Swetismic chords: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 244440849 - Original comment: **
m Improve readability; recategorize
 
(17 intermediate revisions by 7 users not shown)
Line 1: Line 1:
<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>2011-08-05 01:13:10 UTC</tt>.<br>
: The original revision id was <tt>244440849</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">The 11-limit swetismic triad is either the 540/539-tempered version of a 7/6-11/9-7/5 chord or its inversion, an 11/9-7/6-7/5 chord. It is an 11-limit [[dyadic chord|essentially tempered triad]], and can also be characterized as the tempering of 1-7/6-10/7 or 1-11/9-10/7. It can be extended to the 11-limit swetismic tetrad, the 7/6-11/9-7/6-6/5 chord, the swetismic tempering of 1-7/6-10/7-5/3. A 15-limit swetismic otonal tetrad is 1-9/7-3/2-7/4, and a utonal tetrad 1-7/6-3/2-12/7, with steps 9/7-7/6-7/6-8/7 and 7/6-9/7-8/7-7/6.


Equal temperaments with swetismic tetrads include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 224, 354, 537 and 578.</pre></div>
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.
<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;swetismic chords&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 11-limit swetismic triad is either the 540/539-tempered version of a 7/6-11/9-7/5 chord or its inversion, an 11/9-7/6-7/5 chord. It is an 11-limit &lt;a class="wiki_link" href="/dyadic%20chord"&gt;essentially tempered triad&lt;/a&gt;, and can also be characterized as the tempering of 1-7/6-10/7 or 1-11/9-10/7. It can be extended to the 11-limit swetismic tetrad, the 7/6-11/9-7/6-6/5 chord, the swetismic tempering of 1-7/6-10/7-5/3. A 15-limit swetismic otonal tetrad is 1-9/7-3/2-7/4, and a utonal tetrad 1-7/6-3/2-12/7, with steps 9/7-7/6-7/6-8/7 and 7/6-9/7-8/7-7/6.&lt;br /&gt;
There are six swetismic triads, consisting of three pairs of chords in inverse relationship:
&lt;br /&gt;
* 1–7/5–12/7 with steps 7/5, 11/9, 7/6, and its inverse
Equal temperaments with swetismic tetrads include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 224, 354, 537 and 578.&lt;/body&gt;&lt;/html&gt;</pre></div>
* 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 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.

Retrieved from "https://en.xen.wiki/index.php?title=Swetismic_chords&oldid=159470"