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{{ED intro}}
{{ED intro}}
== Theory ==
== Theory ==
Up to this point, all the multiples of 5 have had the 720{{c}} 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 the use of 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. So some may not consider it a valid perfect fifth.
Up to this point, all the multiples of 5 have had the 720{{c}} [[blackwood]] fifth as their best approximation of [[3/2]] apart from [[35edo]], which combines 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 found in the [[Val|patent val]] of an edo, 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 5th, it does not qualify as meantone either, tempering out [[177147/163840]] and [[1053/1024]] in the patent val instead of [[81/80]], which means 4 5ths make a near perfect [[16/13|tridecimal neutral 3rd]] and it takes a full 11 to reach the 5th harmonic.  
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. A simple 5-limit temperament 40edo does support is [[diminished]], equating 6/5 to 300 cents. Diminished temperament is familiar due to also being supported [[12edo]], but 40edo's tuning is quite different from 12edo's (at least with the 12-note or greater MOSes).


81/80 is only tempered out in the 40c alternative val where the aforementioned high neutral 3rd 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. It 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; and [[66/65]] in the 13-limit.
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.  


=== 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 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 have low enough [[harmonic entropy]] to sound [[consonant]] to many listeners.
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"
|-
|-
Line 20: Line 23:
! style="text-align: center;" | Cents
! style="text-align: center;" | Cents
! colspan="3" | Notation
! colspan="3" | Notation
! colspan="2" | Approximate ratios
! Approximate ratios
! Difference
|-
|-
! 0
! 0
! 0¢
| 0¢
| perfect unison
| perfect unison
| P1
| P1
| D
| D
| 1:1
| [[1/1]]
| <small>''0''</small>
| 0
|-
|-
! 1
! 1
! 30
| 30
| augmented 1sn
| augmented 1sn
| A1
| A1
| D#
| D#
| 59:58
| [[33/32]], ''[[36/35]]'', [[49/48]], [[81/80|''80/81'']]
| <small>''29.5944''</small>
| 0.40553
|-
|-
! 2
! 2
! 60
| 60
| double-aug 1sn
| double-aug 1sn
| AA1
| AA1
| Dx
| Dx
| 29:28
| ''[[21/20]]'', ''[[64/63]]''
| <small>''60.7512''</small>
|  -0.75128
|-
|-
! 3
! 3
! 90
| 90
| double-dim 2nd
| triple-aug 1sn, double-dim 2nd
| dd2
| dd2
| D#x, Ebbb
| D#x, Ebbb
| 20:19
| ''[[25/24]]''
| <small>''88.8006''</small>
| 1.19930
|-
|-
! 4
! 4
! 120
| 120
| diminished 2nd
| diminished 2nd
| d2
| d2
| Ebb
| Ebb
| 15:14
|[[14/13]], [[15/14]], [[16/15]]
| <small>''119.4428''</small>
| 0.55719
|-
|-
! 5
! 5
! 150
| 150
| minor 2nd
| minor 2nd
| m2
| m2
| Eb
| Eb
| 12:11
| [[11/10]], [[12/11]], [[13/12]]
| <small>''150.6370''</small>
|  -0.63705
|-
|-
! 6
! 6
! 180
| 180
| major 2nd
| major 2nd
| M2
| M2
| E
| E
| 10:9
| ''[[9/8]]''
| <small>''182.4037''</small>
|  -2.40371
|-
|-
! 7
! 7
! 210
| 210
| augmented 2nd
| augmented 2nd
| A2
| A2
| E#
| E#
| 9:8
| ''[[10/9]]''
| <small>''203.9100''</small>
| 6.08999
|-
|-
! 8
! 8
! 240
| 240
| double-aug 2nd
| double-aug 2nd
| AA2
| AA2
| Ex
| Ex
| 8:7
| [[8/7]], [[15/13]]
| <small>''231.1741''</small>
| 8.82590
|-
|-
! 9
! 9
! 270
| 270
| double-dim 3rd
| double-dim 3rd
| dd3
| dd3
| Fbb
| Fbb
| 7:6
| [[7/6]]
| <small>''266.8709''</small>
| 3.12909
|-
|-
! 10
! 10
! 300
| 300
| diminished 3rd
| diminished 3rd
| d3
| d3
| Fb
| Fb
| 19:16
| ''[[6/5]]''
| <small>''297.5130''</small>
| 2.48698
|-
|-
! 11
! 11
! 330
| 330
| minor 3rd
| minor 3rd
| m3
| m3
| F
| F
| 6:5
| ''[[27/22]]'', [[39/32]]
| <small>''315.6412''</small>
| 14.3587
|-
|-
! 12
! 12
! 360
| 360
| major 3rd
| major 3rd
| M3
| M3
| F#
| F#
| 16:13
| ''[[11/9]]'', [[16/13]]
| <small>''359.4723''</small>
| 0.52766
|-
|-
! 13
! 13
! 390
| 390
| augmented 3rd
| augmented 3rd
| A3
| A3
| Fx
| Fx
| 5:4
| [[5/4]]
| <small>''386.3137''</small>
| 3.68628
|-
|-
! 14
! 14
! 420
| 420
| double-aug 3rd
| double-aug 3rd, triple-dim 4th
| AA3
| AA3
| F#x, Gbbb
| F#x, Gbbb
| 14:11
| ''[[9/7]]''
| <small>''417.5079''</small>
| 2.49203
|-
|-
! 15
! 15
! 450
| 450
| double-dim 4th
| double-dim 4th
| dd4
| dd4
| Gbb
| Gbb
| 22:17
| [[13/10]]
| <small>''446.3625''</small>
| 3.63746
|-
|-
! 16
! 16
! 480
| 480
| diminished 4th
| diminished 4th
| d4
| d4
| Gb
| Gb
| 21:16
| ''[[27/20]]''
| <small>''470.781''</small>
| 9.219
|-
|-
! 17
! 17
! 510
| 510
| perfect 4th
| perfect 4th
| P4
| P4
| G
| G
| 4:3
| [[4/3]]
| <small>''498.0449''</small>
| 11.9550
|-
|-
! 18
! 18
! 540
| 540
| augmented 4th
| augmented 4th
| A4
| A4
| G#
| G#
| 11:8
| [[11/8]], [[15/11]], ''[[18/13]]''
| <small>''551.3179''</small>
|  -11.3179
|-
|-
! 19
! 19
! 570
| 570
| double-aug 4th
| double-aug 4th
| AA4
| AA4
| G##
| G##
| 25:18
| [[7/5]]
| <small>''568.7174''</small>
| 1.2825
|-
|-
! 20
! 20
! 600
| 600
| triple-aug 4th,
| triple-aug 4th,
triple-dim 5th
triple-dim 5th
Line 210: Line 172:
ddd5
ddd5
| Gx#, Abbb
| Gx#, Abbb
| 7:5
| ''[[25/18]]'', ''[[36/25]]''
| <small>''582.5121''</small>
| 17.4878
|-
|-
! 21
! 21
! 630
| 630
| double-dim 5th
| double-dim 5th
| dd5
| dd5
| Abb
| Abb
| 23:16
|  
| <small>''628.2743''</small>
| 1.72565
|-
|-
! 22
! 22
! 660
| 660
| diminished 5th
| diminished 5th
| d5
| d5
| Ab
| Ab
| 16:11
|
| <small>''648.6820''</small>
| 11.3179
|-
|-
! 23
! 23
! 690
| 690
| perfect 5th
| perfect 5th
| P5
| P5
| A
| A
| 3:2
|
| <small>''701.9550''</small>
|  -11.9550
|-
|-
! 24
! 24
! 720
| 720
| augmented 5th
| augmented 5th
| A5
| A5
| A#
| A#
| 32:21
|
| <small>''729.2191''</small>
|  -9.219
|-
|-
! 25
! 25
! 750
| 750
| double-aug 5th
| double-aug 5th
| AA5
| AA5
| Ax
| Ax
| 17:11
|
| <small>''753.6374''</small>
|  -3.63746
|-
|-
! 26
! 26
! 780
| 780
| double-dim 6th
| double-dim 6th
| dd6
| dd6
| A#x, Bbbb
| A#x, Bbbb
| 11:7
|
| <small>''782.4920''</small>
|  -2.49203
|-
|-
! 27
! 27
! 810
| 810
| diminished 6th
| diminished 6th
| d6
| d6
| Bbb
| Bbb
| style="text-align: center;" | 8:5
| style="text-align: center;" |
| <small>''813.6862''</small>
|  -3.68628
|-
|-
! 28
! 28
! 840
| 840
| minor 6th
| minor 6th
| m6
| m6
| Bb
| Bb
| 13:8
|
| <small>''840.5276''</small>
|  -0.52766
|-
|-
! 29
! 29
! 870
| 870
| major 6th
| major 6th
| M6
| M6
| B
| B
| style="text-align: center;" | 5:3
| style="text-align: center;" |
| <small>''884.3587''</small>
|  -14.3587
|-
|-
! 30
! 30
! 900
| 900
| augmented 6th
| augmented 6th
| A6
| A6
| B#
| B#
| style="text-align: center;" | 32:19
| style="text-align: center;" |
| <small>''902.4869''</small>
|  -2.48698
|-
|-
! 31
! 31
! 930
| 930
| double-aug 6th
| double-aug 6th
| AA6
| AA6
| Bx
| Bx
| style="text-align: center;" | 12:7
| style="text-align: center;" |
| <small>''933.1291''</small>
|  -3.12909
|-
|-
! 32
! 32
! 960
| 960
| double-dim 7th
| double-dim 7th
| dd7
| dd7
| Cbb
| Cbb
| style="text-align: center;" | 7:4
| style="text-align: center;" |
| <small>''968.8259''</small>
|  -8.82590
|-
|-
! 33
! 33
! 990
| 990
| diminished 7th
| diminished 7th
| d7
| d7
| Cb
| Cb
| style="text-align: center;" | 16:9
| style="text-align: center;" |
| <small>''996.0899''</small>
|  -6.08999
|-
|-
! 34
! 34
! 1020
| 1020
| minor 7th
| minor 7th
| m7
| m7
| C
| C
| style="text-align: center;" | 9:5
| style="text-align: center;" |
| <small>''1017.5962''</small>
| 2.40371
|-
|-
! 35
! 35
! 1050
| 1050
| major 7th
| major 7th
| M7
| M7
| C#
| C#
| style="text-align: center;" | 11:6
| style="text-align: center;" |
| <small>''1049.3629''</small>
| 0.63705
|-
|-
! 36
! 36
! 1080
| 1080
| augmented 7th
| augmented 7th
| A7
| A7
| Cx
| Cx
| style="text-align: center;" | 28:15
| style="text-align: center;" |
| <small>''1080.5571''</small>
|  -0.55719
|-
|-
! 37
! 37
! 1110
| 1110
| double-aug 7th
| double-aug 7th, triple-dim 8ve
| AA7
| AA7
| C#x, Dbbb
| C#x, Dbbb
| style="text-align: center;" | 19:10
| style="text-align: center;" |
| <small>''1111.1993''</small>
|  -1.19930
|-
|-
! 38
! 38
! 1140
| 1140
| double-dim 8ve
| double-dim 8ve
| dd8
| dd8
| Dbb
| Dbb
| style="text-align: center;" | 56:29
| style="text-align: center;" |
| <small>''1139.2487''</small>
| 0.75128
|-
|-
! 39
! 39
! 1170
| 1170
| diminished 8ve
| diminished 8ve
| d8
| d8
| Db
| Db
| style="text-align: center;" | 116:59
| style="text-align: center;" |
| <small>''1170.4055''</small>
|  -0.40553
|-
|-
! 40
! 40
! 1200
| 1200
| perfect octave
| perfect octave
| P8
| P8
| D
| D
| style="text-align: center;" | 2:1
| style="text-align: center;" |
| <small>''1200''</small>
| 0
|}
|}


Line 407: Line 327:
default [[File:40-EDO_Sagittal.svg]]
default [[File:40-EDO_Sagittal.svg]]
</imagemap>
</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 ==
== Scales ==
Line 418: Line 343:
* 12-tone 4&10edo scale: 4 4 2 2 4 4 4 4 2 2 4 4
* 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
* 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
|}
== Instruments ==
* [[Lumatone mapping for 40edo]]


== Music ==
== Music ==
=== Modern Renderings ===
; [https://www.nts.live/artists/83437-hi-posi Hi-Posi]
* [https://www.youtube.com/shorts/YY8rZ6ZcoXc ''Boku De Artitai''] (1994) – microtonal cover in 40edo by [[Bryan Deister]] (2026)
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/x5cnT4Bw1ZQ ''Balance Beam''] (2026)
* [https://www.youtube.com/shorts/fREuVtq875U ''40edo improv''] (2026)
* [https://www.youtube.com/shorts/JriWAth4vBY ''40edo idea''] (2026)
; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=EMZu6ZE6A3g ''Happy Birthday Canon'', 6-in-1 Canon in 40edo]
* [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])
* [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])


== Instruments ==
; [[Stephen Weigel]]
* [[Lumatone mapping for 40edo]]
* [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:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
[[Category:Todo:add rank 2 temperaments table]]

Latest revision as of 22:43, 9 November 2026

← 39edo 40edo 41edo →
Prime factorization 23 × 5
Step size 30 ¢ 
Fifth 23\40 (690 ¢)
Semitones (A1:m2) 1:5 (30 ¢ : 150 ¢)
Dual sharp fifth 24\40 (720 ¢) (→ 3\5)
Dual flat fifth 23\40 (690 ¢)
Dual major 2nd 7\40 (210 ¢)
Consistency limit 3
Distinct consistency limit 3

40 equal divisions of the octave (abbreviated 40edo or 40ed2), also called 40-tone equal temperament (40tet) or 40 equal temperament (40et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 40 equal parts of exactly 30 ¢ each. Each step represents a frequency ratio of 21/40, or the 40th root of 2.

Theory

Up to this point, all the multiples of 5 have had the 720 ¢ blackwood fifth as their best approximation of 3/2 apart from 35edo, which combines 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 found in the patent val of an edo, only exceeded by 47edo in error, which results in it being inconsistent in the 5-limit - combining the best 5/4 (390 ¢) and the best 6/5 (330 ¢) 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 tridecimal neutral third (16/13) and it takes a full 11 fifths (i.e. at the augmented third) to reach the 5th harmonic. A simple 5-limit temperament 40edo does support is diminished, equating 6/5 to 300 cents. Diminished temperament is familiar due to also being supported 12edo, but 40edo's tuning is quite different from 12edo's (at least with the 12-note or greater MOSes).

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.

Odd harmonics

40edo is most accurate on the 2.9.5.21.33.13.51.19.23 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.

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 elements. Both of its fifths can sound consonant to many listeners.

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

Todo: cleanup
# Cents Notation Approximate ratios
0 0¢ perfect unison P1 D 1/1
1 30 augmented 1sn A1 D# 33/32, 36/35, 49/48, 80/81
2 60 double-aug 1sn AA1 Dx 21/20, 64/63
3 90 triple-aug 1sn, double-dim 2nd dd2 D#x, Ebbb 25/24
4 120 diminished 2nd d2 Ebb 14/13, 15/14, 16/15
5 150 minor 2nd m2 Eb 11/10, 12/11, 13/12
6 180 major 2nd M2 E 9/8
7 210 augmented 2nd A2 E# 10/9
8 240 double-aug 2nd AA2 Ex 8/7, 15/13
9 270 double-dim 3rd dd3 Fbb 7/6
10 300 diminished 3rd d3 Fb 6/5
11 330 minor 3rd m3 F 27/22, 39/32
12 360 major 3rd M3 F# 11/9, 16/13
13 390 augmented 3rd A3 Fx 5/4
14 420 double-aug 3rd, triple-dim 4th AA3 F#x, Gbbb 9/7
15 450 double-dim 4th dd4 Gbb 13/10
16 480 diminished 4th d4 Gb 27/20
17 510 perfect 4th P4 G 4/3
18 540 augmented 4th A4 G# 11/8, 15/11, 18/13
19 570 double-aug 4th AA4 G## 7/5
20 600 triple-aug 4th,

triple-dim 5th

AAA4,

ddd5

Gx#, Abbb 25/18, 36/25
21 630 double-dim 5th dd5 Abb
22 660 diminished 5th d5 Ab
23 690 perfect 5th P5 A
24 720 augmented 5th A5 A#
25 750 double-aug 5th AA5 Ax
26 780 double-dim 6th dd6 A#x, Bbbb
27 810 diminished 6th d6 Bbb
28 840 minor 6th m6 Bb
29 870 major 6th M6 B
30 900 augmented 6th A6 B#
31 930 double-aug 6th AA6 Bx
32 960 double-dim 7th dd7 Cbb
33 990 diminished 7th d7 Cb
34 1020 minor 7th m7 C
35 1050 major 7th M7 C#
36 1080 augmented 7th A7 Cx
37 1110 double-aug 7th, triple-dim 8ve AA7 C#x, Dbbb
38 1140 double-dim 8ve dd8 Dbb
39 1170 diminished 8ve d8 Db
40 1200 perfect octave P8 D

Notation

Sagittal notation

This notation uses the same sagittal sequence as EDOs 30b and 35.

Sagittal notationPeriodic table of EDOs with sagittal notationlimma-fraction notation

Octave stretch or compression

127ed9 optimises 40edo for dual-fifth usage by distributing error evenly between its two fifths. 127ed9 is just 40edo with octaves compressed by 1.9 ¢.

Approximation of harmonics in 127ed9
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.9 +15.0 -3.8 -0.8 +13.1 -14.2 -5.8 +0.0 -2.7 +12.0 +11.1
Relative (%) -6.4 +50.0 -12.8 -2.6 +43.6 -47.4 -19.2 +0.0 -9.0 +40.1 +37.2
Steps
(reduced) 		40
(40) 		64
(64) 		80
(80) 		93
(93) 		104
(104) 		112
(112) 		120
(120) 		127
(0) 		133
(6) 		139
(12) 		144
(17)
Approximation of harmonics in 40edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -12.0 +0.0 +3.7 -12.0 -8.8 +0.0 +6.1 +3.7 -11.3 -12.0
Relative (%) +0.0 -39.9 +0.0 +12.3 -39.9 -29.4 +0.0 +20.3 +12.3 -37.7 -39.9
Steps
(reduced) 		40
(0) 		63
(23) 		80
(0) 		93
(13) 		103
(23) 		112
(32) 		120
(0) 		127
(7) 		133
(13) 		138
(18) 		143
(23)

Scales

  • Amulet[idiosyncratic term], (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 term] (approximated from 66afdo): 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
Approximated from 96edo
Contains 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

Instruments

Music

Modern Renderings

Hi-Posi

21st century

Bryan Deister
Claudi Meneghin
Stephen Weigel
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