two generator Tribonacci scales

GENERALIZING THE SYNTONIC COMMA:

In lieu of the fact that a simple way to find some n-fraction parallel
to adjacent fractions and seeding a Stern-Brocot Tree is proving to be
quite elusive, here's one fruitful way to generalize three-term
"Tribonacci" scales so as to have a built-in ordering rule.

If a three-term index can be seen as a two-term index where "c" and
"b" simply merge, then it would only make sense that you could also
fold that one-dimensional generator back onto itself thereby creating
a new two-dimensional two generator ordering rule where the syntonic
comma is generalized as c-b.

So if a three-term index is defined as [a,b,c], a two-stepsize
cardinality interpretation would [a,(b+c)].

Here's some examples using the "scaled by P" method.

[2, 2, 3], 7, 12, 22, ...

0 215 385 600 707 922 1093 1200
0 170 385 493 707 878 985 1200
0 215 322 537 707 815 1030 1200
0 107 322 493 600 815 985 1200
0 215 385 493 707 878 1093 1200
0 170 278 493 663 878 985 1200

The two generators would be 707.431 and 385.137 cents respectively,
and the faux syntonic comma 44.588 cents.

[3, 1, 3], 7, 11, 21, ...

0 113 340 453 680 794 1020 1200
0 227 340 567 680 907 1087 1200
0 113 340 453 680 860 973 1200
0 227 340 567 747 860 1087 1200
0 113 340 520 633 860 973 1200
0 227 406 520 747 860 1087 1200
0 180 293 520 633 860 973 1200

The generalized two-dimensional generators would be 340.105� and
113.368�, and the generalized two-dimensional comma would be 47.052�.

[1, 1, 3], 5, 9, 17, ...

0 280 559 781 1060 1200
0 280 501 781 920 1200
0 222 501 641 920 1200
0 280 419 699 978 1200
0 140 419 699 920 1200

The two-dimensional generators would be 279.559� and 780.662�, and the
two-dimensional comma would be 58.014�.

[1, 1, 5], 7, 13, 25, ...

0 95 286 477 628 819 1009 1200
0 191 381 533 723 914 1105 1200
0 191 342 533 723 914 1009 1200
0 151 342 533 723 819 1009 1200
0 191 381 572 667 858 1049 1200
0 191 381 477 667 858 1009 1200
0 191 286 477 667 819 1009 1200

The two-dimensional generators would be 1009.296� and 476.759�, and
the two-dimensional comma would be 39.575�.

--Dan Stearns