45edo: Difference between revisions

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TallKite (talk | contribs)
autopatrolled, Sysops
4,198 edits
→Intervals: Changed the gamut to be consistent: dim vmin min mid maj ^maj aug. Moved the up or down to before the note name, vC not Cv.
TallKite (talk | contribs)
autopatrolled, Sysops
4,198 edits
Line 17: Line 17:
! colspan="2" | Just (JI)
! colspan="2" | Just (JI)
! rowspan="2" | Error <br> (ET−JI)
! rowspan="2" | Error <br> (ET−JI)
! rowspan="2" colspan="3" | [[Ups and Downs Notation]]
! colspan="4" rowspan="2" | [[Ups and Downs Notation]]
|-
|-
! Cents
! Cents
Line 29: Line 29:
| 0.000
| 0.000
| Perfect Unison
| Perfect Unison
|P1
| D
| D
|-
|-
Line 37: Line 38:
| -0.174
| -0.174
| Up unison
| Up unison
|^1
| ^D
| ^D
|-
|-
Line 45: Line 47:
| 0.060
| 0.060
| Augmented Unison
| Augmented Unison
|A1
| D#
| D#
|-
|-
Line 53: Line 56:
| -0.537
| -0.537
| Diminished 2nd
| Diminished 2nd
|d2
| Ebb
| Ebb
|-
|-
Line 61: Line 65:
| 1.711
| 1.711
| Downminor 2nd
| Downminor 2nd
|vm2
| vEb
| vEb
|-
|-
Line 69: Line 74:
| 0.095
| 0.095
| Minor 2nd
| Minor 2nd
|m2
| Eb
| Eb
|-
|-
Line 77: Line 83:
| -5.004
| -5.004
| Mid 2nd
| Mid 2nd
|~2
| vE
| vE
|-
|-
Line 85: Line 92:
| 4.262
| 4.262
| Major 2nd
| Major 2nd
|M2
| E
| E
|-
|-
Line 93: Line 101:
| 9.423
| 9.423
| Upmajor 2nd
| Upmajor 2nd
|^M2
| ^E
| ^E
|-
|-
Line 101: Line 110:
| 8.826
| 8.826
| Augmented 2nd
| Augmented 2nd
|A2
| E#
| E#
|-
|-
Line 109: Line 119:
| -0.205
| -0.205
| Diminished 3rd
| Diminished 3rd
|d3
| Fb
| Fb
|-
|-
Line 117: Line 128:
| -0.802
| -0.802
| Downminor 3rd
| Downminor 3rd
|vm3
| vF
| vF
|-
|-
Line 125: Line 137:
| 4.359
| 4.359
| Minor 3rd
| Minor 3rd
|m3
| F
| F
|-
|-
Line 133: Line 146:
| -0.741
| -0.741
| Mid 3rd
| Mid 3rd
|~3
| ^F
| ^F
|-
|-
Line 141: Line 155:
| -12.980
| -12.980
| Major 3rd
| Major 3rd
|M3
| F#
| F#
|-
|-
Line 149: Line 164:
| 13.686
| 13.686
| Upmajor 3rd
| Upmajor 3rd
|^M3
| ^F#
| ^F#
|-
|-
Line 157: Line 173:
| -8.418
| -8.418
| Augmented 3rd
| Augmented 3rd
|A3
| Fx
| Fx
|-
|-
Line 165: Line 182:
| -0.961
| -0.961
| Diminished 4th
| Diminished 4th
|d4
| Gb
| Gb
|-
|-
Line 173: Line 191:
| 9.219
| 9.219
| Down 4th
| Down 4th
|v4
| vG
| vG
|-
|-
Line 181: Line 200:
| 8.622
| 8.622
| Perfect 4th
| Perfect 4th
|P4
| G
| G
|-
|-
Line 189: Line 209:
| -0.409
| -0.409
| Up 4th or Mid 4th
| Up 4th or Mid 4th
|^4, ~4
| ^G
| ^G
|-
|-
Line 197: Line 218:
| -3.382
| -3.382
| Augmented 4th
| Augmented 4th
|A4
| G#
| G#
|-
|-
Line 205: Line 227:
| 4.155
| 4.155
| Upaugmented 4th
| Upaugmented 4th
|^A4
| ^G#
| ^G#
|-
|-
Line 213: Line 236:
| -4.155
| -4.155
| Downdiminshed 5th
| Downdiminshed 5th
|vd5
| vAb
| vAb
|-
|-
Line 221: Line 245:
| 3.382
| 3.382
| Diminished 5th
| Diminished 5th
|d5
| Ab
| Ab
|-
|-
Line 229: Line 254:
| 0.409
| 0.409
| Down 5th or Mid 5th
| Down 5th or Mid 5th
|v5, ~5
| vA
| vA
|-
|-
Line 237: Line 263:
| -8.622
| -8.622
| Perfect 5th
| Perfect 5th
|P5
| A
| A
|-
|-
Line 245: Line 272:
| -9.219
| -9.219
| Up 5th
| Up 5th
|^5
| ^A
| ^A
|-
|-
Line 253: Line 281:
| 0.961
| 0.961
| Augmented 5th
| Augmented 5th
|A5
| A#
| A#
|-
|-
Line 261: Line 290:
| 8.418
| 8.418
| Diminished 6th
| Diminished 6th
|d6
| Bbb
| Bbb
|-
|-
Line 269: Line 299:
| -13.686
| -13.686
| Downminor 6th
| Downminor 6th
|vm6
| vBb
| vBb
|-
|-
Line 277: Line 308:
| 12.980
| 12.980
| Minor 6th
| Minor 6th
|m6
| Bb
| Bb
|-
|-
Line 285: Line 317:
| 0.741
| 0.741
| Mid 6th
| Mid 6th
|~6
| vB
| vB
|-
|-
Line 293: Line 326:
| -4.359
| -4.359
| Major 6th
| Major 6th
|M6
| B
| B
|-
|-
Line 301: Line 335:
| 0.802
| 0.802
| Upmajor 6th
| Upmajor 6th
|^M6
| ^B
| ^B
|-
|-
Line 309: Line 344:
| 0.205
| 0.205
| Augmented 6th
| Augmented 6th
|A6
| B#
| B#
|-
|-
Line 317: Line 353:
| -8.826
| -8.826
| Diminished 7th
| Diminished 7th
|d7
| Cb
| Cb
|-
|-
Line 325: Line 362:
| -9.423
| -9.423
| Downminor 7th
| Downminor 7th
|vm7
| vC
| vC
|-
|-
Line 333: Line 371:
| -4.262
| -4.262
| Minor 7th
| Minor 7th
|m7
| C
| C
|-
|-
Line 341: Line 380:
| 5.004
| 5.004
| Mid 7th
| Mid 7th
|~7
| ^C
| ^C
|-
|-
Line 349: Line 389:
| -0.095
| -0.095
| Major 7th
| Major 7th
|M7
| C#
| C#
|-
|-
Line 357: Line 398:
| -1.711
| -1.711
| Upmajor 7th
| Upmajor 7th
|^M7
| ^C#
| ^C#
|-
|-
Line 365: Line 407:
| 0.537
| 0.537
| Augmented 7th
| Augmented 7th
|A7
| Cx
| Cx
|-
|-
Line 373: Line 416:
| -0.060
| -0.060
| Diminished 8ve
| Diminished 8ve
|d8
| Db
| Db
|-
|-
Line 381: Line 425:
| 0.174
| 0.174
| Down 8ve
| Down 8ve
|v8
| vD
| vD
|-
|-
Line 389: Line 434:
| 0.000
| 0.000
| Perfect Octave
| Perfect Octave
|P8
| D
| D
|}
|}

Revision as of 07:37, 25 June 2022

← 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.

Approximation of odd harmonics in 45edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -8.6 -13.0 -8.8 +9.4 +8.7 +12.8 +5.1 +1.7 -4.2 +9.2 +11.7
Relative (%) -32.3 -48.7 -33.1 +35.3 +32.6 +48.0 +19.0 +6.4 -15.7 +34.6 +44.0
Steps
(reduced) 		71
(26) 		104
(14) 		126
(36) 		143
(8) 		156
(21) 		167
(32) 		176
(41) 		184
(4) 		191
(11) 		198
(18) 		204
(24)

Intervals

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 P1 D
1 26.666 65/64 26.841 -0.174 Up unison ^1 ^D
2 53.333 33/32 53.273 0.060 Augmented Unison A1 D#
3 80.000 22/21 80.537 -0.537 Diminished 2nd d2 Ebb
4 106.666 17/16 104.955 1.711 Downminor 2nd vm2 vEb
5 133.333 27/25 133.238 0.095 Minor 2nd m2 Eb
6 160.000 11/10 165.004 -5.004 Mid 2nd ~2 vE
7 186.666 10/9 182.404 4.262 Major 2nd M2 E
8 213.333 9/8 203.910 9.423 Upmajor 2nd ^M2 ^E
9 240.000 8/7 231.174 8.826 Augmented 2nd A2 E#
10 266.666 7/6 266.871 -0.205 Diminished 3rd d3 Fb
11 293.333 32/27 294.135 -0.802 Downminor 3rd vm3 vF
12 320.000 6/5 315.641 4.359 Minor 3rd m3 F
13 346.666 11/9 347.408 -0.741 Mid 3rd ~3 ^F
14 373.333 5/4- 386.314 -12.980 Major 3rd M3 F#
15 400.000 5/4+ 386.314 13.686 Upmajor 3rd ^M3 ^F#
16 426.666 9/7 435.084 -8.418 Augmented 3rd A3 Fx
17 453.333 13/10 454.294 -0.961 Diminished 4th d4 Gb
18 480.000 21/16 470.781 9.219 Down 4th v4 vG
19 506.666 4/3 498.045 8.622 Perfect 4th P4 G
20 533.333 49/36 533.742 -0.409 Up 4th or Mid 4th ^4, ~4 ^G
21 560.000 18/13 563.382 -3.382 Augmented 4th A4 G#
22 586.666 7/5 582.512 4.155 Upaugmented 4th ^A4 ^G#
23 613.333 10/7 617.488 -4.155 Downdiminshed 5th vd5 vAb
24 640.000 13/9 636.618 3.382 Diminished 5th d5 Ab
25 666.666 72/49 666.258 0.409 Down 5th or Mid 5th v5, ~5 vA
26 693.333 3/2 701.955 -8.622 Perfect 5th P5 A
27 720.000 32/21 729.219 -9.219 Up 5th ^5 ^A
28 746.666 20/13 745.786 0.961 Augmented 5th A5 A#
29 773.333 14/9 764.916 8.418 Diminished 6th d6 Bbb
30 800.000 8/5- 813.686 -13.686 Downminor 6th vm6 vBb
31 826.666 8/5+ 813.686 12.980 Minor 6th m6 Bb
32 853.333 18/11 852.592 0.741 Mid 6th ~6 vB
33 880.000 5/3 884.359 -4.359 Major 6th M6 B
34 906.666 27/16 905.865 0.802 Upmajor 6th ^M6 ^B
35 933.333 12/7 933.129 0.205 Augmented 6th A6 B#
36 960.000 7/4 968.826 -8.826 Diminished 7th d7 Cb
37 986.666 16/9 996.089 -9.423 Downminor 7th vm7 vC
38 1013.333 9/5 1017.596 -4.262 Minor 7th m7 C
39 1040.000 20/11 1034.996 5.004 Mid 7th ~7 ^C
40 1066.666 50/27 1066.762 -0.095 Major 7th M7 C#
41 1093.333 32/17 1095.044 -1.711 Upmajor 7th ^M7 ^C#
42 1120.000 21/11 1119.463 0.537 Augmented 7th A7 Cx
43 1146.666 64/33 1146.727 -0.060 Diminished 8ve d8 Db
44 1173.333 128/65 1173.158 0.174 Down 8ve v8 vD
45 1200.000 2/1 1200.000 0.000 Perfect Octave P8 D
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