Heptacircle chords: Difference between revisions
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'''Heptacircle chords''' are [[ | '''Heptacircle chords''' are [[dyadic chord|essentially tempered chords]] tempered by the heptacircle comma, [[2431/2430]]. | ||
Heptacircle chords are of [[ | Heptacircle chords are of [[dyadic chord/Pattern of essentially tempered chords|pattern 2]] in the 2.3.5.11.13.17 [[subgroup]] [[17-odd-limit]], meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures. | ||
For triads, there are three pairs of chords in inverse relationship: | For triads, there are three pairs of chords in inverse relationship: | ||
* | * 1–11/9–18/13 with steps of 11/9, 17/15, 13/9, and its inverse | ||
* | * 1–13/9–18/11 with steps of 13/9, 17/15, 11/9; | ||
* | * 1–15/11–13/9 with steps of 15/11, 18/17, 18/13, and its inverse | ||
* | * 1–18/17–13/9 with steps of 18/17, 15/11, 18/13; | ||
* | * 1–15/13–11/9 with steps of 15/13, 18/17, 18/11, and its inverse | ||
* | * 1–18/17–11/9 with steps of 18/17, 15/13, 18/11. | ||
For tetrads, there are three palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are | For tetrads, there are three palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are | ||
* | * 1–11/9–18/13–22/13 with steps of 11/9, 17/15, 11/9, 13/11; | ||
* | * 1–15/11–13/9–17/9 with steps of 15/11, 18/17, 17/13, 18/17; | ||
* | * 1–15/13–11/9–17/9 with steps of 15/13, 18/17, 17/11, 18/17. | ||
The inversely related pairs of chords are | The inversely related pairs of chords are | ||
* | * 1–11/9–18/13–17/9 with steps of 11/9, 17/15, 15/11, 18/17, and its inverse | ||
* | * 1–15/11–17/11–17/9 with steps of 15/11, 17/15, 11/9, 18/17; | ||
* | * 1–15/11–5/3–17/9 with steps of 15/11, 11/9, 17/15, 18/17, and its inverse | ||
* | * 1–17/15–18/13–17/9 with steps of 17/15, 11/9, 15/11, 18/17; | ||
* | * 1–18/13–18/11–17/9 with steps of 18/13, 13/11, 15/13, 18/17, and its inverse | ||
* | * 1–15/13–15/11–17/9 with steps of 15/13, 13/11, 18/13, 18/17; | ||
* | * 1–15/13–18/13–17/9 with steps of 15/13, 6/5, 15/11, 18/17, and its inverse | ||
* | * 1–15/11–18/11–17/9 with steps of 15/11, 6/5, 15/13, 18/17; | ||
* | * 1–17/15–18/11–17/9 with steps of 17/15, 13/9, 15/13, 18/17, and its inverse | ||
* | * 1–15/13–5/3–17/9 with steps of 15/13, 13/9, 17/15, 18/17; | ||
* | * 1–13/9–18/11–17/9 with steps of 13/9, 17/15, 15/13, 18/17, and its inverse | ||
* | * 1–15/13–17/13–17/9 with steps of 15/13, 17/15, 13/9, 18/17. | ||
For pentads, there are three pairs of chords in inverse relationship: | For pentads, there are three pairs of chords in inverse relationship: | ||
* | * 1–15/13–15/11–5/3–17/9 with steps of 15/13, 13/11, 11/9, 17/15, 18/17, and its inverse | ||
* | * 1–17/15–18/13–18/11–17/9 with steps of 17/15, 11/9, 13/11, 15/13, 18/17; | ||
* | * 1–15/11–13/9–18/13–17/9 with steps of 15/11, 18/17, 17/15, 15/13, 18/17, and its inverse | ||
* | * 1–15/13–17/13–18/13–17/9 with steps of 15/13, 17/15, 18/17, 15/11, 18/17; | ||
* | * 1–15/13–11/9–18/13–17/9 with steps of 15/13, 18/17, 17/15, 15/11, 18/17, and its inverse | ||
* | * 1–15/11–17/11–18/11–17/9 with steps of 15/11, 17/15, 18/17, 15/13, 18/17. | ||
Equal temperaments with heptacircle chords include {{Optimal ET sequence|24, 34, 53, 58, 72, 87, 94, 111, 159, 183, 217, 270, 277, 311, 494, 581, 605, 677 and 764 }}. | Equal temperaments with heptacircle chords include {{Optimal ET sequence| 24, 34, 53, 58, 72, 87, 94, 111, 159, 183, 217, 270, 277, 311, 494, 581, 605, 677 and 764 }}. | ||
[[Category:17-odd-limit]] | [[Category:17-odd-limit chords]] | ||
[[Category:Essentially tempered chords]] | [[Category:Essentially tempered chords]] | ||
[[Category:Triads]] | [[Category:Triads]] | ||