Magic chords

A magic chord is an essentially tempered chord of the magic temperament; they are included among those listed as chords of magic. Aside from the marvel, sensamagic, ptolemismic, keenanismic, swetismic, pentacircle, apollo, octarod, unimarvel and supermagic chords listed there, there are 23 further essentially magic chords in the 11-odd-limit.

The two magic tetrads are in inverse relationship:

• 1-5/4-14/9-9/5 with steps 5/4-5/4-7/6-10/9 and its inverse 1-7/6-16/11-9/5 with steps 7/6-5/4-5/4-10/9;

For pentads, there are seven pairs of chords in inverse relationship:

• 1-9/8-5/4-14/9-9/5 with steps 9/8-10/9-5/4-7/6-10/9 and its inverse 1-9/8-5/4-16/11-9/5 with steps 9/8-10/9-7/6-5/4-10/9;
• 1-5/4-7/5-14/9-9/5 with steps 5/4-9/8-10/9-7/6-10/9 and its inverse 1-9/8-7/5-14/9-9/5 with steps 9/8-5/4-10/9-7/6-10/9;
• 1-12/11-5/4-14/9-9/5 with steps 12/11-8/7-5/4-7/6-10/9 and its inverse 1-12/11-6/5-7/5-7/4 with steps 12/11-10/9-7/6-5/4-8/7;
• 1-12/11-6/5-14/9-7/4 with steps 12/11-10/9-9/7-9/8-8/7 and its inverse 1-12/11-5/4-7/5-9/5 with steps 12/11-8/7-9/8-9/7-10/9;
• 1-7/6-16/11-18/11-9/5 with steps 7/6-5/4-9/8-10/9-10/9 and its inverse 1-9/8-7/5-18/11-9/5 with steps 9/8-5/4-7/6-10/9-10/9;
• 1-7/6-14/11-16/11-9/5 with steps 7/6-12/11-8/7-5/4-10/9 and its inverse 1-12/11-14/11-7/5-7/4 with steps 12/11-7/6-10/9-5/4-8/7;
• 1-9/8-14/11-14/9-9/5 with steps 9/8-9/8-11/9-7/6-10/9 and its inverse 1-9/8-14/11-7/5-18/11 with steps 9/8-9/8-10/9-7/6-11/9.

For hexads, there are one palindromic chord and three pairs of chords in inverse relationship. The palindromic chord is

• 1-9/8-5/4-7/5-14/9-9/5 with steps 9/8-10/9-9/8-10/9-7/6-10/9.

The inversely related pairs of chords are

• 1-12/11-5/4-7/5-14/9-9/5 with steps 12/11-8/7-9/8-10/9-7/6-10/9 and its inverse 1-12/11-6/5-7/5-14/9-7/4 with steps 12/11-10/9-7/6-10/9-9/8-8/7;
• 1-9/8-14/11-7/5-14/9-9/5 with steps 9/8-9/8-10/9-10/9-7/6-10/9 and its inverse 1-9/8-14/11-7/5-18/11-9/5 with steps 9/8-9/8-10/9-7/6-10/9-10/9;
• 1-12/11-14/11-7/5-14/9-7/4 with steps 12/11-7/6-10/9-10/9-9/8-8/7 and its inverse 1-7/6-14/11-16/11-18/11-9/5 with steps 7/6-12/11-8/7-9/8-10/9-10/9.

The count of magic chords is therefore tetrads: 2, pentads: 14, and hexads: 7.

Equal temperaments with magic chords include 19, 22, 41, 60, 63, 85 and 104.