Consistency and musical context -- for Dan Stearns

Hello, there, Dan Stearns, and thank you for your remarks about
consistency, which move me to the remarks that follow.

First of all, I would say that the "goodness to fit" of a tuning
depends largely on what is stylistically fitting for the music in
question.

Since something like 29-tET has an excellent "goodness to fit" with
many of the styles I love, I'm not shy about advocating and promoting
it here and elsewhere.

If people are looking for "goodness to fit" with a style seeking pure
or near-pure ratios of 5, then there are lots of likelier tunings.
What 29-tET means to me is marching to a different musical and
intonational drummer: "accentuated Pythagorean" thirds and sixths,
also some gentle "submajor/supraminor" thirds and sixths, and those
charming ratios near 13:15, 10:13, and 15:26.[1]

As I often tend to say, and as many of your comments suggest also,
what could helpfully be articulated in many of these discussions are
the often implicit musical asssumptions that go into deciding what a
tuning _should_ fit.

In responding to your comments and Paul's about consistency, I will
attempt something of a synthesis: consistency is a very valuable
concept when used _descriptively_ to appreciate some of the patterns
of a tuning. Either consistency or inconsistency can be musically
attractive, depending on what one is doing.

For example, one obvious and endearing feature of 13-tET is its
"inconsistency" with some basic Pythagorean and other patterns of
interval arithmetic as defined in the Western European tradition of
composition.

In Pythagorean tuning, an 8:9 whole-tone or major second defines the
difference between a 3:4 fourth and a 2:3 fifth. In 13-tET, however,
the best approximation of 3:4 (5/13 octave, ~462 cents) plus the best
approximation of 8:9 (2/13 octave, ~185 cents) does _not_ add up to
the best approximation of 2:3 (8/13 octave, ~738 cents).

Further we find that, to use traditional interval names in order to
demonstrate their dubious applicability, two "fourths" at 5/13 octave
add up to an excellent "major sixth" at 10/13 octave (~923 cents), an
interval which I consider a prime attraction of this tuning. It makes
possible some beautiful coloristic sonorities, as well as guiding a
range of cadential resolutions where it expands to the octave.

Of course, for my typical purposes, 13-tET invites a process of
"Chowningization" to match partials and maximize the consonance of the
8/13-octave and 5/13-octave intervals to get "2:3-like" and "3:4-like"
stability. However, that's also a stylistic preference, and someone
else might prefer a less "concordant" approach.

Here consistency can serve two purposes: to caution against evaluating
the tuning of a "6:8:9" in 13-tET by looking only at the isolated
ratios and not how they fit together; and to celebrate some
distinctive aspects of this tuning which break the usual Western
European patterns, and invite new musical approaches.

For example, I tend to realize a "6:8:9" kind of feeling in 13-tET by
playing a sonority which _looks like_ this pattern on the keyboard,
for example "G3-C4-D4," actually a rounded 0-462-646 cents. To my ear,
it has an overall "quartal/quintal" quality, and I've dubbed it a
"Crunchy Pepper" sonority in honor of Keenan Pepper's famous article
on chords with a "crunchy" quality.

(However, in a 13-tET setting, I'm not sure how "crunchy" this
sonority is in a Pepperian interpretation, where "crunchiness" means
the presence of one dramatically "dissonant" interval. Is 7/13 octave
really such a dissonance in this setting? -- I'm not sure. The overall
effect is something like a usual 6:8:9 in another tuning.)

Since 13-tET is inconsistent for the basic 6:8:9:12 ratios of
Pythagorean tuning, we shouldn't be too surprised that it also turns
out to be "nonunique" for some elementary 14th-century Western
European cadential patterns.

A very effective cadence has a sonority with two adjacent 5-step
intervals -- 0-5-10 steps or 0-462-923 cents -- expanding to the basic
trinic concord of 0-8-13 steps or 0-738-1200 cents.

Interestingly, we might say that the lower 5-step interval in 0-5-10
represents a large "cadential major third" (maybe 7:9 or 10:13), and
the upper 5-step interval a "fourth."

The outer 10-step "major sixth," in contrast, fits a usual paradigm of
neo-Gothic interval sizes just fine. Indeed, just as in medieval
theory, this excellent major sixth is equal to a "tone-plus-fifth"
(_tonus cum diapente_), that is, an 8-step "fifth" plus a 2-step
"whole-tone."

Thus a full appreciation of 13-tET from a neo-medieval kind of
perspective involves recognizing both the inconsistent/nonunique
aspects of the tuning, and some more "conventional" ones. The mix of
familiar and unfamiliar patterns is a big enticement to draw me in.

If we take consistency as a _descriptive_ tool to note some patterns
of a tuning, rather than _prescriptive_ tool to say what's musically
fitting apart from a given style, then it can enhance both our
understanding of the patterns in question and our ability to make more
informed musical choices.

Happily, Paul and Dan, I don't see any inevitable conflict between
your ways of viewing this: when and where one happens to be seeking a
fit with classic "n-limit" JI systems, consistency _is_ one very
relevant facet of "goodness to fit." In a decatonic setting, for
example, it makes sense to verify the best approximations of the
ratios in a 4:5:6:7 tetrad can indeed all be combined at once in a
tuning we are examining -- or that the tuning set is large enough that
this issue may be less significant (a factor you've noted, Paul).

Similarly, I would take harmonic entropy as an approach to musical
consonance/dissonance or simplicity/complexity which itself does not
prescribe whether "dissonance" should be minimized or maximized, but
looks at some of the factors which may affect the perception of a
given interval or sonority in a given type of timbre.

For example, from some of my reading of this theory, I might guess
that I often like ratios at once rather complex and relatively
"concordant," adjusting timbre to "pastelize" (reduce sharply defined
clashing or matching of partials) concord/discord. With 13-tET, I go a
bit further, typically "Chowningizing" to move the "third partial"
close to 8/13 octave or around 738 cents.

This raises an important question: to what degree might my appraisal
of a ratio around 9:7 (~435.08 cents) as "relatively concordant"
reflect my frequent practice of pastelizing to get an effect of
richness and complexity while muting the tension between the fourth
and fifth partials? These remarks might apply with even more force to
something like the 17-tET major third at 6/17 octave or ~423.53 cents,
which can feel quite "rich but relatively concordant" in a pastelized
kind of texture.

Here I should add that something like F#3-Bb3-Eb4 before E3-B3-E4 also
sounds "in tune" in 1/4-comma meantone, with "major thirds" at 25:32
and "major sixths" at 75:128 (~427 cents and ~931 cents), likely in
part because of the cadential context, and maybe in part because the
"near-7:9:12" effect might suggest something like a Partchian
"otonality."

By the way, Paul, maybe your "consistency" concept could be viewed as
one subset of a more general concept of "collocation" -- can all the
relevant ratios of this tuning for a given desired sonority be
combined above the same note at the same time; and, if so, in how many
positions?

It seems to me that the "collocation" concept could be applied to
almost any type of tuning, although "consistency" might be n-tET
specific.

----
Note
----

1. As was mentioned in a recent thread, how one looks at 29-tET may
either make irrelevant or bring to the forefront the possible issue of
a "syntonic comma" equivalent. In a neo-Gothic setting, 29-tET is like
Pythagorean in having a regular or eventone structure: usual major and
minor thirds are defined simply as four fifths up and three fourths
up, respectively, without such complications. In 29-tET, from this
viewpoint, the ~41.38-cent diesis is analogous to the Pythagorean
comma or the enharmonic diesis in meantone, serving like the latter as
a distinct kind of melodic "step." However, if one seeks to use as the
usual thirds the closest approximations of ratios of 5 -- from a
neo-Gothic viewpoint, also approximations of 14:17 and 17:21 or the
like -- then the diesis becomes in effect equivalent to a "syntonic
comma" of 41 cents, not necessarily the most desirable result for
classic 5-limit music, as you commented, Paul.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> This raises an important question: to what degree might my appraisal
> of a ratio around 9:7 (~435.08 cents) as "relatively concordant"
> reflect my frequent practice of pastelizing to get an effect of
> richness and complexity while muting the tension between the fourth
> and fifth partials?

Here's an experiment you can use to answer this question: start with
a 4:5, and gradually increase the size of the interval, listening
carefully every step of the way, to 7:9 and beyond. Do you hear the
interval becoming more concordant as you approach 7:9, and more
discordant as you move beyond it? If not, then pastelization might be
playing a strong role here. Personally, even with strongly harmonic
timbres, I don't find 7:9 as an isolated dyad to be a local minimum
of discordance in low or medium registers, but in high registers I do.

> Here I should add that something like F#3-Bb3-Eb4 before E3-B3-E4
also
> sounds "in tune" in 1/4-comma meantone, with "major thirds" at 25:32
> and "major sixths" at 75:128 (~427 cents and ~931 cents), likely in
> part because of the cadential context, and maybe in part because the
> "near-7:9:12" effect might suggest something like a Partchian
> "otonality."

Yes, I think the ear will tend to pick up on the 7:9:12 pattern.

> By the way, Paul, maybe your "consistency" concept could be viewed
as
> one subset of a more general concept of "collocation" -- can all the
> relevant ratios of this tuning for a given desired sonority be
> combined above the same note at the same time; and, if so, in how
many
> positions?

I don't see how consistency could be viewed as a subset of this
concept.

> It seems to me that the "collocation" concept could be applied to
> almost any type of tuning, although "consistency" might be n-tET
> specific.

Graham Breed (and probably Dave Keenan) have applied the consistency
concept to linear temperaments as well -- but I never understood that
because you can always find better approximations if you go far
enough out in the chain of generators.