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

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This page discusses some common patterns of [[essentially tempered chord]]s for a given [[comma]] and an [[odd limit]].

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Pattern 1 turns up for commas of the form (''n''12''n''2)/(''d''12''d''2) 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

* 1-''d''1/''n''1-''n''2/''d''2 with steps ''d''1/''n''1-''d''1/''n''1-''d''2/''n''2.

* 1-''d''1/''n''1-''n''2/''d''2 with steps of ''d''1/''n''1-''d''1/''n''1-''d''2/''n''2.

=== Pattern 1a ===

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

* 1-''n''1/''d''2-''d''1/''n''1 with steps ''n''1/''d''2-''n''2/''d''1-''n''1/''d''1, and its inverse

* 1-''n''1/''d''2-''d''1/''n''1 with steps of ''n''1/''d''2-''n''2/''d''1-''n''1/''d''1, and its inverse

* 1-''n''2/''d''1-''d''1/''n''1 with steps ''n''2/''d''1-''n''1/''d''2-''n''1/''d''1.

* 1-''n''2/''d''1-''d''1/''n''1 with steps of ''n''2/''d''1-''n''1/''d''2-''n''1/''d''1.

The palindromic tetrads are

* 1-''n''1/''d''2-''d''1/''n''1-''d''1/''d''2 with steps ''n''1/''d''2-''n''2/''d''1-''d''2/''d''1;

* 1-''n''1/''d''2-''d''1/''n''1-''d''1/''d''2 with steps of ''n''1/''d''2-''n''2/''d''1-''n''1/''d''2-''d''2/''d''1;

* 1-''n''2/''d''1-''d''1/''n''1-''n''2/''n''1 with steps ''n''2/''d''1-''n''1/''d''2-''n''1/''n''2.

* 1-''n''2/''d''1-''d''1/''n''1-''n''2/''n''1 with steps of ''n''2/''d''1-''n''1/''d''2-''n''2/''d''1-''n''1/''n''2.

The inversely related pairs of tetrads are

* 1-''d''1/''n''1-''d''1/''d''2-''n''2/''d''2 with steps ''d''1/''n''1-''n''1/''d''2-''n''2/''d''1-''d''2/''n''2, and its inverse

* 1-''d''1/''n''1-''d''1/''d''2-''n''2/''d''2 with steps of ''d''1/''n''1-''n''1/''d''2-''n''2/''d''1-''d''2/''n''2, and its inverse

* 1-''n''2/''d''1-''d''1/''n''1-''n''2/''d''2 with steps ''n''2/''d''1-''n''1/''d''2-''d''1/''n''1-''d''2/''n''2;

* 1-''n''2/''d''1-''d''1/''n''1-''n''2/''d''2 with steps of ''n''2/''d''1-''n''1/''d''2-''d''1/''n''1-''d''2/''n''2;

* 1-''d''1/''n''1-''n''2/''n''1-''n''2/''d''2 with steps ''d''1/''n''1-''n''2/''d''1-''n''1/''d''2-''d''2/''n''2, and its inverse

* 1-''d''1/''n''1-''n''2/''n''1-''n''2/''d''2 with steps of ''d''1/''n''1-''n''2/''d''1-''n''1/''d''2-''d''2/''n''2, and its inverse

* 1-''n''1/''d''2-''d''1/''n''1-''n''2/''d''2 with steps ''n''1/''d''2-''n''2/''d''1-''d''1/''n''1-''d''2/''n''2;

* 1-''n''1/''d''2-''d''1/''n''1-''n''2/''d''2 with steps of ''n''1/''d''2-''n''2/''d''1-''d''1/''n''1-''d''2/''n''2.

The inversely related pair of pentads is

* 1-''n''1/''d''2-''d''1/''n''1-''d''1/''d''2-''n''2/''d''2 with steps ''n''1/''d''2-''n''2/''d''1-''n''1/''d''2-''n''2/''d''1-''d''2/''n''2, and its inverse

* 1-''n''1/''d''2-''d''1/''n''1-''d''1/''d''2-''n''2/''d''2 with steps of ''n''1/''d''2-''n''2/''d''1-''n''1/''d''2-''n''2/''d''1-''d''2/''n''2, and its inverse

* 1-''n''2/''d''1-''d''1/''n''1-''n''2/''n''1-''n''2/''d''2 with steps ''n''2/''d''1-''n''1/''d''2-''n''2/''d''1-''n''1/''d''2-''d''2/''n''2.

* 1-''n''2/''d''1-''d''1/''n''1-''n''2/''n''1-''n''2/''d''2 with steps of ''n''2/''d''1-''n''1/''d''2-''n''2/''d''1-''n''1/''d''2-''d''2/''n''2.

=== Pattern 1b ===

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

* 1-''d''1/''n''1-''n''1/''d''2 with steps ''d''1/''n''1-''d''1/''n''2-''d''2/''n''1, and its inverse

* 1-''d''1/''n''1-''n''1/''d''2 with steps of ''d''1/''n''1-''d''1/''n''2-''d''2/''n''1, and its inverse

* 1-''d''1/''n''2-''n''1/''d''2 with steps ''d''1/''n''2-''d''1/''n''1-''d''2/''n''1.

* 1-''d''1/''n''2-''n''1/''d''2 with steps of ''d''1/''n''2-''d''1/''n''1-''d''2/''n''1.

The palindromic tetrads are

* 1-''d''1/''n''2-''d''1/''n''1-''n''1/''d''2 with steps ''d''1/''n''2-''n''2/''n''1-''d''1/''n''2-''d''2/''n''1;

* 1-''d''1/''n''2-''d''1/''n''1-''n''1/''d''2 with steps of ''d''1/''n''2-''n''2/''n''1-''d''1/''n''2-''d''2/''n''1 (or 1-''d''1/''n''1-''d''1/''n''2-''n''1/''d''2 with steps of ''d''1/''n''1-''n''1/''n''2-''d''1/''n''1-''d''2/''n''1);

* 1-''d''1/''n''1-''n''1/''d''2-''d''1/''d''2 with steps ''d''1/''n''1-''d''1/''n''2-''d''1/''n''1-''d''2/''d''1.

* 1-''d''1/''n''1-''n''1/''d''2-''d''1/''d''2 with steps of ''d''1/''n''1-''d''1/''n''2-''d''1/''n''1-''d''2/''d''1 (or 1-''d''1/''d''2-''d''1/''n''1-''n''1/''d''2 with steps of ''d''1/''d''2-''d''2/''n''1-''d''1/''n''2-''d''2/''n''1).

The inversely related pairs of tetrads are

* 1-''d''1/''n''1-''n''1/''d''2-''n''2/''d''2 with steps ''d''1/''n''1-''d''1/''n''2-''n''2/''n''1-''d''2/''n''2, and its inverse

* 1-''d''1/''n''1-''n''1/''d''2-''n''2/''d''2 with steps of ''d''1/''n''1-''d''1/''n''2-''n''2/''n''1-''d''2/''n''2 (or 1-''d''1/''n''1-''n''2/''d''2-''n''1/''d''2 with steps of ''d''1/''n''1-''d''1/''n''1-''n''1/''n''2-''d''2/''n''1), and its inverse

* 1-''n''2/''n''1-''d''1/''n''1-''n''2/''d''2 with steps ''n''2/''n''1-''d''1/''n''2-''d''1/''n''1-''d''2/''n''2;

* 1-''n''2/''n''1-''d''1/''n''1-''n''2/''d''2 with steps of ''n''2/''n''1-''d''1/''n''2-''d''1/''n''1-''d''2/''n''2 (or 1-''n''1/''n''2-''d''1/''n''2-''n''1/''d''2 with steps of ''n''1/''n''2-''d''1/''n''1-''d''1/''n''1-''d''2/''n''1);

* 1-''d''1/''n''1-''n''2/''d''2-''d''1/''d''2 with steps ''d''1/''n''1-''d''1/''n''1-''d''1/''n''2-''d''2/''d''1, and its inverse

* 1-''d''1/''n''1-''n''2/''d''2-''d''1/''d''2 with steps of ''d''1/''n''1-''d''1/''n''1-''d''1/''n''2-''d''2/''d''1 (or 1-''d''1/''d''2-''d''1/''n''1-''n''2/''d''2 with steps of ''d''1/''d''2-''d''2/''n''1-''d''1/''n''1-''d''2/''n''2), and its inverse

* 1-''d''1/''n''1-''n''2/''d''2-''n''2/''d''1 with steps ''d''1/''n''1-''d''1/''n''1-''d''2/''d''1-''d''1/''n''2.

* 1-''d''1/''n''1-''n''2/''d''2-''n''1/''d''1 with steps of ''d''1/''n''1-''d''1/''n''1-''d''2/''d''1-''d''1/''n''2 (or 1-''d''1/''n''1-''n''2/''d''1-''n''2/''d''2 with steps of ''d''1/''n''1-''d''2/''n''1-''d''1/''d''2-''d''2/''n''2).

The inversely related pair of pentads is

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

* 1-''d''1/''n''2-''d''1/''n''1-''n''1/''d''2-''d''1/''d''2 with steps ''d''1/''n''2-''n''2/''n''1-''d''1/''n''2-''d''1/''n''1-''d''2/''d''1, and its inverse

{| class="wikitable"

* 1-''d''1/''n''2-''d''1/''n''1-''n''1/''d''2-''n''2/''d''1 with steps ''d''1/''n''2-''n''2/''n''1-''d''1/''n''2-''d''2/''d''1-''d''1/''n''1.

|

* 1-''d''1/''n''2-''d''1/''n''1-''n''1/''d''2-''d''1/''d''2 with steps of ''d''1/''n''2-''n''2/''n''1-''d''1/''n''2-''d''1/''n''1-''d''2/''d''1, and its inverse

* 1-''d''1/''n''2-''d''1/''n''1-''n''1/''d''2-''n''2/''d''1 with steps of ''d''1/''n''2-''n''2/''n''1-''d''1/''n''2-''d''2/''d''1-''d''1/''n''1

|-

|

* 1-''d''2/''n''1-''d''1/''n''1-''n''2/''d''1-''n''2/''n''1 with steps of ''d''2/''n''1-''d''1/''d''2-''d''2/''n''1-''d''1/''n''1-''n''1/''n''2, and its inverse

* 1-''d''2/''n''1-''d''1/''n''1-''n''2/''d''1-''n''1/''d''1 with steps of ''d''2/''n''1-''d''1/''d''2-''d''2/''n''1-''n''1/''n''2-''d''1/''n''1

|-

|

* 1-''d''1/''d''2-''d''1/''n''1-''n''1/''d''2-''n''2/''d''2 with steps of ''d''1/''d''2-''d''2/''n''1-''d''1/''n''2-''n''2/''n''1-''d''2/''n''2, and its inverse

* 1-''n''2/''n''1-''d''1/''n''1-''n''2/''d''1-''n''2/''d''2 with steps of ''n''2/''n''1-''d''1/''n''2-''d''2/''n''1-''d''1/''d''2-''d''2/''n''2

|-

|

* 1-''d''1/''n''1-''n''2/''d''2-''n''1/''d''2-''d''1/''d''2 with steps of ''d''1/''n''1-''d''1/''n''1-''n''1/''n''2-''d''1/''n''1-''d''2/''d''1, and its inverse

* 1-''d''1/''n''1-''n''2/''d''2-''n''2/''d''1-''n''2/''n''1 with steps of ''d''1/''n''1-''d''1/''n''1-''d''2/''d''1-''d''1/''n''1-''n''1/''n''2

|}

==~~= Defective =~~==

== Pattern 2 ==

~~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.

Pattern 2 turns up for commas of the form (''n''1''n''2''n''3)/(''d''12''d''2), or (''n''12''n''2)/(''d''1''d''2''d''3) 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 pairs 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 (''n''1''n''2''n''3)/(''d''1''d''2''d''3) up to octave equivalence. It contains six inversely related pairs of triads, eighteen inversely related pairs of tetrads, and nine inversely related pairs of pentads, for a total of 66 distinct chord structures.

~~== Pattern 2 ==~~

Notable examples of this pattern are [[neosatanismic chords]] (19-odd-limit), [[ibnsinmic chords]] (21-odd-limit) and [[neogrendelismic chords]] (21-odd-limit).

~~Pattern 2 turns up for commas~~ of ~~the form (''n''1''n''2''n''3)/~~(~~''d''12''d''2~~), or (~~''n''1''n''22~~)~~/(''d''1''d''2''d''3) 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).~~

[[Category:Essentially tempered chords]]

[[Category:Chords]]

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+{{breadcrumb}}
 This page discusses some common patterns of [[essentially tempered chord]]s for a given [[comma]] and an [[odd limit]].
@@ Line 4: / Line 5: @@
 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
-* 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 ===
 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
-* 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 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>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
-* 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
-* 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 ===
 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
-* 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
-* 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>2</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>.
+|
+* 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>
+|-
+|
+* 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>
+|-
+|
+* 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>
+|-
+|
+* 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>
+|}
-=== Defective ===
+== Pattern 2 ==
-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.
+Pattern 2 turns up for commas of the form (''n''<sub>1</sub>''n''<sub>2</sub>''n''<sub>3</sub>)/(''d''<sub>1</sub><sup>2</sup>''d''<sub>2</sub>), or (''n''<sub>1</sub><sup>2</sup>''n''<sub>2</sub>)/(''d''<sub>1</sub>''d''<sub>2</sub>''d''<sub>3</sub>) 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 pairs 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 (''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 pairs of pentads, for a total of 66 distinct chord structures.
-== Pattern 2 ==
+Notable examples of this pattern are [[neosatanismic chords]] (19-odd-limit), [[ibnsinmic chords]] (21-odd-limit) and [[neogrendelismic chords]] (21-odd-limit).
-Pattern 2 turns up for commas of the form (''n''<sub>1</sub>''n''<sub>2</sub>''n''<sub>3</sub>)/(''d''<sub>1</sub><sup>2</sup>''d''<sub>2</sub>), or (''n''<sub>1</sub>''n''<sub>2</sub><sup>2</sup>)/(''d''<sub>1</sub>''d''<sub>2</sub>''d''<sub>3</sub>) 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).
+[[Category:Essentially tempered chords]]
+[[Category:Chords]]