Ptolemismic chords: Difference between revisions

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'''Ptolemismic chords''' are [[essentially tempered chord]]s tempered by the ptolemisma, [[100/99]].

[[Category:~~11-limit~~]]

[[Category:~~ptolemismic~~]]

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

[[Category:~~triad~~]]

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 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]]

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-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|19edo]], [[22edo|22edo]], [[37edo|37edo]], [[41edo|41edo]] and [[104edo|104edo]].
+'''Ptolemismic chords''' are [[essentially tempered chord]]s tempered by the ptolemisma, [[100/99]].
-[[Category:11-limit]]
-[[Category:ptolemismic]]
+There are one triad, six tetrads and four pentads as 2.3.5.11 subgroup [[11-odd-limit]] essentially tempered chords.
-[[Category:triad]]
+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 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]]