Ptolemismic chords: Difference between revisions

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'''Ptolemismic chords''' are [[essentially tempered chord]]s 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.

~~It can be extended to~~ tetrads as

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

* 1-11/8-5/3-11/6 with steps 11/8-6/5-11/10-12/11;

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* 1-11/10-11/8-5/3 with steps 11/10-5/4-6/5-6/5.

~~They can be extended to~~ pentads as

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

* 1-11/8-3/2-5/3-11/6 with steps 11/8-12/11-11/10-11/10-12/11;

* 1-9/8-5/4-3/2-9/5 with steps 9/8-11/10-6/5-6/5-11/10.

The inversely related pair of chords is

* 1-5/4-11/8-3/2-5/3 with steps 5/4-11/10-12/11-11/10-6/5, and its inverse

* 1-11/10-11/8-5/3-11/6 with steps 11/10-5/4-6/5-11/10-12/11.

Equal divisions with ptolemismic ~~triads~~ include [[19edo]], [[22edo]], [[37edo]], [[41edo]] and [[104edo]].

Equal divisions with ptolemismic chords include [[19edo]], [[22edo]], [[37edo]], [[41edo]] and [[104edo]].

[[Category:11-odd-limit]]

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 '''Ptolemismic chords''' are [[essentially tempered chord]]s 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.
-It can be extended to tetrads as
+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
 * 1-11/8-5/3-11/6 with steps 11/8-6/5-11/10-12/11;
@@ Line 10: / Line 16: @@
 * 1-11/10-11/8-5/3 with steps 11/10-5/4-6/5-6/5.
-They can be extended to pentads as
+For pentads, there are two palindromic chords and a pair of chords in inverse relationship. The palindromic chords are
+* 1-11/8-3/2-5/3-11/6 with steps 11/8-12/11-11/10-11/10-12/11;
+* 1-9/8-5/4-3/2-9/5 with steps 9/8-11/10-6/5-6/5-11/10.
+The inversely related pair of chords is
 * 1-5/4-11/8-3/2-5/3 with steps 5/4-11/10-12/11-11/10-6/5, and its inverse
 * 1-11/10-11/8-5/3-11/6 with steps 11/10-5/4-6/5-11/10-12/11.
-Equal divisions with ptolemismic triads include [[19edo]], [[22edo]], [[37edo]], [[41edo]] and [[104edo]].
+Equal divisions with ptolemismic chords include [[19edo]], [[22edo]], [[37edo]], [[41edo]] and [[104edo]].
 [[Category:11-odd-limit]]