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. 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.

Examples of pattern 1a chords are cuthbert chords (13-odd-limit), aureusmic chords (19-odd-limit) and palingenetic chords (21-odd-limit).

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.

Examples of pattern 1b chords are lambeth chords (13-odd-limit) and sextantonismic chords (17-odd-limit).

Defective

Defective pattern 1 is where some of these chords turn out essentially just. Ptolemismic chords are of this category, as it only has a palindromic triad, two pairs of inversely related tetrads, and a pair of inversely related pentads.