Re: Defining Just intonation -- summing up

Hello, there, everyone, and I would like to thank both Paul Erlich and
Joe Monzo for their responses to my "summing up" remarks on the
question of defining just intonation (JI).

On one level, Paul and Monz, I would like to respond in turn by saying
that we seem in agreement on some important points, and that this
agreement might have been clearer had I written in a more coherent
manner. A main purpose of this post, indeed, is to affirm what appear
to be our points of common understanding, and also to clarify some of
my motives for focusing on the side of my musicmaking which may be
most marginal or "problematic" from the standpoint of this JI debate.

At the same time, I feel impelled by your responses to recognize that
when I was writing that "summing up" statement I was feeling some real
conflict, and that while not necessarily communicating my ideas that
clearly, I may have let some of my actual ambivalence show.
Recognizing this, with your help, has led me to what I find a
comfortable position on these definitional issues, joyfully affirming
that different people can and will take different positions depending
on their musical and philosophical points of view.

First of all, both Paul and Monz, we very strongly agree that
integer-based music where only prime factors of 2 or 3 are basic or
constraining elements is radically different in many ways from music
where such incommensurable primes as 3 and 5, or 5 and 7, are a
constraining part of the musical equation.

An interesting essay might be written on this distinction and its
implications, both historical and practical.

So far in this thread, I have focused mainly on Gothic/neo-Gothic
music not only because it is where my musical and theoretical passion
is at (along with Renaissance/Manneristic music), but because it may
be a marginal or problematic case from many "JI" viewpoints, and thus
possibly of special interest in a definitional discussion.

However, to illustrate the very important point which both of you
made, I need only draw on a bit of practical experience.

Suppose I play a basic kind of 16th-century piece for four voices,
whether closer to 1500 (e.g. a Spanish villancico) or 1600 (say a
brief motet by Lasso). In meantone, my keyboard technique will
generally be quite distinct from that required in Zarlino's JI tuning
(a 15-note version mapped to two keyboards). The problems of combining
prime factors of 3 and 5 -- keeping them both Just on a sustained
basis -- rapidly become clear.

In contrast, suppose that in a neo-Gothic setting I play a medieval
piece or an improvisation first in Pythagorean (a completely
integer-based tuning), then in 29-tone equal temperament (29-tET). My
technique is likely to be much the same, albeit with subtle shifts in
the flavor of some intervals, maybe comparable to shifting from
1/4-comma to 2/7-comma or 1/6-comma meantone for 16th-century music.

This is a difference not only of mathematics or of aural experience --
it is one felt with the hands, the ways they navigate an intonational
geometry.

Another difference which our dialogue may have brought out is that in
neo-Gothic rational intonations, where 2:3:4 represents a full stable
concord, the intonation of most other intervals leaves immense room
for freedom. In a system based on the pure and stable 4:5:6 or
something yet more complex, there are, to borrow Donald Hall's
language, much sparser "degrees of freedom."

Your remarks, Paul and Monz, and also Dave Keenan's focus on locked-in
partials, illustrate a related point: typical "JI" is much more bound
up with the harmonic series (in usual timbres, at least) than
integer-based Gothic/neo-Gothic.

Having relished Dave's humor about designing playing the Hammond organ
with the _intent_ to have its integer ratios construed as JI, I might
point to a more obscure kind of humorous gloss on one of my "summing
up" remarks.

When referring to "modern" ratios such as "28:33:42" or "14:17:21," I
was doing so with a certain wry awareness of using them as
neo-medieval variants on a usual 54:64:81, with equally little
reference to partials in any of these three instances. As Dan Stearns
once suggested by invoking a stirring Latin motto, this is the freedom
of the plateau.

In short, there is an excellent case to be made -- and both of you
have made it -- that a just tuning system constrained by maintaining
pure ratios involving multiple primes, and largely determined by the
harmonic series or relations between partials, might well be placed in
a different category than an integer-based tuning where only such
ratios as 2:3:4 are constraining or "harmonicity-based" elements.

Dave, you have offered a very nice example of this intuitive
conversational distinction in your paper on single chain of fifths
tunings, one well worth quoting, where you distinguish this category
of tunings from others:

"Just tunings are not in this class because their
intervals (other than fifths) are not generated
or approximated by chains of fifths at all."

While you have recently proposed a definition of JI which can include
Pythagorean, I might take this quote not as a position entailing its
exclusion, but as a conversational shorthand for "Just tunings (other
than Pythagorean, not the most typical case, which obviously _is_ a
chain-of-fifths tuning)."

As we have briefly discussed, Monz, there may be an historical reason
for this: "JI" in its typically paradigmatic sense of "5-limit"
emerges in Continental European theory around the later 15th century
(e.g. Ramos 1582), and more definitively in Fogliano (1529) and
Zarlino (1558), as a deliberate departure from Pythagorean intonation.

Modern discussions of vocal intonation, for example, follow this usage
when they speak of "Pythagorean" and "Just" as two contrasting
categories or tendencies, rather than the first as a subset of the
second, which carries the obvious implication "5-limit Just."

Also, Paul, in your comments about "out-of-tuneness," you remark on
the at least arguable tension between the vertical and melodic
dimensions in JI. In Pythagorean or some related Gothic/neo-Gothic
system, the two dimensions generally seem in equilibrium; Carl
Dahlhaus has in fact argued that the breaking of this symmetry is one
feature of the transition from medieval to Renaissance and later
technique.

In my first "summing up" post, my intended theoretical message was
something like this: "The fact that many typical 'JI' concepts feel
uncomfortable to me in a Gothic/neo-Gothic context suggests that maybe
this music is different from that of a usual JI setting, and might fit
better under a distinct paradigm such as 'rational intonation' (RI)."

Now I may be able more coherently, I hope, to list some of the
possible differences:

-----------------------------------------------------------------------
Typical JI (5-limit or higher) Gothic/neo-Gothic RI
-----------------------------------------------------------------------
Complex real-time keyboarding Often simple keyboarding
Nonregular tunings Often regular/quasi-regular tunings
Ratios suggest partials (5:6) Ratios often "nonharmonic" (32:27)
Incommensurate multiple primes Only 2 and 3 essentially constrain
Sparse degrees of freedom (Hall) Most intervals quite free (flavors)
Entropic valleys pervasive Most intervals unstable, on plateaux
Ideal of beatlessness Ideal of diversity and contrast
Preference for small integers Love of large nuanced integers
Vertical/melodic tradeoffs Equilibrium between dimensions
-----------------------------------------------------------------------

Of course, these intonational contrasts typically correlate with very
tangible differences in musical style: compare 13th-century
compositions of Continental Western Europe fitting a Pythagorean
tuning with vocal compositions of the 16th century suggesting an ideal
of 5-limit JI (very possibly adaptive in practice).

A question this kind of approach -- which I give you both credit for,
Paul and Monz, while assuming full responsibility for my gloss here
with whatever imperfections it may have -- raises, is whether both
approaches felicitously can fit under the single tent of "JI."

What they share in common is a basis in integer ratios; but those
ratios may actually mean or imply different things: in one case
partials, in the other case simply whatever numbers or nuances of
positioning on a plateau happen to strike the musician's fancy at
given time or for a given piece.

Similarly, a neo-Gothic use of 29-tET, a Renaissance-like use of
31-tET, and an avant garde serialist use of 11-tET all might fit in
the category of "equal temperament" -- but just how unified is this
category in terms of style or practice?

Given the diversity of musical cultures, styles, and tuning systems,
people will draw lines and associate patterns in different ways. The
problem of defining "JI" is only one illustration.

My own rather serene conclusion is that in discourse settings where
people tend to associate "JI" with integer ratios, I would describe
Gothic/neo-Gothic RI and Renaissance 5-limit music as subsets of JI.

In discourse settings where people tend to associate "JI" with 5-limit
or higher systems, or with a predominantly harmonicity-based outlook,
I would stick with the term "RI" (or something more specific, as the
context may invite, e.g. "Pythagorean") for Gothic/neo-Gothic music,
while reserving "JI" for Renaissance or other contexts where it may
have its expected meaning.

The connection has been drawn between the multiple dictionary entries
for common English words such as "run," and the multiple definitions
of intonational terms. I find this multiplicity natural in both
cases.

To borrow another comparison involving natural languages, I might
suggest that just as the tense-aspect systems of different languages
may draw different lines between categories of time and the temporal
structure of situations, so different people may draw such lines in
grouping integer-based tunings together, or tunings with pervasive
locking-in partials, etc.

It is fascinating to explore the margins or outlying areas of a
concept through a dialogue like this: adaptive tunings where vertical
intervals remain pure (or in some systems close to pure) based on
tempered melodic steps; non-integer-based tunings with Setharean
equivalents of "locking-in partials"; systems based in large part or
even in whole on complex integer ratios; and so on.

Writing around 1100, the theorist known as John of Afflighem or John
Cotton opened a discussion on polyphony by saying that different
people treat it in different ways, but he would share his own approach
based in good part on the concept of contrary motion.

Nine centuries later, I hope that this thread has shown the virtues of
the same pluralistic approach, with much room for recognition of
commonalities and creative differences alike.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@egroups.com, "M. Schulter" <MSCHULTER@V...> wrote,
among many other things:

> http://www.egroups.com/message/tuning/16417
>
>
Typical JI (5-limit or higher) Gothic/neo-Gothic RI
------------------------------------------------------------------
Complex real-time keyboarding Often simple keyboarding
Nonregular tunings Often regular/quasi-regular tunings
Ratios suggest partials (5:6) Ratios often "nonharmonic" (32:27)
Incommensurate multiple primes Only 2 and 3 essentially constrain
Sparse degrees of freedom (Hall) Most intervals quite free (flavors)
Entropic valleys pervasive Most intervals unstable, on plateaux
Ideal of beatlessness Ideal of diversity and contrast
Preference for small integers Love of large nuanced integers
Vertical/melodic tradeoffs Equilibrium between dimensions
------------------------------------------------------------------
>
> Of course, these intonational contrasts typically correlate
> with very tangible differences in musical style: compare
> 13th-century compositions of Continental Western Europe fitting
> a Pythagorean tuning with vocal compositions of the 16th century
> suggesting an ideal of 5-limit JI (very possibly adaptive in
> practice).
>
> A question this kind of approach -- which I give you both
> credit for, Paul and Monz, while assuming full responsibility
> for my gloss here with whatever imperfections it may have --
> raises, is whether both approaches felicitously can fit under
> the single tent of "JI."

Margo, I thank you for this excellent summary of
intonational/stylistic differences centered around the
transition between the predominance of Pythagorean and
that of 5-limit JI theory.

I think this is an important step forward in illuminating
the 'progress' (I hesitate to use that term) of European
music-theory.

I've admitted here already that I too have been 'guilty'
of labelling my own work as 'JI' when 'RI' would be much
more appropriate. Upon further reflection, I suppose that
many composers and theorists have used 'JI' simply to
demarcate a boundary between anything that can be construed
as JI/RI, and work tuned in 12-tET or other temperaments.

I too am very appreciative of the dialog we've had on this
topic lately, and look forward to evolving my Dictionary
entry on 'just intonation' into a very detailed exploration
of the subject. Thanks very much to you and Dave Keenan
above all (but not meaning to slight anyone else's contributions
either) for the progress we've made in clarifying the
various meanings of the term.

And again, I'm very happy that we were able to revive this
discussion in a much more beneficial way this time around,
and to avoid the ill-feeling that surfaced a few months
ago when it was first brought up. And I concur with Dave K.
in lamenting the fact that Kraig Grady was not a part of
this dialog (and also that Daniel Wolf's voice was absent).

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'