40edo: Difference between revisions

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== 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]]. 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 ~~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.

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 ===

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. ~~As a dual-fifth system, it really shines, as both~~ of its fifths ~~have low enough [[harmonic entropy]] to~~ sound [[consonant]] to many listeners~~, giving two consonant intervals for the price of one~~.

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.

== Intervals ==

{| class="wikitable center-all"

|-

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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}}.

== Scales ==

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** 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

|}

== Instruments ==

* [[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])

=~~= 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:Todo:add rank 2 temperaments table]]

@@ Line 2: / Line 2: @@
 {{ED intro}}
 == 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]]. 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 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.
-/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.
+edo 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.
+/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 ===
 edo 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.
-edo 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. As a dual-fifth system, it really shines, as both of its fifths have low enough [[harmonic entropy]] to sound [[consonant]] to many listeners, giving two consonant intervals for the price of one.
+edo 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}}
 == Intervals ==
+{{Todo|cleanup|inline=1}}
 {| class="wikitable center-all"
 |-
@@ Line 407: / Line 410: @@
 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 ==
@@ Line 416: / Line 424: @@
 ** 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
+|}
+== Instruments ==
+* [[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])
-== 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:Todo:add rank 2 temperaments table]]