Augustmic chords: Difference between revisions

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An '''~~augustmic chord~~''' ~~is an~~ [[essentially tempered ~~chord~~]] tempered by the augustma, [[154/153]].

'''Augustmic chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the augustma, [[154/153]].

~~The augustmic temperament has a palindromic tetrad of~~ 1~~-11/9-9/7-11/7 with steps 11/9-18/17-11/9-14/11~~ as ~~an essentially tempered chord, despite that any three of the components are essentially just~~. ~~In this temperament,~~ 7~~:9:~~11 ~~triad has steps of 22/~~17~~-17/14-14/11 or 14:~~17~~:22 triad has steps of 11/9~~-~~9/7~~-~~14/11, so that 7:9:11 and 14:17:22 have an inverse relationship~~.

There are 2 triads, 12 tetrads, 10 pentads and 1 hexad as 2.3.7.11.17 subgroup [[17-odd-limit]] essentially tempered chords.

Equal temperaments with augustmic chords include {{Optimal ET sequence|~~5, 7, 8d, 9,~~ 10, 14, 17g, 22, 24, 27eg, 31, 32, 41 and 46 }}.

For triads, there is a pair of chords in inverse relationship:

* 1-12/11-7/6 with steps of 12/11-17/16-12/7 and its inverse 1-17/16-7/6 with steps of 17/16-12/11-12/7.

For tetrads, there are two palindromic chords and five pairs of chords in inverse relationship. The palindromic chords are

* 1-12/11-7/6-14/11 with steps of 12/11-17/16-12/11-11/7;

* 1-11/9-9/7-11/7 with steps of 11/9-18/17-11/9-14/11.

The inversely related pairs of chords are

* 1-17/16-3/2-11/6 with steps of 17/16-24/17-11/9-12/11 and its inverse 1-24/17-3/2-18/11 with steps of 24/17-17/16-12/11-11/9;

* 1-9/7-3/2-11/6 with steps of 9/7-7/6-11/9-12/11 and its inverse 1-7/6-3/2-18/11 with steps of 7/6-9/7-12/11-11/9;

* 1-9/7-24/17-3/2 with steps of 9/7-12/11-17/16-4/3 and its inverse 1-17/16-7/6-3/2 with steps of 17/16-12/11-9/7-4/3;

* 1-9/7-11/8-3/2 with steps of 9/7-17/16-12/11-4/3 and its inverse 1-12/11-7/6-3/2 with steps of 12/11-17/16-9/7-4/3.

* 1-3/2-12/7-11/6 with steps of 3/2-8/7-17/16-12/11 and its inverse 1-3/2-18/11-7/4 with steps of 3/2-12/11-17/16-8/7;

For pentads, there are five pairs of chords in inverse relationship:

* 1-9/7-11/7-12/7-11/6 with steps of 9/7-11/9-12/11-17/16-12/11 and its inverse 1-11/9-11/7-12/7-11/6 with steps of 11/9-9/7-12/11-17/16-12/11;

* 1-9/7-11/8-3/2-11/6 with steps of 9/7-17/16-12/11-11/9-12/11 and its inverse 1-12/11-7/6-3/2-18/11 with steps of 12/11-17/16-9/7-12/11-11/9;

* 1-9/7-3/2-12/7-11/6 with steps of 9/7-7/6-8/7-17/16-12/11 and its inverse 1-7/6-3/2-18/11-7/4 with steps of 7/6-9/7-12/11-17/16-8/7;

* 1-11/9-9/7-11/7-11/6 with steps of 11/9-18/17-11/9-7/6-12/11 and its inverse 1-11/9-9/7-11/7-12/7 with steps of 11/9-18/17-11/9-12/11-7/6;

* 1-18/17-9/7-24/17-3/2 with steps of 18/17-11/9-12/11-17/16-4/3 and its inverse 1-4/3-17/12-17/11-17/9 with steps of 4/3-17/16-12/11-11/9-18/17.

Finally, there is a palindromic hexad 1-11/9-9/7-11/7-12/7-11/6 with steps of 11/9-18/17-11/9-12/11-17/16-12/11.

Equal temperaments with augustmic chords include {{Optimal ET sequence| 10, 14, 17g, 22, 24, 27eg, 31, 32, 41 and 46 }}.

== Augustmic plurichords ==

The augustmic temperament has two inversely related pairs of triads as plurichords:

* 7:9:11 triad which has steps of 22/17-17/14-14/11 and its inverse 14:17:22 triad which has steps of 11/9-9/7-14/11.

* 14:17:18 triad which has steps of 11/9-18/17-17/11 and its inverse 17:18:22 triad which has steps of 18/17-17/14-14/9.

[[Category:17-odd-limit]]

[[Category:Essentially tempered chords]]

[[Category:Triads]]

[[Category:Tetrads]]

[[Category:Pentads]]

[[Category:Hexads]]

[[Category:Augustmic]]

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-An '''augustmic chord''' is an [[essentially tempered chord]] tempered by the augustma, [[154/153]].
+'''Augustmic chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the augustma, [[154/153]].
-The augustmic temperament has a palindromic tetrad of 1-11/9-9/7-11/7 with steps 11/9-18/17-11/9-14/11 as an essentially tempered chord, despite that any three of the components are essentially just. In this temperament, 7:9:11 triad has steps of 22/17-17/14-14/11 or 14:17:22 triad has steps of 11/9-9/7-14/11, so that 7:9:11 and 14:17:22 have an inverse relationship.
+There are 2 triads, 12 tetrads, 10 pentads and 1 hexad as 2.3.7.11.17 subgroup [[17-odd-limit]] essentially tempered chords.
-Equal temperaments with augustmic chords include {{Optimal ET sequence|5, 7, 8d, 9, 10, 14, 17g, 22, 24, 27eg, 31, 32, 41 and 46 }}.
+For triads, there is a pair of chords in inverse relationship:
+* 1-12/11-7/6 with steps of 12/11-17/16-12/7 and its inverse 1-17/16-7/6 with steps of 17/16-12/11-12/7.
+For tetrads, there are two palindromic chords and five pairs of chords in inverse relationship. The palindromic chords are
+* 1-12/11-7/6-14/11 with steps of 12/11-17/16-12/11-11/7;
+* 1-11/9-9/7-11/7 with steps of 11/9-18/17-11/9-14/11.
+The inversely related pairs of chords are
+* 1-17/16-3/2-11/6 with steps of 17/16-24/17-11/9-12/11 and its inverse 1-24/17-3/2-18/11 with steps of 24/17-17/16-12/11-11/9;
+* 1-9/7-3/2-11/6 with steps of 9/7-7/6-11/9-12/11 and its inverse 1-7/6-3/2-18/11 with steps of 7/6-9/7-12/11-11/9;
+* 1-9/7-24/17-3/2 with steps of 9/7-12/11-17/16-4/3 and its inverse 1-17/16-7/6-3/2 with steps of 17/16-12/11-9/7-4/3;
+* 1-9/7-11/8-3/2 with steps of 9/7-17/16-12/11-4/3 and its inverse 1-12/11-7/6-3/2 with steps of 12/11-17/16-9/7-4/3.
+* 1-3/2-12/7-11/6 with steps of 3/2-8/7-17/16-12/11 and its inverse 1-3/2-18/11-7/4 with steps of 3/2-12/11-17/16-8/7;
+For pentads, there are five pairs of chords in inverse relationship:
+* 1-9/7-11/7-12/7-11/6 with steps of 9/7-11/9-12/11-17/16-12/11 and its inverse 1-11/9-11/7-12/7-11/6 with steps of 11/9-9/7-12/11-17/16-12/11;
+* 1-9/7-11/8-3/2-11/6 with steps of 9/7-17/16-12/11-11/9-12/11 and its inverse 1-12/11-7/6-3/2-18/11 with steps of 12/11-17/16-9/7-12/11-11/9;
+* 1-9/7-3/2-12/7-11/6 with steps of 9/7-7/6-8/7-17/16-12/11 and its inverse 1-7/6-3/2-18/11-7/4 with steps of 7/6-9/7-12/11-17/16-8/7;
+* 1-11/9-9/7-11/7-11/6 with steps of 11/9-18/17-11/9-7/6-12/11 and its inverse 1-11/9-9/7-11/7-12/7 with steps of 11/9-18/17-11/9-12/11-7/6;
+* 1-18/17-9/7-24/17-3/2 with steps of 18/17-11/9-12/11-17/16-4/3 and its inverse 1-4/3-17/12-17/11-17/9 with steps of 4/3-17/16-12/11-11/9-18/17.
+Finally, there is a palindromic hexad 1-11/9-9/7-11/7-12/7-11/6 with steps of 11/9-18/17-11/9-12/11-17/16-12/11.
+Equal temperaments with augustmic chords include {{Optimal ET sequence| 10, 14, 17g, 22, 24, 27eg, 31, 32, 41 and 46 }}.
+== Augustmic plurichords ==
+The augustmic temperament has two inversely related pairs of triads as plurichords:
+* 7:9:11 triad which has steps of 22/17-17/14-14/11 and its inverse 14:17:22 triad which has steps of 11/9-9/7-14/11.
+* 14:17:18 triad which has steps of 11/9-18/17-17/11 and its inverse 17:18:22 triad which has steps of 18/17-17/14-14/9.
 [[Category:17-odd-limit]]
 [[Category:Essentially tempered chords]]
+[[Category:Triads]]
 [[Category:Tetrads]]
+[[Category:Pentads]]
+[[Category:Hexads]]
 [[Category:Augustmic]]