Didymic chords - Revision history

FloraC: Expand

2025-12-15T08:43:06Z

Expand

← Older revision		Revision as of 08:43, 15 December 2025
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	A '''didymic chord''' is an [[essentially tempered chord]] of 5-limit [[meantone]]. The basic form of a didymic chord is the meantone sus2/6 tetrad:		A '''didymic chord''' is an [[essentially tempered chord]] of 5-limit [[meantone]]. The basic form of a didymic chord is the palindromic meantone sus2/6 tetrad:

	* 1–9/8–3/2–5/3 with steps of 9/8, 4/3, 9/8, 6/5.		* 1–9/8–3/2–5/3 with steps of 9/8, 4/3, 9/8, 6/5.

	Every interval is an element of the [[9-odd-limit]]. In diatonic, it can be notated as C–D–G–A when built on C. The tempered essence explains why the common chord progression vi–ii–V–I does not work outside meantone unless one accepts a [[27/16]] major sixth, a [[27/20]] acute fourth, or a [[40/27]] grave fifth.		Every interval is an element of the [[9-odd-limit]]. In diatonic, it can be notated as C–D–G–A when built on C. The tempered essence explains why the common chord progression vi–ii–V–I does not work outside meantone unless one accepts a [[27/16]] major sixth, a [[27/20]] acute fourth, or a [[40/27]] grave fifth. It makes for an interesting comparison with the [[archytas chords\|archy sus4/7]] chord.

	The chord can be extended to a pentad known as the meantone add2/6 pentad, which also happens to be the [[2L 3s\|pentic]] scale [[meantone5\|Meantone[5]]]:		The chord can be extended to a palindromic pentad known as the meantone add2/6 pentad, which also happens to be the [[2L 3s\|pentic]] scale [[meantone5\|Meantone[5]]]:

	* 1–9/8–5/4–3/2–5/3 with steps of 9/8, 9/8, 6/5, 9/8, 6/5.		* 1–9/8–5/4–3/2–5/3 with steps of 9/8, 9/8, 6/5, 9/8, 6/5.
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	Built on C, it is C–D–E–G–A.		Built on C, it is C–D–E–G–A.

	~~Both chords are palindromic~~.		There is also a didymic tetrad involving [[prime interval\|prime]] [[7/1\|7]]:
			* 1–9/8–9/7–10/7 with steps 9/8, 8/7, 9/8, 7/5.

	== Dominant seventh chord ==		== Dominant seventh chord ==

ArrowHead294 at 10:57, 13 September 2024

2024-09-13T10:57:07Z

← Older revision		Revision as of 10:57, 13 September 2024
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	== Septimal meantone chords ==		== Septimal meantone chords ==
	Since [[81/80]] is tempered out, didymic chords are septimal meantone chords. Since [[126/125]] ~~is tempered out, [[starling chords]] are septimal meantone chords. Since~~ [[225/224]] is tempered out, [[marvel chords]] are septimal meantone chords. There are also septimal meantone chords which are none of these, the essentially '''septimal meantone chords'''.		Since [[81/80]] is tempered out, didymic chords are septimal meantone chords. Since [[126/125]] and [[225/224]] are tempered out, [[starling chords\|starling]] and [[marvel chords]] are also septimal meantone chords. There are also septimal meantone chords which are none of these, the essentially '''septimal meantone chords'''.

	The basic form of these chords are pentads, with a unique palindromic chord and a pair of chords in inverse relationship:		The basic form of these chords are pentads, with a unique palindromic chord and a pair of chords in inverse relationship:
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	== External links ==		== External links ==
	* [https://youtu.be/TYhPAbsIqA8 Adam Neely - Benedetti's Puzzle (mathematically impossible music)]~~, a~~ video explanation of the chord in terms of a [[comma pump]]		* [https://youtu.be/TYhPAbsIqA8 Adam Neely - Benedetti's Puzzle (mathematically impossible music)] – A video explanation of the chord in terms of a [[comma pump]]

	[[Category:9-odd-limit chords]]		[[Category:9-odd-limit chords]]

FloraC: Only starling is possible

2024-09-12T15:21:22Z

Only starling is possible

← Older revision		Revision as of 15:21, 12 September 2024
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	* 1–5/4–3/2–9/5 with steps 5/4, 6/5, 6/5, 9/8.		* 1–5/4–3/2–9/5 with steps 5/4, 6/5, 6/5, 9/8.

	Built on G, it is G–B–D–F. Note that ~~septimal meantone~~ is ~~not required~~ to make it a 9-odd-limit concord~~. Either [[starling]] or [[marvel]] can be used~~, ~~depending on what interval~~ the top note ~~is interpreted as, though both are required for it~~ to represent 16/9~9/5.		Built on G, it is G–B–D–F. Note that tempering out the [[126/125\|starling comma (126/125)]] alone is enough to make it a 9-odd-limit concord, though septimal meantone is required for the top note to represent 16/9~9/5.

	Its inverse is the [[half-diminished seventh chord]]:		Its inverse is the [[half-diminished seventh chord]]:

FloraC: /* Dominant seventh chord */ +link to half-diminished seventh chord

2024-09-11T17:58:09Z

Dominant seventh chord: +link to half-diminished seventh chord

← Older revision		Revision as of 17:58, 11 September 2024
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	Built on G, it is G–B–D–F. Note that septimal meantone is not required to make it a 9-odd-limit concord. Either [[starling]] or [[marvel]] can be used, depending on what interval the top note is interpreted as, though both are required for it to represent 16/9~9/5.		Built on G, it is G–B–D–F. Note that septimal meantone is not required to make it a 9-odd-limit concord. Either [[starling]] or [[marvel]] can be used, depending on what interval the top note is interpreted as, though both are required for it to represent 16/9~9/5.

	Its ~~inversion~~ is the half-diminished chord:		Its inverse is the [[half-diminished seventh chord]]:
	* 1–6/5–10/7–9/5 with steps 6/5, 6/5, 5/4, 9/8.		* 1–6/5–10/7–9/5 with steps 6/5, 6/5, 5/4, 9/8.

FloraC: Recategorize

2024-08-24T10:25:50Z

Recategorize

← Older revision		Revision as of 10:25, 24 August 2024
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	* [https://youtu.be/TYhPAbsIqA8 Adam Neely - Benedetti's Puzzle (mathematically impossible music)], a video explanation of the chord in terms of a [[comma pump]]		* [https://youtu.be/TYhPAbsIqA8 Adam Neely - Benedetti's Puzzle (mathematically impossible music)], a video explanation of the chord in terms of a [[comma pump]]

	[[Category:9-odd-limit]]		[[Category:9-odd-limit chords]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
	[[Category:Tetrads]]		[[Category:Tetrads]]

FloraC: Readability and note the 5-limit pentad is a pentic scale

2024-08-24T10:16:36Z

Readability and note the 5-limit pentad is a pentic scale

← Older revision		Revision as of 10:16, 24 August 2024
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	A '''didymic chord''' is an [[essentially tempered chord]] of 5-limit [[meantone]]. The basic form of a didymic chord is the meantone sus2/6 tetrad:		A '''didymic chord''' is an [[essentially tempered chord]] of 5-limit [[meantone]]. The basic form of a didymic chord is the meantone sus2/6 tetrad:

	* ~~1-9~~/~~8-3~~/~~2-5~~/3 with steps of 9/8-4/3-9/8-6/5.		* 1–9/8–3/2–5/3 with steps of 9/8, 4/3, 9/8, 6/5.

	Every interval is an element of the [[9-odd-limit]]. In diatonic, it can be notated as ~~C-D-G-A~~ when built on C. The tempered essence explains why the common chord progression ~~vi-ii-V-I~~ does not work outside meantone unless one accepts a [[27/16]] major sixth, a [[27/20]] acute fourth, or a [[40/27]] grave fifth.		Every interval is an element of the [[9-odd-limit]]. In diatonic, it can be notated as C–D–G–A when built on C. The tempered essence explains why the common chord progression vi–ii–V–I does not work outside meantone unless one accepts a [[27/16]] major sixth, a [[27/20]] acute fourth, or a [[40/27]] grave fifth.

	The chord can be extended to a pentad known as the meantone add2/6 pentad:		The chord can be extended to a pentad known as the meantone add2/6 pentad, which also happens to be the [[2L 3s\|pentic]] scale [[meantone5\|Meantone[5]]]:

	* ~~1-9~~/~~8-5~~/~~4-3~~/~~2-5~~/3 with steps of 9/8-9/8-6/5-9/8-6/5.		* 1–9/8–5/4–3/2–5/3 with steps of 9/8, 9/8, 6/5, 9/8, 6/5.

	Built on C, it is ~~C-D-E-G-A~~.		Built on C, it is C–D–E–G–A.

	Both chords are palindromic.		Both chords are palindromic.
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	== External links ==		== External links ==
	* [https://youtu.be/TYhPAbsIqA8 Adam Neely - Benedetti's Puzzle (mathematically impossible music)], a video explanation of the chord		* [https://youtu.be/TYhPAbsIqA8 Adam Neely - Benedetti's Puzzle (mathematically impossible music)], a video explanation of the chord in terms of a [[comma pump]]

	[[Category:9-odd-limit]]		[[Category:9-odd-limit]]

FloraC: /* Septimal meantone chords */ improve readability. Note the dominant seventh chord doesn't really require septimal meantone. Review the rest and leave only those that require septimal meantone

2024-08-24T10:12:16Z

Septimal meantone chords: improve readability. Note the dominant seventh chord doesn't really require septimal meantone. Review the rest and leave only those that require septimal meantone

← Older revision		Revision as of 10:12, 24 August 2024
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	Both chords are palindromic.		Both chords are palindromic.

	== ~~Septimal meantone chords~~ ==		== Dominant seventh chord ==
	~~The~~ dominant seventh chord is an essentially tempered chord ~~in septimal meantone~~:		In septimal meantone, the [[dominant seventh chord]] is an essentially tempered chord:
	* ~~1-5~~/~~4-3~~/~~2-9~~/5 with steps of 5/4-6/5-6/5-9/8.		* 1–5/4–3/2–9/5 with steps 5/4, 6/5, 6/5, 9/8.

	Built on G, it is G-B-~~D-F~~.		Built on G, it is G–B–D–F. Note that septimal meantone is not required to make it a 9-odd-limit concord. Either [[starling]] or [[marvel]] can be used, depending on what interval the top note is interpreted as, though both are required for it to represent 16/9~9/5.

	Its inversion is the half-diminished chord:		Its inversion is the half-diminished chord:
	* ~~1-6~~/~~5-10~~/~~7-9~~/5 with steps of 6/5-6/5-5/4-9/8.		* 1–6/5–10/7–9/5 with steps 6/5, 6/5, 5/4, 9/8.

	Built on B, it is ~~B-D-F-A~~.		Built on B, it is B–D–F–A.

	~~Other tetrads of septimal~~ meantone ~~are~~		== Septimal meantone chords ==
	* 1-9/~~8-5~~/~~4-7~~/~~5 with steps of 9/8-9/8-9/8-10/7;~~		Since [[81/80]] is tempered out, didymic chords are septimal meantone chords. Since [[126/125]] is tempered out, [[starling chords]] are septimal meantone chords. Since [[225/224]] is tempered out, [[marvel chords]] are septimal meantone chords. There are also septimal meantone chords which are none of these, the essentially '''septimal meantone chords'''.
	* 1-9/8-7/5-14/9 with steps of ~~9/8-5/4-9/8-9/7;~~
	* 1-9/8-5/4-14/9 with steps of 9/8-9/8-5/4-9/7, ~~and its inverse~~
	* 1-5/4-7/5-14/9 with steps of 5/4-9/8-9/8-9/7.

	~~They can be extended to the following~~ pentads, with ~~two~~ palindromic ~~chords~~ and ~~two pairs~~ of chords in inverse relationship:		The basic form of these chords are pentads, with a unique palindromic chord and a pair of chords in inverse relationship:
	* ~~1-9/8-5/4-3/2-9/5 with steps of 9/8-9/8-6/5-6/5-9/8;~~		* 1–9/8–5/4–7/5–14/9 with steps of 9/8, 9/8, 9/8, 9/8, 9/7;
	* 1-9/~~8-5~~/~~4-7~~/~~5-14~~/9 with steps of 9/8-9/8-9/8-9/8-9/7;		* 1–9/8–5/4–14/9–7/4 with steps of 9/8, 9/8, 5/4, 9/8, 8/7, and its inverse
	* ~~1-9~~/~~8-5~~/~~4-7~~/~~5-7~~/4 with steps of 9/8-9/8~~-9/8-~~5/4~~-8/7~~, ~~and its inverse~~		* 1–9/8–7/5–14/9–7/4 with steps of 9/8, 5/4, 9/8, 9/8, 8/7.
	* 1-5/4-7/5-14/9-7/4 with steps of 5/4-9/8~~-9/8-9/8-8/7;~~
	* 1-9/8-5/4-14/9-7/4 with steps of 9/8-9/8-5/4-9/8-8/7, and its inverse
	* ~~1-9~~/~~8-7~~/~~5-14~~/~~9-7~~/4 with steps of 9/8-5/4-9/8-9/8-8/7.

	Finally, there ~~are~~ a palindromic hexad,		Finally, there is a palindromic hexad,
	* ~~1-9~~/~~8-5~~/~~4-7~~/~~5-14~~/~~9-7~~/4 with steps of 9/8-9/8-9/8-9/8-9/8-8/7.		* 1–9/8–5/4–7/5–14/9–7/4 with steps of 9/8, 9/8, 9/8, 9/8, 9/8, 8/7.

	== External links ==		== External links ==

Xenllium: Update the article

2023-11-11T12:44:37Z

Update the article

← Older revision		Revision as of 12:44, 11 November 2023
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	A '''didymic chord''' is an [[essentially tempered chord]] of 5-limit [[meantone]]. The basic form of a didymic chord is the meantone sus2/6 tetrad:		A '''didymic chord''' is an [[essentially tempered chord]] of 5-limit [[meantone]]. The basic form of a didymic chord is the meantone sus2/6 tetrad:

	* 1-9/8-3/2-5/3, with steps 9/8-4/3-9/8-6/5		* 1-9/8-3/2-5/3 with steps of 9/8-4/3-9/8-6/5.

	Every interval is an element of the [[9-odd-limit]]. In diatonic, it can be notated as C-D-G-A when built on C. The tempered essence explains why the common chord progression vi-ii-V-I does not work outside meantone unless one accepts a [[27/16]] major sixth, a [[27/20]] acute fourth, or a [[40/27]] grave fifth.		Every interval is an element of the [[9-odd-limit]]. In diatonic, it can be notated as C-D-G-A when built on C. The tempered essence explains why the common chord progression vi-ii-V-I does not work outside meantone unless one accepts a [[27/16]] major sixth, a [[27/20]] acute fourth, or a [[40/27]] grave fifth.
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	The chord can be extended to a pentad known as the meantone add2/6 pentad:		The chord can be extended to a pentad known as the meantone add2/6 pentad:

	* 1-9/8-5/4-3/2-5/3, with steps 9/8-9/8-6/5-9/8-6/5		* 1-9/8-5/4-3/2-5/3 with steps of 9/8-9/8-6/5-9/8-6/5.

	Built on C, it is C-D-E-G-A.		Built on C, it is C-D-E-G-A.

	Both chords are palindromic.		Both chords are palindromic.

	== Septimal meantone chords ==		== Septimal meantone chords ==
	The dominant seventh chord is an essentially tempered chord in septimal meantone:		The dominant seventh chord is an essentially tempered chord in septimal meantone:
			* 1-5/4-3/2-9/5 with steps of 5/4-6/5-6/5-9/8.

	* 1-5/4-3/2-9/5, with steps 5/4-6/5-6/5-9/8		Built on G, it is G-B-D-F.

	Built on G, it is G-B-D-F.

	Its inversion is the half-diminished chord:		Its inversion is the half-diminished chord:
			* 1-6/5-10/7-9/5 with steps of 6/5-6/5-5/4-9/8.

	* 1-6/5-10/7-9/5, ~~with steps 6/5~~-~~6/5~~-~~5/4~~-~~9/8~~		Built on B, it is B-D-F-A.

	~~Built on B~~, ~~it is B~~-D-F-A.		Other tetrads of septimal meantone are
			* 1-9/8-5/4-7/5 with steps of 9/8-9/8-9/8-10/7;
			* 1-9/8-7/5-14/9 with steps of 9/8-5/4-9/8-9/7;
			* 1-9/8-5/4-14/9 with steps of 9/8-9/8-5/4-9/7, and its inverse
			* 1-5/4-7/5-14/9 with steps of 5/4-9/8-9/8-9/7.

	~~Other tetrads~~ of ~~septimal meantone are~~:		They can be extended to the following pentads, with two palindromic chords and two pairs of chords in inverse relationship:
			* 1-9/8-5/4-3/2-9/5 with steps of 9/8-9/8-6/5-6/5-9/8;
			* 1-9/8-5/4-7/5-14/9 with steps of 9/8-9/8-9/8-9/8-9/7;
			* 1-9/8-5/4-7/5-7/4 with steps of 9/8-9/8-9/8-5/4-8/7, and its inverse
			* 1-5/4-7/5-14/9-7/4 with steps of 5/4-9/8-9/8-9/8-8/7;
			* 1-9/8-5/4-14/9-7/4 with steps of 9/8-9/8-5/4-9/8-8/7, and its inverse
			* 1-9/8-7/5-14/9-7/4 with steps of 9/8-5/4-9/8-9/8-8/7.

	* 1-9/8-5/4-7/5~~, with steps 9/8~~-9/8-9/8-~~10/~~7		Finally, there are a palindromic hexad,
	* 1-9/8-5/4~~-14/9,~~ with steps 9/8~~-9/8-5/4-9/7~~		* 1-9/8-5/4-7/5-14/9-7/4 with steps of 9/8-9/8-9/8-9/8-9/8-8/7.
	* 1-5/4-7/5-14/9, with steps 5/4-9/8-9/8~~-9/7~~
	* 1-9/8-~~7/5-14/9, with steps~~ 9/8-~~5/4-9/~~8-9/7

	== External links ==		== External links ==
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	[[Category:Tetrads]]		[[Category:Tetrads]]
	[[Category:Pentads]]		[[Category:Pentads]]
			[[Category:Hexads]]
	[[Category:Meantone]]		[[Category:Meantone]]

FloraC: Recategorize

2023-05-23T11:56:43Z

Recategorize

← Older revision		Revision as of 11:56, 23 May 2023
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	[[Category:9-odd-limit]]		[[Category:9-odd-limit]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
	[[Category:~~Tetrad~~]]		[[Category:Tetrads]]
	[[Category:~~Pentad~~]]		[[Category:Pentads]]
	[[Category:Meantone]]		[[Category:Meantone]]

FloraC: Categories

2022-02-08T18:15:55Z

← Older revision		Revision as of 18:15, 8 February 2022
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	[[Category:9-odd-limit]]		[[Category:9-odd-limit]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
			[[Category:Tetrad]]
			[[Category:Pentad]]
	[[Category:Meantone]]		[[Category:Meantone]]