40edo - Revision history

Lucius Chiaraviglio: /* Music */ Stephen Weigel's ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!'': Add live performance in Munich, Germany (2026)

2026-05-31T03:25:00Z

Music: Stephen Weigel's ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!'': Add live performance in Munich, Germany (2026)

← Older revision		Revision as of 03:25, 31 May 2026
Line 471:		Line 471:
	; [[Stephen Weigel]]		; [[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=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]]

Overthink: /* Intervals */ table could be better

2026-03-02T00:33:51Z

Intervals: table could be better

← Older revision		Revision as of 00:33, 2 March 2026
Line 17:		Line 17:

	== Intervals ==		== Intervals ==
			{{Todo\|cleanup\|inline=1}}
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|-		\|-

Lériendil: cleaned up some wording

2026-01-31T08:39:55Z

cleaned up some wording

← Older revision		Revision as of 08:39, 31 January 2026
Line 2:		Line 2:
	{{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 5/4 and 6/5 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 ===		=== Odd harmonics ===
	40edo is most accurate on the 2.9.5.21.33.13.51.19.23 [[kN_subgroups\| 240 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 [[kN_subgroups\| 240 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}}

VectorGraphics: bruh "major third" does not equal 5/4

2026-01-31T08:02:09Z

bruh "major third" does not equal 5/4

← Older revision		Revision as of 08:02, 31 January 2026
Line 2:		Line 2:
	{{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 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 5/4 and 6/5 will result in the blackwood 5th 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 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.

Lucius Chiaraviglio: Move Instruments above Music for consistency with most other EDO pages

2026-01-05T16:22:14Z

Move Instruments above Music for consistency with most other EDO pages

← Older revision		Revision as of 16:22, 5 January 2026
Line 454:		Line 454:
	* Underpass: 11 12 8 3 6		* Underpass: 11 12 8 3 6
	\|}		\|}

			== Instruments ==
			* [[Lumatone mapping for 40edo]]

	== Music ==		== Music ==
Line 465:		Line 468:
	; [[Stephen Weigel]]		; [[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=tLmaQK10aYM ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!''] (2025; mostly in 42edo, but also some in 40edo)

	~~== Instruments ==~~
	* [[Lumatone mapping for 40edo]]

	[[Category:Listen]]		[[Category:Listen]]
	[[Category:Todo:add rank 2 temperaments table]]		[[Category:Todo:add rank 2 temperaments table]]

Lucius Chiaraviglio: /* Music */ Add Bryan Deister's ''Balance Beam'' (2026)

2026-01-05T16:18:46Z

Music: Add Bryan Deister's ''Balance Beam'' (2026)

← Older revision		Revision as of 16:18, 5 January 2026
Line 456:		Line 456:

	== Music ==		== Music ==
			; [[Bryan Deister]]
			* [https://www.youtube.com/shorts/x5cnT4Bw1ZQ ''Balance Beam''] (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]

Lucius Chiaraviglio: /* Music */ Add Stephen Weigel's ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!'' (2025; mostly in 42edo, but also some in 40edo)

2025-12-26T21:47:12Z

Music: Add Stephen Weigel's ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!'' (2025; mostly in 42edo, but also some in 40edo)

← Older revision		Revision as of 21:47, 26 December 2025
Line 459:		Line 459:
	* [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])

			; [[Stephen Weigel]]
			* [https://www.youtube.com/watch?v=tLmaQK10aYM ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!''] (2025; mostly in 42edo, but also some in 40edo)

	== Instruments ==		== Instruments ==

BudjarnLambeth: /* Scales */ add several

2025-10-29T05:03:12Z

Scales: add several

@@ Line 423: / Line 423: @@
 * 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
 == Music ==

BudjarnLambeth: /* Octave stretch or compression */

2025-10-29T04:37:42Z

Octave stretch or compression

← Older revision		Revision as of 04:37, 29 October 2025
Line 409:		Line 409:

	== Octave stretch or compression ==		== Octave stretch or compression ==
	'''127ed9''' optimises 40edo for [[dual-fifth]] usage by distributing error evenly between its two fifths:		'''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\|127\|9\|1\|intervals=integer\|collapsed=yes}}
	{{harmonics in equal\|40\|2\|1\|intervals=integer\|collapsed=yes}}		{{harmonics in equal\|40\|2\|1\|intervals=integer\|collapsed=yes}}

BudjarnLambeth: /* Sagittal notation */ octave stretch section

2025-10-29T04:36:34Z

Sagittal notation: octave stretch section

← Older revision		Revision as of 04:36, 29 October 2025
Line 407:		Line 407:
	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:
			{{harmonics in equal\|127\|9\|1\|intervals=integer\|collapsed=yes}}
			{{harmonics in equal\|40\|2\|1\|intervals=integer\|collapsed=yes}}

	== Scales ==		== Scales ==