40edo: Difference between revisions

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→Intervals: changed ups and downs interval names to extended pythagorean, since this is a sharp-1 edo
Line 27: Line 27:
|0
|0
|0
|0
|Unison
|perfect unison
|1
|P1
|D
|D
|-
|-
Line 36: Line 36:
|29.5944
|29.5944
|0.40553
|0.40553
|Up Unison
|augmented 1sn
|
|A1
|D#
|D#
|-
|-
Line 45: Line 45:
|60.7512
|60.7512
| -0.75128
| -0.75128
|Downminor 2nd
|double-aug 1sn
|
|AA1
|D##
|D##
|-
|-
Line 54: Line 54:
|88.8006
|88.8006
|1.19930
|1.19930
|Minor 2nd
|double-dim 2nd
|
|dd2
|D###/Ebbb
|D###/Ebbb
|-
|-
Line 63: Line 63:
|119.4428
|119.4428
|0.55719
|0.55719
|Upminor 2nd
|diminished 2nd
|
|d2
|Ebb
|Ebb
|-
|-
Line 72: Line 72:
|150.6370
|150.6370
| -0.63705
| -0.63705
|Downmajor 2nd
|minor 2nd
|
|m2
|Eb
|Eb
|-
|-
Line 81: Line 81:
|182.4037
|182.4037
| -2.40371
| -2.40371
|Major 2nd
|major 2nd
|
|M2
|E
|E
|-
|-
Line 90: Line 90:
|203.9100
|203.9100
|6.08999
|6.08999
|Upmajor 2nd
|augmented 2nd
|
|A2
|E#
|E#
|-
|-
Line 99: Line 99:
|231.1741
|231.1741
|8.82590
|8.82590
|Augmented 2nd
|double-aug 2nd
|
|AA2
|E##
|E##
|-
|-
Line 108: Line 108:
|266.8709
|266.8709
|3.12909
|3.12909
|Diminished 3rd
|double-dim 3rd
|
|dd3
|Fbb
|Fbb
|-
|-
Line 117: Line 117:
|297.5130
|297.5130
|2.48698
|2.48698
|Downminor 3rd
|diminished 3rd
|
|d3
|Fb
|Fb
|-
|-
Line 126: Line 126:
|315.6412
|315.6412
|14.3587
|14.3587
|Upminor 3rd
|minor 3rd
|
|m3
|F
|F
|-
|-
Line 135: Line 135:
|359.4723
|359.4723
|0.52766
|0.52766
|Neutral 3rd
|major 3rd
|
|M3
|F#
|F#
|-
|-
Line 144: Line 144:
|386.3137
|386.3137
|3.68628
|3.68628
|Major 3rd
|augmented 3rd
|
|A3
|F##
|F##
|-
|-
Line 153: Line 153:
|417.5079
|417.5079
|2.49203
|2.49203
|Augmented 3rd
|double-aug 3rd
|
|AA3
|F###/Gbbb
|F###/Gbbb
|-
|-
Line 162: Line 162:
|446.3625
|446.3625
|3.63746
|3.63746
|Diminished 4th
|double-dim 4th
|
|dd4
|Gbb
|Gbb
|-
|-
Line 171: Line 171:
|470.781
|470.781
|9.219
|9.219
|Blackwood 4th
|diminished 4th
|
|d4
|Gb
|Gb
|-
|-
Line 180: Line 180:
|498.0449
|498.0449
|11.9550
|11.9550
|Diatonic 4th
|perfect 4th
|
|P4
|G
|G
|-
|-
Line 189: Line 189:
|551.3179
|551.3179
| -11.3179
| -11.3179
|Augmented 4th
|augmented 4th
|
|A4
|G#
|G#
|-
|-
Line 198: Line 198:
|568.7174
|568.7174
|1.2825
|1.2825
|Minor Tritone
|double-aug 4th
|
|AA4
|G##
|G##
|-
|-
Line 207: Line 207:
|582.5121
|582.5121
|17.4878
|17.4878
|Perfect Tritone
|triple-aug 4th,
|
triple-dim 5th
|AAA4,
ddd5
|G###/Abbb
|G###/Abbb
|-
|-
Line 216: Line 218:
|628.2743
|628.2743
|1.72565
|1.72565
|Major Tritone
|double-dim 5th
|
|dd5
|Abb
|Abb
|-
|-
Line 225: Line 227:
|648.6820
|648.6820
|11.3179
|11.3179
|Diminished 5th
|diminished 5th
|
|d5
|Ab
|Ab
|-
|-
Line 234: Line 236:
|701.9550
|701.9550
| -11.9550
| -11.9550
|Diatonic 5th
|perfect 5th
|
|P5
|A
|A
|-
|-
Line 243: Line 245:
|729.2191
|729.2191
| -9.219
| -9.219
|Blackwood 5th
|augmented 5th
|
|A5
|A#
|A#
|-
|-
Line 252: Line 254:
|753.6374
|753.6374
| -3.63746
| -3.63746
|Augmented 5th
|double-aug 5th
|
|AA5
|A##
|A##
|-
|-
Line 261: Line 263:
|782.4920
|782.4920
| -2.49203
| -2.49203
|Diminished 6th
|double-dim 6th
|
|dd6
|A###/Bbbb
|A###/Bbbb
|-
|-
Line 270: Line 272:
|813.6862
|813.6862
| -3.68628
| -3.68628
|Minor 6th
|diminished 6th
|
|d6
|Bbb
|Bbb
|-
|-
Line 279: Line 281:
|840.5276
|840.5276
| -0.52766
| -0.52766
|Neutral 6th
|minor 6th
|
|m6
|Bb
|Bb
|-
|-
Line 288: Line 290:
|884.3587
|884.3587
| -14.3587
| -14.3587
|Downmajor 6th
|major 6th
|
|M6
|B
|B
|-
|-
Line 297: Line 299:
|902.4869
|902.4869
| -2.48698
| -2.48698
|Upmajor 6th
|augmented 6th
|
|A6
|B#
|B#
|-
|-
Line 306: Line 308:
|933.1291
|933.1291
| -3.12909
| -3.12909
|Augmented 6th
|double-aug 6th
|
|AA6
|B##
|B##
|-
|-
Line 315: Line 317:
|968.8259
|968.8259
| -8.82590
| -8.82590
|Harmonic 7th
|double-dim 7th
|
|dd7
|Cbb
|Cbb
|-
|-
Line 324: Line 326:
|996.0899
|996.0899
| -6.08999
| -6.08999
|Downminor 7th
|diminished 7th
|
|d7
|Cb
|Cb
|-
|-
Line 333: Line 335:
|1017.5962
|1017.5962
|2.40371
|2.40371
|Minor 7th
|minor 7th
|
|m7
|C
|C
|-
|-
Line 342: Line 344:
|1049.3629
|1049.3629
|0.63705
|0.63705
|Upminor 7th
|major 7th
|
|M7
|C#
|C#
|-
|-
Line 351: Line 353:
|1080.5571
|1080.5571
| -0.55719
| -0.55719
|Downmajor 7th
|augmented 7th
|
|A7
|C##
|C##
|-
|-
Line 360: Line 362:
|1111.1993
|1111.1993
| -1.19930
| -1.19930
|Major 7th
|double-aug 7th
|
|AA7
|C###/Dbbb
|C###/Dbbb
|-
|-
Line 369: Line 371:
|1139.2487
|1139.2487
|0.75128
|0.75128
|Upmajor 7th
|double-dim 8ve
|
|dd8
|Dbb
|Dbb
|-
|-
Line 378: Line 380:
|1170.4055
|1170.4055
| -0.40553
| -0.40553
|Down Octave
|diminished 8ve
|
|d8
|Db
|Db
|-
|-
Line 387: Line 389:
|1200
|1200
|0
|0
|Octave
|perfect octave
|
|P8
|D
|D
|}
|}
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:subgroup]]
[[Category:subgroup]]

Revision as of 07:18, 25 June 2022

40edo is the equal division of the octave into 40 parts of exactly 30 cents each. Up to this point, all the multiples of 5 have had the 720 cent blackwood 5th as their best approximation of 3/2. 35edo combined the small circles of blackwood and whitewood 5ths, almost equally far from just, requiring you to use both to reach all keys. 40edo adds a diatonic 5th that's closer to just. However, it is still the second flattest diatonic 5th, only exceeded by 47edo in error, which results in it being inconsistent in the 5-limit - combining the best major and minor third will result in the blackwood 5th instead. As such, calling it a perfect 5th seems very much a misnomer. Despite all keys being reachable by stacking this 5th, it does not qualify as meantone either, as stacking 4 of them results in a near perfect tridecimal neutral third rather than a major one. The resulting 5L 2s scale has large steps of 6 intervals and small ones of 5, putting sharps and flats right next to letters without any ups or downs in between and requiring a lot of them to notate more distant keys. It tempers out 648/625 in the 5-limit; 225/224 and in the 7-limit; 99/98, 121/120 and 176/175 in the 11-limit; and 66/65 in the 13-limit.

40edo is more accurate on the 2.9.5.21.33.13.51.19 2*40 subgroup, where it offers the same tuning as 80edo, and tempers out the same commas. It is also the first equal temperament to approximate both the 23rd and 19th harmonic, by tempering out the 9 cent comma to 4-edo, with 10 divisions therein.

Approximation of odd harmonics in 40edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -12.0 +3.7 -8.8 +6.1 -11.3 -0.5 -8.3 -15.0 +2.5 +9.2 +1.7
Relative (%) -39.9 +12.3 -29.4 +20.3 -37.7 -1.8 -27.6 -49.9 +8.3 +30.7 +5.8
Steps
(reduced) 		63
(23) 		93
(13) 		112
(32) 		127
(7) 		138
(18) 		148
(28) 		156
(36) 		163
(3) 		170
(10) 		176
(16) 		181
(21)

Intervals

Step # ET Just Difference

(ET minus Just)

Notation
Cents Interval Cents
0 0¢ 1:1 0 0 perfect unison P1 D
1 30 59:58 29.5944 0.40553 augmented 1sn A1 D#
2 60 29:28 60.7512 -0.75128 double-aug 1sn AA1 D##
3 90 20:19 88.8006 1.19930 double-dim 2nd dd2 D###/Ebbb
4 120 15:14 119.4428 0.55719 diminished 2nd d2 Ebb
5 150 12:11 150.6370 -0.63705 minor 2nd m2 Eb
6 180 10:9 182.4037 -2.40371 major 2nd M2 E
7 210 9:8 203.9100 6.08999 augmented 2nd A2 E#
8 240 8:7 231.1741 8.82590 double-aug 2nd AA2 E##
9 270 7:6 266.8709 3.12909 double-dim 3rd dd3 Fbb
10 300 19:16 297.5130 2.48698 diminished 3rd d3 Fb
11 330 6:5 315.6412 14.3587 minor 3rd m3 F
12 360 16:13 359.4723 0.52766 major 3rd M3 F#
13 390 5:4 386.3137 3.68628 augmented 3rd A3 F##
14 420 14:11 417.5079 2.49203 double-aug 3rd AA3 F###/Gbbb
15 450 22:17 446.3625 3.63746 double-dim 4th dd4 Gbb
16 480 21:16 470.781 9.219 diminished 4th d4 Gb
17 510 4:3 498.0449 11.9550 perfect 4th P4 G
18 540 11:8 551.3179 -11.3179 augmented 4th A4 G#
19 570 25:18 568.7174 1.2825 double-aug 4th AA4 G##
20 600 7:5 582.5121 17.4878 triple-aug 4th,

triple-dim 5th

AAA4,

ddd5

G###/Abbb
21 630 23:16 628.2743 1.72565 double-dim 5th dd5 Abb
22 660 16:11 648.6820 11.3179 diminished 5th d5 Ab
23 690 3:2 701.9550 -11.9550 perfect 5th P5 A
24 720 32:21 729.2191 -9.219 augmented 5th A5 A#
25 750 17:11 753.6374 -3.63746 double-aug 5th AA5 A##
26 780 11:7 782.4920 -2.49203 double-dim 6th dd6 A###/Bbbb
27 810 8:5 813.6862 -3.68628 diminished 6th d6 Bbb
28 840 13:8 840.5276 -0.52766 minor 6th m6 Bb
29 870 5:3 884.3587 -14.3587 major 6th M6 B
30 900 32:19 902.4869 -2.48698 augmented 6th A6 B#
31 930 12:7 933.1291 -3.12909 double-aug 6th AA6 B##
32 960 7:4 968.8259 -8.82590 double-dim 7th dd7 Cbb
33 990 16:9 996.0899 -6.08999 diminished 7th d7 Cb
34 1020 9:5 1017.5962 2.40371 minor 7th m7 C
35 1050 11:6 1049.3629 0.63705 major 7th M7 C#
36 1080 28:15 1080.5571 -0.55719 augmented 7th A7 C##
37 1110 19:10 1111.1993 -1.19930 double-aug 7th AA7 C###/Dbbb
38 1140 56:29 1139.2487 0.75128 double-dim 8ve dd8 Dbb
39 1170 116:59 1170.4055 -0.40553 diminished 8ve d8 Db
40 1200 2:1 1200 0 perfect octave P8 D
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