Ragismic chords: Difference between revisions

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'''Ragismic chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the ragisma, [[4375/4374]].

~~The~~ [[~~Ragismic microtemperaments~~|~~ragismic temperament~~]] ~~has 4~~ triads, 11 tetrads, 9 pentads and 2 hexads as essentially tempered chords.

Since the temperament is complex, even the simplest ragismic chords are [[27-odd-limit]], although they contain only [[7-limit|7-prime-limit]] intervals. There are 6 triads, 21 tetrads, 22 pentads and 7 hexads as essentially tempered chords.

For triads, there are two palindromic chords and ~~a pair~~ of chords in inverse relationship. The palindromic chords are

For triads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are

* 1-27/25-7/~~6 chord~~ with steps of 27/25-27/25~~-12~~/7;

* 1-25/18-36/25 with steps of 25/18-28/27-25/18;

* 1-25~~/18~~-36/~~25 chord~~ with steps of 25~~/18~~-28/27-25/18.

* 1-27/25-7/6 with steps of 27/25-27/25-12/7.

The inversely related ~~pair~~ of chords is

The inversely related pairs of chords are

* 1-9/7-25/18 with steps of 9/7-27/25-36/25 and its inverse 1-27/25-25/18 with steps of 27/25-9/7-36/25.

* 1-9/7-25/18 with steps of 9/7-27/25-36/25, and its inverse

* 1-27/25-25/18 with steps of 27/25-9/7-36/25;

* 1-25/24-27/25 with steps of 25/24-28/27-50/27, and its inverse

* 1-28/27-27/25 with steps of 28/27-25/24-50/27.

For tetrads, there are ~~three~~ palindromic chords and ~~four~~ pairs of chords in inverse relationship. The palindromic chords are

For tetrads, there are five palindromic chords and eight pairs of chords in inverse relationship. The palindromic chords are

* 1-27/25-36/25-14/9 ~~chord~~ with steps of 27/25-4/3-27/25-9/7;

* 1-28/25-36/25-14/9 with steps of 28/25-9/7-27/25-9/7;

* 1-27/25-25/18-14/~~9 chord~~ with steps of 27/25-9/7-28/25-9/7;

* 1-27/25-25/18-3/2 with steps of 27/25-9/7-27/25-4/3;

* 1-27/25-9/7-25/18 ~~chord~~ with steps of 27/25-25/21-27/25-36/25.

* 1-27/25-9/7-25/18 with steps of 27/25-25/21-27/25-36/25;

* 1-25/24-27/25-9/8 with steps of 25/24-28/27-25/24-16/9;

* 1-28/27-27/25-28/25 with steps of 28/27-25/24-28/27-25/14.

The inversely related pairs of chords are

* 1-27/25-7/6-14/9 with steps of 27/25-27/25-4/3-9/7 and its inverse 1-27/25-7/6-3/2 with steps of 27/25-27/25-9/7-4/3;

* 1-25/18-36/25-3/2 with steps of 25/18-28/27-25/24-4/3, and its inverse

* 1-27/25-7/6-25/18 with steps of 27/25-27/25-25/21-36/25 and its inverse 1-25/21-9/7-25/18 with steps of 25/21-27/25-27/25-36/25;

* 1-25/24-27/25-3/2 with steps of 25/24-28/27-25/18-4/3;

* 1-27/25-25/18-36/25 with steps of 27/25-9/7-28/27-25/18 and its inverse 1-25/18-36/25-50/27 with steps of ~~25/18-~~28/27-9/7-27/25;

* 1-9/7-25/18-3/2 with steps of 9/7-27/25-27/25-4/3, and its inverse

* 1-25~~/18~~-36/25-14/9 with steps of 25~~/18~~-28/27-27/25-9/7 and its inverse 1-9/7-25~~/18~~-36/25 with steps of ~~9/7-27~~/25-28/27-25/18.

* 1-27/25-7/6-3/2 with steps of 27/25-27/25-9/7-4/3;

* 1-9/7-25/18-36/25 with steps of 9/7-27/25-28/27-25/18, and its inverse

* 1-28/27-28/25-36/25 with steps of 28/27-27/25-9/7-25/18;

* 1-27/25-3/2-14/9 with steps of 27/25-25/18-28/27-9/7, and its inverse

* 1-25/18-3/2-27/14 with steps of 25/18-27/25-9/7-28/27;

* 1-36/25-3/2-14/9 with steps of 36/25-25/24-28/27-9/7, and its inverse

* 1-25/24-3/2-27/14 with steps of 25/24-36/25-9/7-28/27;

* 1-25/21-9/7-25/18 with steps of 25/21-27/25-27/25-36/25, and its inverse

* 1-27/25-7/6-25/18 with steps of 27/25-27/25-25/21-36/25;

* 1-27/25-9/8-7/6 with steps of 27/25-25/24-28/27-12/7, and its inverse

* 1-28/27-27/25-7/6 with steps of 28/27-25/24-27/25-12/7;

* 1-27/25-28/25-7/6 with steps of 27/25-28/27-25/24-12/7, and its inverse

* 1-25/24-27/25-7/6 with steps of 25/24-28/27-27/25-12/7.

For pentads, there are ~~one~~ palindromic ~~chord~~ and ~~four~~ pairs of chords in inverse relationship. The palindromic ~~chord is~~

For pentads, there are two palindromic chords and ten pairs of chords in inverse relationship. The palindromic chords are

* 1-27/25-25/18-36/25-50/~~27 chord~~ with steps of 27/25-9/7-28/27-9/7-27/25.

* 1-27/25-7/6-3/2-14/9 with steps of 27/25-27/25-9/7-28/27-9/7;

* 1-4/3-25/18-36/25-3/2 with steps of 4/3-25/24-28/27-25/24-4/3.

The inversely related pairs of chords are

* 1-27/25-25/18-36/25-14/9 with steps of 27/25-9/7-28/27-27/25-9/7 and its inverse 1-9/7-25/18-36/25-50/27 with steps of 9/7-27/25-28/27-9/7-27/25;

* 1-9/7-25/18-3/2-25/14 with steps of 9/7-27/25-27/25-25/21-28/25, and its inverse

* 1-27/25-9/7-25/18-50/27 with steps of 27~~/25~~-25/21-27/25-4/3~~-27/25~~ and its inverse 1-27/25-36/25-14/9-50/27 with steps of 27/25-4/3-27/25-25/21-27/25;

* 1-27/25-7/6-3/2-42/25 with steps of 27/25-27/25-9/7-28/25-25/21;

* 1-27/25-9/7-36/25-50/27 with steps of 27/25-25/21-28/25-9/7-27/25 and its inverse 1-27/25-25/18-14/9-50/27 with steps of 27/25-9/7-28/25-25/21-27/25;

* 1-27/25-25/18-3/2-14/9 with steps of 27/25-9/7-27/25-28/27-9/7, and its inverse

* 1-27/25-9/7-25/18-36/25 with steps of 27/25-25/21-27/25-28/27-25/18 and its inverse 1-25/18-36/25-14/9-50/27 with steps of 25/18-28/27-27/25-25/21-27/25.

* 1-27/25-25/18-3/2-27/14 with steps of 27/25-9/7-27/25-9/7-28/27;

* 1-9/7-25/18-36/25-3/2 with steps of 9/7-27/25-28/27-25/24-4/3, and its inverse

* 1-25/24-27/25-7/6-3/2 with steps of 25/24-28/27-27/25-9/7-4/3;

* 1-9/7-4/3-25/18-3/2 with steps of 9/7-28/27-25/24-27/25-4/3, and its inverse

* 1-27/25-9/8-7/6-3/2 with steps of 27/25-25/24-28/27-9/7-4/3;

* 1-27/25-25/18-36/25-3/2 with steps of 27/25-9/7-28/27-25/24-4/3, and its inverse

* 1-25/24-27/25-25/18-3/2 with steps of 25/24-28/27-9/7-27/25-4/3;

* 1-27/25-9/7-25/18-3/2 with steps of 27/25-25/21-27/25-27/25-4/3, and its inverse

* 1-27/25-7/6-25/18-3/2 with steps of 27/25-27/25-25/21-27/25-4/3;

* 1-25/18-36/25-3/2-14/9 with steps of 25/18-28/27-25/24-28/27-9/7, and its inverse

* 1-25/24-27/25-3/2-27/14 with steps of 25/24-28/27-25/18-9/7-28/27;

* 1-25/18-3/2-25/14-27/14 with steps of 25/18-27/25-25/21-27/25-28/27, and its inverse

* 1-27/25-3/2-14/9-42/25 with steps of 27/25-25/18-28/27-27/25-25/21;

* 1-36/25-3/2-14/9-42/25 with steps of 36/25-25/24-28/27-27/25-25/21, and its inverse

* 1-25/24-3/2-25/14-27/14 with steps of 25/24-36/25-25/21-27/25-28/27;

* 1-25/24-27/25-9/8-7/6 with steps of 25/24-28/27-25/24-28/27-12/7, and its inverse

* 1-28/27-27/25-28/25-7/6 with steps of 28/27-25/24-28/27-25/24-12/7.

~~They can be extended to~~ hexads as

For hexads, there are one palindromic chord and three pairs of chords in inverse relationship. The palindromic chord is

* 1-27/25-9/7-25/18-36/25-50/27 with steps of 27/25-25/21-27/25-28/27-9/7-27/25 and its inverse 1-27/25-25/18-36/25-14/9-50/27 with steps of 27/25-9/7-28/27-27/25-25/21-27/25.

* 1-27/25-25/18-36/25-3/2-14/9 with steps of 27/25-9/7-28/27-25/24-28/27-9/7.

The inversely related pairs of chords are

* 1-27/25-7/6-25/18-3/2-14/9 with steps of 27/25-27/25-25/21-27/25-28/27-9/7, and its inverse

* 1-27/25-9/7-25/18-3/2-27/14 with steps of 27/25-25/21-27/25-27/25-9/7-28/27;

* 1-9/7-4/3-25/18-36/25-3/2 with steps of 9/7-28/27-25/24-28/27-25/24-4/3, and its inverse

* 1-25/24-27/25-9/8-7/6-3/2 with steps of 25/24-28/27-25/24-28/27-9/7-4/3;

* 1-27/25-9/7-25/18-36/25-3/2 with steps of 27/25-25/21-27/25-28/27-25/24-4/3, and its inverse

* 1-25/24-27/25-7/6-25/18-3/2 with steps of 25/24-28/27-27/25-25/21-27/25-4/3.

Equal temperaments with ragismic chords include {{Optimal ET sequence|19, 53, 72, 99, 118, 152, 171, 224, 270, 323, 441, 494, 612, 665, 783, 935, 1106, 1277 and 1547}}.

@@ Line 1: / Line 1: @@
 '''Ragismic chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the ragisma, [[4375/4374]].
-The [[Ragismic microtemperaments|ragismic temperament]] has 4 triads, 11 tetrads, 9 pentads and 2 hexads as essentially tempered chords.
+Since the temperament is complex, even the simplest ragismic chords are [[27-odd-limit]], although they contain only [[7-limit|7-prime-limit]] intervals. There are 6 triads, 21 tetrads, 22 pentads and 7 hexads as essentially tempered chords.
-For triads, there are two palindromic chords and a pair of chords in inverse relationship. The palindromic chords are
+For triads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
-* 1-27/25-7/6 chord with steps of 27/25-27/25-12/7;
+* 1-25/18-36/25 with steps of 25/18-28/27-25/18;
-* 1-25/18-36/25 chord with steps of 25/18-28/27-25/18.
+* 1-27/25-7/6 with steps of 27/25-27/25-12/7.
-The inversely related pair of chords is
+The inversely related pairs of chords are
-* 1-9/7-25/18 with steps of 9/7-27/25-36/25 and its inverse 1-27/25-25/18 with steps of 27/25-9/7-36/25.
+* 1-9/7-25/18 with steps of 9/7-27/25-36/25, and its inverse
+* 1-27/25-25/18 with steps of 27/25-9/7-36/25;
+* 1-25/24-27/25 with steps of 25/24-28/27-50/27, and its inverse
+* 1-28/27-27/25 with steps of 28/27-25/24-50/27.
-For tetrads, there are three palindromic chords and four pairs of chords in inverse relationship. The palindromic chords are
+For tetrads, there are five palindromic chords and eight pairs of chords in inverse relationship. The palindromic chords are
-* 1-27/25-36/25-14/9 chord with steps of 27/25-4/3-27/25-9/7;
+* 1-28/25-36/25-14/9 with steps of 28/25-9/7-27/25-9/7;
-* 1-27/25-25/18-14/9 chord with steps of 27/25-9/7-28/25-9/7;
+* 1-27/25-25/18-3/2 with steps of 27/25-9/7-27/25-4/3;
-* 1-27/25-9/7-25/18 chord with steps of 27/25-25/21-27/25-36/25.
+* 1-27/25-9/7-25/18 with steps of 27/25-25/21-27/25-36/25;
+* 1-25/24-27/25-9/8 with steps of 25/24-28/27-25/24-16/9;
+* 1-28/27-27/25-28/25 with steps of 28/27-25/24-28/27-25/14.
 The inversely related pairs of chords are
-* 1-27/25-7/6-14/9 with steps of 27/25-27/25-4/3-9/7 and its inverse 1-27/25-7/6-3/2 with steps of 27/25-27/25-9/7-4/3;
+* 1-25/18-36/25-3/2 with steps of 25/18-28/27-25/24-4/3, and its inverse
-* 1-27/25-7/6-25/18 with steps of 27/25-27/25-25/21-36/25 and its inverse 1-25/21-9/7-25/18 with steps of 25/21-27/25-27/25-36/25;
+* 1-25/24-27/25-3/2 with steps of 25/24-28/27-25/18-4/3;
-* 1-27/25-25/18-36/25 with steps of 27/25-9/7-28/27-25/18 and its inverse 1-25/18-36/25-50/27 with steps of 25/18-28/27-9/7-27/25;
+* 1-9/7-25/18-3/2 with steps of 9/7-27/25-27/25-4/3, and its inverse
-* 1-25/18-36/25-14/9 with steps of 25/18-28/27-27/25-9/7 and its inverse 1-9/7-25/18-36/25 with steps of 9/7-27/25-28/27-25/18.
+* 1-27/25-7/6-3/2 with steps of 27/25-27/25-9/7-4/3;
+* 1-9/7-25/18-36/25 with steps of 9/7-27/25-28/27-25/18, and its inverse
+* 1-28/27-28/25-36/25 with steps of 28/27-27/25-9/7-25/18;
+* 1-27/25-3/2-14/9 with steps of 27/25-25/18-28/27-9/7, and its inverse
+* 1-25/18-3/2-27/14 with steps of 25/18-27/25-9/7-28/27;
+* 1-36/25-3/2-14/9 with steps of 36/25-25/24-28/27-9/7, and its inverse
+* 1-25/24-3/2-27/14 with steps of 25/24-36/25-9/7-28/27;
+* 1-25/21-9/7-25/18 with steps of 25/21-27/25-27/25-36/25, and its inverse
+* 1-27/25-7/6-25/18 with steps of 27/25-27/25-25/21-36/25;
+* 1-27/25-9/8-7/6 with steps of 27/25-25/24-28/27-12/7, and its inverse
+* 1-28/27-27/25-7/6 with steps of 28/27-25/24-27/25-12/7;
+* 1-27/25-28/25-7/6 with steps of 27/25-28/27-25/24-12/7, and its inverse
+* 1-25/24-27/25-7/6 with steps of 25/24-28/27-27/25-12/7.
-For pentads, there are one palindromic chord and four pairs of chords in inverse relationship. The palindromic chord is
+For pentads, there are two palindromic chords and ten pairs of chords in inverse relationship. The palindromic chords are
-* 1-27/25-25/18-36/25-50/27 chord with steps of 27/25-9/7-28/27-9/7-27/25.
+* 1-27/25-7/6-3/2-14/9 with steps of 27/25-27/25-9/7-28/27-9/7;
+* 1-4/3-25/18-36/25-3/2 with steps of 4/3-25/24-28/27-25/24-4/3.
 The inversely related pairs of chords are
-* 1-27/25-25/18-36/25-14/9 with steps of 27/25-9/7-28/27-27/25-9/7 and its inverse 1-9/7-25/18-36/25-50/27 with steps of 9/7-27/25-28/27-9/7-27/25;
+* 1-9/7-25/18-3/2-25/14 with steps of 9/7-27/25-27/25-25/21-28/25, and its inverse
-* 1-27/25-9/7-25/18-50/27 with steps of 27/25-25/21-27/25-4/3-27/25 and its inverse 1-27/25-36/25-14/9-50/27 with steps of 27/25-4/3-27/25-25/21-27/25;
+* 1-27/25-7/6-3/2-42/25 with steps of 27/25-27/25-9/7-28/25-25/21;
-* 1-27/25-9/7-36/25-50/27 with steps of 27/25-25/21-28/25-9/7-27/25 and its inverse 1-27/25-25/18-14/9-50/27 with steps of 27/25-9/7-28/25-25/21-27/25;
+* 1-27/25-25/18-3/2-14/9 with steps of 27/25-9/7-27/25-28/27-9/7, and its inverse
-* 1-27/25-9/7-25/18-36/25 with steps of 27/25-25/21-27/25-28/27-25/18 and its inverse 1-25/18-36/25-14/9-50/27 with steps of 25/18-28/27-27/25-25/21-27/25.
+* 1-27/25-25/18-3/2-27/14 with steps of 27/25-9/7-27/25-9/7-28/27;
+* 1-9/7-25/18-36/25-3/2 with steps of 9/7-27/25-28/27-25/24-4/3, and its inverse
+* 1-25/24-27/25-7/6-3/2 with steps of 25/24-28/27-27/25-9/7-4/3;
+* 1-9/7-4/3-25/18-3/2 with steps of 9/7-28/27-25/24-27/25-4/3, and its inverse
+* 1-27/25-9/8-7/6-3/2 with steps of 27/25-25/24-28/27-9/7-4/3;
+* 1-27/25-25/18-36/25-3/2 with steps of 27/25-9/7-28/27-25/24-4/3, and its inverse
+* 1-25/24-27/25-25/18-3/2 with steps of 25/24-28/27-9/7-27/25-4/3;
+* 1-27/25-9/7-25/18-3/2 with steps of 27/25-25/21-27/25-27/25-4/3, and its inverse
+* 1-27/25-7/6-25/18-3/2 with steps of 27/25-27/25-25/21-27/25-4/3;
+* 1-25/18-36/25-3/2-14/9 with steps of 25/18-28/27-25/24-28/27-9/7, and its inverse
+* 1-25/24-27/25-3/2-27/14 with steps of 25/24-28/27-25/18-9/7-28/27;
+* 1-25/18-3/2-25/14-27/14 with steps of 25/18-27/25-25/21-27/25-28/27, and its inverse
+* 1-27/25-3/2-14/9-42/25 with steps of 27/25-25/18-28/27-27/25-25/21;
+* 1-36/25-3/2-14/9-42/25 with steps of 36/25-25/24-28/27-27/25-25/21, and its inverse
+* 1-25/24-3/2-25/14-27/14 with steps of 25/24-36/25-25/21-27/25-28/27;
+* 1-25/24-27/25-9/8-7/6 with steps of 25/24-28/27-25/24-28/27-12/7, and its inverse
+* 1-28/27-27/25-28/25-7/6 with steps of 28/27-25/24-28/27-25/24-12/7.
-They can be extended to hexads as
+For hexads, there are one palindromic chord and three pairs of chords in inverse relationship. The palindromic chord is
-* 1-27/25-9/7-25/18-36/25-50/27 with steps of 27/25-25/21-27/25-28/27-9/7-27/25 and its inverse 1-27/25-25/18-36/25-14/9-50/27 with steps of 27/25-9/7-28/27-27/25-25/21-27/25.
+* 1-27/25-25/18-36/25-3/2-14/9 with steps of 27/25-9/7-28/27-25/24-28/27-9/7.
+The inversely related pairs of chords are
+* 1-27/25-7/6-25/18-3/2-14/9 with steps of 27/25-27/25-25/21-27/25-28/27-9/7, and its inverse
+* 1-27/25-9/7-25/18-3/2-27/14 with steps of 27/25-25/21-27/25-27/25-9/7-28/27;
+* 1-9/7-4/3-25/18-36/25-3/2 with steps of 9/7-28/27-25/24-28/27-25/24-4/3, and its inverse
+* 1-25/24-27/25-9/8-7/6-3/2 with steps of 25/24-28/27-25/24-28/27-9/7-4/3;
+* 1-27/25-9/7-25/18-36/25-3/2 with steps of 27/25-25/21-27/25-28/27-25/24-4/3, and its inverse
+* 1-25/24-27/25-7/6-25/18-3/2 with steps of 25/24-28/27-27/25-25/21-27/25-4/3.
 Equal temperaments with ragismic chords include {{Optimal ET sequence|19, 53, 72, 99, 118, 152, 171, 224, 270, 323, 441, 494, 612, 665, 783, 935, 1106, 1277 and 1547}}.