Re: Tribonacci sequence

This web page may possibly be relevant for those trying to find some type
of Tribonacci analog of the golden ratio
scales:

http://www.mathsoft.com/asolve/constant/gold/gupta.html
and
http://www.mathsoft.com/asolve/constant/pythag/dgm.html

I'm interested to understand your posts Dan. Have got as far
as realising that you show each scale multiple times in all its modes,
which puzzled me at first, and they are indeed Tribonacci (on
testing in SCALA one sometimes gets answer that they are not
because of the rounding to nearest cent means intervals can
differ in size by one cent when actually they are the same one.

But haven't yet read the most recent posts and don't yet understand
what it is you have discovered - will take a look later, and ask if
there is something specific I see that needs clarification.

Robert

Robert Walker wrote,

<< I'm interested to understand your posts Dan. Have got as far as
realising that you show each scale multiple times in all its modes,
which puzzled me at first, and they are indeed Tribonacci >>

Let me see if I can't clarify some of the terms that I often use and
try to make the whole procedure a bit easier to follow...

When I say [2,5], or [2,2,3], or [a,b], or [a,b,c] I'm referring to a
scale's index where the alphabetized variables are small to large
stepsizes. So [a,b] and [a,b,c] are the generalized two and three
stepsize indexes, and [2,5] and [2,2,3] are just a familiar example
of each. So [2,5] means a scale consisting of 2 "small" steps and 5
"large" steps. [2,2,3] means a scale consisting of 2 "small", 2
"medium", and 3 "large" steps.

This is what I mean when I refer to a scale index.

By converting a one-dimensional single generator chain into a two
plane chain it is possible to convert any given [a,b] index into an
[a,b,c] index.

To convert an [a,b] index into a single generator chain, you must
first convert the two terms into adjacent fractions -- fractions that
when cross-multiplied will differ by 1.

The process (or algorithm) I use to convert an [a,b] index into a
single generator chain uses three constants. I refer to these apical
constants -- "apical" roughly meaning 'a bud which terminates a stem';
by analogy this would be the constant terminating the series. Anyway
the constants I use are (1+sqrt(5))/2, 2, and sqrt(2)+1.

Phi is derived from the series

1/1, 1/2, 2/3, 3/5, 5/8, ...

and in my terminology, an [a,b] index weighted by this constant gives
a generalized golden scale.

2 is derived from series

1/1.5, 1.5/3.25, 3.25/6.375, 6.375/12.8125, 12.8125/25.59375, ...

and in my terminology, an [a,b] index weighted by this constant gives
a generalized equal scale.

sqrt(2)+1 is derived from the following series

1/2, 2/5, 5/12, 12/29, 29/70, ...

and in my terminology, an [a,b] index weighted by this
constant gives a generalized silver scale.

Here's the generalized formula for deriving a weighted generator for
any given [a,b] index within a given periodicity.

X = P/((b+W*d))*(a+W*c)

Where:

"P" = any given periodicity

"W" = the weight, constant

"a"/"b", "c"/"d" = the two adjacent fractions of a given [a,b] index

and "X" = the resulting weighted generator

Looking at "X" as a generalized analogue to the fifth in a circle (or
spiral) of fifths really helps, because if you multiply "X" by the
forth term of the two-term/Fibonacci [a,b] index the difference
between this and "P" (a given periodicity, generally the octave) will
give a generalized Pythagorean comma by way of the analogy.

Converting a one-dimensional generator chain into a two plane chain
makes it possible to convert a given [a,b] index into a corresponding
[a,b,c] index.

To do this I simply "fold" the single generator chain and scale the
second plane by the generalized Pythagorean comma... this gives a
second "schismic" plane:

schismic plane
/ \
/ \
/ \
1/1------single generator chain

If the a+b of the [a,b] index is an odd number (when reduced by their
GCD), the single generator chain always has one more interval (or
note) than the schismic plane. These scales are always three-term
Myhill scales.

If the a+b of the [a,b] index is an even number (when reduced by their
GCD), the single generator chain always has the same number of
intervals (or notes) as the schismic plane. These scales are also
three-term scales where the midpoint interval class always consists of
two interval sizes.

Here the normalization rule for the commatic alterations of the second
(or schismic) plane if "W" = sqrt(2)+1:

If a<b then [a,b]=[a,(b+c)], and c/b = (sqrt(2)+1)/2 and b/a =
sqrt(2).

If a>b then [a,b]=[(a+b),c)], and both c/b and b/a = sqrt(2).

I like this weighting because I feel that it helps to better define
the three stepsizes.

But I also like a "W" = sqrt(5) weighting, as this is all but a
generalized syntonic model. Using this sqrt(5) weighting results in
the following proportions:

when a<b,

c/b = Phi-.5
b/a = Phi

when a>b,

c/b = Phi
b/a = sqrt(5)-1

I hope that this might be helpful for anyone who wanted to understand
this method a bit better, but if it is not and anyone cares to, please
do ask any specific questions and I'd be happy to try and answer or
better explain myself.

thanks,

--Dan Stearns

Here are some examples that I had left off of the last post in an
attempt to not overcrowd it so much.

These are all the 3- through 8-tone, 2- to 3-term conversions that
have their sole periodicity at the 1:2 where "W"=sqrt(2)+1.

3-tone:

[1,2] = [1,1,1]

0 291 703 1200
0 412 909 1200
0 497 788 1200

[2,1] = [1,1,1]

0 544 928 1200
0 384 656 1200
0 272 816 1200

4-tone:

[1,3] = [1,1,2]

0 206 557 849 1200
0 351 643 994 1200
0 291 643 849 1200
0 351 557 909 1200

[1,3] = [2,1,1]

0 443 665 978 1200
0 222 535 757 1200
0 313 535 978 1200
0 222 665 887 1200

5-tone:

[1,4] = [1,1,3]

0 159 431 656 928 1200
0 272 497 769 1041 1200
0 225 497 769 928 1200
0 272 544 703 975 1200
0 272 431 703 928 1200

[2,3] = [2,2,1]

0 313 573 757 1016 1200
0 260 443 703 887 1200
0 184 443 627 940 1200
0 260 443 757 1016 1200
0 184 497 757 940 1200

[3,2] = [1,2,2]

0 153 370 677 893 1200
0 217 523 740 1047 1200
0 307 523 830 983 1200
0 217 523 677 893 1200
0 307 460 677 983 1200

[4,1] = [3,1,1]

0 374 561 826 1013 1200
0 187 452 639 826 1200
0 265 452 639 1013 1200
0 187 374 748 935 1200
0 187 561 748 1013 1200

6-tone:

[1,5] = [1,1,4]

0 130 351 573 757 978 1200
0 222 443 627 849 1070 1200
0 222 405 627 849 978 1200
0 184 405 627 757 978 1200
0 222 443 573 795 1016 1200
0 222 351 573 795 978 1200

[5,1] = [4,1,1]

0 324 486 647 876 1038 1200
0 162 324 553 714 876 1200
0 162 391 553 714 1038 1200
0 229 391 553 876 1038 1200
0 162 324 647 809 971 1200
0 162 486 647 809 1038 1200

7-tone:

[1,6] = [1,1,5]

0 110 297 484 639 826 1013 1200
0 187 374 529 716 903 1090 1200
0 187 342 529 716 903 1013 1200
0 155 342 529 716 826 1013 1200
0 187 374 561 671 858 1045 1200
0 187 374 484 671 858 1013 1200
0 187 297 484 671 826 1013 1200

[2,5] = [2,2,3]

0 206 376 582 703 909 1079 1200
0 171 376 497 703 874 994 1200
0 206 326 532 703 824 1029 1200
0 121 326 497 618 824 994 1200
0 206 376 497 703 874 1079 1200
0 171 291 497 668 874 994 1200
0 121 326 497 703 824 1029 1200

[3,4] = [3,3,1]

0 229 419 553 742 876 1066 1200
0 190 324 513 647 837 971 1200
0 134 324 458 647 781 1010 1200
0 190 324 513 647 876 1066 1200
0 134 324 458 687 876 1010 1200
0 190 324 553 742 876 1066 1200
0 134 363 553 687 876 1010 1200

[4,3] = [1,3,3]

0 107 258 471 622 836 987 1200
0 151 364 515 729 880 1093 1200
0 213 364 578 729 942 1049 1200
0 151 364 515 729 836 987 1200
0 213 364 578 685 836 1049 1200
0 151 364 471 622 836 987 1200
0 213 320 471 685 836 1049 1200

[5,2] = [3,2,2]

0 122 295 417 661 783 956 1200
0 173 295 539 661 834 1078 1200
0 122 366 488 661 905 1027 1200
0 244 366 539 783 905 1078 1200
0 122 295 539 661 834 956 1200
0 173 417 539 712 834 1078 1200
0 244 366 539 661 905 1027 1200

[6,1] = [5,1,1]

0 285 428 570 772 915 1057 1200
0 143 285 487 630 772 915 1200
0 143 344 487 630 772 1057 1200
0 202 344 487 630 915 1057 1200
0 143 285 428 713 856 998 1200
0 143 285 570 713 856 1057 1200
0 143 428 570 713 915 1057 1200

8-tone:

[1,7] = [1,1,6]

0 95 257 419 580 714 876 1038 1200
0 162 324 486 620 781 943 1105 1200
0 162 324 458 620 781 943 1038 1200
0 162 296 458 620 781 876 1038 1200
0 134 296 458 620 714 876 1038 1200
0 162 324 486 580 742 904 1066 1200
0 162 324 419 580 742 904 1038 1200
0 162 257 419 580 742 876 1038 1200

[3,5] = [3,3,2]

0 113 272 384 577 736 849 1008 1200
0 159 272 464 623 736 895 1087 1200
0 113 305 464 577 736 928 1041 1200
0 192 351 464 623 816 928 1087 1200
0 159 272 431 623 736 895 1008 1200
0 113 272 464 577 736 849 1041 1200
0 159 351 464 623 736 928 1087 1200
0 192 305 464 577 769 928 1041 1200

[5,3] = [2,3,3]

0 196 335 531 629 767 963 1102 1200
0 139 335 433 571 767 906 1004 1200
0 196 294 433 629 767 865 1061 1200
0 98 237 433 571 669 865 1004 1200
0 139 335 473 571 767 906 1102 1200
0 196 335 433 629 767 963 1061 1200
0 139 237 433 571 767 865 1004 1200
0 98 294 433 629 727 865 1061 1200

[7,1] = [6,1,1]

0 255 382 510 637 818 945 1073 1200
0 127 255 382 563 690 818 945 1200
0 127 255 435 563 690 818 1073 1200
0 127 308 435 563 690 945 1073 1200
0 180 308 435 563 818 945 1073 1200
0 127 255 382 637 765 892 1020 1200
0 127 255 510 637 765 892 1073 1200
0 127 382 510 637 765 945 1073 1200

All the [a,b] indexes in these examples are weighted with the
sqrt(2)+1 constant... so I'll dub these "silver schismic" scales for
now.

Is the Fibonacci to Tribonacci parallel still unclear?

If so, note that if the weighting were 2 then the fourth term of an
[a,b,c] Tribonacci series would be the corresponding subset of the
sixth term, just as the third term of an [a,b] Fibonacci series would
be the corresponding subset of its fourth term... this is the fib to
tri parallel, at least until someone can find a three-term (three
fraction) analogue to adjacent fractions and seeding a Stern-Brocot
Tree!

Here's another (easier) question: can anyone see a uniform way to
change the even numbered scales into a full three-term "Trihills" as
is the case with odd numbered scales?

--Dan Stearns

I wrote,

<< note that if the weighting were 2 then the fourth term of an
[a,b,c] Tribonacci series would be the corresponding subset of the
sixth term, just as the third term of an [a,b] Fibonacci series would
be the corresponding subset of its fourth term >>

This was not quite complete... If an [a,b] index's weighting is 2,
then the third term of an [a,b] fibonacci series will always be the
corresponding subset of its fourth term. But if the same principle is
held to the three-term conversion, the weighting must be 4 if a<b, and
2.5 if a>b. Then the fourth term of a converted [a,b,c] tribonacci
series will always be the corresponding subset of its sixth term.

--Dan Stearns