Didymic chords: Difference between revisions
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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. | ||
Every interval is an element of the [[9-odd-limit]]. In diatonic, it can be notated as | 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: | 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. | ||
Built on C, it is | Built on C, it is C–D–E–G–A. | ||
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 == | ||
In septimal meantone, the [[dominant seventh chord]] is an essentially tempered chord: | |||
* | * 1–5/4–3/2–9/5 with steps 5/4, 6/5, 6/5, 9/8. | ||
Built on G, it is | 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 | 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. | ||
Built on B, it is | Built on B, it is B–D–F–A. | ||
== 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: | |||
* | * 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 | |||
* | * 1–9/8–7/5–14/9–7/4 with steps of 9/8, 5/4, 9/8, 9/8, 8/7. | ||
* | |||
Finally, there | 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. | ||
== External links == | == External links == | ||
* [https://youtu.be/TYhPAbsIqA8 Adam Neely - Benedetti's Puzzle (mathematically impossible music)] | * [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]] | ||
Latest revision as of 08:43, 15 December 2025
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.
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 archy sus4/7 chord.
The chord can be extended to a palindromic pentad known as the meantone add2/6 pentad, which also happens to be the pentic scale Meantone[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.
There is also a didymic tetrad involving prime 7:
- 1–9/8–9/7–10/7 with steps 9/8, 8/7, 9/8, 7/5.
Dominant seventh chord
In septimal meantone, the dominant seventh chord is an essentially tempered chord:
- 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 tempering out the 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:
- 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.
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 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:
- 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
- 1–9/8–7/5–14/9–7/4 with steps of 9/8, 5/4, 9/8, 9/8, 8/7.
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.
External links
- Adam Neely - Benedetti's Puzzle (mathematically impossible music) – A video explanation of the chord in terms of a comma pump