Swetismic chords: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
A '''swetismic chord''' is an [[essentially tempered chord]] tempered by the swetisma, [[540/539]].
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>2011-12-17 17:22:12 UTC</tt>.<br>
: The original revision id was <tt>287093716</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 //swetismic chord// is a swetismic (540/539) 11-odd-limit [[dyadic chord|essentially tempered chord]].  


There are six swetismic triads, consisting of three pairs of chords in inverse relationship: 1-7/5-12/7 with steps 7/5-11/9-7/6 and 1-7/6-10/7 with steps 7/6-11/9-7/5; 1-7/6-9/7 with steps 7/6-11/10-14/9 and 1-14/9-12/7 with steps 14/9-11/10-7/6; and 1-9/7-7/5 with steps 9/7-12/11-10/7 and 1-10/7-14/9 with steps 10/7-12/11-9/7.
Swetismic 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.  


There are fifteen swetismic tetrads, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. Then palindromic tetrads are 1-7/6-9/7-3/2 with steps 7/6-11/10-7/6-4/3; 1-6/5-7/5-12/7 with steps 6/5-7/6-11/9-7/6 and 1-9/7-7/5-9/5 with steps 9/7-12/11-9/7-10/9. The six pairs are 1-9/7-3/2-11/6 with steps 9/7-7/6-11/9-12/11 and 1-7/6-3/2-18/11 with steps 7/6-9/7-12/11-11/9; 1-7/6-9/7-10/7 with steps 7/6-11/10-10/9-7/5 and 1-7/5-14/9-12/7 with steps 7/5-10/9-11/10-7/6; 1-7/6-9/7-18/11 with steps 7/6-11/10-14/11-11/9 and 1-14/11-7/5-18/11 with steps 14/11-11/10-7/6-11/9; 1-7/6-10/7-11/6 with steps 7/6-11/9-9/7-12/11 and 1-9/7-11/7-11/6 with steps 9/7-11/9-7/6-12/11; 1-7/6-9/7-11/6 with steps 7/6-11/10-10/7-12/11 and 1-10/7-11/7-11/6 with steps 10/7-11/10-7/6-12/11; and 1-10/7-14/9-12/7 with steps 10/7-12/11-11/10-7/6 and 1-7/6-9/7-7/5 with steps 7/6-11/10-12/11-10/7.
There are six swetismic triads, consisting of three pairs of chords in inverse relationship:
* 1–7/5–12/7 with steps 7/5, 11/9, 7/6, and its inverse
* 1–7/6–10/7 with steps 7/6, 11/9, 7/5;
* 1–7/6–9/7 with steps 7/6, 11/10, 14/9, and its inverse
* 1–14/9–12/7 with steps 14/9, 11/10, 7/6;
* 1–9/7–7/5 with steps 9/7, 12/11, 10/7, and its inverse
* 1–10/7–14/9 with steps 10/7, 12/11, 9/7.


Finally, there are six swetismic pentads coming in three pairs: 1-7/6-9/7-3/2-11/6 with steps 7/6-11/10-7/6-11/9-12/11 and 1-7/6-9/7-3/2-18/11 with steps 7/6-11/10-7/6-12/11-11/9; 1-7/6-10/7-5/3-11/6 with steps 7/6-11/9-7/6-11/10-12/11 and 1-7/6-10/7-5/3-20/11 with steps 7/6-11/9-7/6-12/11-11/10; and 1-7/6-9/7-10/7-11/6 with steps 7/6-11/10-10/9-9/7-12/11 and 1-9/7-10/7-11/7-11/6 with steps 9/7-10/9-11/10-7/6-12/11.
There are fifteen swetismic tetrads, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. The palindromic tetrads are:


The 11 limit swetismic chords number triads: 6, tetrads: 15, pentads: 6 for a total of 27.
* 1–7/6–9/7–3/2 with steps 7/6, 11/10, 7/6, 4/3;
* 1–6/5–7/5–12/7 with steps 6/5, 7/6, 11/9, 7/6;
* 1–9/7–7/5–9/5 with steps 9/7, 12/11, 9/7, 10/9.


If we are willing to consider the 15-limit, there are also 15-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.
The six pairs are:


Equal temperaments with swetismic chords include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de.</pre></div>
* 1–9/7–3/2–11/6 with steps 9/7, 7/6, 11/9, 12/11, and its inverse
<h4>Original HTML content:</h4>
* 1–7/6–3/2–18/11 with steps 7/6, 9/7, 12/11, 11/9;
<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;swetismic chords&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;swetismic chord&lt;/em&gt; is a swetismic (540/539) 11-odd-limit &lt;a class="wiki_link" href="/dyadic%20chord"&gt;essentially tempered chord&lt;/a&gt;. &lt;br /&gt;
* 1–7/6–9/7–10/7 with steps 7/6, 11/10, 10/9, 7/5, and its inverse
&lt;br /&gt;
* 1–7/5–14/9–12/7 with steps 7/5, 10/9, 11/10, 7/6;
There are six swetismic triads, consisting of three pairs of chords in inverse relationship: 1-7/5-12/7 with steps 7/5-11/9-7/6 and 1-7/6-10/7 with steps 7/6-11/9-7/5; 1-7/6-9/7 with steps 7/6-11/10-14/9 and 1-14/9-12/7 with steps 14/9-11/10-7/6; and 1-9/7-7/5 with steps 9/7-12/11-10/7 and 1-10/7-14/9 with steps 10/7-12/11-9/7.&lt;br /&gt;
* 1–7/6–9/7–18/11 with steps 7/6, 11/10, 14/11, 11/9, and its inverse
&lt;br /&gt;
* 1–14/11–7/5–18/11 with steps 14/11, 11/10, 7/6, 11/9;
There are fifteen swetismic tetrads, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. Then palindromic tetrads are 1-7/6-9/7-3/2 with steps 7/6-11/10-7/6-4/3; 1-6/5-7/5-12/7 with steps 6/5-7/6-11/9-7/6 and 1-9/7-7/5-9/5 with steps 9/7-12/11-9/7-10/9. The six pairs are 1-9/7-3/2-11/6 with steps 9/7-7/6-11/9-12/11 and 1-7/6-3/2-18/11 with steps 7/6-9/7-12/11-11/9; 1-7/6-9/7-10/7 with steps 7/6-11/10-10/9-7/5 and 1-7/5-14/9-12/7 with steps 7/5-10/9-11/10-7/6; 1-7/6-9/7-18/11 with steps 7/6-11/10-14/11-11/9 and 1-14/11-7/5-18/11 with steps 14/11-11/10-7/6-11/9; 1-7/6-10/7-11/6 with steps 7/6-11/9-9/7-12/11 and 1-9/7-11/7-11/6 with steps 9/7-11/9-7/6-12/11; 1-7/6-9/7-11/6 with steps 7/6-11/10-10/7-12/11 and 1-10/7-11/7-11/6 with steps 10/7-11/10-7/6-12/11; and 1-10/7-14/9-12/7 with steps 10/7-12/11-11/10-7/6 and 1-7/6-9/7-7/5 with steps 7/6-11/10-12/11-10/7.&lt;br /&gt;
* 1–7/6–10/7–11/6 with steps 7/6, 11/9, 9/7, 12/11, and its inverse
&lt;br /&gt;
* 1–9/7–11/7–11/6 with steps 9/7, 11/9, 7/6, 12/11;
Finally, there are six swetismic pentads coming in three pairs: 1-7/6-9/7-3/2-11/6 with steps 7/6-11/10-7/6-11/9-12/11 and 1-7/6-9/7-3/2-18/11 with steps 7/6-11/10-7/6-12/11-11/9; 1-7/6-10/7-5/3-11/6 with steps 7/6-11/9-7/6-11/10-12/11 and 1-7/6-10/7-5/3-20/11 with steps 7/6-11/9-7/6-12/11-11/10; and 1-7/6-9/7-10/7-11/6 with steps 7/6-11/10-10/9-9/7-12/11 and 1-9/7-10/7-11/7-11/6 with steps 9/7-10/9-11/10-7/6-12/11.&lt;br /&gt;
* 1–7/6–9/7–11/6 with steps 7/6, 11/10, 10/7, 12/11, and its inverse
&lt;br /&gt;
* 1–10/7–11/7–11/6 with steps 10/7, 11/10, 7/6, 12/11;  
The 11 limit swetismic chords number triads: 6, tetrads: 15, pentads: 6 for a total of 27.&lt;br /&gt;
* 1–10/7–14/9–12/7 with steps 10/7, 12/11, 11/10, 7/6, and its inverse
&lt;br /&gt;
* 1–7/6–9/7–7/5 with steps 7/6, 11/10, 12/11, 10/7.
If we are willing to consider the 15-limit, there are also 15-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.&lt;br /&gt;
 
&lt;br /&gt;
Finally, there are six swetismic pentads coming in three pairs:
Equal temperaments with swetismic chords include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
* 1–7/6–9/7–3/2–11/6 with steps 7/6, 11/10, 7/6, 11/9, 12/11, and its inverse
* 1–7/6–9/7–3/2–18/11 with steps 7/6, 11/10, 7/6, 12/11, 11/9;
* 1–7/6–10/7–5/3–11/6 with steps 7/6, 11/9, 7/6, 11/10, 12/11, and its inverse
* 1–7/6–10/7–5/3–20/11 with steps 7/6, 11/9, 7/6, 12/11, 11/10;  
* 1–7/6–9/7–10/7–11/6 with steps 7/6, 11/10, 10/9, 9/7, 12/11, and its inverse
* 1–9/7–10/7–11/7–11/6 with steps 9/7, 10/9, 11/10, 7/6, 12/11.
 
If we are willing to consider the [[15-odd-limit]], there are also 15-odd-limit swetismic tetrads, including something important for functional harmony. The ''swetismic dominant seventh chord'' is a tempering of  
* 1–9/7–3/2–7/4 with steps 9/7, 7/6, 7/6, 8/7.
 
Its inversion might be called the ''swetismic half-diminished chord'', a tempering of
* 1–7/6–3/2–12/7 with steps 7/6, 9/7, 8/7, 7/6.
 
We also have
* 1–11/9–10/7–5/3 with steps 11/9, 7/6, 7/6, 6/5; and its inverse
* 1–7/6–15/11–5/3 with steps 7/6, 7/6, 11/9, 6/5.
 
[[Equal temperament]]s with swetismic chords include {{EDOs| 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 205, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de }}.
 
[[Category:11-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Swetismic]]

Latest revision as of 13:40, 11 October 2024

A swetismic chord is an essentially tempered chord tempered by the swetisma, 540/539.

Swetismic 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.

There are six swetismic triads, consisting of three pairs of chords in inverse relationship:

  • 1–7/5–12/7 with steps 7/5, 11/9, 7/6, and its inverse
  • 1–7/6–10/7 with steps 7/6, 11/9, 7/5;
  • 1–7/6–9/7 with steps 7/6, 11/10, 14/9, and its inverse
  • 1–14/9–12/7 with steps 14/9, 11/10, 7/6;
  • 1–9/7–7/5 with steps 9/7, 12/11, 10/7, and its inverse
  • 1–10/7–14/9 with steps 10/7, 12/11, 9/7.

There are fifteen swetismic tetrads, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. The palindromic tetrads are:

  • 1–7/6–9/7–3/2 with steps 7/6, 11/10, 7/6, 4/3;
  • 1–6/5–7/5–12/7 with steps 6/5, 7/6, 11/9, 7/6;
  • 1–9/7–7/5–9/5 with steps 9/7, 12/11, 9/7, 10/9.

The six pairs are:

  • 1–9/7–3/2–11/6 with steps 9/7, 7/6, 11/9, 12/11, and its inverse
  • 1–7/6–3/2–18/11 with steps 7/6, 9/7, 12/11, 11/9;
  • 1–7/6–9/7–10/7 with steps 7/6, 11/10, 10/9, 7/5, and its inverse
  • 1–7/5–14/9–12/7 with steps 7/5, 10/9, 11/10, 7/6;
  • 1–7/6–9/7–18/11 with steps 7/6, 11/10, 14/11, 11/9, and its inverse
  • 1–14/11–7/5–18/11 with steps 14/11, 11/10, 7/6, 11/9;
  • 1–7/6–10/7–11/6 with steps 7/6, 11/9, 9/7, 12/11, and its inverse
  • 1–9/7–11/7–11/6 with steps 9/7, 11/9, 7/6, 12/11;
  • 1–7/6–9/7–11/6 with steps 7/6, 11/10, 10/7, 12/11, and its inverse
  • 1–10/7–11/7–11/6 with steps 10/7, 11/10, 7/6, 12/11;
  • 1–10/7–14/9–12/7 with steps 10/7, 12/11, 11/10, 7/6, and its inverse
  • 1–7/6–9/7–7/5 with steps 7/6, 11/10, 12/11, 10/7.

Finally, there are six swetismic pentads coming in three pairs:

  • 1–7/6–9/7–3/2–11/6 with steps 7/6, 11/10, 7/6, 11/9, 12/11, and its inverse
  • 1–7/6–9/7–3/2–18/11 with steps 7/6, 11/10, 7/6, 12/11, 11/9;
  • 1–7/6–10/7–5/3–11/6 with steps 7/6, 11/9, 7/6, 11/10, 12/11, and its inverse
  • 1–7/6–10/7–5/3–20/11 with steps 7/6, 11/9, 7/6, 12/11, 11/10;
  • 1–7/6–9/7–10/7–11/6 with steps 7/6, 11/10, 10/9, 9/7, 12/11, and its inverse
  • 1–9/7–10/7–11/7–11/6 with steps 9/7, 10/9, 11/10, 7/6, 12/11.

If we are willing to consider the 15-odd-limit, there are also 15-odd-limit swetismic tetrads, including something important for functional harmony. The swetismic dominant seventh chord is a tempering of

  • 1–9/7–3/2–7/4 with steps 9/7, 7/6, 7/6, 8/7.

Its inversion might be called the swetismic half-diminished chord, a tempering of

  • 1–7/6–3/2–12/7 with steps 7/6, 9/7, 8/7, 7/6.

We also have

  • 1–11/9–10/7–5/3 with steps 11/9, 7/6, 7/6, 6/5; and its inverse
  • 1–7/6–15/11–5/3 with steps 7/6, 7/6, 11/9, 6/5.

Equal temperaments with swetismic chords include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 205, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de.

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