Dyadic chord/Pattern of essentially tempered chords: Difference between revisions

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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 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. 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).
Pattern 1 has two subpatterns, 1a and 1b, both of whose basic palindromic triads are of the same form, but their basic inversely related pair of triads and 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
The palindromic triad is  
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</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>2</sub>/''n''<sub>2</sub>.
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>.


=== Pattern 1a ===
=== Pattern 1a ===
For pattern 1a, the inversely related pair of triads is
For pattern 1a, the inversely related pair of triads is  
* 1-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</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>1</sub>, and its inverse
* 1-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub> with steps of ''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>1</sub>, and its inverse  
* 1-''n''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>1</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>/''d''<sub>1</sub>.  
* 1-''n''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub> with steps of ''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>1</sub>/''d''<sub>1</sub>.


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>-''n''<sub>1</sub>/''d''<sub>2</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 of ''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>2</sub>/''d''<sub>1</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 of ''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  
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps ''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>, and its inverse
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>, and its inverse  
* 1-''n''<sub>2</sub>/''d''<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>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</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>/''d''<sub>2</sub> with steps of ''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<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>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps ''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>, and its inverse
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>, and its inverse  
* 1-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</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>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>;
* 1-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>.


The inversely related pair of pentads is
The inversely related pair of pentads is  
* 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>-''n''<sub>2</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>-''n''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>, and its inverse
* 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>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>, and its inverse  
* 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>-''n''<sub>2</sub>/''d''<sub>2</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>/''d''<sub>2</sub>-''d''<sub>2</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>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''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>/''d''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>.


=== Pattern 1b ===
=== Pattern 1b ===
For pattern 1b, the inversely related pair of triads are
For pattern 1b, the inversely related pair of triads are  
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub> with steps ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>, and its inverse
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>, and its inverse  
* 1-''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>1</sub>/''d''<sub>2</sub> with steps ''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>.
* 1-''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>1</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>.


The palindromic tetrads are
The palindromic tetrads are  
* 1-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub> with steps ''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>;  
* 1-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub> (or 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>1</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>);  
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</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>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''d''<sub>1</sub>.  
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</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>/''d''<sub>1</sub> (or 1-''d''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>).


The inversely related pairs of tetrads are
The inversely related pairs of tetrads are  
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>, and its inverse
* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub> (or 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub>-''n''<sub>1</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>), and its inverse  
* 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 of ''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> (or 1-''n''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>1</sub>/''d''<sub>2</sub> with steps of ''n''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>1</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 of ''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> (or 1-''d''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>), and its inverse  
* 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>.
* 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 of ''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> (or 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>).


The inversely related pair of pentads is
The inversely related pair of pentads is either one of the following:
* 1-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''d''<sub>2</sub> with steps ''d''<sub>1</sub>/''n''<sub>2</sub>-''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>/''d''<sub>1</sub>, and its inverse
{| class="wikitable"
* 1-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub> with steps ''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>.
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* 1-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>2</sub>-''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>/''d''<sub>1</sub>, and its inverse
* 1-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>1</sub> with steps of ''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''d''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>
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* 1-''d''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>2</sub>/''n''<sub>1</sub> with steps of ''d''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''n''<sub>2</sub>, and its inverse  
* 1-''d''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>1</sub> with steps of ''d''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>
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* 1-''d''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>, and its inverse
* 1-''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub> with steps of ''n''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>2</sub>/''n''<sub>2</sub>
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* 1-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>2</sub>/''d''<sub>2</sub>-''n''<sub>1</sub>/''d''<sub>2</sub>-''d''<sub>1</sub>/''d''<sub>2</sub> with steps of ''d''<sub>1</sub>/''n''<sub>1</sub>-''d''<sub>1</sub>/''n''<sub>1</sub>-''n''<sub>1</sub>/''n''<sub>2</sub>-''d''<sub>1</sub>/''n''<sub>1</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>2</sub>/''d''<sub>1</sub>-''n''<sub>2</sub>/''n''<sub>1</sub> with steps of ''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>1</sub>-''n''<sub>1</sub>/''n''<sub>2</sub>
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== Pattern 2 ==
== Pattern 2 ==
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Notable examples of this pattern are [[keenanismic chords]] (11-odd-limit), [[werckismic chords]] (11-odd-limit), and [[swetismic chords]] (11-odd-limit).
Notable examples of this pattern are [[keenanismic chords]] (11-odd-limit), [[werckismic chords]] (11-odd-limit), and [[swetismic chords]] (11-odd-limit).
== Pattern 3 ==
Pattern 3 turns up for commas of the form (''n''<sub>1</sub>''n''<sub>2</sub>''n''<sub>3</sub>)/(''d''<sub>1</sub>''d''<sub>2</sub>''d''<sub>3</sub>) up to octave equivalence. It contains six inversely related pairs of triads, eighteen inversely related pairs of tetrads, and nine inversely related pair of pentads, for a total of 66 distinct chord structures.
A notable example of this pattern is [[ibnsinmic chords]] (21-odd-limit).

Revision as of 02:45, 26 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 basic inversely related pair of triads and 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 of 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 of n1/d2-n2/d1-n1/d1, and its inverse
  • 1-n2/d1-d1/n1 with steps of n2/d1-n1/d2-n1/d1.

The palindromic tetrads are

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

The inversely related pairs of tetrads are

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

The palindromic tetrads are

  • 1-d1/n2-d1/n1-n1/d2 with steps of d1/n2-n2/n1-d1/n2-d2/n1 (or 1-d1/n1-d1/n2-n1/d2 with steps of d1/n1-n1/n2-d1/n1-d2/n1);
  • 1-d1/n1-n1/d2-d1/d2 with steps of d1/n1-d1/n2-d1/n1-d2/d1 (or 1-d1/d2-d1/n1-n1/d2 with steps of d1/d2-d2/n1-d1/n2-d2/n1).

The inversely related pairs of tetrads are

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

The inversely related pair of pentads is either one of the following:

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

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).

Pattern 3

Pattern 3 turns up for commas of the form (n1n2n3)/(d1d2d3) up to octave equivalence. It contains six inversely related pairs of triads, eighteen inversely related pairs of tetrads, and nine inversely related pair of pentads, for a total of 66 distinct chord structures.

A notable example of this pattern is ibnsinmic chords (21-odd-limit).

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