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

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