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


There are three ''rastmic triads'':  
There are three rastmic triads: the classic neutral triad,  
 
* 1–11/9–3/2 with steps of 11/9, 11/9, 4/3;
the classic neutral triad,  
* 1-11/9-3/2 with steps of 11/9-11/9-4/3,


and an inversely related pair of triads,  
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.
* 1–9/8–11/6 with steps of 9/8, 18/11, 12/11, and its inverse
* 1–9/8–11/9 with steps of 9/8, 12/11, 18/11.


''Rastmic tetrads'' are seven in number:  
Rastmic tetrads are seven in number: three palindromic tetrads,
 
* 1–11/9–3/2–11/6 with steps of 11/9, 11/9, 11/9, 12/11 (→ [[neutral tetrad]]);  
three palindromic tetrads,
* 1–3/2–18/11–11/6 with steps of 3/2, 12/11, 9/8, 12/11;  
* 1-11/9-3/2-11/6 with steps of 11/9-11/9-11/9-12/11 (→[[neutral tetrad]]);  
* 1–9/8–11/9–11/8 with steps of 9/8, 12/11, 9/8, 16/11;  
* 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,  
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;  
* 1–11/9–11/8–3/2 with steps of 11/9, 9/8, 12/11, 4/3, and its inverse
* 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.
* 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 its inverse
There are five ''rastmic pentads'':
* 1–9/8–3/2–11/6 with steps of 9/8, 4/3, 11/9, 12/11.


the palindromic pentad,  
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;  
* 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,  
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;  
* 1–9/8–11/8–3/2–11/6 with steps of 9/8, 11/9, 12/11, 11/9, 12/11, and its inverse
* 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.
* 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 its inverse
* 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'':  
There is also a unique 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.  
* 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}}.
[[Equal temperament]]s 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"
{| class="wikitable"
|-  
|-  
! chord by pitches
! Chord by Pitches
! chord by intervals
! Chord by Intervals
|-  
|-  
| 1 - 11/9 - 3/2
| 1–11/9–3/2
| 11/9 - 11/9 - 4/3
| 11/9, 11/9, 4/3
|-  
|-  
| 1 - 9/8 - 11/6
| 1–9/8–11/6
| 9/8 - 18/11 - 12/11
| 9/8, 18/11, 2/11
|-  
|-  
| 1 - 9/8 - 11/9
| 1–9/8–11/9
| 9/8 - 12/11 - 18/11
| 9/8, 12/11, 18/11
|-  
|-  
| 1 - 11/9 - 3/2 - 11/6
| 1–11/9–3/2–11/6
| 11/9 - 11/9 - 11/9 - 12/11
| 11/9, 11/9, 11/9, 12/11
|-  
|-  
| 1 - 3/2 - 18/11 - 11/6
| 1–3/2–18/11–11/6
| 3/2 - 12/11 - 9/8 - 12/11
| 3/2, 12/11, 9/8, 12/11
|-  
|-  
| 1 - 9/8 - 11/9 - 11/8
| 1–9/8–11/9–11/8
| 9/8 - 12/11 - 9/8 - 16/11
| 9/8, 12/11, 9/8, 16/11
|-  
|-  
| 1 - 11/9 - 11/8 - 3/2
| 1–11/9–11/8–3/2
| 11/9 - 9/8 - 12/11 - 4/3
| 11/9, 9/8, 12/11, 4/3
|-  
|-  
| 1 - 12/11 - 11/9 - 3/2
| 1–12/11–11/9–3/2
| 12/11 - 9/8 - 11/9 - 4/3
| 12/11, 9/8, 11/9, 4/3
|-  
|-  
| 1 - 9/8 - 11/9 - 3/2
| 1–9/8–11/9–3/2
| 9/8 - 12/11 - 11/9 - 4/3
| 9/8, 12/11, 11/9, 4/3
|-  
|-  
| 1 - 9/8 - 3/2 - 11/6
| 1–9/8–3/2–11/6
| 9/8 - 4/3 - 11/9 - 12/11
| 9/8, 4/3, 11/9, 12/11
|-  
|-  
| 1 - 9/8 - 11/9 - 3/2 - 11/6
| 1–9/8–11/9–3/2–11/6
| 9/8 - 12/11 - 11/9 - 11/9 - 12/11
| 9/8, 12/11, 11/9, 11/9, 12/11
|-  
|-  
| 1 - 9/8 - 11/8 - 3/2 - 11/6
| 1–9/8–11/8–3/2–11/6
| 9/8 - 11/9 - 12/11 - 11/9 - 12/11
| 9/8, 11/9, 12/11, 11/9, 12/11
|-  
|-  
| 1 - 11/9 - 11/8 - 3/2 - 11/6
| 1–11/9–11/8–3/2–11/6
| 11/9 - 9/8 - 12/11 - 11/9 - 12/11
| 11/9, 9/8, 12/11, 11/9, 12/11
|-  
|-  
| 1 - 9/8 - 11/9 - 11/8 - 3/2
| 1–9/8–11/9–11/8–3/2
| 9/8 - 12/11 - 9/8 - 12/11 - 4/3
| 9/8, 12/11, 9/8, 12/11, 4/3
|-  
|-  
| 1 - 9/8 - 3/2 - 18/11 - 11/6
| 1–9/8–3/2–18/11–11/6
| 9/8 - 4/3 - 12/11 - 9/8 - 12/11
| 9/8, 4/3, 12/11, 9/8, 12/11
|-  
|-  
| 1 - 9/8 - 11/9 - 11/8 - 3/2 - 11/6
| 1–9/8–11/9–11/8–3/2–11/6
| 9/8 - 12/11 - 9/8 - 12/11 - 11/9 - 12/11
| 9/8, 12/11, 9/8, 12/11, 11/9, 12/11
|}
|}


== Rastgross heptad ==
== 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.
Rastmic chords can be extended to the [[neutralization|neutralized]] [[5L 2s|diatonic]] scale, [[3L 4s|LsLsLss]], which is [[neutral7|Neutral[7]]]. In [[rastmic clan #Neutral|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 Neutral[7], 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]]).
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]]
[[Category:11-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Triads]]

Revision as of 09:34, 10 October 2024

Rastmic chords are 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 of 11/9, 11/9, 4/3;

and an inversely related pair of triads,

  • 1–9/8–11/6 with steps of 9/8, 18/11, 12/11, and its inverse
  • 1–9/8–11/9 with steps of 9/8, 12/11, 18/11.

Rastmic tetrads are seven in number: 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 its inverse
  • 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 its inverse
  • 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 its inverse
  • 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 its inverse
  • 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 unique 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 10, 17, 24, 31, 41, 58, 72, 130, 202, 736be, 938be, 1075be, 1116be, 1277be and 1318be.

Chord by Pitches Chord by Intervals
1–11/9–3/2 11/9, 11/9, 4/3
1–9/8–11/6 9/8, 18/11, 2/11
1–9/8–11/9 9/8, 12/11, 18/11
1–11/9–3/2–11/6 11/9, 11/9, 11/9, 12/11
1–3/2–18/11–11/6 3/2, 12/11, 9/8, 12/11
1–9/8–11/9–11/8 9/8, 12/11, 9/8, 16/11
1–11/9–11/8–3/2 11/9, 9/8, 12/11, 4/3
1–12/11–11/9–3/2 12/11, 9/8, 11/9, 4/3
1–9/8–11/9–3/2 9/8, 12/11, 11/9, 4/3
1–9/8–3/2–11/6 9/8, 4/3, 11/9, 12/11
1–9/8–11/9–3/2–11/6 9/8, 12/11, 11/9, 11/9, 12/11
1–9/8–11/8–3/2–11/6 9/8, 11/9, 12/11, 11/9, 12/11
1–11/9–11/8–3/2–11/6 11/9, 9/8, 12/11, 11/9, 12/11
1–9/8–11/9–11/8–3/2 9/8, 12/11, 9/8, 12/11, 4/3
1–9/8–3/2–18/11–11/6 9/8, 4/3, 12/11, 9/8, 12/11
1–9/8–11/9–11/8–3/2–11/6 9/8, 12/11, 9/8, 12/11, 11/9, 12/11

Rastgross heptad

Rastmic chords can be extended to the neutralized diatonic scale, LsLsLss, which is 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 Neutral[7], 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).

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