Rothenberg's Efficiency

Rick Sanford wrote...

>I agree, but the term 'tonal' means specifically supporting a
tonic-dominant >relationship in the piece, no? If not, we could call many
drone-type things >'tonal', which they are not.

Perhaps this is the distinction Doty suggests:

1) What I've always called "tonality"
- the psychoacoustic phenomenon of
fitting a group of pitches to a harmonic series

2) What I've always called "key"
- the cognitive phenomenon of locating a
stimulus within P (a proper map of pitch space)

The first one, as I understand it, basically depends on the virtual pitch
mechanism, provided roughness is low enough for it to work. To the extent
that this does not imply a harmonic series, by combination tones or perhaps
some hard-wired preference, tonality can be established by fitting a group
to an inharmonic series.

The second one is best explained by Rothenberg. See his series of 5 papers
published to Mathematical Systems Theory in the late 1970's (Myhill was on
the Editorial board at the time, I see). I have read the three of them
that are listed in the Microtonal bibliography...

ftp://ella.mills.edu/ccm/tuning/papers/bib.html

They are the single most important body of theory on melody known to me,
laying down a framework capable of supplying a useful explanation for just
about every melodic phenomenon there is. The strength of the explanations
may be easily over-estimated, and maybe Rothenberg suffers from this a
little bit, but on the whole he is a first-rate theorist who applies his
model with impressive accuracy.

R. calls a stimulus that occupies a unique location in P a "sufficient
set". Sufficient sets which have no subsets that are both sufficient and
proper are called "minimal sets". "Efficiency" is basically the percent of
the scale you need to hear, on average, before a sufficient set occurs.
This amounts to asking when the listener will be able to measure intervals
by scale degrees. In 12tET's 5-limit diatonic, the key-minimal set is the
diminished triad. [It is interesting to speculate on the fact that this
contains the disjuct fifth (and the only ambiguous interval in the scale),
and that scales without modes (like the "wholetone" scale) are necessarily
without ambiguous intervals.]

So, I would say that "drone-type things" tend to meet both of these
criteria quite well. By the first, one might say that the tonality is
static, but certainly not weak.

By the second, OTOH, the tonality is probably shifting more than in most
western music. Even tho the scales used in such are often improper, the
drone provides the reference needed to measure intervals in scale degrees.
But since the scale is improper, the key will constantly be shifting as
various proper subsets are sectioned off. It could be that Mr. Sanford
does not hear this as being tonal because he is used to scales with a
relatively high efficiency. Improper scales may work best with low
efficiency.

Carl

Carl Lumma wrote,

>Perhaps this is the distinction Doty suggests:

Not at all -- Doty was distinguishing between "common-practice tonality"
as defined by Western "art" music from 1600-1900, and a more general,
cross-cultural definition of the term.

>1) What I've always called "tonality"
> - the psychoacoustic phenomenon of
> fitting a group of pitches to a harmonic series

I'll call that harmonic tonality for now.

>2) What I've always called "key"
> - the cognitive phenomenon of locating a
> stimulus within P (a proper map of pitch space)

I'll call that melodic tonality for now.

>The first one, as I understand it, basically depends on the virtual
pitch
>mechanism, provided roughness is low enough for it to work.

Roughness doesn't interfere with the virtual pitch mechanism, though it
may interfere with its ability to create a sensation of consonance.

>To the extent
>that this does not imply a harmonic series, by combination tones or
perhaps
>some hard-wired preference, tonality can be established by fitting a
group
>to an inharmonic series.

I don't understand that.

Anyhow, do you think the minor third in the minor key is not part of the
phenomenon of harmonic tonality?

>R. calls a stimulus that occupies a unique location in P a "sufficient
>set". Sufficient sets which have no subsets that are both sufficient
and
>proper are called "minimal sets". "Efficiency" is basically the
percent of
>the scale you need to hear, on average, before a sufficient set occurs.
>This amounts to asking when the listener will be able to measure
intervals
>by scale degrees.

Why is that? A random scale has 100% efficiency, unless you meant
"minimal" instead of "sufficient" in the second-to-last sentence above.

In 12tET's 5-limit diatonic, the key-minimal set is the
>diminished triad.

In any other regular diatonic tuning, the key-minimal set is the
tritone. Do you think that spells a qualitative difference between the
diatonic scale in 12tET and that in other regular tunings (I sure
don't!)?

What is the point of saying "5-limit" above?

Do you think it consequential that the Pythagorean diatonic scale is not
proper?

>[It is interesting to speculate on the fact that this
>contains the disjuct fifth (and the only ambiguous interval in the
scale),
>and that scales without modes (like the "wholetone" scale) are
necessarily
>without ambiguous intervals.]

The diatonic scale in meantone tuning is proper (not that I think that
matters) and has no ambiguous intervals.

>So, I would say that "drone-type things" tend to meet both of these
>criteria quite well. By the first, one might say that the tonality is
>static, but certainly not weak.

>By the second, OTOH, the tonality is probably shifting more than in
most
>western music. Even tho the scales used in such are often improper,
the
>drone provides the reference needed to measure intervals in scale
degrees.

I don't understand. Did you mean the reference needed to locate your
position within the scale?

>But since the scale is improper, the key will constantly be shifting as
>various proper subsets are sectioned off.

Very vaguely related -- With Mad Duxx I played a lot of dronal things on
the 22-tET guitar. With 0 as the drone, I found two equally-spaced scale
fragments particularly useful: 18 22 26 and 26 29 32 35. These represent
harmonic series fragments 7,8,9 and 9,10,11,12, respectively. Each one
is great for melody but connecting the two sounds awkward.

>>>"Efficiency" is basically the percent of the scale you need to hear, on
>>>average, before a sufficient set occurs. This amounts to asking when
the >>>listener will be able to measure intervals by scale degrees.
>>
>>Why is that? A random scale has 100% efficiency, unless you meant
>>"minimal" instead of "sufficient" in the second-to-last sentence above.

I mean sufficient. There's nothing that says high efficiency is somehow
good...

This involves a distinction I didn't make earlier. All of R.'s measures
apply to codes, not stimuli. That is, a random scale may have 100%
information, but it can not have 100% efficiency. R.'s idea is that people
need a code (I take this to be very hardware-abstracted stuff) to get
key-effects from pitch data. Thus, the random scale might sound like a
mistuned diatonic scale to an average western listener. The idea is that
after enough exposure to the scale, one learns a code just for it. So
scales are evaluated only by the type of code they imply, and the point of
the whole thing is to predict what kind of music best takes advantage of
the code. This is difficult, but I believe R. has made useful predictions
with it.

>>In any other regular diatonic tuning, the key-minimal set is the tritone.
>>Do you think that spells a qualitative difference between the diatonic
>>scale in 12tET and that in other regular tunings (I sure don't!)?

The nature of the minimal set shouldn't matter that much. What should
matter more is the overall efficiency, which Rothenberg claims is stable
for the diatonic scale across all tunings with reasonable fifths.

>>What is the point of saying "5-limit" above?

In this case, it probably didn't matter. But you never know... it's not
quite the same scale at the 5-limit as it is at the 3-limit, is it?

>>Do you think it consequential that the Pythagorean diatonic scale is not
>>proper?

I think things ought to be run thru some sort of fit function, or whatever,
before being subjected to R.'s measures. Something to prevent 1 cent
differences from impropering them...

I've been screaming in favor of the just diatonic scale on this list for
two years, and in general I believe that almost any scale can be
successfully approximated during skillful tours of (pick your limit) JI.

I've also praised the melodic usefulness of harmonic series segments, which
are often improper. Impropriety isn't bad, but I believe it does influence
how these scales are used. Melody almost always involves making proper
subsets of the master scale, wether or not the master scale is proper
itself. With harmonics 8-15, I noticed myself treating 13 and 15 as
passing tones before I knew anything about propriety. They are good
passing tones, tho, and they are fantastic harmonically...

>>>[It is interesting to speculate on the fact that this contains the
disjuct >>>fifth (and the only ambiguous interval in the scale), and that
scales >>>without modes (like the "wholetone" scale) are necessarily
without >>>ambiguous intervals.]
>>
>>The diatonic scale in meantone tuning is proper (not that I think that
>>matters) and has no ambiguous intervals.

It matters for something. What it is might not be as simple or as strong
as you seem to want it to be.

Here's a few items regarding a possible relation between ambiguous
intervals, minimal sets, and the "disjunct" position in MOS chains:

While the augmented fourth and diminished fiths aren't ambiguous in
meantone, they do represent a blurring between the two classes.

The key-minimal set in meantone is the disjoint 5th, according to you. In
12tET it is the disjoint 5th plus a third note, according to R. For MOS's
that are proper but not strictly so, the disjoint fifth was ambiguous in
every case considered by John Chalmers (tho he just looked at it briefly).

>>>By the second, OTOH, the tonality is probably shifting more than in most
>>>western music. Even tho the scales used in such are often improper, the
>>>drone provides the reference needed to measure intervals in scale degrees.
>>
>>I don't understand. Did you mean the reference needed to locate your
>>position within the scale?

Yes. I am a little weak on the locating part, and need to think about it
some more. Obviously, and the point is, composition plays a big role in
all this. If the composer purposely avoids all sufficient sets, then
efficiency is sort of out the window -> except that it determines what must
be avoided and at what cost...

Carl

P.S. I just noticed, both the pentachoral and symmetrical decatonic scales
are strictly proper in 22tET :~)

The thing Rotherberg and the others miss is the second order MOS's.
Scales such as C E F G B or
G B C D F. If these aren't scales what is!
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com