Pentacircle chords: Difference between revisions

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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]]~~

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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]]