Didymic chords

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.

Septimal meantone chords

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.

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

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.

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.

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.

Finally, there are 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