Didymic chords: Difference between revisions
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→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 |
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Both chords are palindromic. | Both chords are palindromic. | ||
== | == 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 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 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]] 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'''. | |||
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 == | ||
Revision as of 10:12, 24 August 2024
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.
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:
- 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.
Both chords are palindromic.
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 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:
- 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 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.
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