Pentacircle chords: Difference between revisions

Revision as of 15:20, 28 April 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.

The pentacircle triads are three in number:

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.

There are six pentacircle tetrads, including 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.

The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.

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.

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-A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[Just intonation subgroup|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]].
-The pentacircle [[Triad|triads]] are three in number:
+The pentacircle [[triad]]s are three in number:
+* 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.
-There are six pentacircle [[Tetrad|tetrads]]:
+There are six pentacircle [[tetrad]]s, including the palindromic
+* 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7;
-* 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.
-* the palindromic 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7
-* the [[inverse pair]] 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8 and 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7
-* the inverse pair 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8 and 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7
-Finally, there are two pentacircle [[Pentad|pentads]], inversely related:
+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.
-* 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8
+Finally, there are two pentacircle [[pentad]]s, inversely related:
-* 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8
+* 1-11/9-11/8-11/7-16/9 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.
 The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.
-[[EDO|Equal divisions of the octave]] with pentacircle chords include [[22edo|22]], [[27edo|27]], [[41edo|41]], [[46edo|46]], [[58edo|58]], [[68edo|68]], [[80edo|80]], [[87edo|87]], [[121edo|121]], [[145edo|145]], [[167edo|167]], [[208edo|208]], [[266edo|266e]] and [[433edo|433bce]].
+[[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]].
 [[Category:11-odd-limit]]
 [[Category:Essentially tempered chords]]
+[[Category:Triad]]
+[[Category:Tetrad]]
+[[Category:Pentad]]
 [[Category:Pentacircle]]
-[[Category:Tetrad]]

Pentacircle chords: Difference between revisions

Revision as of 15:20, 28 April 2023

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