Formulas using phi!

Disclaimer: First you will have to blame/or congratulate John chalmers for tricking me back
on to the list (it might be temporary) with this one! Please direct any comments to Him;-)

Margo!
You might want to take a look at the formulas on the first page of this
http://www.anaphoria.com/hrgm01.html or at the top of the horagrams themselves
Also those interested might notice that the 7 tone- phi scale with the both versions of large
and small have been mapped to the keyboard. http://www.anaphoria.com/key.html see 3/7
keyboard

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Hi Kraig,

Good to have you back! Over the last couple of weeks I've been working
on a generalized "Golden scale" approach that allows for a unique
measure of any given two-stepsize cardinality over any given
periodicity. Here -- for the last time I promise <!> -- is the
algorithm:

X = P/(N*D)

"P" = any given periodicity, "N" = (b+phi*d) and "D" = (a+phi*c) where
a/b, c/d are the adjacent fractions (by "adjacent" I mean fractions
that always differ by 1 when cross-multiplied), and "X" = the
corresponding phi-weighted interval, or "Golden generator"

While this approach is geared towards a complete generalization of
Thorvald Kornerup's idea, here's an example of some Golden scales that
fall within the "magic number 7, plus or minus 2" business district.

5-tones

1s-4L

0 260 520 780 940 1200

0, 161 260, 420 520, 680 780, 940 1039

1L-4s

0 214 427 641 986 1200

0, 214 346, 427 559, 641 773, 854 986

2s-3L

0 283 458 742 1025 1200

0, 175 283, 458 567, 633 742, 917 1025

2L-3s

0 192 504 696 889 1200

0, 192 311, 385 504, 696 815, 889 1008

6-tones

1s-5L

0 214 427 641 854 986 1200

0, 132 214, 346 427, 559 641, 773 854, 986 1068

1L-5s

0 181 363 544 725 1019 1200

0, 181 293, 363 475, 544 656, 725 837, 907 1019

2s-4L

0 229 371 600 829 971 1200

0, 142 229, 371 458, 600, 742 829, 971 1058

2L-4s

0 166 434 600 766 1034 1200

0, 166 268, 332 434, 600, 766 868, 932 1034

3s-3L

0 247 400 647 800 1047 1200

0, 153 247, 400, 553 647, 800, 953 1047

7-tones

1s-6L

0 181 363 544 725 907 1019 1200

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

1L-6s

0 158 315 473 630 788 1042 1200

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

2s-5L

0 192 385 504 696 889 1081 1200

0, 119 192, 311 385, 504 577, 623 696, 815 889, 1008 1081

2L-5s

0 146 291 527 673 819 964 1200

0, 146 236, 291 381, 437 527, 673 763, 819 909, 964 1054

3s-4L

0 205 332 537 663 868 1073 1200

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

3L-4s

0 136 355 490 710 845 981 1200

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

8-tones

1s-7L

0 158 315 473 630 788 945 1042 1200

0, 97 158, 255 315, 412 473, 570 630, 727 788, 885 945, 1042 1103

1L-7s

0 139 278 418 557 696 835 1061 1200

0, 139 225, 278 365, 418 504, 557 643, 696 782, 835 922, 975 1061

2s-6L

0 166 332 434 600 766 932 1034 1200

0, 102 166, 268 332, 434 498, 600, 702 766, 868 932, 1034 1098

2L-6s

0 130 260 470 600 730 860 1070 1200

0, 130 210, 260 340, 390 470, 600, 730 810, 860 940, 990 1070

3s-5L

0 175 283 458 633 742 917 1025 1200

0, 108 175, 283 350, 391 458, 567 633, 742 809, 850 917, 1025 1092

3L-5s

0 122 319 441 562 759 881 1078 1200

0, 122 197, 244 319, 441 516, 562 638, 684 759, 881 956, 1003 1078

4s-4L

0 185 300 485 600 785 900 1085 1200

0, 115 185, 300, 415 485, 600, 715 785, 900, 1015 1085

9-tones

1s-8L

0 139 278 418 557 696 835 975 1061 1200

0, 86 139, 225 278, 365 418, 504 557, 643 696, 782 835, 922 975, 1061
1114

1L-8s

0 125 250 374 499 624 749 873 1075 1200

0, 125 202, 250 327, 374 451, 499 576, 624 701, 749 826, 873 950, 998
1075

2s-7L

0 146 291 437 527 673 819 964 1110 1200

0, 90 146, 236 291, 381 437, 527 583, 617 673, 763 819, 909 964, 1054
1110

2L-7s

0 117 234 352 541 659 776 893 1010 1200

0, 117 190, 234 307, 352 424, 469 541, 659 731, 776 848, 893 966, 1010
1083

3s-6L

0 153 247 400 553 647 800 953 1047 1200

0, 94 153, 247 306, 400, 494 553, 647 706, 800, 894 953, 1047 1106

3L-6s

0 111 289 400 511 689 800 911 1089 1200

0, 111 179, 221 289, 400, 511 579, 621 689, 800, 911 979, 1021 1089

4s-5L

0 161 260 420 520 680 780 940 1101 1200

0, 99 161, 260 321, 359 420, 520 581, 619 680, 780 841, 879 940, 1039
1101

4L-5s

0 105 274 378 548 652 822 926 1031 1200

0, 105 169, 209 274, 378 443, 483 548, 652 717, 757 822, 926 991, 1031
1095

I'm only considering scales where L/s = phi to be "Golden scales", and
all others, such as those that Margo has been beautifully writing
about of late, to be points along a given Phi path of which there are
many many indeed.

I think the one overriding characteristic of these scales is that they
put a given Ls index into an arrangement where interval class
distinction is uniquely "optimized". So if clear, uniquely defined
interval classes are your thing, these generalized Golden scales would
seem to be of interest.

--Dan Stearns