Dyadic chord/Pattern of essentially tempered chords: Difference between revisions
Turns out ptolemismic isn't of pattern 1 since the 3's can be stacked to 9 |
mNo edit summary |
||
| Line 15: | Line 15: | ||
The palindromic tetrads are | The palindromic tetrads are | ||
* 1-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''d''<sub>2</sub> with steps ''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>2</sub>/''d''<sub>1</sub>; | * 1-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''d''<sub>2</sub> with steps ''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>2</sub>/''d''<sub>1</sub>; | ||
* 1-''n''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''n''<sub>1</sub> with steps ''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>1</sub>/''n''<sub>2</sub>. | * 1-''n''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''n''<sub>1</sub> with steps ''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''n''<sub>2</sub>. | ||
The inversely related pairs of tetrads are | The inversely related pairs of tetrads are | ||
| Line 41: | Line 41: | ||
* 1-''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps ''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>; | * 1-''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps ''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>; | ||
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''d''<sub>2</sub> with steps ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''d''<sub>1</sub>, and its inverse | * 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''d''<sub>2</sub> with steps ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''d''<sub>1</sub>, and its inverse | ||
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub>-''n''<sub> | * 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub>-''n''<sub>1</sub>/''d''<sub>1</sub> with steps ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>. | ||
The inversely related pair of pentads is | The inversely related pair of pentads is | ||
Revision as of 01:17, 3 November 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. 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-n1/d2-d2/d1;
- 1-n2/d1-d1/n1-n2/n1 with steps n2/d1-n1/d2-n2/d1-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-n1/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).