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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:17-odd-limit]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:Augustmic]]
[[Category:Augustmic]]

Revision as of 09:48, 19 August 2023

Augustmic chords are essentially tempered chords tempered by the augustma, 154/153.

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

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 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.
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