45edo: Difference between revisions

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[[Category:Theory]]
[[Category:Theory]]


{| class="wikitable center-all"
{| class="wikitable right-all center-3 left-6 center-7"
|-
|-
! rowspan="2" | Step #
! rowspan="2" | Step #
! ET
! ET
! colspan="2" | Just
! colspan="2" | Just (JI)
! rowspan="2" | Difference <br> (ET minus Just)
! rowspan="2" | Error <br> (ET−JI)
! rowspan="2" colspan="3" | [[Ups and Downs Notation]]
! rowspan="2" colspan="3" | [[Ups and Downs Notation]]
|-
|-
Line 26: Line 26:
|-
|-
| 0
| 0
| 0¢
| 0.000
| [[1/1]]
| [[1/1]]
| 0
| 0.000
| 0
| 0.000
| Perfect Unison
| Perfect Unison
| D
| D
Line 50: Line 50:
|-
|-
| 3
| 3
| 80.0
| 80.000
| [[22/21]]
| [[22/21]]
| 80.537
| 80.537
Line 61: Line 61:
| [[17/16]]
| [[17/16]]
| 104.955
| 104.955
| 1.7112
| 1.711
| Downminor 2nd
| Downminor 2nd
| Ebv
| Ebv
Line 74: Line 74:
|-
|-
| 6
| 6
| 160
| 160.000
| [[11/10]]
| [[11/10]]
| 165.004
| 165.004
Line 98: Line 98:
|-
|-
| 9
| 9
| 240
| 240.000
| [[8/7]]
| [[8/7]]
| 231.174
| 231.174
Line 122: Line 122:
|-
|-
| 12
| 12
| 320
| 320.000
| [[6/5]]
| [[6/5]]
| 315.641
| 315.641
Line 146: Line 146:
|-
|-
| 15
| 15
| 400
| 400.000
| [[5/4]]+
| [[5/4]]+
| 386.314
| 386.314
Line 170: Line 170:
|-
|-
| 18
| 18
| 480
| 480.000
| [[21/16]]
| [[21/16]]
| 470.781
| 470.781
Line 194: Line 194:
|-
|-
| 21
| 21
| 560
| 560.000
| [[18/13]]
| [[18/13]]
| 563.382
| 563.382
Line 218: Line 218:
|-
|-
| 24
| 24
| 640
| 640.000
| [[13/9]]
| [[13/9]]
| 636.618
| 636.618
Line 242: Line 242:
|-
|-
| 27
| 27
| 720
| 720.000
| [[32/21]]
| [[32/21]]
| 729.219
| 729.219
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|-
|-
| 30
| 30
| 800
| 800.000
| [[8/5]]-
| [[8/5]]-
| 813.686
| 813.686
Line 290: Line 290:
|-
|-
| 33
| 33
| 880
| 880.000
| [[5/3]]
| [[5/3]]
| 884.359
| 884.359
Line 314: Line 314:
|-
|-
| 36
| 36
| 960
| 960.000
| [[7/4]]
| [[7/4]]
| 968.826
| 968.826
Line 338: Line 338:
|-
|-
| 39
| 39
| 1040
| 1040.000
| [[20/11]]
| [[20/11]]
| 1034.996
| 1034.996
Line 357: Line 357:
| [[32/17]]
| [[32/17]]
| 1095.044
| 1095.044
| -1.7112
| -1.711
| Upmajor 7th
| Upmajor 7th
| C#^
| C#^
|-
|-
| 42
| 42
| 1120
| 1120.000
| [[21/11]]
| [[21/11]]
| 1119.463
| 1119.463
Line 386: Line 386:
|-
|-
| 45
| 45
| 1200
| 1200.000
| [[2/1]]
| [[2/1]]
| 1200
| 1200.000
| 0
| 0.000
| Perfect Octave
| Perfect Octave
| D
| D

Revision as of 18:13, 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 a very inaccurate version of ennealimmal, if you want to view it that way. 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 (JI) Error
(ET−JI) 		Ups and Downs Notation
Cents Interval Cents
0 0.000 1/1 0.000 0.000 Perfect Unison 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 80.000 22/21 80.537 -0.537 Upaugmented Unison D#^
4 106.666 17/16 104.955 1.711 Downminor 2nd Ebv
5 133.333 27/25 133.238 0.095 Minor 2nd Eb
6 160.000 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^
9 240.000 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
12 320.000 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#
15 400.000 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
18 480.000 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^
21 560.000 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
24 640.000 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
27 720.000 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#^
30 800.000 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
33 880.000 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#
36 960.000 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 1040.000 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 -1.711 Upmajor 7th C#^
42 1120.000 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 1200.000 2/1 1200.000 0.000 Perfect Octave D
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