Twosquare chords: Difference between revisions
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'''Twosquare chords''' are [[ | '''Twosquare chords''' are [[essentially tempered chord]]s tempered by the twosquare comma, [[1089/1088]]. | ||
Twosquare chords are of [[ | Twosquare chords are of [[dyadic chord/Pattern of essentially tempered chords|pattern 1b]] in the 2.3.11.17 [[subgroup]] [[17-odd-limit]], meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures. | ||
For triads, there are one palindromic chord and a pair of chords in inverse relationship. The palindromic chord is | For triads, there are one palindromic chord and a pair of chords in inverse relationship. The palindromic chord is | ||
* | * 1–11/8–17/9 with steps of 11/8, 11/8, 18/17. | ||
The inversely related pair of chords is | The inversely related pair of chords is | ||
* | * 1–11/8–17/11 with steps of 11/8, 9/8, 22/17, and its inverse | ||
* | * 1–9/8–17/11 with steps of 9/8, 11/8, 22/17. | ||
For tetrads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are | For tetrads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are | ||
* | * 1–9/8–11/8–17/11 with steps of 9/8, 11/9, 9/8, 22/17; | ||
* | * 1–9/8–16/11–17/11 with steps of 9/8, 22/17, 17/16, 22/17. | ||
The inversely related pairs of chords are | The inversely related pairs of chords are | ||
* | * 1–11/8–17/11–17/9 with steps of 11/8, 9/8, 11/9, 18/17, and its inverse | ||
* | * 1–11/9–11/8–17/9 with steps of 11/9, 9/8, 11/8, 18/17; | ||
* | * 1–11/8–16/9–17/9 with steps of 11/8, 22/17, 17/16, 18/17, and its inverse | ||
* | * 1–17/16–11/8–17/9 with steps of 17/16, 22/17, 11/8, 18/17. | ||
Then there is an inversely related pair of pentads: | Then there is an inversely related pair of pentads: | ||
* | * 1–11/9–11/8–16/9–17/9 with steps of 11/9, 9/8, 22/17, 17/16, 18/17, and its inverse | ||
* | * 1–17/16–11/8–17/11–17/9 with steps of 17/16, 22/17, 9/8, 11/9, 18/17. | ||
Equal temperaments with twosquare chords include {{Optimal ET sequence| 22, 24, 41, 46, 65, 70, 89, 94, 111, 135, 159, 183, 207 and 231 }}. | Equal temperaments with twosquare chords include {{Optimal ET sequence| 22, 24, 41, 46, 65, 70, 89, 94, 111, 135, 159, 183, 207 and 231 }}. | ||
[[Category:17-odd-limit]] | [[Category:17-odd-limit chords]] | ||
[[Category:Essentially tempered chords]] | [[Category:Essentially tempered chords]] | ||
[[Category:Triads]] | [[Category:Triads]] | ||
Latest revision as of 08:21, 3 December 2025
Twosquare chords are essentially tempered chords tempered by the twosquare comma, 1089/1088.
Twosquare chords are of pattern 1b in the 2.3.11.17 subgroup 17-odd-limit, meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.
For triads, there are one palindromic chord and a pair of chords in inverse relationship. The palindromic chord is
- 1–11/8–17/9 with steps of 11/8, 11/8, 18/17.
The inversely related pair of chords is
- 1–11/8–17/11 with steps of 11/8, 9/8, 22/17, and its inverse
- 1–9/8–17/11 with steps of 9/8, 11/8, 22/17.
For tetrads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
- 1–9/8–11/8–17/11 with steps of 9/8, 11/9, 9/8, 22/17;
- 1–9/8–16/11–17/11 with steps of 9/8, 22/17, 17/16, 22/17.
The inversely related pairs of chords are
- 1–11/8–17/11–17/9 with steps of 11/8, 9/8, 11/9, 18/17, and its inverse
- 1–11/9–11/8–17/9 with steps of 11/9, 9/8, 11/8, 18/17;
- 1–11/8–16/9–17/9 with steps of 11/8, 22/17, 17/16, 18/17, and its inverse
- 1–17/16–11/8–17/9 with steps of 17/16, 22/17, 11/8, 18/17.
Then there is an inversely related pair of pentads:
- 1–11/9–11/8–16/9–17/9 with steps of 11/9, 9/8, 22/17, 17/16, 18/17, and its inverse
- 1–17/16–11/8–17/11–17/9 with steps of 17/16, 22/17, 9/8, 11/9, 18/17.
Equal temperaments with twosquare chords include 22, 24, 41, 46, 65, 70, 89, 94, 111, 135, 159, 183, 207 and 231.