40edo - Revision history

CompactStar: /* Theory */

2026-11-09T22:43:21Z

Theory

← Older revision		Revision as of 22:43, 9 November 2026
Line 4:		Line 4:
	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.		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. A simple 5-limit temperament 40edo does support is [[diminished]], equating 6/5 to 300 cents, ~~which could potentially be used as an alternative to meantone in~~ 40edo ~~for reaching simple 5~~-~~limit chords~~.		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).

	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.		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.

CompactStar: /* Theory */

2026-11-09T22:41:58Z

Theory

← Older revision		Revision as of 22:41, 9 November 2026
Line 4:		Line 4:
	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.		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. A simple 5-limit temperament 40edo does support~~, which could potentially be used as an alternative to meantone,~~ is [[diminished]], equating 6/5 to 300 cents.		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, which could potentially be used as an alternative to meantone in 40edo for reaching simple 5-limit chords.

	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.		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.

CompactStar: /* Theory */

2026-11-09T22:40:16Z

Theory

← Older revision		Revision as of 22:40, 9 November 2026
Line 4:		Line 4:
	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.		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.		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, which could potentially be used as an alternative to meantone, is [[diminished]], equating 6/5 to 300 cents.

	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.		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.

CompactStar: /* Intervals */

2026-11-09T21:26:40Z

Intervals

← Older revision		Revision as of 21:26, 9 November 2026
Line 121:		Line 121:
	\| A3		\| A3
	\| Fx		\| Fx
	\|		\| [[5/4]]
	\|-		\|-
	! 14		! 14
Line 128:		Line 128:
	\| AA3		\| AA3
	\| F#x, Gbbb		\| F#x, Gbbb
	\|		\| ''[[9/7]]''
	\|-		\|-
	! 15		! 15
Line 135:		Line 135:
	\| dd4		\| dd4
	\| Gbb		\| Gbb
	\|		\| [[13/10]]
	\|-		\|-
	! 16		! 16
Line 142:		Line 142:
	\| d4		\| d4
	\| Gb		\| Gb
	\|		\| ''[[27/20]]''
	\|-		\|-
	! 17		! 17
Line 149:		Line 149:
	\| P4		\| P4
	\| G		\| G
	\|		\| [[4/3]]
	\|-		\|-
	! 18		! 18
Line 156:		Line 156:
	\| A4		\| A4
	\| G#		\| G#
	\|		\| [[11/8]], [[15/11]], ''[[18/13]]''
	\|-		\|-
	! 19		! 19
Line 163:		Line 163:
	\| AA4		\| AA4
	\| G##		\| G##
	\|		\| [[7/5]]
	\|-		\|-
	! 20		! 20
Line 172:		Line 172:
	ddd5		ddd5
	\| Gx#, Abbb		\| Gx#, Abbb
	\|		\| ''[[25/18]]'', ''[[36/25]]''
	\|-		\|-
	! 21		! 21
Line 179:		Line 179:
	\| dd5		\| dd5
	\| Abb		\| Abb
	\|		\|
	\|-		\|-
	! 22		! 22

CompactStar: /* Intervals */

2026-11-09T21:22:58Z

Intervals

← Older revision		Revision as of 21:22, 9 November 2026
Line 100:		Line 100:
	\| d3		\| d3
	\| Fb		\| Fb
	\|		\| ''[[6/5]]''
	\|-		\|-
	! 11		! 11
Line 107:		Line 107:
	\| m3		\| m3
	\| F		\| F
	\|		\| ''[[27/22]]'', [[39/32]]
	\|-		\|-
	! 12		! 12
Line 114:		Line 114:
	\| M3		\| M3
	\| F#		\| F#
	\|		\| ''[[11/9]]'', [[16/13]]
	\|-		\|-
	! 13		! 13

CompactStar: /* Intervals */

2026-11-09T21:22:07Z

Intervals

← Older revision		Revision as of 21:22, 9 November 2026
Line 65:		Line 65:
	\| m2		\| m2
	\| Eb		\| Eb
	\|		\| [[11/10]], [[12/11]], [[13/12]]
	\|-		\|-
	! 6		! 6
Line 72:		Line 72:
	\| M2		\| M2
	\| E		\| E
	\|		\| ''[[9/8]]''
	\|-		\|-
	! 7		! 7
Line 79:		Line 79:
	\| A2		\| A2
	\| E#		\| E#
	\|		\| ''[[10/9]]''
	\|-		\|-
	! 8		! 8
Line 86:		Line 86:
	\| AA2		\| AA2
	\| Ex		\| Ex
	\|		\| [[8/7]], [[15/13]]
	\|-		\|-
	! 9		! 9
Line 93:		Line 93:
	\| dd3		\| dd3
	\| Fbb		\| Fbb
	\|		\| [[7/6]]
	\|-		\|-
	! 10		! 10

CompactStar: /* Intervals */

2026-11-09T21:21:00Z

Intervals

← Older revision		Revision as of 21:21, 9 November 2026
Line 125:		Line 125:
	! 14		! 14
	\| 420		\| 420
	\| double-aug 3rd		\| double-aug 3rd, triple-dim 4th
	\| AA3		\| AA3
	\| F#x, Gbbb		\| F#x, Gbbb

CompactStar: /* Intervals */

2026-11-09T21:20:47Z

Intervals

← Older revision		Revision as of 21:20, 9 November 2026
Line 48:		Line 48:
	! 3		! 3
	\| 90		\| 90
	\| double-dim 2nd		\| triple-aug 1sn, double-dim 2nd
	\| dd2		\| dd2
	\| D#x, Ebbb		\| D#x, Ebbb
Line 288:		Line 288:
	! 37		! 37
	\| 1110		\| 1110
	\| double-aug 7th		\| double-aug 7th, triple-dim 8ve
	\| AA7		\| AA7
	\| C#x, Dbbb		\| C#x, Dbbb

CompactStar: /* Intervals */

2026-11-09T21:20:23Z

Intervals

← Older revision		Revision as of 21:20, 9 November 2026
Line 89:		Line 89:
	\|-		\|-
	! 9		! 9
	! 270		\| 270
	\| double-dim 3rd		\| double-dim 3rd
	\| dd3		\| dd3
Line 103:		Line 103:
	\|-		\|-
	! 11		! 11
	! 330		\| 330
	\| minor 3rd		\| minor 3rd
	\| m3		\| m3
Line 110:		Line 110:
	\|-		\|-
	! 12		! 12
	! 360		\| 360
	\| major 3rd		\| major 3rd
	\| M3		\| M3

CompactStar: /* Intervals */

2026-11-09T21:20:04Z

Intervals

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