Xenharmonic Wiki

45edo: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 213137236 - Original comment: **
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,262 edits
→Music: add 1
 
(115 intermediate revisions by 32 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-23 07:50:01 UTC</tt>.<br>
 
: The original revision id was <tt>213137236</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
45edo effectively has two approximate [[5/4]] major thirds, each almost equally far from just, but the flat one is slightly closer. Combined with a [[3/2|perfect fifth]] 8.6 cents flat of just, it can be used as a [[meantone]] tuning, forming a good approximation to [[2/5-comma meantone]] (in fact falling into the [[flattone]] range). It is a flat-tending system in the [[7-limit]], with harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] all flat. However, harmonics [[11/1|11]] and [[13/1|13]] are sharp, but this can be fixed with the 45ef val.
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
=== Odd harmonics ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 45 equal temperament divides the octave into 45 equal parts of 26.667 cents. It is the [[optimal patent val]] for [[Meantone family|flattone temperament]] the 7-limit 525/512 planar [[Avicennmic temperaments|avicennmic]] temperament,the 11-limit [[Didymus rank three family|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.</pre></div>
{{Harmonics in equal|45}}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;45edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 45 equal temperament divides the octave into 45 equal parts of 26.667 cents. It is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for &lt;a class="wiki_link" href="/Meantone%20family"&gt;flattone temperament&lt;/a&gt; the 7-limit 525/512 planar &lt;a class="wiki_link" href="/Avicennmic%20temperaments"&gt;avicennmic&lt;/a&gt; temperament,the 11-limit &lt;a class="wiki_link" href="/Didymus%20rank%20three%20family"&gt;calliope&lt;/a&gt; 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.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== As a tuning of other temperaments ===
It tempers out [[81/80]], [[525/512]], [[875/864]], and [[3125/3087]] in the 7-limit, and [[45/44]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7- and 11-limit flattone temperament, and the 45f val is an excellent tuning for [[13-limit]] flattone. It also provides the optimal patent val for the 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 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 2.27.25.63.33.65.17 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.
 
=== Subsets and supersets ===
Since 45 factors into primes as {{nowrap| 3<sup>2</sup> × 5 }}, 45edo has subset edos {{EDOs| 3, 5, 9, and 15 }}. [[135edo]], which triples it, corrects its primes 3, 7, and 11 to near-just qualities, and 270edo offers even more.
 
== Intervals ==
{| class="wikitable center-1 right-2 center-5 center-6"
|-
! #
! Cents
! Approximate ratios*
! colspan="4" | [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| Perfect Unison
| P1
| D
|-
| 1
| 26.7
| [[49/48]], [[50/49]]
| Up unison
| ^1
| ^D
|-
| 2
| 53.3
| [[36/35]], ''[[25/24]]'', ''[[64/63]]''
| Augmented Unison
| A1
| D#
|-
| 3
| 80.0
| [[21/20]]
| Diminished 2nd
| d2
| Ebb
|-
| 4
| 106.7
| [[15/14]]
| Downminor 2nd
| vm2
| vEb
|-
| 5
| 133.3
| [[13/12]], [[14/13]], [[27/25]], ''[[16/15]]''
| Minor 2nd
| m2
| Eb
|-
| 6
| 160.0
| [[54/49]]
| Mid 2nd
| ~2
| vE
|-
| 7
| 186.7
| [[10/9]], ''[[9/8]]''
| Major 2nd
| M2
| E
|-
| 8
| 213.3
|
| Upmajor 2nd
| ^M2
| ^E
|-
| 9
| 240.0
| [[8/7]], [[15/13]]
| Augmented 2nd
| A2
| E#
|-
| 10
| 266.7
| [[7/6]]
| Diminished 3rd
| d3
| Fb
|-
| 11
| 293.3
| [[25/21]]
| Downminor 3rd
| vm3
| vF
|-
| 12
| 320.0
| [[6/5]]
| Minor 3rd
| m3
| F
|-
| 13
| 346.7
| [[49/40]], [[60/49]]
| Mid 3rd
| ~3
| ^F
|-
| 14
| 373.3
| [[5/4]], [[26/21]], ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
| 15
| 400.0
| [[63/50]]
| Upmajor 3rd
| ^M3
| ^F#
|-
| 16
| 426.7
| [[9/7]]
| Augmented 3rd
| A3
| Fx
|-
| 17
| 453.3
| [[13/10]], ''[[21/16]]''
| Diminished 4th
| d4
| Gb
|-
| 18
| 480.0
| ''[[64/49]]''
| Down 4th
| v4
| vG
|-
| 19
| 506.7
| [[4/3]]
| Perfect 4th
| P4
| G
|-
| 20
| 533.3
| [[49/36]]
| Up 4th or Mid 4th
| ^4, ~4
| ^G
|-
| 21
| 560.0
| [[18/13]]
| Augmented 4th
| A4
| G#
|-
| 22
| 586.7
| [[7/5]]
| Upaugmented 4th
| ^A4
| ^G#
|-
| 23
| 613.3
| [[10/7]]
| Downdiminshed 5th
| vd5
| vAb
|-
| 24
| 640.0
| [[13/9]]
| Diminished 5th
| d5
| Ab
|-
| 25
| 666.7
| [[72/49]]
| Down 5th or Mid 5th
| v5, ~5
| vA
|-
| 26
| 693.3
| [[3/2]]
| Perfect 5th
| P5
| A
|-
| 27
| 720.0
| ''[[49/32]]''
| Up 5th
| ^5
| ^A
|-
| 28
| 746.7
| [[20/13]], ''[[32/21]]''
| Augmented 5th
| A5
| A#
|-
| 29
| 773.3
| [[14/9]]
| Diminished 6th
| d6
| Bbb
|-
| 30
| 800.0
| [[100/63]]
| Downminor 6th
| vm6
| vBb
|-
| 31
| 826.7
| [[8/5]], [[21/13]], ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
| 32
| 853.3
| [[49/30]], [[80/49]]
| Mid 6th
| ~6
| vB
|-
| 33
| 880.0
| [[5/3]]
| Major 6th
| M6
| B
|-
| 34
| 906.7
| [[42/25]]
| Upmajor 6th
| ^M6
| ^B
|-
| 35
| 933.3
| [[12/7]]
| Augmented 6th
| A6
| B#
|-
| 36
| 960.0
| [[7/4]], [[26/15]]
| Diminished 7th
| d7
| Cb
|-
| 37
| 986.7
|
| Downminor 7th
| vm7
| vC
|-
| 38
| 1013.3
| [[9/5]], ''[[16/9]]''
| Minor 7th
| m7
| C
|-
| 39
| 1040.0
| [[49/27]]
| Mid 7th
| ~7
| ^C
|-
| 40
| 1066.7
| [[13/7]], [[24/13]], [[50/27]], ''[[15/8]]''
| Major 7th
| M7
| C#
|-
| 41
| 1093.3
| [[28/15]]
| Upmajor 7th
| ^M7
| ^C#
|-
| 42
| 1120.0
| [[40/21]]
| Augmented 7th
| A7
| Cx
|-
| 43
| 1146.7
| [[35/18]], ''[[48/25]]'', ''[[63/32]]''
| Diminished 8ve
| d8
| Db
|-
| 44
| 1173.3
| [[49/25]], [[96/49]]
| Down 8ve
| v8
| vD
|-
| 45
| 1200.0
| [[2/1]]
| Perfect Octave
| P8
| D
|}
<nowiki/>* As a 2.3.5.7.13-subgroup temperament, using the 45f val
 
== Notation ==
 
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{Ups and downs sharpness}}
 
=== 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.
 
== Approximation to JI==
=== Interval mappings ===
{{Q-odd-limit intervals|45}}{{Q-odd-limit intervals|44.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 45ef val mapping}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -71 45 }}
| {{Mapping| 45 71 }}
| +2.72
| 2.73
| 10.2
|-
| 2.3.5
| 81/80, {{monzo| -27 1 11 }}
| {{Mapping| 45 71 104 }}
| +3.68
| 2.61
| 9.75
|-
| 2.3.5.7
| 81/80, 525/512, 2401/2400
| {{Mapping| 45 71 104 126 }}
| +3.55
| 2.27
| 8.49
|-
| 2.3.5.7.13
| 65/64, 81/80, 105/104, 2401/2400
| {{Mapping| 45 71 104 126 166 }} (45f)
| +3.59
| 2.03
| 7.60
|}
 
=== Commas ===
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"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| <abbr title="7629394531250/7625597484987">(26 digits)</abbr>
| {{Monzo| 1 -27 18 }}
| 0.86
| Satritribiyo
| [[Ennealimma]]
|-
| 7
| [[16807/16384]]
| {{Monzo| -14 0 0 5 }}
| 44.13
| Laquinzo
| Cloudy comma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[875/864]]
| {{Monzo| -5 -3 3 1 }}
| 21.90
| Zotrigu
| Keema
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| <abbr title="40353607/40310784">(16 digits)</abbr>
| {{Monzo| -11 -9 0 9 }}
| 1.84
| Tritrizo
| [[Septimal ennealimma]]
|-
| 7
| [[4375/4374]]
| {{Monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[45/44]]
| {{Monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal 1/5-tone
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| <abbr title="117440512/117406179">(18 digits)</abbr>
| {{Monzo| 24 -6 0 1 -5 }}
| 0.51
| Saquinlu-azo
| [[Quartisma]]
|}
<references group="note" />
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 1\45
| 26.7
| 49/48
| [[Sfourth]]
|-
| 1
| 2\45
| 53.3
| 36/35
| [[Chromo]]
|-
| 1
| 7\45
| 186.7
| 10/9
| [[Mintone]]
|-
| 1
| 11\45
| 293.3
| 25/21
| [[Quasitemp]]
|-
| 1
| 14\45
| 373.3
| 5/4
| [[Submerged]]
|-
| 1
| 16\45
| 426.7
| 9/7
| [[Squares]]
|-
| 1
| 23\45
| 453.3
| 13/10
| [[Maja]]
|-
| 1
| 19\45
| 506.7
| 4/3
| [[Flattone]]
|-
| 3
| 19\45<br>(4\45)
| 506.7<br>(106.7)
| 4/3<br>(15/14)
| [[Lithium]]
|-
| 5
| 19\45<br>(1\45)
| 506.7<br>(26.7)
| 4/3<br>(49/48)
| [[Cloudtone]]
|-
| 9
| 12\45<br>(2\45)
| 320.0<br>(53.3)
| 6/5<br>(36/35)
| [[Ennealimmal]]
|-
| 15
| 19\45<br>(1\45)
| 506.7<br>(26.7)
| 4/3<br>(126/125)
| [[Pentadecal]]
|}
<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 and compression ==
45edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by a [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed12|161ed12]] or [[ed6|116ed6]]. The trade-off is a slightly worse 2/1. [[207zpi]] also improves on all of those harmonics except for 17/1.
 
The tuning [[equal tuning|183ed17]] may also be used, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1 (with different mappings for many) but at the cost of a noticeably worse 2/1 than the others.
 
== Scales ==
* [[Cloudtone]][10] - recommended by [[Maeve Gutierrez]]: 8 1 8 1 8 1 8 1 8 1
* [[JUMBLE]]'s "moment of chaos scale": 3 9 6 1 4 7 2 5 8 (used in several works including [https://www.youtube.com/watch?v=WqEOi4cd1Og ''Archipelago Arpeggio''] and [https://www.youtube.com/watch?v=4iwJFVIWEII ''FERAL (45edo microtonal ambient track)''])
* 13-tone 5&9edo scale: 5 4 1 5 3 2 5 2 3 5 1 4 5
* 12-tone 5&9edo scale{{idio}}: 5 4 1 5 3 2 5 2 3 5 5 5
 
== Instruments ==
'''Lumatone'''
 
See [[Lumatone mapping for 45edo]]
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/33tKBiWZvXM ''(short clip) Fantasy in 45edo''] (2025)
* [https://www.youtube.com/watch?v=Xblr-4aGBtM ''<nowiki>Twin Arrows [45edo]</nowiki>''] (2026)
 
; [[JUMBLE]]
* [https://www.youtube.com/watch?v=tbc_OxHp-ec ''Fishbowl''] (2023)
* [https://www.youtube.com/watch?v=WqEOi4cd1Og ''Archipelago Arpeggio''] (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=24gnhAbHtiw ''Qúchze úzeq Qávka''] (2025)
* [https://www.youtube.com/watch?v=K2p7HOI3TUE ''Sodium Light (45edo Microtonal Chillwave)''] (2026)
* [https://www.youtube.com/watch?v=ex9WfmWVibY ''Yēú Zee Kiidhai (45edo microtonal ambient)''] (2026)
* [https://www.youtube.com/watch?v=4iwJFVIWEII ''FERAL (45edo microtonal ambient track)''] (2026)
* [https://www.youtube.com/watch?v=cXZ3RkTDE-I ''Chmelui-Múzeq - Haoýoze (45edo Microtonal Ambient)''] (2026)
 
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Meantone]]
[[Category:Ennealimmal]]
[[Category:Quartismic]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/45edo"