Marveltwin chords: Difference between revisions

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A '''marveltwin chord''' is a [[13-odd-limit]] [[essentially tempered dyadic chord]] tempered by the marveltwin comma, [[325/324]].
'''Marveltwin chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the marveltwin comma, [[325/324]].


One marveltwin triad consists of a stack of two [[6/5]] minor thirds followed by an [[18/13]], that is,
There are 6 triads, 18 tetrads, 14 pentads and 3 hexads as 2.3.5.13 subgroup [[13-odd-limit]] essentially tempered chords.
* 1-6/5-[[13/9]] with steps 6/5-6/5-18/13,


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.  
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]].


Another triad consists of a stack of two [[10/9]] whole tones followed by a [[16/13]], that is,
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.
* 1-10/9-16/13 with steps 10/9-10/9-[[13/8]],  


as well as those built by
Another palindromic triad consists of [[16/13]] found as a stack of two [[10/9]] whole tones, that is,  
* 1-[[5/4]]-18/13 with steps 5/4-10/9-13/9, and its inverse
* 1–10/9–16/13 with steps of 10/9, 10/9, [[13/8]].  
* 1-10/9-18/13 with steps 10/9-5/4-13/9.
* 1-[[5/3]]-[[24/13]], with steps 5/3-10/9-[[13/12]], and its inverse
* 1-10/9-24/13 with steps 10/9-5/3-13/12.  


Each of these has multiple ways to extend to tetrads, such as using the [[3/2|perfect fifth]] as an anchor:
We also have those built by
* 1-5/4-18/13-[[3/2]] with steps 5/4-10/9-13/12-[[4/3]], and its inverse
* 1–[[5/4]]–18/13 with steps of 5/4, 10/9, 13/9, and its inverse  
* 1-13/12-6/5-3/2 with steps 13/12-10/9-5/4-4/3.
* 1–10/9–18/13 with steps of 10/9, 5/4, 13/9;
* 1-3/2-5/3-24/13 with steps 3/2-10/9-10/9-13/12, and its inverse
* 1–[[5/3]]–[[24/13]] with steps of 5/3, 10/9, [[13/12]], and its inverse  
* 1-3/2-13/8-9/5 with steps 3/2-13/12-10/9-10/9.  
* 1–10/9–24/13 with steps of 10/9, 5/3, 13/12.  


A palindrome exists as
For tetrads, there are four palindromic chords and seven pairs of chords in inverse relationship. The palindromic chords are
* 1-3/2-5/3-9/5 with steps 3/2-10/9-13/12-10/9.  
* 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.


In the 15-odd-limit we may have
The inversely related pairs of chords are
* 1-6/5-13/9-[[26/15]] with steps 6/5-6/5-6/5-[[15/13]].  
* 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.


The marveltwin tonestack is a hexad where each interval belongs to the 13-odd-limit consonance set:
For pentads, there are two palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are
* 1-9/8-5/4-3/2-13/8-9/5, with steps 9/8-10/9-6/5-13/12-10/9-10/9, its inverse being
* 1–9/8–5/4–3/2–9/5 with steps of 9/8, 10/9, 6/5, 6/5, 10/9;
* 1-6/5-4/3-3/2-8/5-24/13, with steps 6/5-10/9-9/8-10/9-10/9-13/12.  
* 1–13/12–3/2–13/8–9/5 with steps of 13/12, 18/13, 13/12, 10/9, 10/9.


Equal temperaments with marveltwin triads include {{EDOs| 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 246, 299 and 333 }}, with 333edo giving the [[optimal patent val]].
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.


[[Category:13-odd-limit]]
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
* 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 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 chords]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:Hexads]]
[[Category:Marveltwin]]
[[Category:Marveltwin]]

Latest revision as of 14:05, 19 March 2025

Marveltwin chords are essentially tempered chords tempered by the marveltwin comma, 325/324.

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 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, 1–6/5–10/7 in starling, and might also be called a tridecimal diminished triad.

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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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 chords include 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 246, 299 and 333, with 333edo giving the optimal patent val.

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