Dyadic chord/Pattern of essentially tempered chords: Difference between revisions
Created page with "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..." |
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== Pattern 1 == | == Pattern 1 == | ||
Pattern 1 turns up for commas of the form (''n''<sub>1</sub><sup>2</sup>''n''<sub>2</sub>)/(''d''<sub>1</sub><sup>2</sup>''d''<sub>2</sub>) up to [[octave equivalence]]. It contains a palindromic triad and an inversely related pair of triads, two palindromic tetrads and two inversely related | Pattern 1 turns up for commas of the form (''n''<sub>1</sub><sup>2</sup>''n''<sub>2</sub>)/(''d''<sub>1</sub><sup>2</sup>''d''<sub>2</sub>) 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 | 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 | ||
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Examples of pattern 1b chords are [[lambeth chords]] (13-odd-limit) and [[sextantonismic chords]] (17-odd-limit). | 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. | |||
Revision as of 05:07, 27 July 2023
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.