Marveltwin chords: Difference between revisions
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There are 6 triads, 18 tetrads, 14 pentads and 3 hexads as 2.3.5.13 subgroup [[13-odd-limit]] essentially tempered chords. | There are 6 triads, 18 tetrads, 14 pentads and 3 hexads as 2.3.5.13 subgroup [[13-odd-limit]] essentially tempered chords. | ||
One marveltwin triad | One marveltwin triad is a palindrome since it identifies [[13/9]] by a stack of two [[6/5]] minor thirds, that is, | ||
* | * 1–6/5–13/9 with steps of 6/5, 6/5, [[18/13]]. | ||
It makes for an instructive comparison with the septimal diminished triad, | It makes for an instructive comparison with the septimal diminished triad, 1–6/5–[[10/7]] in [[starling]], and might also be called a tridecimal diminished triad. | ||
Another triad consists of a stack of two [[10/9]] whole tones | Another palindromic triad consists of [[16/13]] found as a stack of two [[10/9]] whole tones, that is, | ||
* | * 1–10/9–16/13 with steps of 10/9, 10/9, [[13/8]]. | ||
We also have those built by | |||
* | * 1–[[5/4]]–18/13 with steps of 5/4, 10/9, 13/9, and its inverse | ||
* | * 1–10/9–18/13 with steps of 10/9, 5/4, 13/9; | ||
* | * 1–[[5/3]]–[[24/13]] with steps of 5/3, 10/9, [[13/12]], and its inverse | ||
* | * 1–10/9–24/13 with steps of 10/9, 5/3, 13/12. | ||
For tetrads, there are four palindromic chords and seven pairs of chords in inverse relationship. The palindromic chords are | For tetrads, there are four palindromic chords and seven pairs of chords in inverse relationship. The palindromic chords are | ||
* | * 1–10/9–18/13–20/13 with steps of 10/9, 5/4, 10/9, 13/10; | ||
* | * 1–10/9–5/4–18/13 with steps of 10/9, 9/8, 10/9, 13/9; | ||
* | * 1–10/9–6/5–4/3 with steps of 10/9, 13/12, 10/9, 3/2; | ||
* | * 1–13/12–6/5–13/10 with steps of 13/12, 10/9, 13/12, 20/13. | ||
The inversely related pairs of chords are | The inversely related pairs of chords are | ||
* | * 1–18/13–3/2–5/3 with steps of 18/13, 13/12, 10/9, 6/5, and its inverse | ||
* | * 1–13/12–3/2–9/5 with steps of 13/12, 18/13, 6/5, 10/9; | ||
* | * 1–5/4–3/2–9/5 with steps of 5/4, 6/5, 6/5, 10/9, and its inverse | ||
* | * 1–6/5–3/2–5/3 with steps of 6/5, 5/4, 10/9, 6/5; | ||
* | * 1–5/4–18/13–3/2 with steps of 5/4, 10/9, 13/12, 4/3, and its inverse | ||
* | * 1–13/12–6/5–3/2 with steps of 13/12, 10/9, 5/4, 4/3; | ||
* | * 1–10/9–16/13–4/3 with steps of 10/9, 10/9, 13/12, 3/2, and its inverse | ||
* | * 1–13/12–6/5–4/3 with steps of 13/12, 10/9, 10/9, 3/2; | ||
* | * 1–5/4–18/13–20/13 with steps of 5/4, 10/9, 10/9, 13/10, and its inverse | ||
* | * 1–10/9–16/13–20/13 with steps of 10/9, 10/9, 5/4, 13/10; | ||
* | * 1–6/5–13/10–13/9 with steps of 6/5, 13/12, 10/9, 18/13, and its inverse | ||
* | * 1–10/9–6/5–13/9 with steps of 10/9, 13/12, 6/5, 18/13; | ||
* | * 1–9/8–5/4–18/13 with steps of 9/8, 10/9, 10/9, 13/9, and its inverse | ||
* | * 1–10/9–16/13–18/13 with steps of 10/9, 10/9, 9/8, 13/9. | ||
For pentads, there are two palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are | For pentads, there are two palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are | ||
* | * 1–9/8–5/4–3/2–9/5 with steps of 9/8, 10/9, 6/5, 6/5, 10/9; | ||
* | * 1–13/12–3/2–13/8–9/5 with steps of 13/12, 18/13, 13/12, 10/9, 10/9. | ||
The inversely related pairs of chords are | The inversely related pairs of chords are | ||
* | * 1–18/13–3/2–5/3–9/5 with steps of 18/13, 13/12, 10/9, 13/12, 10/9, and its inverse | ||
* | * 1–13/12–3/2–5/3–9/5 with steps of 13/12, 18/13, 10/9, 13/12, 10/9; | ||
* | * 1–5/4–3/2–5/3–9/5 with steps of 5/4, 6/5, 10/9, 13/12, 10/9, and its inverse | ||
* | * 1–6/5–3/2–5/3–9/5 with steps of 6/5, 5/4, 10/9, 13/12, 10/9; | ||
* | * 1–5/4–3/2–13/8–9/5 with steps of 5/4, 6/5, 13/12, 10/9, 10/9, and its inverse | ||
* | * 1–6/5–3/2–5/3–24/13 with steps of 6/5, 5/4, 10/9, 10/9, 13/12; | ||
* | * 1–5/4–18/13–3/2–9/5 with steps of 5/4, 10/9, 13/12, 6/5, 10/9, and its inverse | ||
* | * 1–13/12–6/5–3/2–5/3 with steps of 13/12, 10/9, 5/4, 10/9, 6/5; | ||
* | * 1–9/8–5/4–18/13–3/2 with steps of 9/8, 10/9, 10/9, 13/12, 4/3, and its inverse | ||
* | * 1–13/12–6/5–4/3–3/2 with steps of 13/12, 10/9, 10/9, 9/8, 4/3; | ||
* | * 1–9/8–5/4–13/8–9/5 with steps of 9/8, 10/9, 13/10, 10/9, 10/9, and its inverse | ||
* | * 1–9/8–5/4–18/13–9/5 with steps of 9/8, 10/9, 10/9, 13/10, 10/9. | ||
For hexads, there are one palindromic chord and a pair of chords in inverse relationship. The palindromic chord is | For hexads, there are one palindromic chord and a pair of chords in inverse relationship. The palindromic chord is | ||
* | * 1–5/4–18/13–3/2–5/3–9/5 with steps of 5/4, 10/9, 13/12, 10/9, 13/12, 10/9. | ||
The inversely related pair of chords is | The inversely related pair of chords is | ||
* | * 1–6/5–4/3–3/2–5/3–24/13 with steps of 6/5, 10/9, 9/8, 10/9, 10/9, 13/12, and its inverse | ||
* | * 1–9/8–5/4–3/2–13/8–9/5 with steps of 9/8, 10/9, 6/5, 13/12, 10/9, 10/9. | ||
Equal temperaments with marveltwin | Equal temperaments with marveltwin chords include {{Optimal ET sequence| 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 246, 299 and 333 }}, with 333edo giving the [[optimal patent val]]. | ||
[[Category:13-odd-limit]] | [[Category:13-odd-limit chords]] | ||
[[Category:Essentially tempered chords]] | [[Category:Essentially tempered chords]] | ||
[[Category:Triads]] | [[Category:Triads]] | ||