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, 11 tetrads, 8 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.

[[Category:17-odd-limit]]

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

* 1–12/11–7/6–14/11 with steps of 12/11, 17/16, 12/11, 11/7.

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 four 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;

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

* 1–17/16–7/6–17/12–3/2 with steps of 17/16, 12/11, 11/9, 18/17, 4/3.

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 and one palindromic tetrad as plurichords.

The palindromic tetrad is both otonal and utonal in the 17-odd-limit:

* 14:17:18:22 tetrad which has steps of 11/9, 18/17, 11/9, 14/11 or 17/14, 18/17, 17/14, 14/11.

The inversely related pairs of triads are any three of the components of the tetrad:

* 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 chords]]

[[Category:Essentially tempered chords]]

[[Category:Triads]]

[[Category:Tetrads]]

[[Category:Pentads]]

[[Category:Hexads]]

[[Category:Augustmic]]

[[Category:Plurichords]]

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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, 11 tetrads, 8 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.
-[[Category:17-odd-limit]]
+For tetrads, there are one palindromic chord and five pairs of chords in inverse relationship. The palindromic chord is
+* 1–12/11–7/6–14/11 with steps of 12/11, 17/16, 12/11, 11/7.
+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 four 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;
+* 1–18/17–9/7–24/17–3/2 with steps of 18/17, 11/9, 12/11, 17/16, 4/3, and its inverse
+* 1–17/16–7/6–17/12–3/2 with steps of 17/16, 12/11, 11/9, 18/17, 4/3.
+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 and one palindromic tetrad as plurichords.
+The palindromic tetrad is both otonal and utonal in the 17-odd-limit:
+* 14:17:18:22 tetrad which has steps of 11/9, 18/17, 11/9, 14/11 or 17/14, 18/17, 17/14, 14/11.
+The inversely related pairs of triads are any three of the components of the tetrad:
+* 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 chords]]
 [[Category:Essentially tempered chords]]
+[[Category:Triads]]
 [[Category:Tetrads]]
+[[Category:Pentads]]
+[[Category:Hexads]]
 [[Category:Augustmic]]
+[[Category:Plurichords]]