Re: A drone with all the syntonic j.i . scale ratios as intervals.

Hi Dan,

I'm thinking over what to say, and here is as far as I've got:

Realised while writing it that the original search results I did was incomplete -
some scales included without their inversions. But the most recent ones I
sent you look okay

Also some of the scales are shown in various rotations, which confuses the
picture a bit.

So, I've programmed it to show each scale and then its inversion, if non-symmetrical,
and to show one only rotation of each scale.

Here are some new searches for various values of n-tet varying the number of notes,
attached.

-----------------------------

Dan Stearns asked me to find the solution to a mathematical problem,
to find all the scales with n notes that fold k-tet for any n,k.

Here a scale that folds k-tet is defined as one that has all the
intervals of k-tet as diads.

The example he gave me to try was this beautiful 20-tet diatonic scale:
0 3 7 8 12 15 18 20

In fact there are 15 proper 7 note scales that fold 20-tet
(or 30 including inversions)

There's also a proper six note scale that folds 20-tet:
1 4 5 2 3 5 (plus inversion)

The complete list of strictly proper scales that fold 20-tet for all numbers of notes is:
* 1 3 3 2 3 2 3 3 !8 notes
* 1 2 2 1 2 2 1 2 2 1 2 2 ! 12 notes
* 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 20 notes

I found these results by programming a computer search.
All the strictly proper scales found so far that fold n-tet are
symmetrical, and would be interesting to see if this
can be proved as a general result.

Similar lists can be made for any n-tet, and I've done a few
of them - may post the results later.

This lead me to think about application to j.i.

The list of scales for 12-tet and 4 notes is:

[6 in cl. 1 and 3] 1 3 2 6 inv. 1 6 2 3
1 2 4 5 inv. 1 5 4 2
2 scales found for 12-tet and 4 notes

Legend:
the [] indicates that a scale is ambiguous,
Any proper scales are shown with an * (neither of these are)
then []* = proper, * = strictly proper, neither = unambiguous
improper and [] = ambiguous improper.

Had the thought to try these out in the syntonic j.i. scale

1 2 4 5 is:

1/1 16/15 6/5 3/2 2/1

This had the interesting property that all the syntonic j.i. scale ratios occur as
intervals between pairs of notes, and there are no
other intervals except for the inversion of the j.i. augmented fourth.

Rotations are:
1/1 16/15 6/5 3/2 2/1
1/1 4/3 64/45 8/5 2/1
1/1 5/4 5/3 16/9 2/1
1/1 9/8 45/32 15/8 2/1

Uses the j.i. scale with 16/9 instead of 9/5
1/1 16/15 9/8 6/5 5/4 4/3 45/32 64/45 3/2 8/5 5/3 16/9 15/8 2/1

(example fractal tune)

Obviously if we include the augmented fourth we have to include its complement,
so what about trying for a symmetrical eleven note scale instead:
1/1 16/15 6/5 8/5 2/1
1/1 5/4 4/3 3/2 2/1
1/1 4/3 5/3 16/9 2/1
1/1 9/8 3/2 15/8 2/1
makes
1/1 16/15 9/8 6/5 5/4 4/3 3/2 8/5 5/3 16/9 15/8 2/1

Then if you play in
1/1 16/15 6/5 8/5 2/1
all the intervals you get will be intervals of this
scale and you can make any of them by playing two of
the notes.

(example fractal tune)

Is there any other four note scale that does this?

Here is the complete list of all the four note scales
that fold 11-tet

1 2 2 6 inv. 1 6 2 2
[5 in cl. 1 and 2] 1 3 2 5 inv. 1 5 2 3
1 6 3 1 inv. 1 1 3 6
1 2 4 4 inv. 1 4 4 2
[3 in cl. 1 and 2] 1 2 5 3 inv. 1 3 5 2
1 3 6 1 inv. 1 1 6 3

So it will have to be one of these.

A check will confirm that none of them work.

....

then plan to do computer search to find all the
chords that fold the 11-tet j.i. scale.

Might also try some other scales.

Any other suggestions for a symmetrical j.i.
scale that might be nice to try to fold
in this way?

Robert