave of 45edo by around 1{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|26.695|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning}}
+{{Harmonics in cet|26.695|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning (continued)}}
+; [[ed12|161ed12]]
+* Step size: Octave size: 1202.4{{c}}
+Stretching the octave of 45edo by around 2.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 12.2{{c}}. The tuning 161ed12 does this.
+{{Harmonics in equal|161|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 161ed12}}
+{{Harmonics in equal|161|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 161ed12 (continued)}}
+; [[ed6|116ed6]]
+* Step size: Octave size: 1203.3{{c}}
+Stretching the octave of 45edo by around 3{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 116ed6 does this. So does [[ed7|126ed7]] whose octave is identical within 0.1{{c}}.
+{{Harmonics in equal|116|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 116ed6}}
+{{Harmonics in equal|116|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 116ed6 (continued)}}
+; [[WE|45et, 7-limit WE tuning]]
+* Step size: 26.745{{c}}, octave size: 1203.5{{c}}
+Stretching the octave of 45edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.6{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
+{{Harmonics in cet|26.745|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning}}
+{{Harmonics in cet|26.745|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning (continued)}}
+; [[zpi|207zpi]]
+* Step size: 26.762{{c}}, octave size: 1204.3{{c}}
+Stretching the octave of 45edo by around 4{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 207zpi does this.
+{{Harmonics in cet|26.762|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 207zpi}}
+{{Harmonics in cet|26.762|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 207zpi (continued)}}
+; [[71edt]]
+* Step size: 26.788{{c}}, octave size: 1205.5{{c}}
+Stretching the octave of 45edo by around 5.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 71edt does this. So do the tunings [[ed5|104ed5]] and [[equal tuning|155ed11]] whose octave is identical within 0.3{{c}}.
+{{Harmonics in equal|71|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 71edt}}
+{{Harmonics in equal|71|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 71edt (continued)}}
+; [[equal tuning|183ed17]]
+* Octave size: 1206.1{{c}}
+Stretching the octave of 45edo by around 6{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 13.3{{c}}. The tuning 183ed17 does this.
+{{Harmonics in equal|183|17|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 183ed17}}
+{{Harmonics in equal|183|17|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 183ed17 (continued)}}
 == Instruments ==
@@ Line 541: / Line 650: @@
 == Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/33tKBiWZvXM ''(short clip) Fantasy in 45edo''] (2025)
 ; [[JUMBLE]]
 * [https://www.youtube.com/watch?v=tbc_OxHp-ec ''Fishbowl''] (2023)
@@ Line 546: / Line 658: @@
 * [https://www.youtube.com/watch?v=Kd4t_iKiKMA ''Fallen Angel''] (2024)
 * [https://www.youtube.com/watch?v=DPztb8W6ykY ''Solar Guardian''] (2024)
-== Notes ==
-<references group="note" />
 [[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->