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]].

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 consists of a stack of two [[6/5]] minor thirds followed by an [[18/13]], that is,

* 1-6/5-[[13/9]] with steps 6/5-6/5-18/13,

* 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 triad consists of a stack of two [[10/9]] whole tones followed by a [[16/13]], that is,

* 1-10/9-16/13 with steps 10/9-10/9-[[13/8]],

* 1-10/9-16/13 with steps of 10/9-10/9-[[13/8]],

as well as 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.

~~as well as those built by~~

The inversely related pairs of chords are

* 1-[[5/4]]-18/13 with steps 5/4-10/9-13/9, and its inverse

* 1-18/13-3/2-5/3 with steps of 18/13-13/12-10/9-6/5, and its inverse

* 1-10/9-18/13 with steps 10/9-5/4-13/9.

* 1-13/12-3/2-9/5 with steps of 13/12-18/13-6/5-10/9;

* 1-[[5/~~3]]~~-~~[[24~~/13]], with steps 5/3-10/9-[[13/~~12]]~~, and its inverse

* 1-5/4-3/2-9/5 with steps of 5/4-6/5-6/5-10/9, and its inverse

* 1-10/9-24/13 with steps 10/9-5/3-13/12.

* 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.

~~Each~~ of ~~these has multiple ways to extend to tetrads, such as using the [[3/2|perfect fifth]] as an anchor:~~

For pentads, there are two palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are

* 1-5/4-18/13-~~[[3~~/~~2]]~~ with steps 5/4-10/9-13/~~12-[[4/3]], and its inverse~~

* 1-9/8-5/4-3/2-9/5 with steps of 9/8-10/9-6/5-6/5-10/9;

* 1-13/12-6/5~~-3/2 with steps 13/12~~-10/9~~-5/4-4/3.~~

* 1-13/12-3/2-13/8-9/5 with steps of 13/12-18/13-13/12-10/9-10/9.

* 1~~-3/2-5/3-24/13 with steps 3/2-10/9-10/9~~-13/12~~, and its inverse~~

* 1-3/2-13/8-9/5 with steps 3/2-13/12-10/9-10/9.

~~A palindrome exists as~~

The inversely related pairs of chords are

* 1-3/2-5/3-9/5 with steps 3/2-10/9-13/12-10/9.

* 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.

~~In the 15-odd-limit we may have~~

For hexads, there are one palindromic chord and a pair of chords in inverse relationship. The palindromic chord is

* 1-6/5-13/9~~-[[26~~/~~15]]~~ with steps 6/5-6/5-6/5-~~[[15~~/13]].

* 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 ~~marveltwin tonestack~~ is ~~a hexad where each interval belongs to the 13-odd-limit consonance set:~~

The inversely related pair of chords is

* 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~~

* 1-6/5-4/3-3/2-5/3-24/13 with steps of 6/5-10/9-9/8-10/9-10/9-13/12, and its inverse

* 1-6/5-4/3-3/2-8/5~~-24/13,~~ with steps 6/5-10/9-9/8-10/9-10/9~~-13/12~~.

* 1-9/8-5/4-3/2-13/8-9/5 with steps of 9/8-10/9-6/5-13/12-10/9-10/9.

Equal temperaments with marveltwin 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]].

Equal temperaments with marveltwin triads 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]]

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[[Category:Triads]]

[[Category:Tetrads]]

[[Category:Pentads]]

[[Category:Hexads]]

[[Category:Marveltwin]]

@@ Line 1: / Line 1: @@
-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]].
+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 consists of a stack of two [[6/5]] minor thirds followed by an [[18/13]], that is,
-* 1-6/5-[[13/9]] with steps 6/5-6/5-18/13,
+* 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 triad consists of a stack of two [[10/9]] whole tones followed by a [[16/13]], that is,
-* 1-10/9-16/13 with steps 10/9-10/9-[[13/8]],
+* 1-10/9-16/13 with steps of 10/9-10/9-[[13/8]],
+as well as 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.
-as well as those built by
+The inversely related pairs of chords are
-* 1-[[5/4]]-18/13 with steps 5/4-10/9-13/9, and its inverse
+* 1-18/13-3/2-5/3 with steps of 18/13-13/12-10/9-6/5, and its inverse
-* 1-10/9-18/13 with steps 10/9-5/4-13/9.
+* 1-13/12-3/2-9/5 with steps of 13/12-18/13-6/5-10/9;
-* 1-[[5/3]]-[[24/13]], with steps 5/3-10/9-[[13/12]], and its inverse
+* 1-5/4-3/2-9/5 with steps of 5/4-6/5-6/5-10/9, and its inverse
-* 1-10/9-24/13 with steps 10/9-5/3-13/12.
+* 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.
-Each of these has multiple ways to extend to tetrads, such as using the [[3/2|perfect fifth]] as an anchor:
+For pentads, there are two palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are
-* 1-5/4-18/13-[[3/2]] with steps 5/4-10/9-13/12-[[4/3]], and its inverse
+* 1-9/8-5/4-3/2-9/5 with steps of 9/8-10/9-6/5-6/5-10/9;
-* 1-13/12-6/5-3/2 with steps 13/12-10/9-5/4-4/3.
+* 1-13/12-3/2-13/8-9/5 with steps of 13/12-18/13-13/12-10/9-10/9.
-* 1-3/2-5/3-24/13 with steps 3/2-10/9-10/9-13/12, and its inverse
-* 1-3/2-13/8-9/5 with steps 3/2-13/12-10/9-10/9.
-A palindrome exists as
+The inversely related pairs of chords are
-* 1-3/2-5/3-9/5 with steps 3/2-10/9-13/12-10/9.
+* 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.
-In the 15-odd-limit we may have
+For hexads, there are one palindromic chord and a pair of chords in inverse relationship. The palindromic chord is
-* 1-6/5-13/9-[[26/15]] with steps 6/5-6/5-6/5-[[15/13]].
+* 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 marveltwin tonestack is a hexad where each interval belongs to the 13-odd-limit consonance set:
+The inversely related pair of chords is
-* 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
+* 1-6/5-4/3-3/2-5/3-24/13 with steps of 6/5-10/9-9/8-10/9-10/9-13/12, and its inverse
-* 1-6/5-4/3-3/2-8/5-24/13, with steps 6/5-10/9-9/8-10/9-10/9-13/12.
+* 1-9/8-5/4-3/2-13/8-9/5 with steps of 9/8-10/9-6/5-13/12-10/9-10/9.
-Equal temperaments with marveltwin 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]].
+Equal temperaments with marveltwin triads 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]]
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 [[Category:Triads]]
 [[Category:Tetrads]]
+[[Category:Pentads]]
 [[Category:Hexads]]
 [[Category:Marveltwin]]