Augustmic chords: Difference between revisions
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'''Augustmic chords''' are [[ | '''Augustmic chords''' are [[dyadic chord|essentially tempered chords]] tempered by the augustma, [[154/153]]. | ||
There are 2 triads, 11 tetrads, 8 pentads and 1 hexad as 2.3.7.11.17 subgroup [[17-odd-limit]] essentially tempered chords. | There are 2 triads, 11 tetrads, 8 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: | 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 one palindromic chord and five pairs of chords in inverse relationship. The palindromic chord is | 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 | 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: | 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, | 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 }}. | Equal temperaments with augustmic chords include {{Optimal ET sequence| 10, 14, 17g, 22, 24, 27eg, 31, 32, 41 and 46 }}. | ||
== Augustmic plurichords == | == Augustmic plurichords == | ||
The augustmic temperament has two inversely related pairs of triads and one palindromic tetrad as 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: | The palindromic tetrad is both otonal and utonal in the 17-odd-limit: | ||
* 14:17:18:22 tetrad which has steps of 11/9 | * 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: | 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 | * 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 | * 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 chords]] | ||
[[Category:Essentially tempered chords]] | [[Category:Essentially tempered chords]] | ||
[[Category:Triads]] | [[Category:Triads]] | ||