Pentacircle chords: Difference between revisions
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A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[subgroup]] in the [[11-odd-limit]], [[tempering out]] the pentacircle comma, [[896/891]]. | A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[subgroup]] in the [[11-odd-limit]], [[tempering out]] the pentacircle comma, [[896/891]]. | ||
Pentacircle chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 1b]] in the 2.9.7.11 [[subgroup]] [[11-odd-limit]], meaning that there are 3 [[triad]]s, 6 [[tetrad]]s and 2 [[pentad]]s, for a total of 11 distinct chord structures. | |||
The pentacircle triads are | |||
* 1-9/7-16/9 with steps 9/7-11/8-9/8; | * 1-9/7-16/9 with steps 9/7-11/8-9/8; | ||
* 1-9/7-16/11 with steps 9/7-9/8-11/8; | * 1-9/7-16/11 with steps 9/7-9/8-11/8; | ||
* 1-11/7-16/9 with steps 11/7-9/8-9/8. | * 1-11/7-16/9 with steps 11/7-9/8-9/8. | ||
The tetrads include the palindromic | |||
* 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7; | * 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7; | ||
* 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7. | * 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7. | ||
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* 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7. | * 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7. | ||
Finally, there are two pentacircle | Finally, there are two pentacircle pentads, inversely related: | ||
* 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8, and its inverse | * 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8, and its inverse | ||
* 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8. | * 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8. | ||
[[EDO|Equal divisions of the octave]] with pentacircle chords include {{EDOs| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]]. | [[EDO|Equal divisions of the octave]] with pentacircle chords include {{EDOs| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]]. | ||
Revision as of 05:19, 27 July 2023
A pentacircle chord is an essentially tempered dyadic chord in the 2.9.7.11 subgroup in the 11-odd-limit, tempering out the pentacircle comma, 896/891.
Pentacircle chords are of pattern 1b in the 2.9.7.11 subgroup 11-odd-limit, meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.
The pentacircle triads are
- 1-9/7-16/9 with steps 9/7-11/8-9/8;
- 1-9/7-16/11 with steps 9/7-9/8-11/8;
- 1-11/7-16/9 with steps 11/7-9/8-9/8.
The tetrads include the palindromic
- 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7;
- 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7.
And the inversely related pairs
- 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8, and its inverse
- 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7;
- 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8, and its inverse
- 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7.
Finally, there are two pentacircle pentads, inversely related:
- 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8, and its inverse
- 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.
Equal divisions of the octave with pentacircle chords include 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208, with 208edo giving the optimal patent val.