Catadictmic chords

Catadictmic chords are essentially tempered dyadic chords tempered by the catadictma, 1287/1280.

Catadictmic chords are of pattern 2 in the 13-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

For triads, there are three pairs of chords in inverse relationship:

• 1–16/13–11/8 with steps of 16/13, 10/9, 16/11, and its inverse
• 1–16/11–13/8 with steps of 16/11, 10/9, 16/13;
• 1–13/10–16/11 with steps of 13/10, 9/8, 11/8, and its inverse
• 1–11/8–20/13 with steps of 11/8, 9/8, 13/10;
• 1–11/10–16/13 with steps of 11/10, 9/8, 13/8, and its inverse
• 1–9/8–16/13 with steps of 9/8, 11/10, 13/8.

For tetrads, there are three palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are

• 1–16/13–16/11–9/5 with steps of 16/13, 13/11, 16/13, 10/9;
• 1–9/8–11/8–20/13 with steps of 9/8, 11/9, 9/8, 13/10;
• 1–9/8–16/13–18/13 with steps of 9/8, 11/10, 9/8, 13/9.

The inversely related pairs of chords are

• 1–13/10–16/11–9/5 with steps of 13/10, 9/8, 16/13, 10/9, and its inverse
• 1–16/13–18/13–9/5 with steps of 16/13, 9/8, 13/10, 10/9;
• 1–9/8–16/11–9/5 with steps of 9/8, 13/10, 16/13, 10/9, and its inverse
• 1–16/13–8/5–9/5 with steps of 16/13, 13/10, 9/8, 10/9;
• 1–13/11–13/10–16/11 with steps of 13/11, 11/10, 9/8, 11/8, and its inverse
• 1–9/8–16/13–16/11 with steps of 9/8, 11/10, 13/11, 11/8;
• 1–13/10–16/11–8/5 with steps of 13/10, 9/8, 11/10, 5/4, and its inverse
• 1–11/10–16/13–8/5 with steps of 11/10, 9/8, 13/10, 5/4;
• 1–11/10–16/13–11/8 with steps of 11/10, 9/8, 10/9, 16/11, and its inverse
• 1–10/9–5/4–11/8 with steps of 10/9, 9/8, 11/10, 16/11;
• 1–11/10–16/13–9/5 with steps of 11/10, 9/8, 16/11, 10/9, and its inverse
• 1–16/11–18/11–9/5 with steps of 16/11, 9/8, 11/10, 10/9.

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

• 1–16/13–16/11–8/5–9/5 with steps of 16/13, 13/11, 11/10, 9/8, 10/9, and its inverse
• 1–9/8–16/13–16/11–9/5 with steps of 9/8, 11/10, 13/11, 16/13, 10/9;
• 1–9/8–16/11–18/11–9/5 with steps of 9/8, 13/10, 9/8, 11/10, 10/9, and its inverse
• 1–11/10–16/13–8/5–9/5 with steps of 11/10, 9/8, 13/10, 9/8, 10/9;
• 1–13/10–16/11–8/5–9/5 with steps of 13/10, 9/8, 11/10, 9/8, 10/9, and its inverse
• 1–9/8–16/13–18/13–9/5 with steps of 9/8, 11/10, 9/8, 13/10, 10/9.

Equal temperaments with catadictmic chords include 7, 12, 19, 22, 24, 26, 29, 31, 43, 46, 50, 53, 72 and 96.