Blumeyer chords

Blumeyer chords are essentially tempered chords tempered by the Blumeyer comma, 2432/2431.

Blumeyer chords are of pattern 2 in the 2.11.13.17.19 subgroup 19-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:

• 1–26/19–16/11 with steps of 26/19–17/16–11/8, and its inverse
• 1–17/16–16/11 with steps of 17/16–26/19–11/8;
• 1–16/13–11/8 with steps of 16/13–19/17–16/11, and its inverse
• 1–19/17–11/8 with steps of 19/17–16/13–16/11;
• 1–22/19–16/13 with steps of 22/19–17/16–13/8, and its inverse
• 1–17/16–16/13 with steps of 17/16–22/19–13/8.

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

• 1–13/11–16/11–13/8 with steps of 13/11–16/13–19/17–16/13;
• 1–17/16–11/8–19/13 with steps of 17/16–22/17–17/16–26/19;
• 1–17/16–16/13–17/13 with steps of 17/16–22/19–17/16–26/17.

The inversely related pairs of chords are

• 1–16/13–11/8–19/13 with steps of 16/13–19/17–17/16–26/19, and its inverse
• 1–17/16–19/16–19/13 with steps of 17/16–19/17–16/13–26/19;
• 1–19/16–11/8–19/13 with steps of 19/16–22/19–17/16–26/19, and its inverse
• 1–17/16–16/13–19/13 with steps of 17/16–22/19–19/16–26/19;
• 1–19/17–11/8–19/13 with steps of 19/17–16/13–17/16–26/19, and its inverse
• 1–17/16–17/13–19/13 with steps of 17/16–16/13–19/17–26/19;
• 1–13/11–26/19–16/11 with steps of 13/11–22/19–17/16–11/8, and its inverse
• 1–17/16–16/13–16/11 with steps of 17/16–22/19–13/11–11/8;
• 1–22/19–16/13–11/8 with steps of 22/19–17/16–19/17–16/11, and its inverse
• 1–19/17–19/16–11/8 with steps of 19/17–17/16–22/19–16/11;
• 1–19/17–22/17–11/8 with steps of 19/17–22/19–17/16–16/11, and its inverse
• 1–17/16–16/13–11/8 with steps of 17/16–22/19–19/17–16/11.

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

• 1–13/11–26/19–16/11–13/8 with steps of 13/11–22/19–17/16–19/17–16/13, and its inverse
• 1–19/17–19/16–11/8–13/8 with steps of 19/17–17/16–22/19–13/11–16/13;
• 1–19/17–19/16–11/8–19/13 with steps of 19/17–17/16–22/19–17/16–26/19, and its inverse
• 1–17/16–16/13–17/13–19/13 with steps of 17/16–22/19–17/16–19/17–26/19;
• 1–17/16–16/13–11/8–19/13 with steps of 17/16–22/19–19/17–17/16–26/19, and its inverse
• 1–17/16–19/16–11/8–19/13 with steps of 17/16–19/17–22/19–17/16–26/19.

Equal temperaments with Blumeyer chords include 13, 20, 24, 33, 37, 46, 57, 67, 70, 80, 100, 104, 113, 124, 137, 150, 170, 194 and 307.