Re: trivalent modes from the SCALA archive

Hi Dan,

Preliminary result of search of modes in n-tet from the SCALA modes archive, for trivalent
scales:

Most of the trivalent modes with an even number of notes have duplicated intervals - by
which I mean, intervals in the same interval class that are also the same size, and have
varying composition in terms of L M S.

However, there are a few exceptions.

For 6 note scales in the modes list, there are three exceptions (out of 27 6-note
trivalent modes).

Here is one of them:

Raga Bhinna Pancama

0 200 500 700 800 1100 1200

steps 2 3 2 1 3 1

M L M S L S

L M S
LM LS MS
LMM LMS LSS
LLMS LMSS LMMS
LLMMS LLMSS LMMSS

Since it has no duplicated intervals, we are free to make micro-adjustments to move it
away from 12-tet.

It has two intervals of each size.

To make a scale which isn't ET, add a small irrational number of cents x to the Ls and Ms,
and subtract 2*x cents from the Ss.

Let's try adding g cents to the Ls and Ms, and subtracting 2*g cents from the Ss:

(just because g is irrational)

steps: 201.618 cents 301.618 cents 201.618 cents 96.7639 cents 301.618 cents 96.7639 cents

Then that also is a trivalent scale with an even number of notes.

However, we can't make it rational in this way, because one of its intervals is
LMS = the midpoint.

A bit more generally, no 6 note scale with [a,b,c] index [2,2,2] can be rational,
because one would want LLMMSS = 2
and so (LMS)^2 = 2

One can make two of the step sizes rational, then the third has to be irrational.

Incidentally, this is also an example of a trivalent scale that is not a subset of
any n-tet scale, and which includes its midpoint.

The other two 6 note trivalent modes in the list are:

Prometheus Neapolitan
0 1 4 6 9 10 12
steps:
1 3 2 3 1 2
S L M L S M

and

Raga Vijayanagari
0 2 3 6 7 9 12
steps:
0 2 1 3 1 2 3
M S L S M L

These three modes all follow the same pattern:
mode of Raga Vijayanagari MLMSLS
mode of Prometheus Neapolitan LMLSLS
mode of Raga Bhinna Pancama LMLSLS

All three have [a,b,c] index [2,2,2], so none
of them can be made into rational scales.

The 8 note trivalent modes in the list all have duplicated intervals in LMS notation
(anyone think of
a better way of expressing this, or is there a name for it).

That doesn't mean that they all are pinned down to being et scales - one would
need to look into it more carefully, to see what equalities they set up
between multiples of L, M and S.

Then, let's look at the trivalent 12 tone mode in 24-tet from the SCALA scales archive:
0 2 4 6 7 10 12 14 16 18 20 21 24
steps:
2 2 2 1 3 2 2 2 2 2 1 3
MMMSLMMMMMSL

LMMMS MMMMM MMMMS MMMSS = 10 10 9 8
works because LS = MM.
This then fixes M to be 2^(1/12)

So this pattern of Ls, Ms, and Ss has to be irrational.

Conclusion so far - scales with an even number of notes needn't be subsets of n-tet.

Also it is possible for such a scale to have two out of the three step sizes rational.

However it's possible that they could all have at least some irrational
notes, on evidence so far.

Also, so far, your first conjecture that every such scale includes its midpoint
is holding up!

Robert

Realised part of what I wrote wasn't very clearly expressed:

> Let's try adding g cents to the Ls and Ms, and subtracting 2*g cents from the Ss:

> (just because g is irrational)

> steps: 201.618 cents 301.618 cents 201.618 cents 96.7639 cents 301.618 cents 96.7639
cents

> Then that also is a trivalent scale with an even number of notes.

> However, we can't make it rational in this way, because one of its intervals is
> LMS = the midpoint.

Idea is that if you add a rational number of cents to each interval, then all the
intervals
are of the form 2^(r/1200) for r rational, so you end up with a subset of n-tet for some n

(where n = lcm of the quotients of all the rational powers that 2 is raised by to make
notes of the scale)

So to make sure it isn't a subset of n-tet, you need at least one note which is defined
using an irrational number of cents.

No n-tet scale can contain an interval with an irrational number of cents.

Having established in this way that the scale doesn't have to be n-tet,

Next, idea is to see if it is possible to make all the steps rational.

For instance, one could try
9/8 32/27 9/8 S 32/27 S
where the S has to be figured out.

You then get
S^2 = 9/8

Very close!

But on taking the square root, you get
3/(2*sqrt(2))
which is irrational.

Then following Dan's idea, one observes that since the scale includes its midpoint LMS,
then there is no solution in which all three intervals are rational.

I've just made a couple of files that list all the trivalent scales and modes in the SCALA
archive, but they aren't quite ready yet - I'll do them again tomorrow, and post a url
for them.

Robert

Robert Walker wrote,

<< Since it has no duplicated intervals, we are free to make
micro-adjustments to move it away from 12-tet. >>

Right, and I'm pretty sure this is what Paul had in mind earlier as
well. I have a suspicion that these types of indexes -- ones that are
not reduced by their GCD (note that these are [2,2,2,] indexes) --
parallel their two-term cousins in as much as I bet that they're not
constructed in the same fashion generator wise. In other words I think
they would be yet another class of scales... but I don't understand
enough about how arbitrary, freestanding (i.e., non two-term
conversion) three-term index is constructed yet to be able to say so.

Anyway, it's nice to know, and thanks for all the wonderful sleuth
work!

--Dan Stearns