Ptolemismic chords: Difference between revisions

Latest revision as of 10:23, 10 October 2024

Ptolemismic chords are essentially tempered chords tempered by the ptolemisma, 100/99.

There are one triad, six tetrads and four pentads as 2.3.5.11 subgroup 11-odd-limit essentially tempered chords.

The ptolemismic triad consists of 11/8 and a stack of two 6/5's closing at the octave:

1–11/8–5/3 with steps of 11/8, 6/5, 6/5.

For tetrads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are

1–9/8–5/4–9/5 with steps 9/8, 11/10, 16/11, 11/10;
1–3/2–5/3–9/5 with steps 3/2, 11/10, 12/11, 11/10.

The inversely related pairs of chords are

1–11/8–3/2–5/3 with steps 11/8, 12/11, 11/10, 6/5, and its inverse
1–12/11–3/2–9/5 with steps 12/11, 11/8, 6/5, 11/10;
1–5/4–3/2–9/5 with steps 5/4, 6/5, 6/5, 11/10, and its inverse
1–6/5–3/2–5/3 with steps 6/5, 5/4, 11/10, 6/5.

For pentads, there are two palindromic chords and a pair of chords in inverse relationship. The palindromic chords are

1–9/8–5/4–3/2–9/5 with steps 9/8, 11/10, 6/5, 6/5, 11/10;
1–11/8–3/2–5/3–11/6 with steps 11/8, 12/11, 11/10, 11/10, 12/11.

The inversely related pair of chords is

1–5/4–3/2–5/3–9/5 with steps 5/4, 6/5, 11/10, 12/11, 11/10, and its inverse
1–6/5–3/2–5/3–9/5 with steps 6/5, 5/4, 11/10, 12/11, 11/10.

Equal temperaments with ptolemismic chords include 19, 22, 37, 41, and 104.

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+'''Ptolemismic chords''' are [[essentially tempered chord]]s tempered by the ptolemisma, [[100/99]].
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-10-19 18:10:09 UTC</tt>.<br>
+There are one triad, six tetrads and four pentads as 2.3.5.11 subgroup [[11-odd-limit]] essentially tempered chords.
-: The original revision id was <tt>266579256</tt>.<br>
-: The revision comment was: <tt></tt><br>
+The ptolemismic triad consists of [[11/8]] and a stack of two [[6/5]]'s closing at the octave:
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
+* 1–11/8–5/3 with steps of 11/8, 6/5, 6/5.
-<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 **ptolemismic triad** is an [[Dyadic chord|essentially tempered chord]] which is the [[100_99|100/99]] tempering of 1-[[11_8|11/8]]-[[5_3|5/3]], with steps of 11/8-[[6_5|6/5]]-6/5. Equal divisions with ptolemismic triads include [[19edo]], [[22edo]], [[37edo]], [[41edo]] and [[104edo]].</pre></div>
+For tetrads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
-<h4>Original HTML content:</h4>
+* 1–9/8–5/4–9/5 with steps 9/8, 11/10, 16/11, 11/10;
-<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;ptolemismic triad&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;strong&gt;ptolemismic triad&lt;/strong&gt; is an &lt;a class="wiki_link" href="/Dyadic%20chord"&gt;essentially tempered chord&lt;/a&gt; which is the &lt;a class="wiki_link" href="/100_99"&gt;100/99&lt;/a&gt; tempering of 1-&lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt;-&lt;a class="wiki_link" href="/5_3"&gt;5/3&lt;/a&gt;, with steps of 11/8-&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;-6/5. Equal divisions with ptolemismic triads include &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37edo&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt; and &lt;a class="wiki_link" href="/104edo"&gt;104edo&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
+* 1–3/2–5/3–9/5 with steps 3/2, 11/10, 12/11, 11/10.
+The inversely related pairs of chords are
+* 1–11/8–3/2–5/3 with steps 11/8, 12/11, 11/10, 6/5, and its inverse
+* 1–12/11–3/2–9/5 with steps 12/11, 11/8, 6/5, 11/10;
+* 1–5/4–3/2–9/5 with steps 5/4, 6/5, 6/5, 11/10, and its inverse
+* 1–6/5–3/2–5/3 with steps 6/5, 5/4, 11/10, 6/5.
+For pentads, there are two palindromic chords and a pair of chords in inverse relationship. The palindromic chords are
+* 1–9/8–5/4–3/2–9/5 with steps 9/8, 11/10, 6/5, 6/5, 11/10;
+* 1–11/8–3/2–5/3–11/6 with steps 11/8, 12/11, 11/10, 11/10, 12/11.
+The inversely related pair of chords is
+* 1–5/4–3/2–5/3–9/5 with steps 5/4, 6/5, 11/10, 12/11, 11/10, and its inverse
+* 1–6/5–3/2–5/3–9/5 with steps 6/5, 5/4, 11/10, 12/11, 11/10.
+[[Equal temperament]]s with ptolemismic chords include {{EDOs| 19, 22, 37, 41, and 104 }}.
+[[Category:11-odd-limit chords]]
+[[Category:Essentially tempered chords]]
+[[Category:Triads]]
+[[Category:Tetrads]]
+[[Category:Pentads]]
+[[Category:Ptolemismic]]

Ptolemismic chords: Difference between revisions

Latest revision as of 10:23, 10 October 2024

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