40edo: Difference between revisions

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== Theory ==

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.

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 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.

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== Intervals ==

{| class="wikitable center-all"

|-

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== 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]]

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; [[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]]

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 {{ED intro}}
 == Theory ==
-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.
+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 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.
@@ Line 17: / Line 17: @@
 == Intervals ==
+{{Todo|cleanup|inline=1}}
 {| class="wikitable center-all"
 |-
@@ Line 461: / Line 462: @@
 == 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]]
@@ Line 470: / Line 478: @@
 ; [[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]]