Xenharmonic Wiki

40edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Wikispaces>FREEZE
No edit summary
Yourmusic Productions (talk | contribs)
autopatrolled, emailconfirmed
891 edits
Create intervals table.
Line 1: Line 1:
'''40edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 40 parts of exactly 30 [[cent|cent]]s each. It has a generally flat tendency, with fifths 12 cents flat. It [[tempering_out|tempers out]] 648/625 in the [[5-limit|5-limit]]; 225/224 and in the [[7-limit|7-limit]]; 99/98, 121/120 and 176/175 in the [[11-limit|11-limit]]; and 66/65 in the [[13-limit|13-limit]].
'''40edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 40 parts of exactly 30 [[cent|cent]]s 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 5L2S scale has large steps of 6 intervals and small ones of 5, putting sharps and flats right next to letters and requiring a lot of them to notate more distant keys. It [[tempering_out|tempers out]] 648/625 in the [[5-limit|5-limit]]; 225/224 and in the [[7-limit|7-limit]]; 99/98, 121/120 and 176/175 in the [[11-limit|11-limit]]; and 66/65 in the [[13-limit|13-limit]].


40edo is more accurate on the 2.9.5.21.33.13.51.19 [[k*N_subgroups| 2*40 subgroup]], where it offers the same tuning as [[80edo|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.
40edo is more accurate on the 2.9.5.21.33.13.51.19 [[k*N_subgroups| 2*40 subgroup]], where it offers the same tuning as [[80edo|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.
{| class="wikitable center-all"
|-
| rowspan="2" |Step #
| style="text-align:center;" |ET
| colspan="2" |Just
|Difference
(ET minus Just)
| colspan="3" |[[Ups and Downs Notation]]
|-
|Cents
|Interval
|Cents
|
|
|
|
|-
|0
|0¢
|1:1
|0
|0
|Unison
|1
|D
|-
|1
|30
|59:58
|29.5944
|0.40553
|Up Unison
|
|D#
|-
|2
|60
|29:28
|60.7512
| -0.75128
|Downminor 2nd
|
|D##
|-
|3
|90
|20:19
|88.8006
|1.19930
|Minor 2nd
|
|D###/Ebbb
|-
|4
|120
|15:14
|119.4428
|0.55719
|Upminor 2nd
|
|Ebb
|-
|5
|150
|12:11
|150.6370
| -0.63705
|Downmajor 2nd
|
|Eb
|-
|6
|180
|10:9
|182.4037
| -2.40371
|Major 2nd
|
|E
|-
|7
|210
|9:8
|203.9100
|6.08999
|Upmajor 2nd
|
|E#
|-
|8
|240
|8:7
|231.1741
|8.82590
|Augmented 2nd
|
|E##
|-
|9
|270
|7:6
|266.8709
|3.12909
|Diminished 3rd
|
|Fbb
|-
|10
|300
|19:16
|297.5130
|2.48698
|Downminor 3rd
|
|Fb
|-
|11
|330
|6:5
|315.6412
|14.3587
|Upminor 3rd
|
|F
|-
|12
|360
|16:13
|359.4723
|0.52766
|Neutral 3rd
|
|F#
|-
|13
|390
|5:4
|386.3137
|3.68628
|Major 3rd
|
|F##
|-
|14
|420
|14:11
|417.5079
|2.49203
|Augmented 3rd
|
|F###/Gbbb
|-
|15
|450
|22:17
|446.3625
|3.63746
|Diminished 4th
|
|Gbb
|-
|16
|480
|21:16
|470.781
|9.219
|Blackwood 4th
|
|Gb
|-
|17
|510
|4:3
|498.0449
|11.9550
|Diatonic 4th
|
|G
|-
|18
|540
|11:8
|551.3179
| -11.3179
|Augmented 4th
|
|G#
|-
|19
|570
|25:18
|568.7174
|1.2825
|Minor Tritone
|
|G##
|-
|20
|600
|7:5
|582.5121
|17.4878
|Perfect Tritone
|
|G###/Abbb
|-
|21
|630
|23:16
|628.2743
|1.72565
|Major Tritone
|
|Abb
|-
|22
|660
|16:11
|648.6820
|11.3179
|Diminished 5th
|
|Ab
|-
|23
|690
|3:2
|701.9550
| -11.9550
|Diatonic 5th
|
|A
|-
|24
|720
|32:21
|729.2191
| -9.219
|Blackwood 5th
|
|A#
|-
|25
|750
|17:11
|753.6374
| -3.63746
|Augmented 5th
|
|A##
|-
|26
|780
|11:7
|782.4920
| -2.49203
|Diminished 6th
|
|A###/Bbbb
|-
|27
|810
| style="text-align:center;" |8:5
|813.6862
| -3.68628
|Minor 6th
|
|Bbb
|-
|28
|840
|13:8
|840.5276
| -0.52766
|Neutral 6th
|
|Bb
|-
|29
|870
| style="text-align:center;" |5:3
|884.3587
| -14.3587
|Downmajor 6th
|
|B
|-
|30
|900
| style="text-align:center;" |32:19
|902.4869
| -2.48698
|Upmajor 6th
|
|B#
|-
|31
|930
| style="text-align:center;" |12:7
|933.1291
| -3.12909
|Augmented 6th
|
|B##
|-
|32
|960
| style="text-align:center;" |7:4
|968.8259
| -8.82590
|Harmonic 7th
|
|Cbb
|-
|33
|990
| style="text-align:center;" |16:9
|996.0899
| -6.08999
|Downminor 7th
|
|Cb
|-
|34
|1020
| style="text-align:center;" |9:5
|1017.5962
|2.40371
|Minor 7th
|
|C
|-
|35
|1050
| style="text-align:center;" |11:6
|1049.3629
|0.63705
|Upminor 7th
|
|C#
|-
|36
|1080
| style="text-align:center;" |28:15
|1080.5571
| -0.55719
|Downmajor 7th
|
|C##
|-
|37
|1110
| style="text-align:center;" |19:10
|1111.1993
| -1.19930
|Major 7th
|
|C###/Dbbb
|-
|38
|1140
| style="text-align:center;" |56:29
|1139.2487
|0.75128
|Upmajor 7th
|
|Dbb
|-
|39
|1170
| style="text-align:center;" |116:59
|1170.4055
| -0.40553
|Down Octave
|
|Db
|-
|40
|1200
| style="text-align:center;" |2:1
|1200
|0
|Octave
|
|D
|-
|
|
| style="text-align:center;" |
|
|
|
|
|
|-
|
|
| style="text-align:center;" |
|
|
|
|
|
|-
|
|
| style="text-align:center;" |
|
|
|
|
|
|-
|
|
| style="text-align:center;" |
|
|
|
|
|
|-
|
|
| style="text-align:center;" |
|
|
|
|
|
|}
[[Category:edo]]
[[Category:edo]]
[[Category:subgroup]]
[[Category:subgroup]]
[[Category:theory]]
[[Category:theory]]
Retrieved from "https://en.xen.wiki/w/40edo"