Biyatismic chords: Difference between revisions

Revision as of 11:30, 7 November 2021

Biyatismic chords are essentially tempered dyadic chords tempered by the biyatisma, 121/120.

Biyatismic temperament contains many possibilities for essentially tempered dyadic chords including five triads and 33 tetrads in the 2.3.5.11 subgroup.

The biyatismic triads consist of a palindromic chord and two pairs of chords in inverse relationship. The palindromic chord is 1-11/8-16/11 chord with steps of 11/8-16/15-11/8. The inversely related pairs of chords are 1-11/8-11/6 chord with steps of 11/8-4/3-11/10 and 1-11/10-16/11 chord with steps of 11/10-4/3-11/8; and 1-11/8-5/3 chord with steps of 11/8-11/9-6/5 and 1-6/5-16/11 chord with steps of 6/5-11/9-11/8.

The biyatismic tetrads consist of three palindromic chords and fifteen pairs of chords in inverse relationship. The palindromic chords are 1-9/8-11/8-18/11 chord with steps of 9/8-11/9-6/5-11/9, 1-5/4-11/8-11/6 chord with steps of 5/4-11/10-4/3-11/10, and 1-11/10-3/2-18/11 chord with steps of 11/10-11/8-11/10-11/9. The inversely related pairs of chords are 1-4/3-5/3-11/6 chord with steps of 4/3-5/4-11/10-11/10 and 1-11/10-6/5-3/2 chord with steps of 11/10-11/10-5/4-4/3; 1-11/9-11/8-18/11 chord with steps of 11/9-9/8-6/5-11/9 and 1-11/9-16/11-18/11 chord with steps of 11/9-6/5-9/8-11/9; 1-11/8-5/3-11/6 chord with steps of 11/8-11/9-11/10-11/10 and 1-11/10-6/5-16/11 chord with steps of 11/10-11/10-11/9-11/8; 1-4/3-3/2-11/6 chord with steps of 4/3-9/8-11/9-11/10 and 1-11/10-4/3-3/2 chord with steps of 11/10-11/9-9/8-4/3; 1-9/8-11/8-11/6 chord with steps of 9/8-11/9-4/3-11/10 and 1-11/10-16/11-16/9 chord with steps of 11/10-4/3-11/9-9/8; 1-4/3-3/2-18/11 chord with steps of 4/3-9/8-11/10-11/9 and 1-11/9-4/3-3/2 chord with steps of 11/9-11/10-9/8-4/3; 1-5/4-3/2-11/6 chord with steps of 5/4-6/5-11/9-11/10 and 1-11/10-4/3-8/5 chord with steps of 11/10-11/9-6/5-5/4; 1-6/5-3/2-11/6 chord with steps of 6/5-5/4-11/9-11/10 and 1-11/10-4/3-5/3 chord with steps of 11/10-11/9-5/4-6/5; 1-6/5-16/11-11/6 chord with steps of 6/5-11/9-5/4-11/10 and 1-11/10-11/8-5/3 chord with steps of 11/10-5/4-11/9-6/5; 1-5/4-11/8-16/11 chord with steps of 5/4-11/10-16/15-11/8 and 1-11/8-16/11-8/5 chord with steps of 11/8-16/15-11/10-5/4; 1-5/4-4/3-16/11 chord with steps of 5/4-16/15-11/10-11/8 and 1-11/8-3/2-8/5 chord with steps of 11/8-11/10-16/15-5/4; 1-5/4-4/3-11/6 chord with steps of 5/4-16/15-11/8-11/10 and 1-11/10-3/2-8/5 chord with steps of 11/10-11/8-16/15-5/4; 1-11/9-11/8-16/11 chord with steps of 11/9-9/8-16/15-11/8 and 1-11/8-16/11-18/11 chord with steps of 11/8-16/15-9/8-11/9; 1-11/8-5/3-16/9 chord with steps of 11/8-11/9-16/15-9/8 and 1-9/8-6/5-16/11 chord with steps of 9/8-16/15-11/9-11/8; and 1-11/8-16/11-16/9 chord with steps of 11/8-16/15-11/9-9/8 and 1-9/8-11/8-16/11 chord with steps of 9/8-11/9-16/15-11/8.

Equal temperaments with biyatismic chords include 7, 9, 15, 16, 22, 24, 29, 31, 38, 46, and 53 EDOs.

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 '''Biyatismic chords''' are [[Dyadic chord|essentially tempered dyadic chords]] tempered by the biyatisma, [[121/120]].
-Biyatismic temperament contains many possibilities for essentially tempered dyadic chords including five triads in the 2.3.5.11 subgroup.
+Biyatismic temperament contains many possibilities for essentially tempered dyadic chords including five triads and 33 tetrads in the 2.3.5.11 subgroup.
 The biyatismic triads consist of a palindromic chord and two pairs of chords in inverse relationship. The palindromic chord is 1-[[11/8]]-[[16/11]] chord with steps of 11/8-[[16/15]]-11/8. The inversely related pairs of chords are 1-11/8-[[11/6]] chord with steps of 11/8-[[4/3]]-[[11/10]] and 1-11/10-16/11 chord with steps of 11/10-4/3-11/8; and 1-11/8-[[5/3]] chord with steps of 11/8-[[11/9]]-[[6/5]] and 1-6/5-16/11 chord with steps of 6/5-11/9-11/8.
-<!-- The biyatismic pentads consist of two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are 1-5/4-11/8-16/11-8/5 with steps 5/4-11/10-16/15-11/10-5/4 and 1-11/10-11/8-16/11-20/11 with steps 11/10-5/4-16/15-5/4-11/10. The inversely related pairs of chords are 1-5/4-11/8-16/11-20/11 with steps 5/4-11/10-16/15-5/4-11/10 and 1-11/10-11/8-16/11-8/5 with steps 11/10-5/4-16/15-11/10-5/4; and 1-11/10-6/5-3/2-15/8 with steps 11/10-11/10-5/4-5/4-16/15 and 1-16/15-4/3-5/3-11/6 with steps 16/15-5/4-5/4-11/10-11/10. -->
+The biyatismic tetrads consist of three palindromic chords and fifteen pairs of chords in inverse relationship. The palindromic chords are 1-9/8-11/8-18/11 chord with steps of 9/8-11/9-6/5-11/9, 1-5/4-11/8-11/6 chord with steps of 5/4-11/10-4/3-11/10, and 1-11/10-3/2-18/11 chord with steps of 11/10-11/8-11/10-11/9. The inversely related pairs of chords are 1-4/3-5/3-11/6 chord with steps of 4/3-5/4-11/10-11/10 and 1-11/10-6/5-3/2 chord with steps of 11/10-11/10-5/4-4/3; 1-11/9-11/8-18/11 chord with steps of 11/9-9/8-6/5-11/9 and 1-11/9-16/11-18/11 chord with steps of 11/9-6/5-9/8-11/9; 1-11/8-5/3-11/6 chord with steps of 11/8-11/9-11/10-11/10 and 1-11/10-6/5-16/11 chord with steps of 11/10-11/10-11/9-11/8; 1-4/3-3/2-11/6 chord with steps of 4/3-9/8-11/9-11/10 and 1-11/10-4/3-3/2 chord with steps of 11/10-11/9-9/8-4/3; 1-9/8-11/8-11/6 chord with steps of 9/8-11/9-4/3-11/10 and 1-11/10-16/11-16/9 chord with steps of 11/10-4/3-11/9-9/8; 1-4/3-3/2-18/11 chord with steps of 4/3-9/8-11/10-11/9 and 1-11/9-4/3-3/2 chord with steps of 11/9-11/10-9/8-4/3; 1-5/4-3/2-11/6 chord with steps of 5/4-6/5-11/9-11/10 and 1-11/10-4/3-8/5 chord with steps of 11/10-11/9-6/5-5/4; 1-6/5-3/2-11/6 chord with steps of 6/5-5/4-11/9-11/10 and 1-11/10-4/3-5/3 chord with steps of 11/10-11/9-5/4-6/5; 1-6/5-16/11-11/6 chord with steps of 6/5-11/9-5/4-11/10 and 1-11/10-11/8-5/3 chord with steps of 11/10-5/4-11/9-6/5; 1-5/4-11/8-16/11 chord with steps of 5/4-11/10-16/15-11/8 and 1-11/8-16/11-8/5 chord with steps of 11/8-16/15-11/10-5/4; 1-5/4-4/3-16/11 chord with steps of 5/4-16/15-11/10-11/8 and 1-11/8-3/2-8/5 chord with steps of 11/8-11/10-16/15-5/4; 1-5/4-4/3-11/6 chord with steps of 5/4-16/15-11/8-11/10 and 1-11/10-3/2-8/5 chord with steps of 11/10-11/8-16/15-5/4; 1-11/9-11/8-16/11 chord with steps of 11/9-9/8-16/15-11/8 and 1-11/8-16/11-18/11 chord with steps of 11/8-16/15-9/8-11/9; 1-11/8-5/3-16/9 chord with steps of 11/8-11/9-16/15-9/8 and 1-9/8-6/5-16/11 chord with steps of 9/8-16/15-11/9-11/8; and 1-11/8-16/11-16/9 chord with steps of 11/8-16/15-11/9-9/8 and 1-9/8-11/8-16/11 chord with steps of 9/8-11/9-16/15-11/8.
 Equal temperaments with biyatismic chords include 7, 9, 15, 16, 22, 24, 29, 31, 38, 46, and 53 [[EDO]]s.
-== Zeus chords ==
-'''Zeus chords''' are essentially tempered dyadic chords of the [[Porwell family|Zeus temperament]]. Aside from the biyatismic, valinorsmic, and keenanismic chords, there are 43 further essentially Zeus chords - eighteen pentads, ten hexads, and fifteen heptads.
-The Zeus pentads consist of two palindromic chords and eight pairs of chords in inverse relationship.
-The two palindromic pentads are: 1-6/5-11/8-16/11-5/3 with steps 6/5-8/7-16/15-8/7-6/5 and 1-8/7-11/8-16/11-7/4 with steps 8/7-6/5-16/15-6/5-8/7.
-The eight inversely related pairs of pentads are: 1-8/7-11/8-16/11-5/3 with steps 8/7-6/5-16/15-8/7-6/5 and 1-6/5-11/8-16/11-7/4 with steps 6/5-8/7-16/15-6/5-8/7; 1-8/7-11/9-16/11-7/4 with steps 8/7-16/15-6/5-6/5-8/7 and 1-8/7-11/8-18/11-7/4 with steps 8/7-6/5-6/5-16/15-8/7; 1-8/7-11/8-16/11-8/5 with steps 8/7-6/5-16/15-11/10-5/4 and 1-5/4-11/8-16/11-7/4 with steps 5/4-11/10-16/15-6/5-8/7; 1-8/7-11/8-16/11-11/6 with steps 8/7-6/5-16/15-5/4-11/10 and 1-11/10-11/8-16/11-7/4 with steps 11/10-5/4-16/15-6/5-8/7; 1-5/4-11/8-16/11-5/3 with steps 5/4-11/10-16/15-8/7-6/5 and 1-6/5-11/8-16/11-8/5 with steps 6/5-8/7-16/15-11/10-5/4; 1-11/10-11/8-16/11-5/3 with steps 11/10-5/4-16/15-8/7-6/5 and 1-6/5-11/8-16/11-11/6 with steps 6/5-8/7-16/15-5/4-11/10; 1-5/4-3/2-8/5-11/6 with steps 5/4-6/5-16/15-8/7-11/10 and 1-11/10-5/4-4/3-8/5 with steps 11/10-8/7-16/15-6/5-5/4; and 1-5/4-11/8-18/11-7/4 with steps 5/4-11/10-6/5-16/15-8/7 and 1-8/7-11/9-16/11-8/5 with steps 8/7-16/15-6/5-11/10-5/4.
-The Zeus hexads consist of two palindromic chords and four pairs of chords in inverse relationship.
-The two palindromic hexads are: 1-11/10-5/4-11/8-11/7-12/7 with steps 11/10-8/7-11/10-8/7-11/10-7/6 and 1-8/7-5/4-11/8-3/2-12/7 with steps 8/7-11/10-11/10-11/10-8/7-7/6.
-The four inversely related pairs of hexads are: 1-8/7-5/4-11/8-3/2-7/4 with steps 8/7-11/10-11/10-11/10-7/6-8/7 and 1-8/7-4/3-16/11-8/5-7/4 with steps 8/7-7/6-11/10-11/10-11/10-8/7; 1-8/7-5/4-10/7-11/7-12/7 with steps 8/7-11/10-8/7-11/10-11/10-7/6 and 1-7/6-14/11-7/5-8/5-7/4 with steps 7/6-11/10-11/10-8/7-11/10-8/7; 1-7/6-14/11-16/11-8/5-7/4 with steps 7/6-11/10-8/7-11/10-11/10-8/7 and 1-8/7-5/4-11/8-11/7-12/7 with steps 8/7-11/10-11/10-8/7-11/10-7/6; and 1-8/7-5/4-16/11-8/5-7/4 with steps 8/7-11/10-7/6-11/10-11/10-8/7 and 1-8/7-5/4-11/8-8/5-7/4 with steps 8/7-11/10-11/10-7/6-11/10-8/7.
-The Zeus heptads consist of three palindromic chords and six pairs of chords in inverse relationship.
-The three palindromic heptads are: 1-11/10-6/5-11/8-16/11-5/3-11/6 with steps 11/10-11/10-8/7-16/15-8/7-11/10-11/10, 1-11/10-5/4-11/8-16/11-8/5-11/6 with steps 11/10-8/7-11/10-16/15-11/10-8/7-11/10, and 1-8/7-5/4-11/8-16/11-8/5-7/4 with steps 8/7-11/10-11/10-16/15-11/10-11/10-8/7.
-The six inversely related pairs of heptads are: 1-11/10-6/5-11/8-16/11-8/5-7/4 with steps 11/10-11/10-8/7-16/15-11/10-11/10-8/7 and 1-8/7-5/4-11/8-16/11-5/3-11/6 with steps 8/7-11/10-11/10-16/15-8/7-11/10-11/10; 1-11/10-6/5-11/8-16/11-8/5-11/6 with steps 11/10-11/10-8/7-16/15-11/10-8/7-11/10 and 1-11/10-5/4-11/8-16/11-5/3-11/6 with steps 11/10-8/7-11/10-16/15-8/7-11/10-11/10; 1-11/10-5/4-11/8-16/11-8/5-7/4 with steps 11/10-8/7-11/10-16/15-11/10-11/10-8/7 and 1-8/7-5/4-11/8-16/11-8/5-11/6 with steps 8/7-11/10-11/10-16/15-11/10-8/7-11/10; 1-11/10-6/5-14/11-16/11-5/3-11/6 with steps 11/10-11/10-16/15-8/7-8/7-11/10-11/10 and 1-11/10-6/5-11/8-11/7-5/3-11/6 with steps 11/10-11/10-8/7-8/7-16/15-11/10-11/10; 1-11/10-6/5-14/11-16/11-8/5-11/6 with steps 11/10-11/10-16/15-8/7-11/10-8/7-11/10 and 1-11/10-5/4-11/8-11/7-5/3-11/6 with steps 11/10-8/7-11/10-8/7-16/15-11/10-11/10; and 1-8/7-5/4-4/3-16/11-8/5-7/4 with steps 8/7-11/10-16/15-11/10-11/10-11/10-8/7 and 1-8/7-5/4-11/8-3/2-8/5-7/4 with steps 8/7-11/10-11/10-11/10-16/15-11/10-8/7.
 [[Category:Chords]]
 [[Category:Dyadic]]
 [[Category:Triad]]
+[[Category:Tetrad]]

Biyatismic chords: Difference between revisions

Revision as of 11:30, 7 November 2021

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