Cuthbert chords

Cuthbert chords are essentially tempered dyadic chords tempered by the cuthbert comma, 847/845.

Cuthbert chords are of pattern 1a in the 2.5.7.11.13 subgroup 13-odd-limit, meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.

The most basic cuthbert triad is a palindrome, consisting of two 13/11's making up 7/5, which implies tempering by cuthbert, the 847/845 comma. It is, in other words, the 847/845-tempered version of

• 1–13/11–7/5 chord with steps of 13/11, 13/11, 10/7.

There is an inversely related pair which is more squeezed and fit for a sort of secundal harmony:

• 1–11/10–13/11 with steps of 11/10, 14/13, 22/13, and its inverse
• 1–14/13–13/11 with steps of 14/13, 11/10, 22/13.

They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic tetrads are

• 1–11/10–13/11–13/10 chord with steps of 11/10, 14/13, 11/10, 20/13;
• 1–14/13–13/11–14/11 chord with steps of 14/13, 11/10, 14/13, 11/7.

The inversely related pairs of tetrads are

• 1–13/11–14/11–7/5 with steps of 13/11, 14/13, 11/10, 10/7, and its inverse
• 1–11/10–13/11–7/5 with steps of 11/10, 14/13, 13/11, 10/7;
• 1–13/11–13/10–7/5 with steps of 13/11, 11/10, 14/13, 10/7, and its inverse
• 1–14/13–13/11–7/5 with steps of 14/13, 11/10, 13/11, 10/7.

Then there is an inversely related pair of pentads:

• 1–11/10–13/11–13/10–7/5 with steps of 11/10, 14/13, 11/10, 14/13, 10/7, and its inverse
• 1–14/13–13/11–14/11–7/5 with steps of 14/13, 11/10, 14/13, 11/10, 10/7.

Equal temperaments with cuthbert triads include 29, 33, 37, 41, 46, 50, 53, 58, 70, 87, 94, 99, 103, 111, 128, 140, 149, 177, 190, 198, 205, 227, 264, 284 and 388.

Garibert tetrad

The first cuthbert triad can be extended to the garibert tetrad, which is the {275/273, 847/845} garibert tempering of a tetrad,

• 1–13/11–7/5–5/3 with steps of size 13/11, 13/11, 13/11, 6/5.

Equal temperaments with the garibert tetrad include 16, 29, 37, 41, 53 and 94; and it is a characteristic chord of 13-limit garibaldi temperament.