45edo: Difference between revisions

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m Added EDO info box, changed interval names to match the note spelling, corrected 25\45 just cents value from 66.258 to 666.258
Line 1: Line 1:
{{Infobox ET
| Prime factorization = 3<sup>2</sup> x 5
| Step size = 26.667¢
| Fifth = 26\45 = 693.333¢
| Major 2nd = 7\45 = 186.667¢
| Minor 2nd = 5\45 = 133.333¢
| Augmented 1sn = 2\45 = 53.333¢
}}
'''45edo''' divides the octave into 45 equal parts of 26.667 [[cent]]s. It 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. Also supports messed-up [[ennealimmal]], if you want. Since 45 is a multiple of 5 and 9, it can be used to model [[Indonesian]] music in both [[wikipedia:Slendro|Slendro]] (~ 5edo) and [[wikipedia:Pelog|Pelog]] (~ modes of 9edo) tunings.
'''45edo''' divides the octave into 45 equal parts of 26.667 [[cent]]s. It 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. Also supports messed-up [[ennealimmal]], if you want. Since 45 is a multiple of 5 and 9, it can be used to model [[Indonesian]] music in both [[wikipedia:Slendro|Slendro]] (~ 5edo) and [[wikipedia:Pelog|Pelog]] (~ modes of 9edo) tunings.


Line 6: Line 15:
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
| rowspan="2" | Step #
| rowspan="2" |Step #
| style="text-align:center;" | ET
| style="text-align:center;" |ET
| colspan="2" | Just  
| colspan="2" |Just
| Difference <br> (ET minus Just)
|Difference <br> (ET minus Just)
| colspan="3" | [[Ups and Downs Notation]]
| colspan="3" |[[Ups and Downs Notation]]
|-
|-
| Cents
|Cents
| Interval
|Interval
| Cents
|Cents
|
|
|
|
Line 20: Line 29:
|
|
|-
|-
| 0
| 0¢
|1:1
| 0
|0
|0
| Unison
|0¢
| 1
| 1:1
| D
|0
|0
|Perfect Unison
|1
|D
|-
|-
| 1
|1
| 26.666
|26.666
| 65:64
|65:64
|26.841
|26.841
| -0.174
| -0.174
|Up unison
|Up unison
|  
|
| D^
|D^
|-
|-
| 2
|2
| 53.333
|53.333
|33:32
|33:32
|53.273
|53.273
|0.060
|0.060
|Downminor 2nd
|Augmented Unison
|
|
|D#
|D#
|-
|-
| 3
|3
|80.0
|80.0
|22:21
|22:21
|80.537
|80.537
| -0.537
| -0.537
|Minor 2nd
|Upaugmented Unison
|
|
|D#^
|D#^
|-
|-
| 4
|4
|106.666
|106.666
|17:16
|17:16
|104.955
|104.955
|1.7112
|1.7112
|Upminor 2nd
|Downminor 2nd
|
|
|Ebv
|Ebv
|-
|-
| 5
|5
| 133.333
|133.333
|27:25
|27:25
|133.238
|133.238
|0.095
|0.095
|Downmid 2nd
| Minor 2nd
|
|
|Eb
|Eb
|-
|-
| 6
|6
|160
|160
|11:10
|11:10
Line 83: Line 92:
|Ev
|Ev
|-
|-
| 7
|7
|186.666
|186.666
|10:9
|10:9
Line 92: Line 101:
|E
|E
|-
|-
| 8
|8
|213.333
|213.333
|9:8
|9:8
Line 101: Line 110:
|E^
|E^
|-
|-
| 9
|9
|240
|240
|8:7
|8:7
|231.174
|231.174
|8.826
|8.826
|2nd/3rd
|Augmented 2nd
|
|
|E#
|E#
|-
|-
| 10
|10
|266.666
|266.666
|7:6
|7:6
|266.871
|266.871
|<nowiki>-0.205</nowiki>
|<nowiki>-0.205</nowiki>
|Flat Third
|Diminished 3rd
|
|
|Fb
|Fb
|-
|-
| 11
|11
|293.333
|293.333
|32:27
|32:27
|294.135
|294.135
| -0.802
| -0.802
|Downminor Third
|Downminor 3rd
|
|
|Fv
|Fv
|-
|-
| 12
|12
|320
|320
|6:5
|6:5
|315.641
|315.641
|4.359
|4.359
|Minor Third
|Minor 3rd
|  
|  
|F
|F
|-
|-
| 13
|13
|346.666
|346.666
|11:9
|11:9
|347.408
|347.408
| -0.741
| -0.741
|Mid Third
|Mid 3rd
|
|
|F^
| F^
|-
|-
| 14
|14
|373.333
|373.333
|5:4-
|5:4-
|386.314
|386.314
|<nowiki>-12.980</nowiki>
|<nowiki>-12.980</nowiki>
|Downmajor Third
|Major 3rd
|
|
|F#
|F#
|-
|-
| 15
|15
|400
|400
|5:4+
|5:4+
|386.314
|386.314
|13.686
|13.686
|Upmajor Third
| Upmajor 3rd
|
|
|F#^
|F#^
|-
|-
| 16
|16
|426.666
|426.666
|9:7
|9:7
|435.084
|435.084
| -8.418
| -8.418
|Sharp Third
|Downdiminshed 4th
|
|
|Gbv
|Gbv
|-
|-
| 17
|17
|453.333
|453.333
|13:10
|13:10
Line 182: Line 191:
|Gb
|Gb
|-
|-
| 18
|18
|480
|480
|21:16
|21:16
|470.781
|470.781
|9.219
|9.219
|Blackwood 4th
|Down 4th
|
|
|Gv
|Gv
|-
|-
| 19
|19
|506.666
|506.666
|4:3
|4:3
|498.045
|498.045
|8.622
|8.622
|perfect 4th
|Perfect 4th
|
|
|G
|G
|-
|-
| 20
|20
|533.333
|533.333
|49:36
|49:36
Line 209: Line 218:
|G^
|G^
|-
|-
| 21
|21
|560
|560
|18:13
|18:13
|563.382
|563.382
|<nowiki>-3.382</nowiki>
|<nowiki>-3.382</nowiki>
|Downmid 4th
|Augmented 4th
|
|
|G#
|G#
|-
|-
| 22
|22
|586.666
|586.666
|7:5
|7:5
|582.512
|582.512
|4.155
| 4.155
|Mid 4th
|Upaugmented 4th
|
|
|G#^
|G#^
|-
|-
| 23
|23
|613.333
|613.333
|10:7
|10:7
|617.488
|617.488
|<nowiki>-4.155</nowiki>
|<nowiki>-4.155</nowiki>
|Mid 5th
|Downdiminshed 5th
|
|
|Abv
|Abv
|-
|-
| 24
|24
|640
|640
|13:9
|13:9
|636.618
|636.618
|3.382
|3.382
|Upmid 5th
|Diminished 5th
|
|
|Ab
|Ab
|-
|-
| 25
|25
|666.666
|666.666
|72:49
|72:49
|66.258
|666.258
|0.409
|0.409
|Down 5th
|Down 5th
Line 254: Line 263:
|Av
|Av
|-
|-
| 26
|26
|693.333
| 693.333
|3:2
| 3:2
|701.955
|701.955
| -8.622
| -8.622
Line 263: Line 272:
|A
|A
|-
|-
| 27
|27
| 720
| 720
| style="text-align:center;" |32:21
| style="text-align:center;" |32:21
|729.219
|729.219
|<nowiki>-9.219</nowiki>
|<nowiki>-9.219</nowiki>
|Blackwood 5th
|Up 5th
|
|
|A^
|A^
|-
|-
| 28
|28
|746.666
|746.666
| 20:13
|20:13
|745.786
|745.786
| 0.961
|0.961
|Augmented 5th
|Augmented 5th
|  
|
| A#
|A#
|-
|-
| 29
|29
|773.333
|773.333
| style="text-align:center;" |14:9
| style="text-align:center;" |14:9
|764.916
|764.916
|8.418
|8.418
|Flat 6th
|Upaugmented 5th
|
|
|A#^
|A#^
|-
|-
| 30
|30
|800
|800
| style="text-align:center;" |8:5-
| style="text-align:center;" |8:5-
Line 299: Line 308:
|Bbv
|Bbv
|-
|-
| 31
|31
|826.666
|826.666
| style="text-align:center;" |8:5+
| style="text-align:center;" |8:5+
|813.686
| 813.686
|12.980
|12.980
|Upminor 6th
|Minor 6th
|
|
|Bb
|Bb
|-
|-
| 32
|32
|853.333
|853.333
| style="text-align:center;" |18:11
| style="text-align:center;" |18:11
Line 315: Line 324:
|Mid 6th
|Mid 6th
|
|
|Bv
| Bv
|-
|-
| 33
|33
|880
|880
| style="text-align:center;" |5:3
| style="text-align:center;" |5:3
Line 326: Line 335:
|B
|B
|-
|-
| 34
|34
|906.666
|906.666
| style="text-align:center;" |27:16
| style="text-align:center;" |27:16
Line 335: Line 344:
|B^
|B^
|-
|-
| 35
|35
|933.333
|933.333
| style="text-align:center;" |12:7
| style="text-align:center;" |12:7
|933.129
|933.129
|0.205
|0.205
|Sharp 6th
|Augmented 6th
|
|
|B#
|B#
|-
|-
| 36
|36
|960
|960
| style="text-align:center;" |7:4
| style="text-align:center;" | 7:4
|968.826
|968.826
|<nowiki>-8.826</nowiki>
|<nowiki>-8.826</nowiki>
|6th/7th
|Diminished 7th
|
|
|Cb
|Cb
|-
|-
| 37
|37
|986.666
| 986.666
| style="text-align:center;" |16:9
| style="text-align:center;" | 16:9
|996.089
|996.089
|<nowiki>-9.423</nowiki>
|<nowiki>-9.423</nowiki>
Line 362: Line 371:
|Cv
|Cv
|-
|-
| 38
|38
|1013.333
|1013.333
| style="text-align:center;" |9:5
| style="text-align:center;" |9:5
| 1017.596
|1017.596
|<nowiki>-4.262</nowiki>
|<nowiki>-4.262</nowiki>
|Minor 7th
|Minor 7th
Line 371: Line 380:
|C
|C
|-
|-
| 39
|39
|1040
|1040
| style="text-align:center;" |20:11
| style="text-align:center;" |20:11
|1034.996
|1034.996
|5.004
|5.004
|Upminor 7th
| Upminor 7th
|
|
|C^
|C^
|-
|-
| 40
|40
|1066.666
| 1066.666
| style="text-align:center;" |50:27
| style="text-align:center;" |50:27
|1066.762
|1066.762
|<nowiki>-0.095</nowiki>
|<nowiki>-0.095</nowiki>
|Mid 7th
|Major 7th
|
|
|C#
|C#
|-
|-
| 41
|41
|1093.333
|1093.333
| style="text-align:center;" |32:17
| style="text-align:center;" |32:17
|1095.044
|1095.044
| -1.7112
| -1.7112
|Downmajor 7th
|Upmajor 7th
|
|
|C#^
|C#^
|-
|-
| 42
|42
|1120
|1120
| style="text-align:center;" |21:11
| style="text-align:center;" |21:11
|1119.463
|1119.463
|0.537
|0.537
|Major 7th
|Downdiminshed 8ve
|
|
|Dbv
|Dbv
|-
|-
| 43
|43
|1146.666
|1146.666
| style="text-align:center;" |64:33
| style="text-align:center;" |64:33
|1146.727
|1146.727
|<nowiki>-0.060</nowiki>
|<nowiki>-0.060</nowiki>
|Upmajor 7th
|Diminished 8ve
|
|
|Db
|Db
|-
|-
| 44
|44
|1173.333
|1173.333
| style="text-align:center;" |128:65
| style="text-align:center;" |128:65
Line 425: Line 434:
|Dv
|Dv
|-
|-
| 45
|45
|1200
|1200
| style="text-align:center;" |2:1
| style="text-align:center;" |2:1
|1200
|1200
|0
|0
|Octave
|Perfect Octave
|
|
|D
|D
|}
|}
[[Category:Meantone]]
[[Category:Meantone]]

Revision as of 15:54, 24 February 2021

← 44edo 45edo 46edo →
Prime factorization 32 x 5
Step size 26.6667 ¢ 
Fifth 26\45 (693.333 ¢)
Semitones (A1:m2) 2:5 (53.33 ¢ : 133.3 ¢)
Consistency limit 7
Distinct consistency limit 7

45edo divides the octave into 45 equal parts of 26.667 cents. It 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 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. Also supports messed-up ennealimmal, if you want. Since 45 is a multiple of 5 and 9, it can be used to model Indonesian music in both Slendro (~ 5edo) and Pelog (~ modes of 9edo) tunings.

Step # ET Just Difference
(ET minus Just) 		Ups and Downs Notation
Cents Interval Cents
0 0¢ 1:1 0 0 Perfect Unison 1 D
1 26.666 65:64 26.841 -0.174 Up unison D^
2 53.333 33:32 53.273 0.060 Augmented Unison D#
3 80.0 22:21 80.537 -0.537 Upaugmented Unison D#^
4 106.666 17:16 104.955 1.7112 Downminor 2nd Ebv
5 133.333 27:25 133.238 0.095 Minor 2nd Eb
6 160 11:10 165.004 -5.004 Downmajor 2nd Ev
7 186.666 10:9 182.404 4.262 Major 2nd E
8 213.333 9:8 203.910 9.423 Upmajor 2nd E^
9 240 8:7 231.174 8.826 Augmented 2nd E#
10 266.666 7:6 266.871 -0.205 Diminished 3rd Fb
11 293.333 32:27 294.135 -0.802 Downminor 3rd Fv
12 320 6:5 315.641 4.359 Minor 3rd F
13 346.666 11:9 347.408 -0.741 Mid 3rd F^
14 373.333 5:4- 386.314 -12.980 Major 3rd F#
15 400 5:4+ 386.314 13.686 Upmajor 3rd F#^
16 426.666 9:7 435.084 -8.418 Downdiminshed 4th Gbv
17 453.333 13:10 454.294 -0.961 Diminished 4th Gb
18 480 21:16 470.781 9.219 Down 4th Gv
19 506.666 4:3 498.045 8.622 Perfect 4th G
20 533.333 49:36 533.742 -0.409 Up 4th G^
21 560 18:13 563.382 -3.382 Augmented 4th G#
22 586.666 7:5 582.512 4.155 Upaugmented 4th G#^
23 613.333 10:7 617.488 -4.155 Downdiminshed 5th Abv
24 640 13:9 636.618 3.382 Diminished 5th Ab
25 666.666 72:49 666.258 0.409 Down 5th Av
26 693.333 3:2 701.955 -8.622 Perfect 5th A
27 720 32:21 729.219 -9.219 Up 5th A^
28 746.666 20:13 745.786 0.961 Augmented 5th A#
29 773.333 14:9 764.916 8.418 Upaugmented 5th A#^
30 800 8:5- 813.686 -13.686 Downminor 6th Bbv
31 826.666 8:5+ 813.686 12.980 Minor 6th Bb
32 853.333 18:11 852.592 0.741 Mid 6th Bv
33 880 5:3 884.359 -4.359 Major 6th B
34 906.666 27:16 905.865 0.802 Upmajor 6th B^
35 933.333 12:7 933.129 0.205 Augmented 6th B#
36 960 7:4 968.826 -8.826 Diminished 7th Cb
37 986.666 16:9 996.089 -9.423 Downminor 7th Cv
38 1013.333 9:5 1017.596 -4.262 Minor 7th C
39 1040 20:11 1034.996 5.004 Upminor 7th C^
40 1066.666 50:27 1066.762 -0.095 Major 7th C#
41 1093.333 32:17 1095.044 -1.7112 Upmajor 7th C#^
42 1120 21:11 1119.463 0.537 Downdiminshed 8ve Dbv
43 1146.666 64:33 1146.727 -0.060 Diminished 8ve Db
44 1173.333 128:65 1173.158 0.174 Down 8ve Dv
45 1200 2:1 1200 0 Perfect Octave D
Retrieved from "https://en.xen.wiki/index.php?title=45edo&oldid=62299"