Yahoo Tuning Groups Ultimate Backup tuning Re: Linear tunings, dual-purpose sonorities: beyond "odd-limits"

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Linear tuning systems and dual-purpose sonorities:
Aesthetic judgments beyond the stability limit
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One approach to consonance/dissonance is to take these categories as a
simple dichotomy, linking them to the concept of an "odd limit" in a
just intonation (JI) system. In such an approach, for example, all
ratios involving odd integers greater than 3 define "dissonances" in a
3-limit (Pythagorean) JI system, e.g. the major third at 81:64; and
likewise all ratios involving odd integers greater than 5 in a 5-limit
system, e.g. the "proximate minor third" or "neutral third" at 11:9.

While this approach may indeed fit the practice and theory of some
musics, I would like to argue here that at least two historical
systems may invite another approach: the Pythagorean JI of the 13th
and 14th centuries in Continental Western Europe; and Nicola
Vicentino's music in the mid-16th century based on a division of the
octave into 31 equal or nearly equal intervals of about 1/5-tone.

If we realize Vicentino's system as 1/4-comma meantone with pure major
thirds, as his own remark that his archicembalo ("superharpsichord")
has a tuning based on that of ordinary keyboard instruments may
suggest, then our two tuning systems here in question are both "linear
tunings." That is, each derives its scheme from the just tuning of the
most choice concord: the Gothic fifth at 3:2, or the Renaissance major
third at 5:4.

Sometimes such tunings have been called "one-dimensional" or "1-D"
tunings, a term which may be from one perspective descriptive (the
tuning can be mapped as a series of intervals along a single line) and
from another possibly unfortunate, since Pythagorean and 31-note
meantone tunings offer many "dimensions" of complexity and subtlety.

Here I would like to argue that much of this complexity and subtlety
may be found not only in the "hardware," the notes of the tuning
themselves, but in the "software," the musical styles to which the
tuning is applied.

Specifically, I will suggest that a two-fold distinction between
"consonance" and "dissonance" may be insufficient to express some of
the fine distinctions in Gothic "3-limit" polyphony and in Vicentino's
modified "5-limit" polyphony. Interestingly, Western European
theorists of the 13th century often recognize five or more gradations
of vertical tension, and Vicentino also notes distinctions between
intervals of his tuning system falling outside the usual "5-odd-limit"
of stability.

In such musics and theories, it is quite possible for an interval to
be "unstable but relatively concordant," or for a relatively
"discordant" interval to be part of a relatively concordant sonority.
If we carry such an approach yet further, it may be possible to make
distinctions even between unequivocally strong discords, and to ask
how well an alternate tuning system fits such written and unwritten
categories of style and theory.

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1. Ludmila Ulehla's "dual-purpose" sonorities
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In her textbook on _Contemporary Harmony_, the modern composer and
theorist Ludmila Ulehla proposes a threefold scheme for analyzing
intervals and sonorities: concord/dual-purpose/discord. While concords
serve as points of repose, and discords seek resolution to concords,
dual-purpose sonorities may be at once active and independently
euphonious.

To refine this approach further, Ulehla proposes to use the lines and
spaces of a musical staff for a visual graph showing the fluctuating
levels of tension represented by the significant vertical sonorities
in a given composition as judged in its own stylistic context. If we
place a complete stable sonority on the lowest line of the staff, and
what we regard as the most acute discords on the highest line, this
graphing technique permits nine grades of tension.

While she applies these concepts to European and related compositions
from the Classic era (c. 1750-1800) to contemporary practice, the same
concepts might equally apply to medieval and Renaissance music, and
have a special kinship to the subtle and sophisticated scales of
"concord/discord" presented by theorists of the 13th-century tradition
from Johannes de Garlandia to Jacobus of Liege.

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2. Odd-limits as stability limits: the dual-purpose complication
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As interesting as Ulehla's concepts and their historical antecedents
might be from the viewpoint of musical composition and performance,
they are equally significant in shaping our concepts of tuning
systems. Specifically, the concept of a "dual-purpose" sonority raises
the possibility that some intervals in a given system may be
relatively concordant but unstable: the medieval 81:64 and 32:27, for
example, or Vicentino's ~11:9 and ~18:11.

Such intervals have ratios beyond the "odd-limit" of a 3-limit or
5-limit system, but nevertheless are treated as relatively concordant
in practice and theory. This suggests that the odd-limit might best be
defined as the _stability limit_ of a system, not necessarily the
"consonance limit," if one joins medieval and Renaissance theorists in
recognizing degrees of consonance or concord which may apply to
unstable as well as stable intervals.

An implication is that the tuning of "dual-purpose" intervals, as well
as that of stable concords, may be quite significant in evaluating an
alternative tuning system for music of a given period or style. If we
were to perform medieval polyphony in 17-tone equal temperament
(17-tet), or Vicentino's fifthtone compositions in 53-tet, we might
thus wish to evaluate the tuning not only of stable 3-odd-limit or
5-odd-limit intervals, but of "dual-purpose" intervals which may add
both euphonious vertical color and subtly gradated vertical tension to
the music.

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3. Pythagorean dual-purpose sonorities (1200-1420)
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Although classical Pythagorean theory, as codified by Boethius
(c. 480-524), draws what appears to be a simple "3-odd-limit"
distinction between consonance and dissonance, medieval theorists
recognize various sonorities beyond this limit to be "imperfect
concords" or to "concord well," although they are treated as
unstable.

As early as around 1030, Guido d'Arezzo lists acceptable intervals in
polyphony as including not only stable unisons and fourths, but also
the whole-tone (9:8), semiditone or minor third (32:27), and ditone or
major third (81:64). In contrast, he excludes the minor second
(256:243) and tritone (729:512).

Theorists of the 13th-century tradition beginning with Johannes de
Garlandia (c. 1240?) recognize major and minor thirds as "imperfect
concords" -- or, in the case of Jacobus of Liege (c. 1325), as "medial
concords." These intervals may be used, for example, to open pieces.
Jacobus additionally describes the _quinta fissa_ or "split fifth," a
three-voice sonority "splitting" an outer fifth into two thirds,
either with the major third above and the minor third below (string-
ratio 81:64:54), or the converse arrangement (string-ratio 96:81:64).
While he prefers the first arrangement, Jacobus recognizes that the
second is also permissible, citing the opening sonority of a motet
found in the Bamberg and Montpellier Codices (A3-C4-E4, with C4 as
middle C, and higher numbers showing higher octaves).

Also, both Jacobus and Coussemaker's Anonymous I (c. 1300, possibly
Jacobus at a younger age?) recommend the three-voice combination of a
major ninth "split" by the third voice into two fifths (e.g. G3-D4-A4)
as "concording well," and this fits the style of certain 13th-century
and 14th-century dialects of polyphony. This sonority, with its ratio
of 9:6:4, represents another sonority beyond the 3-odd-limit deemed
_relatively_ concordant. A similar status may apply to the minor
seventh "split" into two fourths, e.g. G3-C4-F4 (16:12:9), and to a
fifth "split" into fourth and major second, e.g. G3-C4-D4 (string-
ratio 12:9:8) or G3-A3-D4 (string-ratio 9:8:6).

Thus it follows that while the relatively concordant major and minor
thirds of the Gothic era at 81:64 and 32:27 are outside a 3-odd-limit
of stability, their tuning is not a matter of indifference. An
alternate system realizing them as 5:4 and 6:5, or as 9:7 and 7:6,
would be changing the vertical color and tension of the music in a
very significant way.

Similarly, although the major ninth (9:4), minor seventh (16:9), and
major second (9:8) are outside the 3-odd-limit, their participation in
relatively concordant sonorities with a majority of fifths and/or
fourths makes them of considerable interest in evaluating any
alternate tuning for the Gothic music of Continental Western Europe.

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4. Vicentino's fifthtone music: beyond the 5-odd-limit
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The practice and theory of the Renaissance, more than that of the
Gothic, might appear to fit a dichotomous view of "consonance and
dissonance." Generally speaking, intervals within the 5-odd-limit
(unisons, octaves, fifths, thirds, and sixths) are "concordant," while
seconds, sevenths, and augmented or diminished intervals are
"discordant." The fourth (4:3), treated as a concord between upper
voices but as at least a quasi-discord in relation to the lowest
voice, somewhat complicates this otherwise fairly consistent scheme.

However, in his treatise of 1555, Nicola Vicentino (1511-1576) takes
us into an alternative universe of Renaissance music based largely on
the common practice of the period, and yet radically altered by the
introduction of new melodic and vertical intervals resulting from a
division of the tone into five equal or nearly-equal parts..

For the most part, Vicentino's vertical sonorities in practice feature
the usual 5-odd-limit concords and conventional discords of the epoch:
the radically new quality of the music results from the melodic use of
such intervals as the chromatic semitone and the enharmonic diesis or
fifthtone.

However, he also recognizes vertical intervals beyond the 5-odd-limit
either new to his tuning, or already found in a conventional 12-note
meantone tuning but now occurring at more locations in the gamut.

Of special interest is the "proximate minor third" of "nine minor
dieses," i.e. ~9/5-tone. Vicentino gives its "irrational ratio" as
being approximately 5-1/2:4-1/2, i.e. 11:9. He finds it a good
concord, being located between the minor and major third, and somewhat
leaning toward the ideal concord of the latter interval.

Likewise he finds the "proximate minor sixth" -- which would have an
approximate ratio of ~18:11 as the octave complement of the proximate
minor third -- acceptably concordant..

At least if we may judge from Vicentino's rather small sample of
fifthtone compositions and examples, however, "consonant" in this
sense does not necessarily imply "stable." The proximate minor third,
for example, occurs as a cadential interval when a voice progresses by
fifthtone steps from a minor to a closing major third, the raising of
a note by a diesis or fifthtone here being shown by an asterisk (*):

C4
Ab3 Ab*3 A3
F3

However, it does not to the best of my knowledge occur as a closing
interval, or even as the point of repose for an internal cadence. Thus
Ulehla's category of a "dual-purpose" sonority might best fit
Vicentino's theory and practice.

In contrast, Vicentino finds the "proximate major third" a diesis
larger than the usual concord, for which he gives the approximate
ratio of 4-1/2:3-1/2, i.e. 9:7, as "not so good," although tolerable
in running passages. He explains that this interval is between the
major third and the fourth, leaning toward the latter interval with
its equivocal status, and thus tending somewhat toward discord.

He also finds the "proximate major sixth" a diesis wider than the
usual major sixth (also known as a diminished seventh in 12-note
meantone tunings) to tend toward a "seventh," and thus to lean toward
discord.

Interestingly, he suggests a similarity between this last interval and
the "minimal third" a diesis less than minor (also known as an
augmented second in 12-tone meantone tunings). For Vicentino, the
normal minor third is already narrow enough that any further narrowing
will make it "resemble a second," and therefore lean toward
dissonance. He advises that at least from a usual viewpoint of
vertical euphony, this interval like the proximate major sixth might
best be "set aside."

Such judgments are indeed general guidelines, not categorical rules,
since Vicentino himself uses a sonority including both a proximate
major third (or tenth) above the bass and a proximate major sixth:
B2-F#2-B3-Eb4. He asserts that intervals and progressions not so
pleasing in themselves may be used where the sense of a text requires,
an axiom also invoked about 50 years later to justify the bold
dissonances of a Monteverdi or Gesualdo.

From a viewpoint of tuning theory and practice, it is interesting that
Vicentino seems to follow the 5-odd-limit of stability usual for
Renaissance music, but recognizes the proximate minor third of ~11:9
and proximate minor sixth of ~18:11 as relatively concordant, finding
the proximate major third (~9:7), and the proximate major sixth or
minimal third (with ratios of ~12:7 and ~7:6) to tend more toward
"dissonance."

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5. Some tentative conclusions
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Since linear tunings, specifically Pythagorean or 3-limit JI and
meantone, are so important in medieval and Renaissance music, the use
in this music of "unstable but relatively concordant" intervals beyond
the applicable "odd-limit" would seem a significant feature to be
noted in designing or evaluating tuning systems for such musical
styles.

This means that measures of "fit" should consider not only stable
concords within a given odd-limit, but "dual-purpose" sonorities
beyond it: for example 81:64 and 32:27 thirds and 9:6:4 sonorities in
a Gothic setting, or ~11:9 proximate minor thirds in Vicentino.

Of course, optimizing stable concords is a first priority; but
relative concords may also play a vital role in defining the color and
directed motion of a vertical style.

Carrying such distinctions a step further, we might argue that even
shades of difference in tension between unequivocal discords in a
given style might be significant. One example suggested by Easley
Blackwood is the comparatively "milder" quality of the regular
meantone minor second (~117.11 cents) as a vertical dissonance
vis-a-vis the chromatic semitone (~76.05 cents).

While this distinction might hardly seem one of first priority for
Renaissance music, it may yet be a significant nuance of the style. At
least one theorist suggests that augmented or diminished octaves were
regarded as especially strong discords, although this did not prevent
their use as vertical intervals in practice.

To conclude, at least in a stylistic and theoretical setting such as
Gothic music or Vicentino's music, our approach to tunings might take
into account not only stable concords within a given odd-limit, but
also relative concords or "dual-purpose" sonorities beyond this limit,
and even distinctions between grades of clear discord.

How one might weigh such many-sided aspects of a tuning system,
quantitatively or otherwise, remains an interesting question: my
purpose here is not to suggest a comprehensive solution, only to open
a discussion.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Linear tunings, dual-purpose sonorities: beyond "odd-limits"

🔗M. Schulter <mschulter@xxxxx.xxxx>

🔗William S. Annis <wsannis@xxxxxx.xxxx>