Yahoo Tuning Groups Ultimate Backup tuning Julia Werntz: A sisterly response

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To Julia Werntz: A sisterly response
On the reasons for a plurality of tunings
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Dear Julia Werntz:

What a pleasure it is, however unmerited, to share reflections on the
nature and motivating reasons for musical systems of intonation with a
theorist, composer, and educator such as yourself. In such dialogue,
each idea can be a springboard to another, and one felicitous
observation may call forth the next, in a delightful kind of
conceptual antiphony recalling the splendor of the Venetian school
from Willaert and Zarlino to the Gabrielis and Aleoti.

Having so far had an opportunity to meet your published statements
only at second hand, it is with a certain cautious temerity that I
address a few ideas which I have seen quoted or paraphrased from your
writings, often ideas that prompt me to reflection regarding some of
my own musical goals and their possible antecedents in this most rich
art of ours.

To the following remarks I attach the qualification, always prudent
but here especially imperative, that I warmly invite correction of any
errors I make regarding your writings and views, be they matters of
fact, interpretation, or more subtle distortions of emphasis and
context.

In this endeavor, however tentative, I would follow the good counsel
of approaching aesthetic issues hypothetically or suppositionally, as
it were, rather than absolutely. That is, one may take a certain
aesthetic or stylistic postulate as the basis for a given manner of
music -- or to explain the beauty of that manner -- while leaving open
the merits of very different or even apparently contradictory
hypotheses which might lead to equally beautiful musics.

For example, while much of my music and intonational practice is based
on the postulate that "a ratio of 14:11 or 9:7 is an excellent size
for a major third," this postulate supposes a certain manner of
treating these major thirds, one in which they tend to be relatively
concordant but active intervals which often expand to stable fifths at
or near 3:2.

To do intonational justice to the exquisite music of the 16th century
in Western Europe, indeed a tradition marked by the masterly
chromaticism of a Lasso, Vicentino, Marenzio, or Gesualdo, we must
entertain the different hypothesis that a major third is ideally tuned
at or quite near to the ratio of 5:4.

Here there is no absolute contradiction, but rather a contrast between
different styles, each with its own assumptions, which on a grander
scale may be taken alike as joined in a common concentus of reason and
beauty.

Similarly, Harry Partch favors a "monophonic" system of intonation,
here referring not to the performance of a single melody (as opposed
to polyphony), but to a scheme where all pitches are seen, if I
understand correctly, as related to a single central note.

Yet in what I shall term focal pluriphonic harmony, we typically have
tuning systems based on a set of notes forming intervals and vertical
combinations "pleasantly related to each other," but not necessarily
to a single central pitch. I shall discuss this kind of style in what
follows, since it may illustrate a most engaging theme of your
writing: the attractions of a contrast between concord/discord or
stability/instability which may be achieved in a range of tuning
systems.

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1. Just intonation and its rationales
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As a just intonation advocate who often favors a diverting mixture of
simple and complex ratios, and the most attractive contrasts in the
concentus which such mixtures offer, I would like to clarify two
distinct aspects of a "just" worldview -- a worldview which may also
find expression, to a considerable extent, in many irrational
temperaments also.

Although the _purity_ of intervals is a frequently emphasized theme of
a "just" viewpoint, I would like to focus at least equally on the
often complex _rationality_ of intervals as an element, both musical
and extramusical, in the lure of a just intonation system.

Of course, purity is a part of the picture, both historical and
current: I would find it a usual assumption that a "just" tuning
should include many instances of pure fifths and fourths at 3:2 and
4:3, as is true both in the Pythagorean tunings of Gothic Europe, the
font of focal harmony, and in some modern systems mixing these ratios
with pure ratios of 7, 11, 13, or 17, etc.

However, as the writings and instruments of such an illustrious
musician as Kathleen Schlesinger amply demonstrate, a delight in a
complex rational structure, for example an arithmetic division of the
monochord, can play an equally vital role in shaping a just system.

There is a special pleasure, at once musical and intellectual, in
including in a single tuning, along with stable ratios of 3:2 and 4:3,
diverse unstable concords such as major thirds at 81:64, 33:26, 14:11,
23:18, and 9:7. Musically, of course, all of these sizes can fit the
same basic musical "grammar" of a major third striving to expand to a
fifth as its perfect goal, yet there is an elegant diversity of
ratios.

Moreover, a complex ratio such as 33:26 may have the special charm of
at once drawing in the intellect with its rational proportion, and yet
presenting to the ear a kind of intriguing complexity grasped only in
part. Thus the beauty perceived and enjoyed by the senses, albeit
imperfectly, may serve as a worthy ladder of contemplation.

Thus just intonation may fulfill two aesthetic criteria which, if I
understand correctly, your own writings look on as causes of delight:
a mixture of familiar and less familiar intervals; and a contrast
between "concordance/discordance" or "simplicity/complexity," however
these concepts or categories might be defined in a given style.

For those of us following an approach of focal harmony, drawing its
roots from the compositions of such artists as Machaut and Landini,
and enriched by the resources of the Arabic, Persian, Kurdish, and
Turkish traditions, the contrast is often one between the stable 3:2
or 4:3 and the many kaleidoscopic shades of relative concord,
assonance, or discord provided by a congenial tuning system, and
especially a larger one of 17 or 24 notes, for example.[1]

Two aspects of many of these just systems should be frankly
acknowledged. First, for example in systems based on a diverse set of
superparticular ratios, not all regular fifths are pure: some may be
"tempered by ratio" a few cents wide, rather like those in meantone
(albeit there in the opposite direction), while others may have ratios
removed from 3:2 by as much as a full 64:63 or so, that is, at or
around 32:21.

Secondly, a just and fully circulating system would seem to require at
least 53 notes, since this is the size of the cycle of pure fifths
which "virtually closes" when one tunes a chain of 52 such fifths or
fourths and adds the octave of the starting note. If one requires
musical circularity as well as purity of fifths and fourths -- apart
from the last "odd fifth" about 3.62 cents narrow -- then such a cycle
might provide one ideal just solution.

Of course, one might argue that this just and pure tuning is, in
effect, a "quasi-equal" division of the octave; I mention it because
its copious resources may serve, as it were, as a bridge between the
"just" and "equal" schools of intonation.

This leads me to another type of tuning which is neither just nor
equal, but either strictly regular, or else irregularly well-tempered,
and may also admirably serve the needs of focal harmony as an
alternative to either conventional tonality or atonality.

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2. Open regular tunings
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Although just tunings provide one ready approach to a mixture of
simplicity, complexity, and contrast, another approach is an
irrational temperament based on a regular chain of fifths somewhat
larger than pure.

For example, I have found a regular tuning with fifths at about 704.61
cents as a very pleasant scheme in 24 notes. Here we have regular
major and minor thirds close to 14:11 and 33:28, plus middle thirds
not too far from 17:14 and 21:17, and large major and small minor
thirds close to 9:7 and 7:6.

From a melodic viewpoint, we have regular diatonic semitones at about
77 cents; some pleasant chromatic semitones at around 132 cents; and
small semitones or dieses at about 55 cents. There are also some
distinctively "microtonal" steps of around 22 cents for subtle shifts
and nuances.

Such a system, although open, is freely transposible within the range
of the gamut, and permits a diversity at once vertical and melodic. It
fits especially what I might term a _combination-based_ technique, in
which assorted intervals unit in a pluralistic set of multi-voice
combinations or sonority. One can observe this kind of technique in
the music of a Perotin or Machaut, and it takes on new possibilities
in the intonational atmosphere of the 21st century.

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3. New well-temperaments
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In addition to regular temperaments with fifths gently larger than
pure, we have such a compendious system as George Secor's 17-note
well-temperament of 1978, which almost 25 years later has a wealth of
beckoning options to offer a creative composer or improviser.

The scheme of this microtonal master combines tempered ratios of 3 and
7 with some near-just ratios of 11 and 13 in a most ingenious and
musically fertile manner, one admirably seeming as if custom-designed
for a focal style of harmony with stable fifths and fourths and active
regular thirds and sixths, but in fact supporting a wealth of styles
drawing inspiration from many world musics.

Here I should also mention that Paul Erlich has applied the device of
unequal and irregular temperament to a 22-note cycle, and that the
possibilities here for a variety of cycle sizes are still largely
unexplored. As is sometimes said in the natural sciences: "In this
area, almost everything remains to be done."

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4. "Unusual" tunings and new concords
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Given the reality of stylistic relativity in a world filled with
diversified musics and tuning systems, I would like to emphasize that
an "exotic" tuning means one unfamiliar to a specific artist -- and
possibly the classical and canonic norm for another.

Thus while an equal or near-equal fivefold division of the octave
seems common in a range of world musics, the tuning of 20 equal steps
to the octave might be "exotic," with its 720-cent fifths, to a
musician accustomed to Western European systems where fifths are
customarily for the most part within about 7 cents of 3:2.

The purpose, at least from my point of view, in using such a "novel"
system is not to seek out sheer discord, but rather to explore
tuning/timbre combinations yielding new patterns of concord, tension,
and subtle contrast.

For example, 20-equal has a major third of 420 cents, close to 14:11
or 23:18 and thus a commonplace in focal harmony; but the 720-cent
fifths and 480-cent fourths, quite concordant in timbres chosen to
this view, result in a different kind of harmonic geometry, and the
interval of 240 cents may delightfully serve as either a large major
second (quite close to 8:7) or kind of "quasi-minor-third" contracting
to a unison by stepwise contrary motion.

As Dan Stearns, a distinguished composer in this temperament, has
observed, it has some of the attributes of 5-equal -- which, I might
add, is a rough approximation of the slendro scale of gamelan.

The search for new consonance, achieved through the kind of timbral
adjustment discussed by such musicians and theorists as Pierce,
Darreg, Chowning, Carlos, and Sethares, has a particular allure when
applied to tunings such as the division of the octave into 13 equal
limmas, or yet more radically into 11 equal apotomes.[2]

In a more "usual" kind of tuning for focal harmony, the presence of
fifths and fourths at or rather close to 3:2 and 4:3 provides a kind
of "natural gravitation," as it were, toward these intervals as stable
concords of resolution.

In 13-equal, however, the equivalent of the fifth is at around 738
cents; and in 11-equal, at around 655 cents. The sense of stability
and resolution provided by such intervals seems an "artificial
gravity" of timbre, artistic convention, and sheer historical
association. Here, at least in my kind of focal or "quasi-focal"
approach to these tunings, the goal is not maximal discord but rather
enchanting ambiguity: a 462-cent interval in 13-equal might be taken
as either a stable "quasi-3:2" or an active cadential "quasi-9:7" or
"quasi-13:10." In 11-equal, timbred to maximize the concord of the
6-step or 655-cent interval, this interval might represent either a
"quasi-3:2" or a kind of tritone -- which it literally is.

Such "new consonance" is only one approach to such tunings, and I
would emphasize that some other styles seem an attractive counterpart
to 12-tone serialism or pantonalism. While some critics might describe
this kind of music as pursuit of arbitrary dissonance for its own sake
-- consider Schoenberg's alleged "indifference to beauty" in
dodecaphony -- musicians knowing and practicing such an art may have a
very different view.

One might argue, of course, that any "novel" tuning or music will
eventually become familiar; but in the process, the range of the
familiar and the scope for further innovation have been alike
enlarged.

Thus music has its multitude of reasons, each enriching and
ornamenting the overall art.

In conclusion, please let me emphasize that I have not sought here to
address your presentation on the specific attractions which have led
you and other most artful colleagues to a 72-note equal temperament,
but rather to present a kind of counterpoint, possibly combining
elements of similar and contrary motion, which may find its
realization in an agreeable texture of dialogue and friendship.

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Notes
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1. Focal harmony combines standard 14th-century European progressions
with "less conventional" European elements such as the direct
chromaticism and intriguing if often problematic intonational nuances
advocated by Marchettus of Padua (1318), and also resources such as
neutral steps and intervals documented, for example, in medieval Near
Eastern practice and theory as well as other world traditions.

2. These descriptions somewhat playfully suggest that the 13-equal
step of 1/13 octave or ~92.31 cents is quite close to the Pythagorean
limma or diatonic semitone at 256:243 (~90.22 cents); while the
11-equal step of 1/11 octave or ~109.09 cents is not too far from the
Pythagorean apotome or chromatic semitone at 2187:2048 (~113.69
cents).

Most appreciatively,

Margo Schulter
mschulter@value.net

Julia Werntz: A sisterly response

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