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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]]
Retrieved from "https://en.xen.wiki/w/45edo"