more objective

Disregard my last message, I think this is the best yet:

http://lumma.org/gd3.txt

I'd be interested Graham, if you think this is an improvement.

-C.

In-Reply-To: <4.2.2.20020416172331.01f61e50@lumma.org>
Carl Lumma wrote:

> Disregard my last message, I think this is the best yet:

You have a nice 404 page, anyway.

> http://lumma.org/gd3.txt
>
> I'd be interested Graham, if you think this is an improvement.

It's certainly more objective. Other than that, not so bad.

(1b) Immediately fails for any scale without a 3:2. Can we have a range
of recognisable fifths?

What does "scales with only 1 or 2 unique keys" mean? Is this like whole
tone and octatonic scales?

I thought something like (2b) should be added, but why mix it with
stability?

After "Note that stability only applies to proper scales" Rothenberg goes
on to say "Also, it does not really measure the degree to which a motif at
a given pitch of a scale may be identified with (i.e. recognized as
composed of the same interval as) a `modal transposition' of that motif to
another pitch in the scale (i.e. a sequence)." So I don't see what it's
doing under modal transposition here. He proposed "consistency" as an
alternative, although I haven't worked out what he means by it.

Whatever, Lumma stability is better than Rothenberg stability for this
purpose. Rothenberg stability is certainly important, but should be moved
to a different section. I reckon make it (2b) and move the current (2b)
to (4b) without the "appears in only one interval class throughout the
scale". And maybe without the "strong" as well. What was the point of
that?

Actually, I think I'll write my own (4b). The ratio f/K where f is the
number of consonant intervals within the scale and K is the total number
of intervals in the scale, excluding unisons.

So for 5-limit diatonic, we have 7 thirds and 6 fourths, plus octave
complements. So f/K is 13/21. For 7-limit decimal, 10 2-steps, 3
3-steps, 4 4-steps and 10 5-steps plus complements for everything except
5-steps, giving 44/90. For 11-limit neutral thirds, of course, we have
100%.

I think K=k*(k-1).

If you are going to use Rothenberg and Lumma stability as alternatives,
Lumma stability should be given more weight. It tends to give lower
values, doesn't it?

I'd also like to see a "tonality" category, where we have something about
those characteristic dissonances, and the smallest sufficient (and
distinctive) subsets. Efficiency is really the opposite of this. Both
are important.

Graham

>(1b) Immediately fails for any scale without a 3:2. Can we have a range
>of recognisable fifths?

You're allowed to approximate intervals, according to harmonic entropy.

>What does "scales with only 1 or 2 unique keys" mean? Is this like
>whole tone and octatonic scales?

Yep.

>I thought something like (2b) should be added, but why mix it with
>stability?

I don't, I mix it with efficiency.

>After "Note that stability only applies to proper scales" Rothenberg
>goes on to say "Also, it does not really measure the degree to which a
>motif at a given pitch of a scale may be identified with (i.e.
>recognized as composed of the same interval as) a `modal transposition'
>of that motif to another pitch in the scale (i.e. a sequence)."

Huh- what does he suggest instead?

>Actually, I think I'll write my own (4b). The ratio f/K where f is
>the number of consonant intervals within the scale and K is the total
>number of intervals in the scale, excluding unisons.

That would be a rough measure of how consonant the scale is.

>If you are going to use Rothenberg and Lumma stability as alternatives,
>Lumma stability should be given more weight. It tends to give lower
>values, doesn't it?

Yes. I've been thinking about normalizing all these to their values
in the diatonic scale.

>I'd also like to see a "tonality" category, where we have something
>about those characteristic dissonances, and the smallest sufficient
>(and distinctive) subsets. Efficiency is really the opposite of this.
>Both are important.

I punish ambiguous keys, and I allow mode recognition by strong
consonance. Other than that, I'm not willing to do anything for
tonalness.

-Carl