Didymic chords: Difference between revisions

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 [[archytas chords|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 [[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 C–D–E–G–A.

* 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:  

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]]:

* 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.

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:

* 1–9/8–7/5–14/9–7/4 with steps of 9/8, 5/4, 9/8, 9/8, 8/7.

* 1–9/8–5/4–7/5–14/9–7/4 with steps of 9/8, 9/8, 9/8, 9/8, 9/8, 8/7.

* [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]]