Dyadic chord/Pattern of essentially tempered chords

This page discusses some common patterns of essentially tempered chords for a given comma and an odd limit.

Pattern 1

Pattern 1 turns up for commas of the form (n₁²n₂)/(d₁²d₂) up to octave equivalence. It contains a palindromic triad and an inversely related pair of triads, two palindromic tetrads and two inversely related pairs of tetrads, and an inversely related pair of pentads, for a total of 11 distinct chord structures.

Pattern 1 has two subpatterns, 1a and 1b, both of whose basic palindromic triads are of the same form, but their final pentad extensions differ. Examples of pattern 1a chords are sensamagic chords (9-odd-limit), cuthbert chords (13-odd-limit), and aureusmic chords (19-odd-limit). Examples of pattern 1b chords are marvel chords (9-odd-limit), lambeth chords (13-odd-limit) and sextantonismic chords (17-odd-limit).

The palindromic triad is

• 1-d₁/n₁-n₂/d₂ with steps d₁/n₁-d₁/n₁-d₂/n₂.

Pattern 1a

For pattern 1a, the inversely related pair of triads is

• 1-n₁/d₂-d₁/n₁ with steps n₁/d₂-n₂/d₁-n₁/d₁, and its inverse
• 1-n₂/d₁-d₁/n₁ with steps n₂/d₁-n₁/d₂-n₁/d₁.

The palindromic tetrads are

• 1-n₁/d₂-d₁/n₁-d₁/d₂ with steps n₁/d₂-n₂/d₁-d₂/d₁;
• 1-n₂/d₁-d₁/n₁-n₂/n₁ with steps n₂/d₁-n₁/d₂-n₁/n₂.

The inversely related pairs of tetrads are

• 1-d₁/n₁-d₁/d₂-n₂/d₂ with steps d₁/n₁-n₁/d₂-n₂/d₁-d₂/n₂, and its inverse
• 1-n₂/d₁-d₁/n₁-n₂/d₂ with steps n₂/d₁-n₁/d₂-d₁/n₁-d₂/n₂;
• 1-d₁/n₁-n₂/n₁-n₂/d₂ with steps d₁/n₁-n₂/d₁-n₁/d₂-d₂/n₂, and its inverse
• 1-n₁/d₂-d₁/n₁-n₂/d₂ with steps n₁/d₂-n₂/d₁-d₁/n₁-d₂/n₂;

The inversely related pair of pentads is

• 1-n₁/d₂-d₁/n₁-d₁/d₂-n₂/d₂ with steps n₁/d₂-n₂/d₁-n₁/d₂-n₂/d₁-d₂/n₂, and its inverse
• 1-n₂/d₁-d₁/n₁-n₂/n₁-n₂/d₂ with steps n₂/d₁-n₁/d₂-n₂/d₁-n₁/d₂-d₂/n₂.

Pattern 1b

For pattern 1b, the inversely related pair of triads are

• 1-d₁/n₁-n₁/d₂ with steps d₁/n₁-d₁/n₂-d₂/n₁, and its inverse
• 1-d₁/n₂-n₁/d₂ with steps d₁/n₂-d₁/n₁-d₂/n₁.

The palindromic tetrads are

• 1-d₁/n₂-d₁/n₁-n₁/d₂ with steps d₁/n₂-n₂/n₁-d₁/n₂-d₂/n₁;
• 1-d₁/n₁-n₁/d₂-d₁/d₂ with steps d₁/n₁-d₁/n₂-d₁/n₁-d₂/d₁.

The inversely related pairs of tetrads are

• 1-d₁/n₁-n₁/d₂-n₂/d₂ with steps d₁/n₁-d₁/n₂-n₂/n₁-d₂/n₂, and its inverse
• 1-n₂/n₁-d₁/n₁-n₂/d₂ with steps n₂/n₁-d₁/n₂-d₁/n₁-d₂/n₂;
• 1-d₁/n₁-n₂/d₂-d₁/d₂ with steps d₁/n₁-d₁/n₁-d₁/n₂-d₂/d₁, and its inverse
• 1-d₁/n₁-n₂/d₂-n₂/d₁ with steps d₁/n₁-d₁/n₁-d₂/d₁-d₁/n₂.

The inversely related pair of pentads is

• 1-d₁/n₂-d₁/n₁-n₁/d₂-d₁/d₂ with steps d₁/n₂-n₂/n₁-d₁/n₂-d₁/n₁-d₂/d₁, and its inverse
• 1-d₁/n₂-d₁/n₁-n₁/d₂-n₂/d₁ with steps d₁/n₂-n₂/n₁-d₁/n₂-d₂/d₁-d₁/n₁.

Pattern 2

Pattern 2 turns up for commas of the form (n₁n₂n₃)/(d₁²d₂), or (n₁n₂²)/(d₁d₂d₃) up to octave equivalence. It contains three inversely related pairs of triads, three palindromic tetrads and six inversely related pairs of tetrads, and three inversely related pair of pentads, for a total of 27 distinct chord structures.

Notable examples of this pattern are keenanismic chords (11-odd-limit), werckismic chords (11-odd-limit), and swetismic chords (11-odd-limit).