Yahoo Tuning Groups Ultimate Backup tuning Re: N-limits, intervals, and affects

Hello, there, and after seeing some of the recent comments about the
perceived qualities and affects of intervals built from ratios with
various prime factors, I'd like to offer a few possible cautions and
stray observations.

First, a caution to my cautions: if someone hears an interval in a
certain way, or feels a certain aesthetic or emotional reaction,
that's just as much a part of reality as anything I'm about to
state. My concern is not to say what is the "right" or "wrong" way to
hear a 3-based interval, for example, but rather to question a
possible assumption that "everyone" necessarily hears it the same way,
especially when musical settings and expectations change.

Secondly, I'd like to agree with a point which I recall Gary Morrison
made: a lot of this discussion may be focusing on categorical
perceptions of common intervals in certain musical traditions
(e.g. Western European composition), as opposed to the fine nuances of
tuning systems. For example, as this previous contributor pointed out,
someone discussing "3-ness" may actually be focusing on the generic
qualities of fifths (3:2) and fourths (4:3), rather than on alleged
qualities that these intervals share with 27:16 (M6) or 243:128 (M7).

Having offered these opening remarks, I'll now turn to my main point:
the impression a given interval based on a given prime factor (or
factors) makes can depend both on the immediate musical context and on
the listener's expectations.

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1. Intervallic relativity: context counts
-----------------------------------------

For example, the most characteristic intervals of either the 3-limit
(3:2, 4:3) or the 5-limit (5:4, 6:5) may sound quite "simple" or even
"bare" when set in a context where more complex sonorities are the
rule. Rich and full 3-limit Gothic harmonies can sound "empty" when
placed in the environment of a 5-limit Renaissance texture; and the
5-limit harmony of Monteverdi or Mozart can sound "bland" or
"restrained and formal" when surrounded by the ninth and eleventh
chords of contemporary jazz or pop, or for that matter by the
saturated tetrads of 7-limit just intonation (JI).

I would propose that this effect might best be seen as an attribute,
not of 3-limit or 5-limit harmony, but of the musical situation in
which a given sonority is judged in the context of a setting replete
with more complex stable sonorities. So understood, this effect can
be exploited by composers, as it has been at least since the 16th
century.

The converse of this is that a new interval tuning can have a special
enchantment, maybe especially when the categorical interval itself is
on the brink of becoming stable in a given style. Thus a sonority of
4:5:6 seems "odd" in a Gothic 3-limit system, and likewise 4:5:6:7 in
a Classical 5-limit context, despite some theoretical discussion in
both eras (e.g. Walter Odington, c. 1300, for 5:4 and 6:5; Kirnberger,
maybe in his 1771 treatise(?), for the "harmonic seventh").

New sonorities of this kind seem to have a siren-like charm, gently
seducing the attuned listener to accept not only a new interval or
prime factor in tuning but a new way of listening to and making music.
A possible danger of this experience is that one may draw the
conclusion that only through obtuseness or even deliberate
obscurantism could previous theorists or composers have _failed_ to
take advantage of such an "obvious" resource!

"How could Jacobus of Liege have failed to perceive the beauty of a
pure 5:4 as opposed to an 81:64, or Zarlino neglected to describe the
beauty of a pure 7:4 as opposed to a 9:5, except by stubbornly
adhering to a concept of music theory elevating mathematical dogma
above direct sense experience?"

In fact, part of the process of "acculturation" to Gothic or
Renaissance/Manneristic harmony is the discovery of how a "less
consonant" tuning for an interval such as 81:64 or 9:5 nicely fits
within the style. Such cases suggest that from a musical point of
view, the most significant "mistuning" of an interval may be a
listener's extra-stylistic and thus "mistuned" expectations.

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2. Intervals and impressions: variety is the spice of life
----------------------------------------------------------

Turning to ways in which we can response to an interval or
combination, I'd like to say that indeed different people can hear
things in different ways -- not to mention the same person at
different times.

For example, considering the example mentioned in earlier posts of the
pure 3-limit fifth or fourth (3:2, 4:3), I might describe the effect
as one of "spaciousness" or "opulent clarity." Maybe this response
isn't unrelated to the fact that I really like these intervals, and
for some 30 years have often tended to seek out music favoring them
(e.g. European Gothic, some Chinese and African polyphonies, some
20th-century compositions by Debussy and Bartok, etc.).

However, 20th-century harmony texts confirm that these intervals _can_
be used to convey such impressions as "coldness" and "emptiness," as
well as more neutral "vastness." This might raise a question: is my
perception of "spaciousness" some kind of permutation of "vastness
plus delight"?

Then, again, I've noticed that a 5:4, or better yet a 5:1 (as can
occur with certain ambient sounds, maybe even the running of a
computer's fan), can have an "ethereal" feeling maybe not unkindred to
this, or even an "eerie" quality.

Such responses may reinforce the association of such pure intervals
and intonation systems with spirituality: Perotin and Notre Dame (the
cathedral being under construction during his lifetime), or Palestrina
and the Sistine Chapel, or Rossi and the synagogues of Venice.

At the same time, these same intervals and intonation systems can take
on the color of music and text in a variety of genres. Thus the
3-limit harmony of the 13th century feels "intimate" and "sweet" to me
in a love song such as _Je n'amerai autre_ (actually a vocal line
added to a three-voice hocket), and likewise "friendly and cheerful"
in an instrumental hocket such as _PORTARE_. Another likely
instrumental piece of the era, _In seculum viellatoris_ (an early
example of explicit instrumentation?), has a "sweet" quality for me,
somehow inviting an association with "Appalachian fiddle music" -- a
style I'm actually much less acquainted with than Gothic music.

In fact, maybe since around 1968, I've considered _Je n'amerai autre_
a great theme song for some movie, a kind of "13th-century jazz."

Similarly, 5-limit music of the 16th century has an incredible range
of expressive possibilities, and theorists such as Vicentino, Zarlino,
and Morley seem much occupied with the use of intervals to reinforce
the affections of a text. Songs of this era can be quite plangent or
even bawdy, tertian JI and all.

---------------------------------------
3. Systemic and polysystemic approaches
---------------------------------------

Specific subjective responses to specific intervals and tunings are,
of course, valid and valuable -- relativity doesn't imply irrelevance.
Of special interest to me are cases where a tuning seems to bring out
a categorical potential of an interval or combination in an ideal way.

For example, focusing on 3-limit intervals, I might find a pure 6:8:9
"a resonant and energetic fusion," and a 16:24:27 "_vibrant_." In the
first case, the justness not only of the fifth and fourth but of the
9:8 major second contribute to the total effect of relative blend or
concord. In the second case, the "bouncy" quality of the major sixth
at 27:16 adds vibrancy or color, as well as the tension of the 9:8.
This isn't to say that the same sonorities might not have a similar
effect in 12-tone equal temperament (12-tet), or even in 1/4-comma
meantone; but a pure 3-limit tuning seems to "optimize" the effect.

Also, I find it maybe more helpful, rather than seeking a single ethos
for a whole system such as 3-limit, to explore some of the specific
qualities of particular intervals within such a system. Such qualities
will often be linked to specific musical styles and usages, leaving
open the possibility of yet-undiscovered qualities to be evoked by
styles not yet invented.

For example, as already noted, I find a 27:16 pleasantly "bouncy" or
"vibrant," an effect likely tied to the Gothic expectation that the
interval will typically expand to an octave, or resolve obliquely to a
fifth. The 13th-century description _tonus cum diapente_ or
"whole-tone plus fifth" suggests some qualities of what might be
called "9-ness" (3:2 x 9:8), and indeed M6 and M2 are often ranked
together as mild discords or (in Jacobus) "imperfect concords."

As in many musical settings, it's difficult to separate harmony from
melody. An interval is not only how it sounds alone, but how it tends
to move. What might "unify" such diverse Gothic 3-limit intervals as
81:64 (M3), 27:16 (M6), and 16:9 (m7) is the way they can all resolve
to a stable and conclusive fifth or octave by conjunct contrary motion
involving a concise 256:243 melodic semitone in one of the voices:

a b e' f' d' c'
f e g f e f
81:64 3:2 27:16 2:1 16:9 3:2

The 243:128 (M7) can pack quite a bit of "zap," and its oblique
resolution to an octave by a 256:243 semitonal ascent in the upper
voice is an attention-catching opening of some organa by Leonin and
Perotin:

e' - f'
f
243:128 2:1

Likewise, at or near this more dissonant end of the spectrum, the
128:81 (often ranked with or near 243:128) has a very expressive
resolution by a descending 256:243 semitone sometimes used in a final
cadence:

c' - b
e
128:81 3:2

At the same time, a variety of intervals such as 81:64, 128:81, and
243:128 also invite resolutions where both voices move by generously
large 9:8 whole-tones, and these progressions lend their special
flavor to 13th-century harmony:

a g c' d' e' d' c' d
f g e d f g b a
81:64 1:1 128:81 2:1 243:128 3:2 256:243 4:3

Such nuances, of course, are shaped by style as well as by the
inherent "geometries" of the intervals themselves; but exploring
specific tunings and styles in this way may be more revealing than
simply seeking one adjective to characterize "3-limit" or "5-limit,"
etc.

Another way of looking at this kind of approach is that each tuning
has its own potential contrasts, as well as affinities. Thus while a
3:2 or 4:3 in 3-limit may be "spacious," a 256:243 is "tense" or
"keen," the first adjective maybe fitting its quality as a
simultaneous interval and the second as a melodic semitone. In
5-limit, 5:4 and 6:5 may be "sweet," but 9:5 is rather "tense." In
7-limit, a 7:4 may have an "enchanting mildness" and a 7:6 may be
"Bluesy," but a 9:7 (especially as an interval to the lowest part,
e.g. 14:18:21 as opposed to 6:7:9) is "strident."

In JI compositions drawing resources from several n-limit systems, we
get into the dilemmas and opportunities of "polysystemic" analysis:
how do these systems, and the intervals within a given system, relate
when mixed? It's easy to demonstrate that 3-limit intervals can sound
"empty" when the listener's ear has been saturated with 7-limit
tetrads -- but could a composer artfully combine both systems, making
them both seem "rich" or "saturated," but in different ways?

Just as harmony and counterpoint have focused on an artful progression
of intervals or combinations within a given n-limit system, so
polysystemic analysis would seek out artful progressions or mixtures
of sonorities derived from diverse n-limit systems. Maybe such
explorations could be the subject of a new thread.

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4. Conclusion
-------------

At its best, the sharing of impressions about various n-limit
intervals may itself enrich our sense of diversity: both the diversity
of perceptions possible, and our diversity as a community of people
involved in many different tuning systems and musics.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: N-limits, intervals, and affects

🔗M. Schulter <mschulter@xxxxx.xxxx>

🔗vog <vog@xxxxx.xxxx>

🔗Kraig Grady <kraiggrady@xxxxxxxxx.xxxx>