even numbered trivalence

A while back I wrote,

<< 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? >>

No takers 'ey?, well here's something... anyone see a rule?

4-tones

[1,2,1] 4-out-of-12

0 200 500 800 1200
0 300 600 1000 1200
0 300 700 900 1200
0 400 600 900 1200

6-tones

[1,2,3] 6-out-of-20

0 120 360 540 720 960 1200
0 240 420 600 840 1080 1200
0 180 360 600 840 960 1200
0 180 420 660 780 1020 1200
0 240 480 600 840 1020 1200
0 240 360 600 780 960 1200

[3,2,1] 6-out-of-16

0 300 450 675 900 1050 1200
0 150 375 600 750 900 1200
0 225 450 600 750 1050 1200
0 225 375 525 825 975 1200
0 150 300 600 750 975 1200
0 150 450 600 825 1050 1200

8-tones

[1,2,5] 8-out-of-28

0 86 257 429 557 686 857 1029 1200
0 171 343 471 600 771 943 1114 1200
0 171 300 429 600 771 943 1029 1200
0 129 257 429 600 771 857 1029 1200
0 129 300 471 643 729 900 1071 1200
0 171 343 514 600 771 943 1071 1200
0 171 343 429 600 771 900 1029 1200
0 171 257 429 600 729 857 1029 1200

[3,4,1] 8-out-of-22

0 109 273 382 545 709 818 982 1200
0 164 273 436 600 709 873 1091 1200
0 109 273 436 545 709 927 1036 1200
0 164 327 436 600 818 927 1091 1200
0 164 273 436 655 764 927 1036 1200
0 109 273 491 600 764 873 1036 1200
0 164 382 491 655 764 927 1091 1200
0 218 327 491 600 764 927 1036 1200

[1,4,3] 8-out-of-26

0 185 323 508 646 785 969 1108 1200
0 138 323 462 600 785 923 1015 1200
0 185 323 462 646 785 877 1062 1200
0 138 277 462 600 692 877 1015 1200
0 138 323 462 554 738 877 1062 1200
0 185 323 415 600 738 923 1062 1200
0 138 231 415 554 738 877 1015 1200
0 92 277 415 600 738 877 1062 1200

[5,2,1] 8-out-of-20

0 240 360 480 660 840 960 1080 1200
0 120 240 420 600 720 840 960 1200
0 120 300 480 600 720 840 1080 1200
0 180 360 480 600 720 960 1080 1200
0 180 300 420 540 780 900 1020 1200
0 120 240 360 600 720 840 1020 1200
0 120 240 480 600 720 900 1080 1200
0 120 360 480 600 780 960 1080 1200

10-tones

[1,2,7] 10-out-of-36

0 67 200 333 467 567 667 800 933 1067 1200
0 133 267 400 500 600 733 867 1000 1133 1200
0 133 267 367 467 600 733 867 1000 1067 1200
0 133 233 333 467 600 733 867 933 1067 1200
0 100 200 333 467 600 733 800 933 1067 1200
0 100 233 367 500 633 700 833 967 1100 1200
0 133 267 400 533 600 733 867 1000 1100 1200
0 133 267 400 467 600 733 867 967 1067 1200
0 133 267 333 467 600 733 833 933 1067 1200
0 133 200 333 467 600 700 800 933 1067 1200

[3,4,3] 10-out-of-30

0 160 280 400 480 600 760 840 1000 1120 1200
0 120 240 320 440 600 680 840 960 1040 1200
0 120 200 320 480 560 720 840 920 1080 1200
0 80 200 360 440 600 720 800 960 1080 1200
0 120 280 360 520 640 720 880 1000 1120 1200
0 160 240 400 520 600 760 880 1000 1080 1200
0 80 240 360 440 600 720 840 920 1040 1200
0 160 280 360 520 640 760 840 960 1120 1200
0 120 200 360 480 600 680 800 960 1040 1200
0 80 240 360 480 560 680 840 920 1080 1200

[7,2,1] 10-out-of-24

0 200 300 400 500 650 800 900 1000 1100 1200
0 100 200 300 450 600 700 800 900 1000 1200
0 100 200 350 500 600 700 800 900 1100 1200
0 100 250 400 500 600 700 800 1000 1100 1200
0 150 300 400 500 600 700 900 1000 1100 1200
0 150 250 350 450 550 750 850 950 1050 1200
0 100 200 300 400 600 700 800 900 1050 1200
0 100 200 300 500 600 700 800 950 1100 1200
0 100 200 400 500 600 700 850 1000 1100 1200
0 100 300 400 500 600 750 900 1000 1100 1200

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
> A while back I wrote,
>
> << 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? >>
>
> No takers 'ey?, well here's something... anyone see a rule?

Dan,

I can see there is some sort of subsetting rule based on the
sequence, but don't quite get it. How about a "Trihills for Dummies"?

This is to generate subsets from even numbered divisions of the
octave, with 3 step sizes? Right - Wrong???

How do you do it Dan?

Jacky Ligon

>
> 4-tones
>
> [1,2,1] 4-out-of-12
>
> 0 200 500 800 1200
> 0 300 600 1000 1200
> 0 300 700 900 1200
> 0 400 600 900 1200
>

Jacky Ligon wrote,

<< This is to generate subsets from even numbered divisions of the
octave, with 3 step sizes? Right - Wrong??? >>

The even numbered ETs are a byproduct of trying to find scales with an
even number of notes, i.e., 4, 6 , 8, etc., that are also Trihill.

To back up a bit, I think the easiest way to frame all this is by
generalizing the syntonic scale (I actually use the "schismatic"
version of this as the generalization model, but the syntonic scale is
very easy to frame in your mind and visualize what's going on). This
means that you have a scale constructed of two planes; the usual
generator chain (2:3s in the syntonic scale):

1/1-----generator-----generator-----etc.,...

and the commaticlly altered second plane or dimension (4:5s in the
syntonic scale):

the second commaticlly altered dimension
/
/
1/1-----generator

Now all of this is tied to converting an [a,b] index into an [a,b,c]
index which I have posted on before, and 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.

But 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. And therein lies the "even numbered trivalence"
problem.

<< How do you do it Dan? >>

What I did was alter the transformation from a two-term index into a
three-term index so as to make the transformations symmetrical. Here's
an example...

A [1,5] index becomes a [1,1,4] index where abbbbb becomes
accbcc. Changing accbcc into acbbcc turns the [1,1,4] into a [1,2,3]
where

0 114 343 571 743 971 1200
0 229 457 629 857 1086 1200
0 229 400 629 857 971 1200
0 171 400 629 743 971 1200
0 229 457 571 800 1029 1200
0 229 343 571 800 971 1200

becomes

0 120 360 540 720 960 1200
0 240 420 600 840 1080 1200
0 180 360 600 840 960 1200
0 180 420 660 780 1020 1200
0 240 480 600 840 1020 1200
0 240 360 600 780 960 1200

Now it's easy to see that this solves the problem by introducing the
half-octave to the scales midpoint, but what I'm wondering is does
anyone see handy rule here?

I would think that it would be a consistent single chain two generator
rule of some sort where the second generator is a commatic alteration
of the main generator.

Any ideas out there?

--Dan Stearns

Dan Stearns wrote:

> Now all of this is tied to converting an [a,b] index into an [a,b,c]
> index which I have posted on before, and if the a+b of the [a,b] ...

Dan,

Could you please explain what you mean by an [a,b] or [a,b,c] index?
What are the two and three dimensional arrays that they index into?

Regards,
-- Dave Keenan

Dave Keenan wrote,

<< Could you please explain what you mean by an [a,b] or [a,b,c]
index? >>

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 just 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
where [2,5] means a scale consisting of 2 "small" steps and 5 "large"
steps, and [2,2,3] means a scale consisting of 2 "small", 2 "medium",
and 3 "large" steps. This is all I mean when I refer to an index here.

<< What are the two and three dimensional arrays that they index into?
>>

Well by converting a one-dimensional two-term single generator chain
into a two plane -- or two generator, or two-dimensional -- three-term
chain, it is possible to convert a given [a,b] index into an [a,b,c]
index. So there are only one- and two-dimensions, but two- and
three-term sequences.

Does any of this help?

thanks,

--Dan Stearns