Wilsonian institute

Dan- do you remember when Wilson was accused of leaving cryptic papers
behind? Well... could you explain what you're doing here? Maybe just
a good clue...

>0 125 250 374 499 701 826 950 1075 1200
>
>
>The 2s7L/7L2s:
>
>1/2 4/7
> 5/9
> 6/11 9/16
>
>Golden Mediant = ~672.85
>LLsLLLsLL
>
>0 146 291 381 527 673 819 909 1054 1200
>0 146 236 381 527 673 763 909 1054 1200
>0 90 236 381 527 617 763 909 1054 1200
>0 146 291 437 527 673 819 964 1110 1200
>0 146 291 381 527 673 819 964 1054 1200
>0 146 236 381 527 673 819 909 1054 1200
>0 90 236 381 527 673 763 909 1054 1200
>0 146 291 437 583 673 819 964 1110 1200
>0 146 291 437 527 673 819 964 1054 1200
>0 146 291 381 527 673 819 909 1054 1200
>
>Golden Mediant = ~658.62
>ssLsssLss
>
>0 117 234 424 541 659 776 966 1083 1200
>0 117 307 424 541 659 848 966 1083 1200
>0 190 307 424 541 731 848 966 1083 1200
>0 117 234 352 541 659 776 893 1010 1200
>0 117 234 424 541 659 776 893 1083 1200
>0 117 307 424 541 659 776 966 1083 1200
>0 190 307 424 541 659 848 966 1083 1200
>0 117 234 352 469 659 776 893 1010 1200
>0 117 234 352 541 659 776 893 1083 1200
>0 117 234 424 541 659 776 966 1083 1200

-Carl

Carl Lumma wrote,

> could you explain what you're doing here?

Essentially I'm converting the x+y mediant of the Stern-Brocot tree
into a "Phi-weighted" MOS generator where x and y are adjacent
fractions. Looking at x and y as "L" and "s" makes this all a lot
easier to see when considering two-stepsize mappings. As an example of
this I'll use the 7-tone 3L4s/4L3s tree mapping:

3 4
7
10 11

This results in the following adjacent fractions of an octave (where
"adjacent" simply means that the difference between a
cross-multiplication of adjoining fractions on the tree is 1):

2/3 3/4
5/7
7/10 8/11

By phi-weighting the two initial fractions -- a/b, c/d and c/d, a/b --
you achieve a "special" proportional rendering of the two basic
two-stepsize mappings where L/s=phi.

The algorithm I use is a simple X = P/(n*d), where "P" = a given
periodicity, "n" = (b+phi*d), "d" = (a+phi*c), and "X" is the
corresponding phi-weighted MOS generator.

Here's the two 7-tone 3L4s/4L3s phi-weighted scales with the adjacent
fractions in their simplest a/b c/d form...

X = 845.1768�, (2/3 5/7) 3L4s:

0 136 355 490 710 845 1064 1200
0 219 355 574 710 929 1064 1200
0 136 355 490 710 845 981 1200
0 219 355 574 710 845 1064 1200
0 136 355 490 626 845 981 1200
0 219 355 490 710 845 1064 1200
0 136 271 490 626 845 981 1200

X = 868.3282�, (2/3 3/4) 4L3s:

0 205 332 537 663 868 995 1200
0 127 332 458 663 790 995 1200
0 205 332 537 663 868 1073 1200
0 127 332 458 663 868 995 1200
0 205 332 537 742 868 1073 1200
0 127 332 537 663 868 995 1200
0 205 410 537 742 868 1073 1200

And here's the two 7-tone 1L6s/6L1s phi-weighted scales...

1/1 5/6
6/7
7/8 11/13

X = 1042.4790�, (1/1 6/7) 1L6s:

0 158 315 473 727 885 1042 1200
0 158 315 570 727 885 1042 1200
0 158 412 570 727 885 1042 1200
0 255 412 570 727 885 1042 1200
0 158 315 473 630 788 945 1200
0 158 315 473 630 788 1042 1200
0 158 315 473 630 885 1042 1200

X = 1018.6773�, (1/1 5/6) 6L1s:

0 181 363 544 656 837 1019 1200
0 181 363 475 656 837 1019 1200
0 181 293 475 656 837 1019 1200
0 112 293 475 656 837 1019 1200
0 181 363 544 725 907 1088 1200
0 181 363 544 725 907 1019 1200
0 181 363 544 725 837 1019 1200

To my mind this method of scale generation can be seen as a
generalization of Thorvald Kornerup's Golden Meantone, and as a
weighted mediant approach to Erv Wilson's MOS/scale-tree.

dan

Dan wrote,

>To my mind this method of scale generation can be seen as a
>generalization of Thorvald Kornerup's Golden Meantone, and as a
>weighted mediant approach to Erv Wilson's MOS/scale-tree.

Not to mention, identical to Wilson's horagrams when used as scales.