Rastmic chords: Difference between revisions

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A '''~~rastmic chord~~''' ~~is an~~ [[essentially tempered ~~dyadic chord~~]] tempered by the rastma, [[243/242]], in the 2.3.11 [[subgroup]] in the [[11-odd-limit]].

'''Rastmic chords''' are [[dyadic chord|essentially tempered chords]] tempered by the rastma, [[243/242]], in the 2.3.11 [[subgroup]] in the [[11-odd-limit]].

The count of chords is 3 triads, 7 tetrads, 5 pentads, and 1 hexad, for a total of 16:

* There are three ''rastmic triads'': ~~the classic neutral triad 1-11/9-3/2 with steps 11/9-11/9-4/3, and an inversely related pair of triads, 1-9/8-11/6 with steps 9/8-18/11-12/11 and 1-9/8-11/9 with steps 9/8-12/11-18/11.~~

There are three ''rastmic triads'':

* ''Rastmic tetrads'' are seven in number: the ~~palindromic~~ classic [[neutral tetrad]] 1-11/9-3/2-11/6 with steps 11/9-11/9-11/9-12/11; the palindromic 1-3/2-18/11-11/6 with steps 3/2-12/11-9/8-12/11; the palindromic 1-9/8-11/9-11/8 with steps 9/8-12/11-9/8-16/11; the inverse pair 1-11/9-11/8-3/2 with steps 11/9-9/8-12/11-4/3 and 1-12/11-11/9-3/2 with steps 12/11-9/8-11/9-4/3; and the inverse pair 1~~-9/8~~-11/9-3/2 with steps ~~9/8-12/~~11~~-11/9-4~~/~~3 and 1-~~9~~/8-3/2~~-11/~~6 with steps~~ 9/8-4/3~~-11/9-12/11.~~

the classic neutral triad,

* 1-11/9-3/2 with steps of 11/9-11/9-4/3,

* ~~There are five ''rastmic pentads'': the palindromic pentad~~ 1-9/8~~-11/9-3/2~~-11/6 with steps 9/8-~~12/11-11/9-11/9-12/11; the pair 1-9/8-11/8-3/2-11/6 with steps 9/8-11/9-12/11-11/9-12/11 and 1-11/9-11/8-3~~/2-11~~/6 with steps 11/9-9/8~~-12/11~~-11/9-12/11;~~ and ~~the pair~~ 1-9/8-11/9~~-11/8-3/2~~ with steps 9/8-12/11~~-9/8-12/11-4/3 and 1-9/8-3/2~~-18~~/11-11/6 with steps 9/8-4/3-12/11-9/8-12~~/11.

and an inversely related pair of triads,

* 1-9/8-11/6 with steps of 9/8-18/11-12/11 and 1-9/8-11/9 with steps of 9/8-12/11-18/11.

* There is also a ''~~rastmic hexad~~'': 1-9/8-11/9-11/8-3/2-11/6, with steps 9/8-12/11-9/8-12/11-11/9-12/11. This can be extended to the neutral diatonic scale, LsLsLss, which is [[Chromatic_pairs#Neutral|Neutral[7]]]. In neutral, with the neutral third (~11/9) as generator, the rastmic hexad is a chain of five neutral thirds rather than the six which give [[neutral7]], which therefore has two rastmic hexads and of course many more smaller rastmic chords.

''Rastmic tetrads'' are seven in number:

~~Equal divisions~~ of the ~~octave~~ with rastmic chords include 10, 17, 24, 31, 41, 58, 72, 130, 202, 736be, 938be, 1075be, 1116be, 1277be, 1318be.

three palindromic tetrads,

* 1-11/9-3/2-11/6 with steps of 11/9-11/9-11/9-12/11 (→[[neutral tetrad]]);

* 1-3/2-18/11-11/6 with steps of 3/2-12/11-9/8-12/11;

* 1-9/8-11/9-11/8 with steps of 9/8-12/11-9/8-16/11;

and two inversely related pairs of tetrads,

* 1-11/9-11/8-3/2 with steps of 11/9-9/8-12/11-4/3 and 1-12/11-11/9-3/2 with steps of 12/11-9/8-11/9-4/3;

* 1-9/8-11/9-3/2 with steps of 9/8-12/11-11/9-4/3 and 1-9/8-3/2-11/6 with steps of 9/8-4/3-11/9-12/11.

There are five ''rastmic pentads'':

the palindromic pentad,

* 1-9/8-11/9-3/2-11/6 with steps of 9/8-12/11-11/9-11/9-12/11;

and two inversely related pairs of pentads,

* 1-9/8-11/8-3/2-11/6 with steps of 9/8-11/9-12/11-11/9-12/11 and 1-11/9-11/8-3/2-11/6 with steps of 11/9-9/8-12/11-11/9-12/11;

* 1-9/8-11/9-11/8-3/2 with steps of 9/8-12/11-9/8-12/11-4/3 and 1-9/8-3/2-18/11-11/6 with steps of 9/8-4/3-12/11-9/8-12/11.

There is also a ''rastmic hexad'':

* 1-9/8-11/9-11/8-3/2-11/6 with steps of 9/8-12/11-9/8-12/11-11/9-12/11.

Equal temperaments with rastmic chords include {{Optimal ET sequence|10, 17, 24, 31, 41, 58, 72, 130, 202, 736be, 938be, 1075be, 1116be, 1277be and 1318be}}.

{| class="wikitable"

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| 9/8 - 12/11 - 9/8 - 12/11 - 11/9 - 12/11

|}

== Rastgross heptad ==

Rastmic chords can be extended to the neutral diatonic scale, LsLsLss, which is [[Chromatic_pairs#Neutral|Neutral[7]]]. In neutral, with the neutral third (~11/9) as generator, the rastmic hexad is a chain of five neutral thirds rather than the six which give [[neutral7]], which therefore has two rastmic hexads and of course many more smaller rastmic chords.

In the 2.3.11.13 subgroup, this scale is interpreted as an essentially tempered heptad, the '''rastgross heptad''', tempered by [[144/143]] (grossma) and 243/242. This heptad is 1-9/8-11/9-11/8-3/2-22/13-11/6 with steps of 9/8-12/11-9/8-12/11-9/8-12/11-12/11 (→[[rastgross1]]).

[[Category:11-odd-limit]]

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-A '''rastmic chord''' is an [[essentially tempered dyadic chord]] tempered by the rastma, [[243/242]], in the 2.3.11 [[subgroup]] in the [[11-odd-limit]].
+'''Rastmic chords''' are [[dyadic chord|essentially tempered chords]] tempered by the rastma, [[243/242]], in the 2.3.11 [[subgroup]] in the [[11-odd-limit]].
 The count of chords is 3 triads, 7 tetrads, 5 pentads, and 1 hexad, for a total of 16:
-* There are three ''rastmic triads'': the classic neutral triad 1-11/9-3/2 with steps 11/9-11/9-4/3, and an inversely related pair of triads, 1-9/8-11/6 with steps 9/8-18/11-12/11 and 1-9/8-11/9 with steps 9/8-12/11-18/11.
+There are three ''rastmic triads'':
-* ''Rastmic tetrads'' are seven in number: the palindromic classic [[neutral tetrad]] 1-11/9-3/2-11/6 with steps 11/9-11/9-11/9-12/11; the palindromic 1-3/2-18/11-11/6 with steps 3/2-12/11-9/8-12/11; the palindromic 1-9/8-11/9-11/8 with steps 9/8-12/11-9/8-16/11; the inverse pair 1-11/9-11/8-3/2 with steps 11/9-9/8-12/11-4/3 and 1-12/11-11/9-3/2 with steps 12/11-9/8-11/9-4/3; and the inverse pair 1-9/8-11/9-3/2 with steps 9/8-12/11-11/9-4/3 and 1-9/8-3/2-11/6 with steps 9/8-4/3-11/9-12/11.
+the classic neutral triad,
+* 1-11/9-3/2 with steps of 11/9-11/9-4/3,
-* There are five ''rastmic pentads'': the palindromic pentad 1-9/8-11/9-3/2-11/6 with steps 9/8-12/11-11/9-11/9-12/11; the pair 1-9/8-11/8-3/2-11/6 with steps 9/8-11/9-12/11-11/9-12/11 and 1-11/9-11/8-3/2-11/6 with steps 11/9-9/8-12/11-11/9-12/11; and the pair 1-9/8-11/9-11/8-3/2 with steps 9/8-12/11-9/8-12/11-4/3 and 1-9/8-3/2-18/11-11/6 with steps 9/8-4/3-12/11-9/8-12/11.
+and an inversely related pair of triads,
+* 1-9/8-11/6 with steps of 9/8-18/11-12/11 and 1-9/8-11/9 with steps of 9/8-12/11-18/11.
-* There is also a ''rastmic hexad'': 1-9/8-11/9-11/8-3/2-11/6, with steps 9/8-12/11-9/8-12/11-11/9-12/11. This can be extended to the neutral diatonic scale, LsLsLss, which is [[Chromatic_pairs#Neutral|Neutral[7]]]. In neutral, with the neutral third (~11/9) as generator, the rastmic hexad is a chain of five neutral thirds rather than the six which give [[neutral7]], which therefore has two rastmic hexads and of course many more smaller rastmic chords.
+''Rastmic tetrads'' are seven in number:
-Equal divisions of the octave with rastmic chords include 10, 17, 24, 31, 41, 58, 72, 130, 202, 736be, 938be, 1075be, 1116be, 1277be, 1318be.
+three palindromic tetrads,
+* 1-11/9-3/2-11/6 with steps of 11/9-11/9-11/9-12/11 (→[[neutral tetrad]]);
+* 1-3/2-18/11-11/6 with steps of 3/2-12/11-9/8-12/11;
+* 1-9/8-11/9-11/8 with steps of 9/8-12/11-9/8-16/11;
+and two inversely related pairs of tetrads,
+* 1-11/9-11/8-3/2 with steps of 11/9-9/8-12/11-4/3 and 1-12/11-11/9-3/2 with steps of 12/11-9/8-11/9-4/3;
+* 1-9/8-11/9-3/2 with steps of 9/8-12/11-11/9-4/3 and 1-9/8-3/2-11/6 with steps of 9/8-4/3-11/9-12/11.
+There are five ''rastmic pentads'':
+the palindromic pentad,
+* 1-9/8-11/9-3/2-11/6 with steps of 9/8-12/11-11/9-11/9-12/11;
+and two inversely related pairs of pentads,
+* 1-9/8-11/8-3/2-11/6 with steps of 9/8-11/9-12/11-11/9-12/11 and 1-11/9-11/8-3/2-11/6 with steps of 11/9-9/8-12/11-11/9-12/11;
+* 1-9/8-11/9-11/8-3/2 with steps of 9/8-12/11-9/8-12/11-4/3 and 1-9/8-3/2-18/11-11/6 with steps of 9/8-4/3-12/11-9/8-12/11.
+There is also a ''rastmic hexad'':
+* 1-9/8-11/9-11/8-3/2-11/6 with steps of 9/8-12/11-9/8-12/11-11/9-12/11.
+Equal temperaments with rastmic chords include {{Optimal ET sequence|10, 17, 24, 31, 41, 58, 72, 130, 202, 736be, 938be, 1075be, 1116be, 1277be and 1318be}}.
 {| class="wikitable"
@@ Line 66: / Line 89: @@
 | 9/8 - 12/11 - 9/8 - 12/11 - 11/9 - 12/11
 |}
+== Rastgross heptad ==
+Rastmic chords can be extended to the neutral diatonic scale, LsLsLss, which is [[Chromatic_pairs#Neutral|Neutral[7]]]. In neutral, with the neutral third (~11/9) as generator, the rastmic hexad is a chain of five neutral thirds rather than the six which give [[neutral7]], which therefore has two rastmic hexads and of course many more smaller rastmic chords.
+In the 2.3.11.13 subgroup, this scale is interpreted as an essentially tempered heptad, the '''rastgross heptad''', tempered by [[144/143]] (grossma) and 243/242. This heptad is 1-9/8-11/9-11/8-3/2-22/13-11/6 with steps of 9/8-12/11-9/8-12/11-9/8-12/11-12/11 (→[[rastgross1]]).
 [[Category:11-odd-limit]]