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Rastmic chords: Difference between revisions

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Adjust the rotation of certain chords to better contrast their inverses
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
'''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]].
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-13 14:40:26 UTC</tt>.<br>
: The original revision id was <tt>285600748</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A //rastmic chord// is an [[Dyadic chord#Essentially%20tempered%20dyadic%20chords|essentially tempered dyadic chord]] tempered by the rastma, 243/242, in the 2.3.11 subgroup of the 11-limit. 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. 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. 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.


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 [[http://xenharmonic.wikispaces.com/Chromatic+pairs#Neutraltet|Neutraltet\[7\]]]. In neutraltet, with the neutral third (~11/9) as generator, the rastmic hexad is a chain of five neutral thirds rather than the six which give [[neutraltet7]].</pre></div>
The count of chords is 3 triads, 7 tetrads, 5 pentads, and 1 hexad, for a total of 16.  
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;rastmic chords&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;rastmic chord&lt;/em&gt; is an &lt;a class="wiki_link" href="/Dyadic%20chord#Essentially%20tempered%20dyadic%20chords"&gt;essentially tempered dyadic chord&lt;/a&gt; tempered by the rastma, 243/242, in the 2.3.11 subgroup of the 11-limit. 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. Rastmic tetrads are seven in number: the palindromic classic &lt;a class="wiki_link" href="/neutral%20tetrad"&gt;neutral tetrad&lt;/a&gt; 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. 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.&lt;br /&gt;
There are three rastmic triads: the classic neutral triad,
&lt;br /&gt;
* 1–11/9–3/2 with steps of 11/9, 11/9, 4/3;
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 &lt;a href="http://xenharmonic.wikispaces.com/Chromatic+pairs#Neutraltet"&gt;Neutraltet\[7\&lt;/a&gt;]. In neutraltet, with the neutral third (~11/9) as generator, the rastmic hexad is a chain of five neutral thirds rather than the six which give &lt;a class="wiki_link" href="/neutraltet7"&gt;neutraltet7&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
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
* 1–11/9–4/3–3/2 with steps of 11/9, 12/11, 9/8, 4/3.
 
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
* 1–12/11–4/3–3/2–18/11 with steps of 12/11, 11/9, 9/8, 12/11, 11/9;  
* 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
* 1–12/11–11/9–4/3–3/2 with steps of 12/11, 9/8, 12/11, 9/8, 4/3.
 
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 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"
|-
! 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
|-  
| 1–11/9–4/3–3/2
| 11/9, 12/11, 9/8, 4/3
|-  
| 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
|-
| 1–12/11–4/3–3/2–18/11
| 12/11, 11/9, 9/8, 12/11, 11/9
|-  
| 1–9/8–11/9–11/8–3/2
| 9/8, 12/11, 9/8, 12/11, 4/3
|-  
| 1–12/11–11/9–4/3–3/2
| 12/11, 9/8, 12/11, 9/8, 4/3
|-  
| 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 [[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]]).
 
[[Category:11-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:Rastmic]]
Retrieved from "https://en.xen.wiki/w/Rastmic_chords"