Werckismic chords: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
A '''werckismic chord''' is an [[essentially tempered dyadic chord]] tempered by the werckisma, [[441/440]].  
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>2011-07-31 16:41:55 UTC</tt>.<br>
Werckismic chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 2]] in the [[11-odd-limit]], meaning that there are 6 [[triad]]s, 15 [[tetrad]]s and 6 [[pentad]]s, for a total of 27 distinct chord structures.  
: The original revision id was <tt>243617377</tt>.<br>
 
: The revision comment was: <tt></tt><br>
The werckismic triads consist of three pairs of inversely related chords:  
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
* 1–11/9–7/5 with steps 11/9, 8/7, 10/7, and its inverse
<h4>Original Wikitext content:</h4>
* 1–11/9–7/4 with steps 11/9, 10/7, 8/7;
<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 //werckismic triad// is one of two 11-limit [[dyadic chord|essentially tempered dyadic chords]] in inverse relation, constructed from the intervals 8/7, 11/9 and 10/7 which make up an octave under werckismic (441/440) tempering. That is, tempered versions of either 1-8/7-7/5 or 1-11/9-7/5. The triads can be extended to the //werckismic tetrad//, 1-8/7-7/5-8/5, with steps of size 8/7-11/9-8/7-5/4. Equal temperaments with werckismic triads and tetrads include 31, 41, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289 and 320.</pre></div>
* 1–11/7–7/4 with steps 11/7, 10/9, 8/7, and its inverse
<h4>Original HTML content:</h4>
* 1–11/7–9/5 with steps 11/7, 8/7, 10/9;
<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;werckismic chords&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;werckismic triad&lt;/em&gt; is one of two 11-limit &lt;a class="wiki_link" href="/dyadic%20chord"&gt;essentially tempered dyadic chords&lt;/a&gt; in inverse relation, constructed from the intervals 8/7, 11/9 and 10/7 which make up an octave under werckismic (441/440) tempering. That is, tempered versions of either 1-8/7-7/5 or 1-11/9-7/5. The triads can be extended to the &lt;em&gt;werckismic tetrad&lt;/em&gt;, 1-8/7-7/5-8/5, with steps of size 8/7-11/9-8/7-5/4. Equal temperaments with werckismic triads and tetrads include 31, 41, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289 and 320.&lt;/body&gt;&lt;/html&gt;</pre></div>
* 1–7/5–11/7 with steps 7/5, 9/8, 14/11, and its inverse
* 1–14/11–10/7 with steps 14/11, 9/8, 7/5.
 
The werckismic tetrads consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are
* 1–5/4–10/7–7/4 with steps 5/4, 8/7, 11/9, 8/7;
* 1–9/8–10/7–11/7 with steps 9/8, 14/11, 11/10, 14/11;
* 1–14/11–7/5–16/9 with steps 14/11, 11/10, 14/11, 9/8.
 
The pairs of chords are:
* 1–11/9–11/7–7/4 with steps 11/9, 9/7, 10/9, 8/7, and its inverse
* 1–9/7–11/7–9/5 with steps 9/7, 11/9, 8/7, 10/9;
* 1–10/7–11/7–7/4 with steps 10/7, 11/10, 10/9, 8/7, and its inverse
* 1–8/7–14/11–7/5 with steps 8/7, 10/9, 11/10, 10/7;
* 1–9/8–10/7–7/4 with steps 9/8, 14/11, 11/9, 8/7, and its inverse
* 1–11/9–14/9–7/4 with steps 11/9, 14/11, 9/8, 8/7;
* 1–11/9–11/8–7/4 with steps 11/9, 9/8, 14/11, 8/7, and its inverse
* 1–14/11–10/7–7/4 with steps 14/11, 9/8, 11/9, 8/7;
* 1–9/8–5/4–10/7 with steps 9/8, 10/9, 8/7, 7/5, and its inverse
* 1–10/9–5/4–7/4 with steps 10/9, 9/8, 7/5, 8/7;
* 1–9/8–9/7–10/7 with steps 9/8, 8/7, 10/9, 7/5, and its inverse
* 1–9/8–11/7–7/4 with steps 9/8, 7/5, 10/9, 8/7.
 
The werckismic pentads consist of three pairs of chords in an inverse relationship:
* 1–9/8–5/4–10/7–7/4 with steps 9/8, 10/9, 8/7, 11/9, 8/7, and its inverse
* 1–11/9–7/5–14/9–7/4 with steps 11/9, 8/7, 10/9, 9/8, 8/7;
* 1–11/9–11/8–11/7–7/4 with steps 11/9, 9/8, 8/7, 10/9, 8/7, and its inverse
* 1–9/8–11/8–11/7–7/4 with steps 9/8, 11/9, 8/7, 10/9, 8/7;  
* 1–9/8–9/7–10/7–11/7 with steps 9/8, 8/7, 10/9, 11/10, 14/11, and its inverse
* 1–9/8–10/7–11/7–7/4 with steps 9/8, 14/11, 11/10, 10/9, 8/7.
 
[[Equal temperament]]s with werckismic chords include {{EDOs| 31, 41, 43, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289, and 320 }}, with [[320edo]] giving the [[optimal patent val]].
 
[[Category:11-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Werckismic]]

Latest revision as of 13:35, 11 October 2024

A werckismic chord is an essentially tempered dyadic chord tempered by the werckisma, 441/440.

Werckismic chords are of pattern 2 in the 11-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

The werckismic triads consist of three pairs of inversely related chords:

  • 1–11/9–7/5 with steps 11/9, 8/7, 10/7, and its inverse
  • 1–11/9–7/4 with steps 11/9, 10/7, 8/7;
  • 1–11/7–7/4 with steps 11/7, 10/9, 8/7, and its inverse
  • 1–11/7–9/5 with steps 11/7, 8/7, 10/9;
  • 1–7/5–11/7 with steps 7/5, 9/8, 14/11, and its inverse
  • 1–14/11–10/7 with steps 14/11, 9/8, 7/5.

The werckismic tetrads consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are

  • 1–5/4–10/7–7/4 with steps 5/4, 8/7, 11/9, 8/7;
  • 1–9/8–10/7–11/7 with steps 9/8, 14/11, 11/10, 14/11;
  • 1–14/11–7/5–16/9 with steps 14/11, 11/10, 14/11, 9/8.

The pairs of chords are:

  • 1–11/9–11/7–7/4 with steps 11/9, 9/7, 10/9, 8/7, and its inverse
  • 1–9/7–11/7–9/5 with steps 9/7, 11/9, 8/7, 10/9;
  • 1–10/7–11/7–7/4 with steps 10/7, 11/10, 10/9, 8/7, and its inverse
  • 1–8/7–14/11–7/5 with steps 8/7, 10/9, 11/10, 10/7;
  • 1–9/8–10/7–7/4 with steps 9/8, 14/11, 11/9, 8/7, and its inverse
  • 1–11/9–14/9–7/4 with steps 11/9, 14/11, 9/8, 8/7;
  • 1–11/9–11/8–7/4 with steps 11/9, 9/8, 14/11, 8/7, and its inverse
  • 1–14/11–10/7–7/4 with steps 14/11, 9/8, 11/9, 8/7;
  • 1–9/8–5/4–10/7 with steps 9/8, 10/9, 8/7, 7/5, and its inverse
  • 1–10/9–5/4–7/4 with steps 10/9, 9/8, 7/5, 8/7;
  • 1–9/8–9/7–10/7 with steps 9/8, 8/7, 10/9, 7/5, and its inverse
  • 1–9/8–11/7–7/4 with steps 9/8, 7/5, 10/9, 8/7.

The werckismic pentads consist of three pairs of chords in an inverse relationship:

  • 1–9/8–5/4–10/7–7/4 with steps 9/8, 10/9, 8/7, 11/9, 8/7, and its inverse
  • 1–11/9–7/5–14/9–7/4 with steps 11/9, 8/7, 10/9, 9/8, 8/7;
  • 1–11/9–11/8–11/7–7/4 with steps 11/9, 9/8, 8/7, 10/9, 8/7, and its inverse
  • 1–9/8–11/8–11/7–7/4 with steps 9/8, 11/9, 8/7, 10/9, 8/7;
  • 1–9/8–9/7–10/7–11/7 with steps 9/8, 8/7, 10/9, 11/10, 14/11, and its inverse
  • 1–9/8–10/7–11/7–7/4 with steps 9/8, 14/11, 11/10, 10/9, 8/7.

Equal temperaments with werckismic chords include 31, 41, 43, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289, and 320, with 320edo giving the optimal patent val.

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