Ramanujanismic chords: Difference between revisions

(One intermediate revision by one other user not shown)

Line 1:

'''Ramanujanismic chords''' are [[essentially tempered ~~dyadic chord~~]]s tempered by the ramanujanisma, [[1729/1728]].

'''Ramanujanismic chords''' are [[dyadic chord|essentially tempered chords]] tempered by the ramanujanisma, [[1729/1728]].

Ramanujanismic chords are numerous. A basic ramanujanismic triad in the 2.3.7.13.19 [[subgroup]] [[19-odd-limit]] consists of [[7/6]], [[13/12]], and [[19/12]] closing at the [[octave]]:

Line 116:

* 1-19/16-3/2-7/4-24/13 with steps 19/16-24/19-7/6-19/18-13/12.

For hexads, the ~~known ones are~~

These chords put two otonal 19's against a utonal 7 or 13, or the other way around:

* 1-19/16-3/2-19/12-12/7 with steps 19/16-24/19-19/18-13/12-7/6, and its inverse

* 1-24/19-3/2-7/4-36/19 with steps 24/19-19/16-7/6-13/12-19/18;

* 1-19/16-3/2-19/12-24/13 with steps 19/16-24/19-19/18-7/6-13/12, and its inverse

* 1-24/19-3/2-13/8-36/19 with steps 24/19-19/16-13/12-7/6-19/18.

These chords put two otonal 7's against a utonal 13 or 19, or the other way around:

* 1-7/6-24/19-3/2-7/4 with steps 7/6-13/12-19/16-7/6-8/7, and its inverse

* 1-19/16-9/7-3/2-12/7 with steps 19/16-13/12-7/6-8/7-7/6;

* 1-7/6-3/2-7/4-24/13 with steps 7/6-9/7-7/6-19/18-13/12, and its inverse

* 1-9/7-3/2-13/8-12/7 with steps 9/7-7/6-13/12-19/18-7/6.

These chords put two otonal 13's against a utonal 7 or 19, or the other way around:

* 1-13/12-24/19-3/2-13/8 with steps 13/12-7/6-19/16-13/12-16/13, and its inverse

* 1-19/16-18/13-3/2-24/13 with steps 19/16-7/6-13/12-16/13-13/12;

* 1-18/13-3/2-7/4-24/13 with steps 18/13-13/12-7/6-19/18-13/12, and its inverse

* 1-13/12-3/2-19/12-12/7 with steps 13/12-18/13-13/12-19/18-7/6.

For hexads, the following chords put two otonal 19's against utonal 7 and 13, or the other way around:

* 1-19/16-3/2-19/12-12/7-24/13 with steps 19/16-24/19-19/18-13/12-14/13-13/12, and its inverse

* 1-24/19-3/2-13/8-7/4-36/19 with steps 24/19-19/16-13/12-14/13-13/12-19/18.

These chords put two otonal 7's against utonal 13 and 19, or the other way around:

* 1-7/6-24/19-3/2-7/4-24/13 with steps 7/6-13/12-19/16-7/6-19/18-13/12, and its inverse

* 1-19/16-9/7-3/2-13/8-12/7 with steps 19/16-13/12-7/6-13/12-19/18-7/6.

These chords put two otonal 13's against utonal 7 and 19, or the other way around:

* 1-13/12-24/19-3/2-13/8-12/7 with steps 13/12-7/6-19/16-13/12-19/18-7/6, and its inverse

* 1-19/16-18/13-3/2-7/4-24/13 with steps 19/16-7/6-13/12-7/6-19/18-13/12.

These chords put otonal 9 and 19 against utonal 7 and 13, or the other way around:

* 1-9/8-19/16-9/7-18/13-3/2 with steps 9/8-19/18-13/12-14/13-13/12-4/3, and its inverse

* 1-13/12-7/6-24/19-4/3-3/2 with steps 13/12-14/13-13/12-19/18-9/8-4/3;

* 1-13/12-7/6-24/19-4/3-3/2 with steps 13/12-14/13-13/12-19/18-9/8-4/3.

* 1-19/16-9/7-3/2-13/8-12/7 with steps 19/16-13/12-7/6-13/12-19/18-7/6, and its inverse

* 1-7/6-24/19-3/2-7/4-24/13 with steps 7/6-13/12-19/16-7/6~~-19/18~~-13/12;

These chords put otonal 9 and 7 against utonal 13 and 19, or the other way around:

* 1-9/8-18/13-3/2-7/4-36/19 with steps 9/8-16/13-13/12-7/6-13/12-19/18, and its inverse

* 1-13/12-4/3-3/2-19/12-12/7 with steps 13/12-16/13-9/8-19/18-13/12-7/6.

These chords put otonal 9 and 13 against utonal 7 and 19, or the other way around:

* 1-9/8-9/7-3/2-13/8-36/19 with steps 9/8-8/7-7/6-13/12-7/6-19/18, and its inverse

* 1-7/6-4/3-3/2-19/12-24/13 with steps 7/6-8/7-9/8-19/18-7/6-13/12.

Equal temperaments with ramanujanismic chords include {{Optimal ET sequence|17, 27, 36, 41, 53, 77, 94, 104, 113, 130, 140, 166, 207, 234, 243, 270, 347, 400 and 477}}.

[[Category:21-odd-limit]]

[[Category:19-odd-limit]]

[[Category:Essentially tempered chords]]

[[Category:Triads]]

@@ Line 1: / Line 1: @@
-'''Ramanujanismic chords''' are [[essentially tempered dyadic chord]]s tempered by the ramanujanisma, [[1729/1728]].
+'''Ramanujanismic chords''' are [[dyadic chord|essentially tempered chords]] tempered by the ramanujanisma, [[1729/1728]].
 Ramanujanismic chords are numerous. A basic ramanujanismic triad in the 2.3.7.13.19 [[subgroup]] [[19-odd-limit]] consists of [[7/6]], [[13/12]], and [[19/12]] closing at the [[octave]]:
@@ Line 116: / Line 116: @@
 * 1-19/16-3/2-7/4-24/13 with steps 19/16-24/19-7/6-19/18-13/12.
-For hexads, the known ones are
+These chords put two otonal 19's against a utonal 7 or 13, or the other way around:
+* 1-19/16-3/2-19/12-12/7 with steps 19/16-24/19-19/18-13/12-7/6, and its inverse
+* 1-24/19-3/2-7/4-36/19 with steps 24/19-19/16-7/6-13/12-19/18;
+* 1-19/16-3/2-19/12-24/13 with steps 19/16-24/19-19/18-7/6-13/12, and its inverse
+* 1-24/19-3/2-13/8-36/19 with steps 24/19-19/16-13/12-7/6-19/18.
+These chords put two otonal 7's against a utonal 13 or 19, or the other way around:
+* 1-7/6-24/19-3/2-7/4 with steps 7/6-13/12-19/16-7/6-8/7, and its inverse
+* 1-19/16-9/7-3/2-12/7 with steps 19/16-13/12-7/6-8/7-7/6;
+* 1-7/6-3/2-7/4-24/13 with steps 7/6-9/7-7/6-19/18-13/12, and its inverse
+* 1-9/7-3/2-13/8-12/7 with steps 9/7-7/6-13/12-19/18-7/6.
+These chords put two otonal 13's against a utonal 7 or 19, or the other way around:
+* 1-13/12-24/19-3/2-13/8 with steps 13/12-7/6-19/16-13/12-16/13, and its inverse
+* 1-19/16-18/13-3/2-24/13 with steps 19/16-7/6-13/12-16/13-13/12;
+* 1-18/13-3/2-7/4-24/13 with steps 18/13-13/12-7/6-19/18-13/12, and its inverse
+* 1-13/12-3/2-19/12-12/7 with steps 13/12-18/13-13/12-19/18-7/6.
+For hexads, the following chords put two otonal 19's against utonal 7 and 13, or the other way around:
+* 1-19/16-3/2-19/12-12/7-24/13 with steps 19/16-24/19-19/18-13/12-14/13-13/12, and its inverse
+* 1-24/19-3/2-13/8-7/4-36/19 with steps 24/19-19/16-13/12-14/13-13/12-19/18.
+These chords put two otonal 7's against utonal 13 and 19, or the other way around:
+* 1-7/6-24/19-3/2-7/4-24/13 with steps 7/6-13/12-19/16-7/6-19/18-13/12, and its inverse
+* 1-19/16-9/7-3/2-13/8-12/7 with steps 19/16-13/12-7/6-13/12-19/18-7/6.
+These chords put two otonal 13's against utonal 7 and 19, or the other way around:
+* 1-13/12-24/19-3/2-13/8-12/7 with steps 13/12-7/6-19/16-13/12-19/18-7/6, and its inverse
+* 1-19/16-18/13-3/2-7/4-24/13 with steps 19/16-7/6-13/12-7/6-19/18-13/12.
+These chords put otonal 9 and 19 against utonal 7 and 13, or the other way around:
 * 1-9/8-19/16-9/7-18/13-3/2 with steps 9/8-19/18-13/12-14/13-13/12-4/3, and its inverse
-* 1-13/12-7/6-24/19-4/3-3/2 with steps 13/12-14/13-13/12-19/18-9/8-4/3;
+* 1-13/12-7/6-24/19-4/3-3/2 with steps 13/12-14/13-13/12-19/18-9/8-4/3.
-* 1-19/16-9/7-3/2-13/8-12/7 with steps 19/16-13/12-7/6-13/12-19/18-7/6, and its inverse
-* 1-7/6-24/19-3/2-7/4-24/13 with steps 7/6-13/12-19/16-7/6-19/18-13/12;
+These chords put otonal 9 and 7 against utonal 13 and 19, or the other way around:
+* 1-9/8-18/13-3/2-7/4-36/19 with steps 9/8-16/13-13/12-7/6-13/12-19/18, and its inverse
+* 1-13/12-4/3-3/2-19/12-12/7 with steps 13/12-16/13-9/8-19/18-13/12-7/6.
+These chords put otonal 9 and 13 against utonal 7 and 19, or the other way around:
+* 1-9/8-9/7-3/2-13/8-36/19 with steps 9/8-8/7-7/6-13/12-7/6-19/18, and its inverse
+* 1-7/6-4/3-3/2-19/12-24/13 with steps 7/6-8/7-9/8-19/18-7/6-13/12.
+Equal temperaments with ramanujanismic chords include {{Optimal ET sequence|17, 27, 36, 41, 53, 77, 94, 104, 113, 130, 140, 166, 207, 234, 243, 270, 347, 400 and 477}}.
-[[Category:21-odd-limit]]
+[[Category:19-odd-limit]]
 [[Category:Essentially tempered chords]]
 [[Category:Triads]]