Xenharmonic Wiki

40edo: Difference between revisions

← Older edit
VisualWikitext
CompactStar (talk | contribs)
autopatrolled
5,084 edits
No edit summary
→Music: Stephen Weigel's ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!'': Add live performance in Munich, Germany (2026)
 
(37 intermediate revisions by 13 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|40}}
{{ED intro}}
== Theory ==
== Theory ==
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. This fifth qualifies for [[flattone]], a variant of meantone with flat fifths, although 40edo's fifth is a bit extreme even for flattone. 40edo's fifth is flat enough that the meantone major third falls into submajor or even sharp neutral third territory at 360 cents, while the minor third is supraminor although not quite high enough to be considered neutral at 330 cents. 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 up to 3 of them to notate more distant keys. It [[tempering_out|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]].
Up to this point, all the multiples of 5 have had the 720{{c}} [[blackwood]] fifth as their best approximation of [[3/2]]. 35edo combined the small circles of blackwood and whitewood fifths, almost equally far from just, requiring the use of both to reach all keys. 40edo adds a diatonic fifth that's closer to just. However, it is still the second flattest diatonic fifth, only exceeded by 47edo in error, which results in it being inconsistent in the [[5-limit]] - combining the best 5/4 (390{{c}}) and the best 6/5 (330{{c}}) will result in the blackwood fifth instead. So some may not consider it a valid perfect fifth.
 
Despite all keys being reachable by stacking this fifth, it does not qualify as meantone either. Instead, it supports [[deeptone]], which tempers out [[177147/163840]] and [[1053/1024]] in the patent val instead of [[81/80]], meaning that four fifths make a near perfect [[16/13|tridecimal neutral third (16/13)]] and it takes a full 11 fifths (i.e. at the augmented third) to reach the 5th harmonic.  
 
40edo tempers out [[648/625]] in the 5-limit; [[225/224]] and [[16807/16384]] in the [[7-limit]]; [[99/98]], [[121/120]] and [[176/175]] in the [[11-limit]] - tuning [[orwell]] though highly suboptimally; and [[66/65]] in the 13-limit.
 
81/80 is only tempered out in the 40c alternative [[val]] where the aforementioned high neutral third is equated with 5/4 instead of the much more accurate 390-cent interval. 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.  


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.
=== Odd harmonics ===
=== Odd harmonics ===
40edo is most accurate on the 2.9.5.21.33.13.51.19.23 [[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 can be treated as a [[dual-fifth system]] in the 2.3+.3-.5.7.11 subgroup, or the 2.3+.3-.5.7.11.13.19.23 subgroup for those who aren’t intimidated by lots of [[basis element]]s. Both of its fifths can sound [[consonant]] to many listeners.
{{harmonics in equal|40}}
{{harmonics in equal|40}}


== Intervals ==
== Intervals ==
{{Todo|cleanup|inline=1}}
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! #
! #
! style="text-align:center;" |Cents
! style="text-align: center;" | Cents
! colspan="2" |Approximate ratios
! colspan="3" | Notation
!Difference
! colspan="2" | Approximate ratios
!colspan="3" |Notation
! Difference
|-
|-
|0
! 0
|0¢
! 0¢
|1:1
| perfect unison
|0
| P1
|0
| D
|perfect unison
| 1:1
|P1
| <small>''0''</small>
|D
| 0
|-
|-
|1
! 1
|30
! 30
|59:58
| augmented 1sn
|29.5944
| A1
|0.40553
| D#
|augmented 1sn
| 59:58
|A1
| <small>''29.5944''</small>
|D#
| 0.40553
|-
|-
|2
! 2
|60
! 60
|29:28
| double-aug 1sn
|60.7512
| AA1
| -0.75128
| Dx
|double-aug 1sn
| 29:28
|AA1
| <small>''60.7512''</small>
|Dx
| -0.75128
|-
|-
|3
! 3
|90
! 90
|20:19
| double-dim 2nd
|88.8006
| dd2
|1.19930
| D#x, Ebbb
|double-dim 2nd
| 20:19
|dd2
| <small>''88.8006''</small>
|D#x, Ebbb
| 1.19930
|-
|-
|4
! 4
|120
! 120
|15:14
| diminished 2nd
|119.4428
| d2
|0.55719
| Ebb
|diminished 2nd
| 15:14
|d2
| <small>''119.4428''</small>
|Ebb
| 0.55719
|-
|-
|5
! 5
|150
! 150
|12:11
| minor 2nd
|150.6370
| m2
| -0.63705
| Eb
|minor 2nd
| 12:11
|m2
| <small>''150.6370''</small>
|Eb
| -0.63705
|-
|-
|6
! 6
|180
! 180
|10:9
| major 2nd
|182.4037
| M2
| -2.40371
| E
|major 2nd
| 10:9
|M2
| <small>''182.4037''</small>
|E
| -2.40371
|-
|-
|7
! 7
|210
! 210
|9:8
| augmented 2nd
|203.9100
| A2
|6.08999
| E#
|augmented 2nd
| 9:8
|A2
| <small>''203.9100''</small>
|E#
| 6.08999
|-
|-
|8
! 8
|240
! 240
|8:7
| double-aug 2nd
|231.1741
| AA2
|8.82590
| Ex
|double-aug 2nd
| 8:7
|AA2
| <small>''231.1741''</small>
|Ex
| 8.82590
|-
|-
|9
! 9
|270
! 270
|7:6
| double-dim 3rd
|266.8709
| dd3
|3.12909
| Fbb
|double-dim 3rd
| 7:6
|dd3
| <small>''266.8709''</small>
|Fbb
| 3.12909
|-
|-
|10
! 10
|300
! 300
|19:16
| diminished 3rd
|297.5130
| d3
|2.48698
| Fb
|diminished 3rd
| 19:16
|d3
| <small>''297.5130''</small>
|Fb
| 2.48698
|-
|-
|11
! 11
|330
! 330
|6:5
| minor 3rd
|315.6412
| m3
|14.3587
| F
|minor 3rd
| 6:5
|m3
| <small>''315.6412''</small>
|F
| 14.3587
|-
|-
|12
! 12
|360
! 360
|16:13
| major 3rd
|359.4723
| M3
|0.52766
| F#
|major 3rd
| 16:13
|M3
| <small>''359.4723''</small>
|F#
| 0.52766
|-
|-
|13
! 13
|390
! 390
|5:4
| augmented 3rd
|386.3137
| A3
|3.68628
| Fx
|augmented 3rd
| 5:4
|A3
| <small>''386.3137''</small>
|Fx
| 3.68628
|-
|-
|14
! 14
|420
! 420
|14:11
| double-aug 3rd
|417.5079
| AA3
|2.49203
| F#x, Gbbb
|double-aug 3rd
| 14:11
|AA3
| <small>''417.5079''</small>
|F#x, Gbbb
| 2.49203
|-
|-
|15
! 15
|450
! 450
|22:17
| double-dim 4th
|446.3625
| dd4
|3.63746
| Gbb
|double-dim 4th
| 22:17
|dd4
| <small>''446.3625''</small>
|Gbb
| 3.63746
|-
|-
|16
! 16
|480
! 480
|21:16
| diminished 4th
|470.781
| d4
|9.219
| Gb
|diminished 4th
| 21:16
|d4
| <small>''470.781''</small>
|Gb
| 9.219
|-
|-
|17
! 17
|510
! 510
|4:3
| perfect 4th
|498.0449
| P4
|11.9550
| G
|perfect 4th
| 4:3
|P4
| <small>''498.0449''</small>
|G
| 11.9550
|-
|-
|18
! 18
|540
! 540
|11:8
| augmented 4th
|551.3179
| A4
| -11.3179
| G#
|augmented 4th
| 11:8
|A4
| <small>''551.3179''</small>
|G#
| -11.3179
|-
|-
|19
! 19
|570
! 570
|25:18
| double-aug 4th
|568.7174
| AA4
|1.2825
| G##
|double-aug 4th
| 25:18
|AA4
| <small>''568.7174''</small>
|G##
| 1.2825
|-
|-
|20
! 20
|600
! 600
|7:5
| triple-aug 4th,
|582.5121
|17.4878
|triple-aug 4th,
triple-dim 5th
triple-dim 5th
|AAA4,
| AAA4,
ddd5
ddd5
|Gx#, Abbb
| Gx#, Abbb
| 7:5
| <small>''582.5121''</small>
| 17.4878
|-
|-
|21
! 21
|630
! 630
|23:16
| double-dim 5th
|628.2743
| dd5
|1.72565
| Abb
|double-dim 5th
| 23:16
|dd5
| <small>''628.2743''</small>
|Abb
| 1.72565
|-
|-
|22
! 22
|660
! 660
|16:11
| diminished 5th
|648.6820
| d5
|11.3179
| Ab
|diminished 5th
| 16:11
|d5
| <small>''648.6820''</small>
|Ab
| 11.3179
|-
|-
|23
! 23
|690
! 690
|3:2
| perfect 5th
|701.9550
| P5
| -11.9550
| A
|perfect 5th
| 3:2
|P5
| <small>''701.9550''</small>
|A
| -11.9550
|-
|-
|24
! 24
|720
! 720
|32:21
| augmented 5th
|729.2191
| A5
| -9.219
| A#
|augmented 5th
| 32:21
|A5
| <small>''729.2191''</small>
|A#
| -9.219
|-
|-
|25
! 25
|750
! 750
|17:11
| double-aug 5th
|753.6374
| AA5
| -3.63746
| Ax
|double-aug 5th
| 17:11
|AA5
| <small>''753.6374''</small>
|Ax
| -3.63746
|-
|-
|26
! 26
|780
! 780
|11:7
| double-dim 6th
|782.4920
| dd6
| -2.49203
| A#x, Bbbb
|double-dim 6th
| 11:7
|dd6
| <small>''782.4920''</small>
|A#x, Bbbb
| -2.49203
|-
|-
|27
! 27
|810
! 810
| style="text-align:center;" |8:5
| diminished 6th
|813.6862
| d6
| -3.68628
| Bbb
|diminished 6th
| style="text-align: center;" | 8:5
|d6
| <small>''813.6862''</small>
|Bbb
| -3.68628
|-
|-
|28
! 28
|840
! 840
|13:8
| minor 6th
|840.5276
| m6
| -0.52766
| Bb
|minor 6th
| 13:8
|m6
| <small>''840.5276''</small>
|Bb
| -0.52766
|-
|-
|29
! 29
|870
! 870
| style="text-align:center;" |5:3
| major 6th
|884.3587
| M6
| -14.3587
| B
|major 6th
| style="text-align: center;" | 5:3
|M6
| <small>''884.3587''</small>
|B
| -14.3587
|-
|-
|30
! 30
|900
! 900
| style="text-align:center;" |32:19
| augmented 6th
|902.4869
| A6
| -2.48698
| B#
|augmented 6th
| style="text-align: center;" | 32:19
|A6
| <small>''902.4869''</small>
|B#
| -2.48698
|-
|-
|31
! 31
|930
! 930
| style="text-align:center;" |12:7
| double-aug 6th
|933.1291
| AA6
| -3.12909
| Bx
|double-aug 6th
| style="text-align: center;" | 12:7
|AA6
| <small>''933.1291''</small>
|Bx
| -3.12909
|-
|-
|32
! 32
|960
! 960
| style="text-align:center;" |7:4
| double-dim 7th
|968.8259
| dd7
| -8.82590
| Cbb
|double-dim 7th
| style="text-align: center;" | 7:4
|dd7
| <small>''968.8259''</small>
|Cbb
| -8.82590
|-
|-
|33
! 33
|990
! 990
| style="text-align:center;" |16:9
| diminished 7th
|996.0899
| d7
| -6.08999
| Cb
|diminished 7th
| style="text-align: center;" | 16:9
|d7
| <small>''996.0899''</small>
|Cb
| -6.08999
|-
|-
|34
! 34
|1020
! 1020
| style="text-align:center;" |9:5
| minor 7th
|1017.5962
| m7
|2.40371
| C
|minor 7th
| style="text-align: center;" | 9:5
|m7
| <small>''1017.5962''</small>
|C
| 2.40371
|-
|-
|35
! 35
|1050
! 1050
| style="text-align:center;" |11:6
| major 7th
|1049.3629
| M7
|0.63705
| C#
|major 7th
| style="text-align: center;" | 11:6
|M7
| <small>''1049.3629''</small>
|C#
| 0.63705
|-
|-
|36
! 36
|1080
! 1080
| style="text-align:center;" |28:15
| augmented 7th
|1080.5571
| A7
| -0.55719
| Cx
|augmented 7th
| style="text-align: center;" | 28:15
|A7
| <small>''1080.5571''</small>
|Cx
| -0.55719
|-
|-
|37
! 37
|1110
! 1110
| style="text-align:center;" |19:10
| double-aug 7th
|1111.1993
| AA7
| -1.19930
| C#x, Dbbb
|double-aug 7th
| style="text-align: center;" | 19:10
|AA7
| <small>''1111.1993''</small>
|C#x, Dbbb
| -1.19930
|-
|-
|38
! 38
|1140
! 1140
| style="text-align:center;" |56:29
| double-dim 8ve
|1139.2487
| dd8
|0.75128
| Dbb
|double-dim 8ve
| style="text-align: center;" | 56:29
|dd8
| <small>''1139.2487''</small>
|Dbb
| 0.75128
|-
|-
|39
! 39
|1170
! 1170
| style="text-align:center;" |116:59
| diminished 8ve
|1170.4055
| d8
| -0.40553
| Db
|diminished 8ve
| style="text-align: center;" | 116:59
|d8
| <small>''1170.4055''</small>
|Db
| -0.40553
|-
|-
|40
! 40
|1200
! 1200
| style="text-align:center;" |2:1
| perfect octave
|1200
| P8
|0
| D
|perfect octave
| style="text-align: center;" | 2:1
|P8
| <small>''1200''</small>
|D
| 0
|}
 
== Notation ==
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[30edo#Second-best fifth notation|30b]] and [[35edo#Sagittal notation|35]].
 
<imagemap>
File:40-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 647 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:40-EDO_Sagittal.svg]]
</imagemap>
 
== Octave stretch or compression ==
'''127ed9''' optimises 40edo for [[dual-fifth]] usage by distributing error evenly between its two fifths. 127ed9 is just 40edo with [[octave shrinking|octaves compressed]] by 1.9{{c}}.
{{harmonics in equal|127|9|1|intervals=integer|collapsed=yes}}
{{harmonics in equal|40|2|1|intervals=integer|collapsed=yes}}
 
== Scales ==
* Amulet{{idiosyncratic}}, (approximated from [[magic]] in [[25edo]]): 3 2 3 3 2 3 5 3 3 2 3 5 3
* [[Equipentatonic]]: 8 8 8 8 8
* Evacuated planet{{idiosyncratic}} (approximated from [[66afdo|66]][[afdo]]): 4 13 6 12 5
* Approximations of [[gamelan]] scales:
** 5-tone pelog: 4 5 14 3 14
** 7-tone pelog: 4 5 8 6 3 10 4
** 5-tone slendro: 8 8 8 8 8
* 12-tone 4&10edo scale: 4 4 2 2 4 4 4 4 2 2 4 4
* 12-tone 5&8edo scale: 5 3 2 5 1 4 4 1 5 2 3 5
 
{| class="wikitable mw-collapsible mw-collapsed"
|+Approximated from [[96edo]]
|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
* Flattened major: 7 6 4 6 7 7 3
* Sharpened minor: 7 3 7 6 4 7 6
* Sharpened harmonic minor: 7 3 7 7 3 11 2
* Flattened major pentatonic: 6 7 10 7 10
* Sharpened minor pentatonic: 10 7 6 11 6
* Evened minor hexatonic: 6 4 7 6 10 7
* Roughened augmented: 11 3 10 3 10 3
* Evened dominant pentatonic: 7 7 9 10 7
* Sharpened Dorian: 7 3 7 7 7 3 6
* Flattened Ionian pentatonic: 13 3 7 13 4
* Sharpened Dorian harmonic: 7 3 10 4 7 3 6
* Evened Mixolydian pentatonic: 13 4 6 10 7
* Evened Phrygian dominant: 4 9 4 6 4 6 7
* Evened Phrygian dominant hexatonic: 3 10 4 6 10 7
* Sharpened Phrygian pentatonic: 4 7 13 3 13
* Sharpened minor harmonic pentatonic I: 7 3 14 13 3
* Evened hirajoshi: 7 4 12 4 13
* Sharpened hirajoshi: 7 4 13 4 12
* Extra-sharp hirajoshi: 8 3 13 4 12
* Evened akebono I: 6 5 12 6 11
* Sharpened akebono I: 7 3 14 6 10
* Extra-sharp akebono I: 7 4 13 7 9
* Evened Javanese pentachordal: 4 7 9 4 16
* Moonbeam: 7 4 12 14 3
* Palace (type of [[equiheptatonic]]): 5 6 6 6 6 6 5
* Underpass: 11 12 8 3 6
|}
|}


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
== Instruments ==
[[Category:Subgroup]]
* [[Lumatone mapping for 40edo]]
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/x5cnT4Bw1ZQ ''Balance Beam''] (2026)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=EMZu6ZE6A3g ''Happy Birthday Canon'', 6-in-1 Canon in 40edo]
* [https://www.youtube.com/watch?v=eu854Ld_uLE ''Canon on Twinkle Twinkle Little Star'', for Baroque Oboe & Viola] (2023) – ([https://www.youtube.com/watch?v=l7vDHwsboLE for Organ])
 
; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=tLmaQK10aYM ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!''] (2025; mostly in 42edo, but also some in 40edo)
** [https://www.youtube.com/watch?v=NE77rwCsGHw live performance of the above in Munich, Germany] (2026)
 
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/40edo"