Ptolemismic chords: Difference between revisions
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'''Ptolemismic chords''' are [[essentially tempered chord]]s tempered by the ptolemisma, [[100/99]]. | '''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: | 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 | [[Equal temperament]]s with ptolemismic chords include {{EDOs| 19, 22, 37, 41, and 104 }}. | ||
[[Category:11-odd-limit]] | [[Category:11-odd-limit chords]] | ||
[[Category:Essentially tempered chords]] | [[Category:Essentially tempered chords]] | ||
[[Category:Triads]] | [[Category:Triads]] | ||
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.