Werckismic chords: Difference between revisions

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A '''werckismic chord''' is an [[essentially tempered dyadic chord]] in werckismic (441/440) tempering in the 2.9.5.7.11 subgroup in the 11-odd-limit.

A '''werckismic chord''' is an [[essentially tempered dyadic chord]] in [[Werckismic temperaments|werckismic]] ([[441/440]]) tempering in the 2.9.5.7.11 [[Just intonation subgroup|subgroup]] in the [[11-odd-limit]].

The werckismic triads consist of three pairs of inversely related chords:

The werckismic [[Triad|triads]] consist of three pairs of inversely related chords:

* 1-11/9-7/5 with steps 11/9-8/7-10/7 and 1-11/9-7/4 with steps 11/9-10/7-8/7;

* 1-11/9-7/5 with [[Step|steps]] 11/9-8/7-10/7 and 1-11/9-7/4 with steps 11/9-10/7-8/7;

* 1-11/7-7/4 with steps 11/7-10/9-8/7 and 1-11/7-9/5 with steps 11/7-8/7-10/9; and

* 1-7/5-11/7 with steps 7/5-9/8-14/11 and 1-14/11-10/7 with steps 14/11-9/8-7/5.

The werckismic tetrads consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are

The werckismic [[Tetrad|tetrads]] consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are

* 1-5/4-10/7-7/4 with steps 5/4-8/7-11/9-8/7;

* 1-9/8-10/7-11/7 with steps 9/8-14/11-11/10-14/11; and

* 1-14/11-7/5-16/9 with steps 14/11-11/10-14/11-9/8.

The pairs of chords are 1-11/9-11/7-7/4 with steps 11/9-9/7-10/9-8/7 and 1-9/7-11/7-9/5 with steps 9/7-11/9-8/7-10/9;

The pairs of chords are:

* 1-11/9-11/7-7/4 with steps 11/9-9/7-10/9-8/7 and 1-9/7-11/7-9/5 with steps 9/7-11/9-8/7-10/9;

* 1-10/7-11/7-7/4 with steps 10/7-11/10-10/9-8/7 and 1-8/7-14/11-7/5 with steps 8/7-10/9-11/10-10/7;

* 1-9/8-10/7-7/4 with steps 9/8-14/11-11/9-8/7 and 1-11/9-14/9-7/4 with steps 11/9-14/11-9/8-8/7;

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* 1-9/8-9/7-10/7 with steps 9/8-8/7-10/9-7/5 and 1-9/8-11/7-7/4 with steps 9/8-7/5-10/9-8/7.

The werckismic pentads consist of three pairs of chords in an inverse relationship:

The werckismic [[Pentad|pentads]] consist of three pairs of chords in an inverse relationship:

* 1-9/8-5/4-10/7-7/4 with steps 9/8-10/9-8/7-11/9-8/7 and 1-11/9-7/5-14/9-7/4 with steps 11/9-8/7-10/9-9/8-8/7;

* 1-11/9-11/8-11/7-7/4 with steps 11/9-9/8-8/7-10/9-8/7 and 1-9/8-11/8-11/7-7/4 with steps 9/8-11/9-8/7-10/9-8/7;

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The count of chords is therefore triads: 3, tetrads: 13, pentads: 6, for a total of 22.

~~Equal temperaments with werckismic chords include {{EDOs| 31, 41, 43, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289, and 320 }}, with [[320edo]] giving the [[optimal patent val]].~~

[[Category:11-odd-limit]]

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[[Category:Werckismic]]

[[EDO|Equal temperaments]] with werckismic chords include {{EDOs| 31, 41, 43, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289, and 320 }}, with [[320edo]] giving the [[optimal patent val]].

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-A '''werckismic chord''' is an [[essentially tempered dyadic chord]] in werckismic (441/440) tempering in the 2.9.5.7.11 subgroup in the 11-odd-limit.
+A '''werckismic chord''' is an [[essentially tempered dyadic chord]] in [[Werckismic temperaments|werckismic]] ([[441/440]]) tempering in the 2.9.5.7.11 [[Just intonation subgroup|subgroup]] in the [[11-odd-limit]].
-The werckismic triads consist of three pairs of inversely related chords:
+The werckismic [[Triad|triads]] consist of three pairs of inversely related chords:
-* 1-11/9-7/5 with steps 11/9-8/7-10/7 and 1-11/9-7/4 with steps 11/9-10/7-8/7;
+* 1-11/9-7/5 with [[Step|steps]] 11/9-8/7-10/7 and 1-11/9-7/4 with steps 11/9-10/7-8/7;
 * 1-11/7-7/4 with steps 11/7-10/9-8/7 and 1-11/7-9/5 with steps 11/7-8/7-10/9; and
 * 1-7/5-11/7 with steps 7/5-9/8-14/11 and 1-14/11-10/7 with steps 14/11-9/8-7/5.
-The werckismic tetrads consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are
+The werckismic [[Tetrad|tetrads]] consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are
 * 1-5/4-10/7-7/4 with steps 5/4-8/7-11/9-8/7;
 * 1-9/8-10/7-11/7 with steps 9/8-14/11-11/10-14/11; and
 * 1-14/11-7/5-16/9 with steps 14/11-11/10-14/11-9/8.
-The pairs of chords are 1-11/9-11/7-7/4 with steps 11/9-9/7-10/9-8/7 and 1-9/7-11/7-9/5 with steps 9/7-11/9-8/7-10/9;
+The pairs of chords are:
+* 1-11/9-11/7-7/4 with steps 11/9-9/7-10/9-8/7 and 1-9/7-11/7-9/5 with steps 9/7-11/9-8/7-10/9;
 * 1-10/7-11/7-7/4 with steps 10/7-11/10-10/9-8/7 and 1-8/7-14/11-7/5 with steps 8/7-10/9-11/10-10/7;
 * 1-9/8-10/7-7/4 with steps 9/8-14/11-11/9-8/7 and 1-11/9-14/9-7/4 with steps 11/9-14/11-9/8-8/7;
@@ Line 18: / Line 20: @@
 * 1-9/8-9/7-10/7 with steps 9/8-8/7-10/9-7/5 and 1-9/8-11/7-7/4 with steps 9/8-7/5-10/9-8/7.
-The werckismic pentads consist of three pairs of chords in an inverse relationship:
+The werckismic [[Pentad|pentads]] consist of three pairs of chords in an inverse relationship:
 * 1-9/8-5/4-10/7-7/4 with steps 9/8-10/9-8/7-11/9-8/7 and 1-11/9-7/5-14/9-7/4 with steps 11/9-8/7-10/9-9/8-8/7;
 * 1-11/9-11/8-11/7-7/4 with steps 11/9-9/8-8/7-10/9-8/7 and 1-9/8-11/8-11/7-7/4 with steps 9/8-11/9-8/7-10/9-8/7;
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 The count of chords is therefore triads: 3, tetrads: 13, pentads: 6, for a total of 22.
-Equal temperaments with werckismic chords include {{EDOs| 31, 41, 43, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289, and 320 }}, with [[320edo]] giving the [[optimal patent val]].
 [[Category:11-odd-limit]]
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 [[Category:Werckismic]]
-{{Todo| add links }}
+[[EDO|Equal temperaments]] with werckismic chords include {{EDOs| 31, 41, 43, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289, and 320 }}, with [[320edo]] giving the [[optimal patent val]].