Yahoo Tuning Groups Ultimate Backup tuning "Lumma Stability"

Hi Jake,

Did you see them in Scala's "show data" output?

In one sentence, Lumma stability is the portion of a
scale's period which is not covered by its interval classes,
and the "Lumma impropriety factor" is the portion which is
more than singly-covered. Longer version:

Rothenberg assumes a process whereby a listener hears
melodic intervals between sounds and sorts them by size.
This produces an "interval matrix" (IM) in the listener's
mind. The listener continually refines this IM as stimuli
arrive, and if a source is consistent for long enough he
may remember its IM for later use.

The IM for a scale is simply a list of all its intervals
(dyads) grouped by the modes in which they appear. Scala
will display it with "show/line intervals". Every fixed
scale corresponds to one and only one IM. So if we assume
that listeners will eventually perfect their picture of
a scale's IM, we can infer things about the scale from
its IM.

Imagine a log-frequency ruler whose total length is the
interval of equivalence of our periodic scale (e.g. 1200
cents). Mark off each interval in the IM on this ruler.
Now we'll draw line segments on the ruler with colored
pencil, using the marks as endpoints. We'll connect all
marks belonging to the same interval class with a single
line, using a different color for each interval class.
Lumma stability is the portion of the ruler that has no
pencil on it. The impropriety factor is the portion
that's more than singly covered -- where different colors
overlap. The idea being that when two interval classes
overlap, listeners will not be able to distinguish them
in all cases. Lumma stability measures how easily
distinguishable the non-overlapping classes will be.

Rothenberg stability is very similar, except it counts
the number of overlaps, so it can't distinguish a gross
overlap (in cents) from a small one, or detect when two
interval classes are arbitrarily close to overlapping.

-Carl

🔗genewardsmith <genewardsmith@...>

🔗Jake Freivald <jdfreivald@...>

Carl,

Thanks for your response. This is great.

> Did you see them in Scala's "show data" output?

Yes. As part of my slow-and-stubborn education, I'm trying to learn what many of those things mean. I found some discussion on tuning-math from 5 years ago, but didn't "get it". Your explanation here was much easier for me to digest.

Since recently I've been focused on proper and strictly proper scales, I hadn't seen the Lumma impropriety factor, but I've looked for it in improper scales, I see it now, too.

> In one sentence, Lumma stability is the portion of a scale's period
> which is not covered by its interval classes

If I understand what follows, you mean that, say, the collection of one-step intervals in a scale (i.e., step 1 to 2, 18 to 19, and generally n to n-1) will have a certain variation, from (say) 60 cents to 70 cents, covering 10 cents. All of the two-step intervals will have a range as well, (say) 130 to 135 cents, covering 5 cents. And so on. If I add up to total of these distances and divide it by the period, I get the Lumma stability.

So, for instance, all EDOs will have a Lumma stability of 1, because the interval classes have no variation and are thus represented by points, which have size zero. Now that I think of it, all, and only, equal divisions of any period will have a Lumma stability of 1.

I would expect most regular- and well-temperaments to have a Lumma stability relatively close to 1. What does "relatively close" mean? In practice:
* Werckmeister III = 0.86
* Prinz = 0.84
* pajara_mm.scl (Pajara minimax 22-tone scale from the Scala list) = 0.95
* Meantone[12] = 0.70

The traditional 12-tET modes have a Lumma Stability of 0.5, and it looks like traditional diatonic modes of well-temperaments are around that, too.

12-tET harmonic minor, by gaining that 3-half-step interval, earns a Lumma Stability of 0.167. (It's still proper, though, so the Impropriety Factor is zero.) That appears to be pretty low. By comparison, Blackjack[21], which is not proper, has a Lumma Stability of 0.31 (and a Lumma Impropriety Factor of 0.14).

Generally speaking, improper scales seem to have relatively low Stability Factors, in the 0.3 range, sometimes going to 0.1 or lower.

> and the "Lumma impropriety factor" is the portion which is
> more than singly-covered.

Only improper scales will have an impropriety factor (big surprise). It's not the inverse of the Lumma Stability Factor, though -- many improper scales (which will have a positive impropriety factor) will have a low but non-zero Stability Factor. You can have a low stability and a low impropriety, or a low stability and a high impropriety, but not a high stability and a high impropriety.

These seem like pretty useful measures. Melodically, I don't know that I want to have a stability too close to 1, but a scale with a stability of close to 1 might be good raw material from which I could make a mode, analogous to a major or minor mode, that gets me to a lower stability factor, closer to 0.5. Or not. Something to keep an eye on, anyway.

Thanks for the explanation.

Regards,
Jake

🔗Carl Lumma <carl@...>

Gene wrote:

> I'm thinking of editing this article and sticking it on the
> Xenwiki,

Please do.

> but I wonder if there might be a better starting point
> around.

I took this as a nudge to convert my FAQ draft to html with
anchor links. So I can suggest
http://lumma.org/music/theory/TuningFAQ.html#propriety
and
http://lumma.org/music/theory/TuningFAQ.html#meanvariety

You can also try it straight from the horse's mouth:
http://lumma.org/music/theory/RothenbergExcerpts.txt

(these URLs should be long-lived)

> By the way, someone needs to write an article on
> harmonic entropy.

You might start by copying the Sounds of India page
http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937

or maybe find something of use in
http://lumma.org/music/theory/TuningFAQ.html#harmonicentropy
and
http://lumma.org/music/theory/TuningFAQ.html#2HE

-Carl

🔗Carl Lumma <carl@...>

Hi Jake,

> If I understand what follows, you mean that, say, the
> collection of one-step intervals in a scale (i.e., step
> 1 to 2, 18 to 19, and generally n to n-1) will have a
> certain variation, from (say) 60 cents to 70 cents,
> covering 10 cents. All of the two-step intervals will have
> a range as well, (say) 130 to 135 cents, covering 5 cents.
> And so on. If I add up to total of these distances and
> divide it by the period, I get the Lumma stability.

...Assuming there's no overlap. If the scale is improper,
this sum will be too big.

> So, for instance, all EDOs will have a Lumma stability
> of 1, because the interval classes have no variation and
> are thus represented by points, which have size zero.

Right!

> 12-tET harmonic minor, by gaining that 3-half-step interval,
> earns a Lumma Stability of 0.167. (It's still proper, though,
> so the Impropriety Factor is zero.) That appears to be
> pretty low.

The Lumma stability may not mater much once a scale is
learned, but might indicate how hard it was to learn...

> Generally speaking, improper scales seem to have relatively
> low Stability Factors, in the 0.3 range, sometimes going
> to 0.1 or lower.

Yes, should be. Thanks for reporting on this.

The impropriety factor is expected to be the more important
of the two. I did try combining them once. See item (3)
here http://lumma.org/music/theory/gd/gd3-spec.txt

> Melodically, I don't know that I want to have a stability
> too close to 1,

Yeah, ETs tend to be too regular. It's hard to know what
the tonic is supposed to be. That said, they can produce
an interesting unanchored feeling.

> but a scale with a stability of close to 1 might be good
> raw material from which I could make a mode, analogous
> to a major or minor mode, that gets me to a lower stability
> factor,

You bet. Happy exploring!

-Carl

🔗Michael <djtrancendance@...>

Carl>"If the tones of the chord are complex tones, the timbre will influence their relative likelihoods of winning the contest (since the hearing system considers all partials). If 4 or 8 win, the brain is interpreting 10:12:15 literally, as a segment of harmonics relatively high in the series. It is somewhat more likely that 5 or 10 will win; the brain hears 10:15 = 3:2 and dismisses 12 as an artifact.
The above appears to concern how louder partials in timbre can make certain notes in a chord loud enough to "mask" other tones in the chord and shift the sense of the root tone, correct?

>"The same tradeoff exists in the case of 16:19:24, but here the literal interpretation is less likely because it relies on even higher harmonics. However, this time both the literal and 'outer fifth only' interpretations point to the same pitch, namely 16 (or 4 or 8)."
-http://lumma.org/music/theory/TuningFAQ.html#harmonicentropy

This idea seems to make a lot of sense...if I have it right IE that 16 and 24 in the above chord share the factors of 4 and 8 and thus point to those tones, which comes across well as they are both octave-equivalents of the root.

The explanation is also unique in that it (finally!) appears to take into account how critical band dissonance matters (IE in 6:7:9), along with how virtual pitch matters (the above example), and how Harmonic Entropy matters....rather than saying one matters some bizarre/biased amount more than any of the others.