39edo - Revision history

Overthink: /* As a tuning of other temperaments */ link here

2026-06-12T19:11:54Z

As a tuning of other temperaments: link here

← Older revision		Revision as of 19:11, 12 June 2026
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	39edo, with its 400{{c}} major third, [[tempering out\|tempers out]] the [[128/125\|diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo\|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].		39edo, with its 400{{c}} major third, [[tempering out\|tempers out]] the [[128/125\|diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo\|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].

	Alternatively, the patent val tempers out [[49/48]] to yield [[semaphore]], and provides a reasonable tuning of [[triforce]] beyond [[15edo]], and optimizes both its semaphore and augmented components by tuning the fifth sharp. The 39c val supports [[negri]] and 5-limit [[sixix]].		Alternatively, the patent val tempers out [[49/48]] to yield [[semaphore]], and provides a reasonable tuning of [[triforce]] beyond [[15edo]], and optimizes both its semaphore and augmented components by tuning the fifth sharp. The 39c val supports [[negri]] and 5-limit [[Syntonic–chromatic_equivalence_continuum#Sixix_(5-limit)\|sixix]].

	If we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]] through the 39bc val, and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s\|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]] and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.		If we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]] through the 39bc val, and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s\|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]] and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.

Overthink: /* As a tuning of other temperaments */ also this

2026-06-12T19:09:26Z

As a tuning of other temperaments: also this

← Older revision		Revision as of 19:09, 12 June 2026
Line 14:		Line 14:
	39edo, with its 400{{c}} major third, [[tempering out\|tempers out]] the [[128/125\|diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo\|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].		39edo, with its 400{{c}} major third, [[tempering out\|tempers out]] the [[128/125\|diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo\|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].

	Alternatively, the patent val tempers out [[49/48]] to yield [[semaphore]], and provides a reasonable tuning of [[triforce]] beyond [[15edo]], and optimizes both its semaphore and augmented components by tuning the fifth sharp. The 39c val supports [[negri]].		Alternatively, the patent val tempers out [[49/48]] to yield [[semaphore]], and provides a reasonable tuning of [[triforce]] beyond [[15edo]], and optimizes both its semaphore and augmented components by tuning the fifth sharp. The 39c val supports [[negri]] and 5-limit [[sixix]].

	If we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]] through the 39bc val, and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s\|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]] and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.		If we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]] through the 39bc val, and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s\|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]] and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.

Overthink: /* Intervals */ add these

2026-06-12T00:46:37Z

Intervals: add these

← Older revision		Revision as of 00:46, 12 June 2026
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	\| 2		\| 2
	\| 61.5		\| 61.5
	\| [[33/32]]		\| [[33/32]], ''[[256/243]]''
	\|		\|
	\| ''[[21/20]]'', [[36/35]]		\| ''[[21/20]]'', [[36/35]]
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	\| 9		\| 9
	\| 276.9		\| 276.9
	\|		\| ''[[32/27]]''
	\|		\|
	\|		\|
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	\| 14		\| 14
	\| 430.8		\| 430.8
	\|		\| ''[[81/64]]''
	\|		\|
	\| ''[[35/27]]''		\| ''[[35/27]]''
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	\| 25		\| 25
	\| 769.2		\| 769.2
	\|		\| ''[[128/81]]''
	\|		\|
	\| ''[[54/35]]''		\| ''[[54/35]]''
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	\| 30		\| 30
	\| 923.1		\| 923.1
	\|		\| ''[[27/16]]''
	\|		\|
	\|		\|
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	\| 37		\| 37
	\| 1138.5		\| 1138.5
	\| [[64/33]]		\| [[64/33]], ''[[243/128]]''
	\|		\|
	\| [[35/18]], ''[[40/21]]''		\| [[35/18]], ''[[40/21]]''

FloraC: /* Intervals */ + intervals of 13

2026-06-11T10:18:54Z

Intervals: + intervals of 13

← Older revision		Revision as of 10:18, 11 June 2026
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	! #		! #
	! Cents		! Cents
	! Approximate ratios		! Approximate ratios*
	\|-		\|-
	\| 0		\| 0
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	\| 2		\| 2
	\| 61.5		\| 61.5
	\| [[28/27]], [[33/32]], ''[[49/48]]''		\| ''[[22/21]]'', [[28/27]], [[33/32]], ''[[49/48]]''
	\|-		\|-
	\| 3		\| 3
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	\| 4		\| 4
	\| 123.1		\| 123.1
	\| [[15/14]]		\| [[14/13]], [[15/14]]
	\|-		\|-
	\| 5		\| 5
	\| 153.8		\| 153.8
	\| [[11/10]], [[12/11]]		\| [[11/10]], [[12/11]], [[13/12]]
	\|-		\|-
	\| 6		\| 6
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	\| 8		\| 8
	\| 246.2		\| 246.2
	\| [[81/70]]		\| [[15/13]]
	\|-		\|-
	\| 9		\| 9
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	\| 11		\| 11
	\| 338.5		\| 338.5
	\| [[11/9]]		\| [[11/9]], ''[[16/13]]''
	\|-		\|-
	\| 12		\| 12
	\| 369.2		\| 369.2
	\| [[27/22]]		\| [[26/21]], [[27/22]]
	\|-		\|-
	\| 13		\| 13
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	\| 15		\| 15
	\| 461.5		\| 461.5
	\| [[35/27]]		\| [[13/10]]
	\|-		\|-
	\| 16		\| 16
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	\| 18		\| 18
	\| 553.8		\| 553.8
	\| [[11/8]]		\| [[11/8]], [[18/13]], ''[[15/11]]''
	\|-		\|-
	\| 19		\| 19
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	\| 21		\| 21
	\| 646.2		\| 646.2
	\| [[16/11]]		\| [[13/9]], [[16/11]], ''[[22/15]]''
	\|-		\|-
	\| 22		\| 22
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	\| 24		\| 24
	\| 738.5		\| 738.5
	\| [[54/35]]		\| [[20/13]]
	\|-		\|-
	\| 25		\| 25
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	\| 27		\| 27
	\| 830.8		\| 830.8
	\| [[44/27]]		\| [[21/13]], [[44/27]]
	\|-		\|-
	\| 28		\| 28
	\| 861.5		\| 861.5
	\| [[18/11]]		\| [[18/11]], ''[[13/8]]''
	\|-		\|-
	\| 29		\| 29
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	\| 31		\| 31
	\| 953.8		\| 953.8
	\| [[~~140~~/81]]		\| [[26/15]]
	\|-		\|-
	\| 32		\| 32
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	\| 34		\| 34
	\| 1046.2		\| 1046.2
	\| [[11/6]], [[20/11]]		\| [[11/6]], [[20/11]], [[24/13]]
	\|-		\|-
	\| 35		\| 35
	\| 1076.9		\| 1076.9
	\| [[28/15]]		\| [[13/7]], [[28/15]]
	\|-		\|-
	\| 36		\| 36
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	\| 37		\| 37
	\| 1138.5		\| 1138.5
	\| [[27/14]], [[64/33]], ''[[96/49]]''		\| ''[[21/11]]'', [[27/14]], [[64/33]], ''[[96/49]]''
	\|-		\|-
	\| 38		\| 38
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	\| [[2/1]]		\| [[2/1]]
	\|}		\|}
			<nowiki/>* As a 13-limit temperament

	{\| class="wikitable center-1 right-2"		{\| class="wikitable center-1 right-2"

FloraC: /* Intervals */ split (see talk)

2026-06-11T09:53:57Z

Intervals: split (see talk)

Show changes

FloraC: /* Intervals */ break off the tri-column of color notation for a clean table

2026-06-11T08:55:50Z

Intervals: break off the tri-column of color notation for a clean table

Show changes

FloraC: Explain exactly why 39d is better. Move other mappings to their subsection

2026-06-11T08:46:39Z

Explain exactly why 39d is better. Move other mappings to their subsection

← Older revision		Revision as of 08:46, 11 June 2026
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	== Theory ==		== Theory ==
	39edo's [[3/2\|perfect fifth]] is 5.8~~{{c}}~~ sharp~~. Together~~ with its best [[5/4\|classical major third]] which is the familiar 400~~{{c}}~~ of [[12edo]]. We have two choices for a tuning of [[7/1\|7]]~~. The~~ sharp one ~~yields~~ [[~~superpyth~~]] ~~temperament, while~~ the ~~flat (patent) one yields~~ [[~~semaphore~~]] (and also [[~~hemifamity~~]]~~) temperament~~.		39edo's [[3/2\|perfect fifth]] is 5.8 cents sharp, together with its best [[5/4\|classical major third]] which is the familiar 400 cents of [[12edo]]. We have two choices for a tuning of [[7/1\|7]], but the sharp one blends better with the sharp 3 and 5 as their errors cancel with their ratios, thus yielding significantly higher average accuracy than the [[patent val]] in the [[7-odd-limit\|7-]] and [[9-odd-limit]]. It also has a fine [[11/1\|11]], and adding it to consideration the best choice for 39et is the sharp-tending 39df val {{val\| 39 62 91 '''110''' 135 '''145''' }}.

	As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap\|(39 {{=}} 17 + 22)}}; its fifth thus falls in the "shrub region" where the diatonic thirds are between standard neogothic thirds and septimal thirds. The specific 7-limit variant supported by 39d is [[quasisuper]]~~. Alternatively, it can be seen as a [[hemifamity]] semaphore (that is, immunity) system in the patent val~~. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody assuming that the MOS diatonic is used, because the [[diatonic semitone]] is [[quartertone]]-sized. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.		As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap\|(39 {{=}} 17 + 22)}}; its fifth thus falls in the "shrub region" where the diatonic thirds are between standard neogothic thirds and septimal thirds. The specific 7-limit variant supported by 39d is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody assuming that the MOS diatonic is used, because the [[diatonic semitone]] is [[quartertone]]-sized. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.

	Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s\|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.

	39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).

	~~39edo is a reasonable tuning of [[triforce]] beyond 15edo, and optimizes both its semaphore and augmented components by tuning the fifth sharp~~.

	=== Odd harmonics ===		=== Odd harmonics ===
Line 18:		Line 12:

	=== As a tuning of other temperaments ===		=== As a tuning of other temperaments ===
	39edo, with its 400{{c}} major third, [[tempering out\|tempers out]] the [[diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list~~. Alternatively, the patent val tempers out 49/48 to yield semaphore~~. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo\|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].		39edo, with its 400{{c}} major third, [[tempering out\|tempers out]] the [[128/125\|diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo\|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].

			Alternatively, the patent val tempers out [[49/48]] to yield [[semaphore]], and provides a reasonable tuning of [[triforce]] beyond [[15edo]], and optimizes both its semaphore and augmented components by tuning the fifth sharp. The 39c val supports [[negri]].

			If we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]] through the 39bc val, and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s\|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]] and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.

			39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate mavila as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).

	=== Subsets and supersets ===		=== Subsets and supersets ===

VectorGraphics: added more patent val discussion

2026-06-03T09:10:56Z

added more patent val discussion

← Older revision		Revision as of 09:10, 3 June 2026
Line 10:		Line 10:

	39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).		39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).

			39edo is a reasonable tuning of [[triforce]] beyond 15edo, and optimizes both its semaphore and augmented components by tuning the fifth sharp.

	=== Odd harmonics ===		=== Odd harmonics ===

Overthink: /* Intervals */ - obsolete note

2026-05-30T22:00:18Z

Intervals: - obsolete note

← Older revision		Revision as of 22:00, 30 May 2026
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	\| D		\| D
	\|}		\|}
	~~<nowiki/>* 11-limit in the 39d val, inconsistent intervals in ''italic''~~

	Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].		Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].

Overthink: /* Intervals */ left align those

2026-05-30T21:59:50Z

Intervals: left align those

← Older revision		Revision as of 21:59, 30 May 2026
Line 22:		Line 22:

	== Intervals ==		== Intervals ==
	{\| class="wikitable center-all right-2 left-3 right-9 right-10"		{\| class="wikitable center-all right-2 left-3 left-4 left-5 right-9 right-10"
	\|-		\|-
	! rowspan="2" \| Steps		! rowspan="2" \| Steps
Line 392:		Line 392:

	Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].		Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].


	== Notation ==		== Notation ==