Yahoo Tuning Groups Ultimate Backup tuning JI and Sethares: terrains and textures

Hello, there, everyone, and in response to recent threads on classic
and adaptive forms of just intonation (JI) and related issues of
consonance which mention the approach of xentonality and "sensory
consonance" developed by Bill Sethares, I'd like to suggest that both
approaches can serve as the basis for beautiful musics.

Personally I embrace both types of approaches, at one moment rejoicing
in a JI system with pure sonorities of 2:3:4 and 14:18:21:24; and at
the next, taking amazed delight in a meantone "well-timbrement" where
the 738-cent diminished sixth G#-Eb of 1/4-comma can serve as a stable
and consonant fifth.

My own musical experiences with the wonders of JI systems and
Setharean tuning/timbre systems alike leads me to favor a subtle and
complex approach to questions of consonance/dissonance. Such concepts
as sensory consonance, Paul Erlich's harmonic entropy, and stylistic
concord/discord may all address important dimensions of the question.

Here I would like especially to address two basic points. The first is
that the "sensory consonance" of Sethares -- a factor or dimension I
might term smoothness/roughness -- can interact with Erlich's harmonic
entropy or what I'll call simplicity/complexity in assorted
permutations.

A second important point is to distinguish between the general theme
of "reasonably low integer ratios" in various world musics, and the
specifics often associated with "JI" in many discussions, for example
a style where 4:5:6 or 4:5:6:7 serves as a favorite sonority. My own
frequent leaning toward other kinds of sonorities with simple as well
as complex ratios maybe gives me another perspective on culturally or
stylistically specific assumptions.

Above all, I'd like to emphasize that what I say about timbre/tuning
reflects my own pragmatic experience, however idiosyncratic or
otherwise, as well as theory.

Performances like those on Jacky Ligon's CD _Galunlati_ show the
beauties of xenharmonic and xentonal scale design realized in
exquisite musical practice, with human voices, acoustical instruments,
and synthesizer sounds.

The artistry of a Balinese or Javanese gamelan ensemble, or of an
ensemble performing medieval European music in Pythagorean JI, or
Renaissance music in 5-limit JI (subtly adaptive, maybe), also show
how theory can take shape in beautiful practice.

While often focusing on JI systems and the xentonality of Sethares,
I'd like to emphasize that various kinds of historical or other
temperaments may bridge a continuum between these two contrasting
models, with striking musical patterns sometimes possible when new
timbres are applied to familiar tunings.

--------------------------------------------------------
1. Terrains and textures: Entropy and sensory consonance
--------------------------------------------------------

In approaching the fruitful and often astonishing universe of
timbre/tuning interactions, I would like to suggest a two-dimensional
model in which the sensory consonance of Bill Sethares and the
harmonic entropy of Paul Erlich or Dave Keenan may complement each
other, with myriad permutations and choices open for musicians.

Here I will address "sensory consonance" first, because its musical
salience and power is so dramatically tangible. Very simply, as
demonstrated by traditional gamelan ensembles as well as by composers
of new musics such as Ivor Darreg or Bill Sethares, the factor of
timbre can radically alter perceptions of consonance/dissonance or
stability/instability.

For me, an early and effective demonstration of this sonorous reality
was with 17-tET, a tuning where the regular major third has a size of
6/17 octave or ~423.53 cents. In the synthesizer timbres I was using,
this interval seemed a bit too "strange" or "tense" to serve routinely
as a Gothic-style major third, which in a 13th-14th century kind of
European style should be active and unstable, but _relatively_
concordant.

At some point, however, I happened to try 17-tET using a more gentle
timbre on the Yahama TX-802 called "Puff Pipes" (voice A56), and found
this same interval and tuning absolutely delightful for a 14th-century
style in the manner of Guillaume de Machaut. Now those major thirds
had a quality of "partial concord," at once rather blending and
intriguing, like Pythagorean major thirds at 81:64 (~407.82 cents) can
have in assorted somewhat brighter timbres.

Recently I have been relishing a more dramatic application of sensory
consonance: a "well-timbred" 1/4-comma meantone in which a diminished
sixth at around 738 cents is musically interchangeable with a regular
fifth at around 697 cents; and likewise a diminished fourth at 32:25
(around 427 cents) with a usual major third at a pure 5:4 (around 386
cents).

As it happens, the "Piccolo" timbre of the TX-802 (voice A27) can give
1/4-comma meantone this quality of a "well-timbrement," with sensorily
concordant forms of the usual 16th-century consonances (at least to my
ears) available from every step of a 12-note tuning. Two such
circulating 12-note keyboards a diesis apart (128:125, ~41.06 cents)
make available a gamut of "modal color," with many opportunities for
creative musical ambiguity.

For example, an interval of 32:25 might represent now a remote
transposition of a usual Renaissance major third; now a
Renaissance-style diminished fourth with a resolution typical of that
era; and now a bold neo-Gothic major third inviting a 14th-century
kind of resolution to some variety of stable fifth.

For more on meantone well-timbrement, and some sound files most helpfully
and skillfully produced by Jacky Ligon, please see the latest issue of
_The Microtonal Activist_ or TMA, volume 2, an edition featuring articles
on topics ranging from the related theme of tunings/timbres with stretched
or compressed intervals (Jacky and Jeff Scott) to Joseph Pehrson's lecture
at Moscow Conservatory's Theremin Center (transcription by Anton Rovner)
to Robert Walker's invitation to compose for the spontaneous musician:

<http://www.geocities.com/jacky_ligon/TMA_archive_index.htm>

Having illustrated both a "moderate" application of Sethareanization
with 17-tET in a neo-Gothic setting, and a dramatic application with
1/4-comma meantone, I would now like to turn to a harmonic entropy
kind of approach as one kind of strategy for putting these phenomena
in a larger perspective.

While Setharean "sensory consonance" often focuses on the "smoothness"
of an interval or sonority, the harmonic entropy of Erlich or Keenan
focuses on what I might term the "simplicity" of an interval.

For example, as we have seen, a 17-tET major third at around 423.5
cents may be either "rough" or "smooth" depending on the timbre; from
a viewpoint of harmonic entropy, it is in any event "complex," located
in a "plateau" region between the two simple ratios of 5:4 (~386.31
cents) or 9:7 (~435.08 cents).

Similarly, the 737.64-cent diminished sixth of 1/4-comma meantone is
historically famed as a "Wolf fifth" with its "howling" beats in a
typical Renaissance or Baroque timbre -- the essence of "roughness" --
but can become intriguingly smooth when "well-timbred." Again,
however, from a perspective of harmonic entropy, it remains a complex
or plateau interval, in contrast to the usual meantone fifth at about
696.58 cents located in the "valley" around the simple ratio of 3:2
(~701.955 cents).

Combining the perspectives of sensory consonance _and_ harmonic
entropy, we find ourselves in a tapestry of terrains and textures.
While traditional JI systems focus on smooth valleys like 3:2 or 5:4,
Setharean xentonality often explores timbres bringing about smooth
plateaux: a relatively concordant major third at 424 cents, for
example, or even a stable one at 427 cents.

What harmonic entropy may tell us is that while a smooth valley and a
smooth plateau may be in some ways analogous or even musically
interchangeable, they are not musical identical. Of course, this
conclusion is quite concordant with the outlook of Bill Sethares
himself, who presents xentonality as an alternative approach to
conventional JI or other kinds of tunings, not a replacement for these
systems.

To borrow the valley/plateau imagery, which Dave Keenan has developed
in very poetic ways, Setharean tuning/timbre combinations may place us
in a garden-like park -- at an elevation of 4,000 meters or 13,000
feet. While the local texture of the landscape might remind us of a
familiar valley, the atmospheric pressure and other features of our
environment are quite different.

In a circulating meantone well-timbrement with 12 notes per octave in
1/4-comma, for example, the difference of some 41 cents in the size of
the diatonic and chromatic semitones will lend a different effect than
in an unequal well-temperament such as Werckmeister where these sizes
are not so disparate, let alone 12-tET.

To conclude for the moment these remarks on a topic inviting open
dialogue, I mean to suggest that an outlook recognizing both
smoothness/roughness and simplicity/complexity or valley/plateau
distinctions can give us a perspective including the approaches of
xentonality and traditional "valley-oriented" systems, including the
JI systems of various cultures.

----------------------------------------------------------
2. Low integer ratios: some cultural aspects of JI systems
----------------------------------------------------------

While some of us define "JI" in the most general sense as a system of
music based on integer ratios, large or small, there has recently been
a tendency to use the term "Rational Intonation" (RI) for this kind of
integer-based concept, with JI suggesting more specifically a focus on
_small_ integer ratios.

One very creative theorist, Dave Keenan, has suggested that JI should
be defined not in terms of integer ratios, but in terms of its
tangible musical effect: a "locking in" of partials, which can provide
a basis for tuning an interval (e.g. 7:6) by ear. Keenan emphasizes
that a sonority including complex intevals or dyads not in themselves
"just" may have an overall and recognizable justness, e.g. 16:19:24

Cautioning that in what follows I speak only for myself, not for
Keenan and others who have contributed to the theory of JI and
tempered tunings, I find in this kind of approach a suggestion that
there may be various kinds of just sonorities.

In many musical cultures, for example, we have sonorities based on
pure fifths, fourths, and octaves, e.g. 2:3:4 or 3:4:6. A Pythagorean
JI system makes these concords available in pure ratios, while also
making available pleasing contrasts of melodic steps (pentatonic,
diatonic, chromatic, or more complex).

Such a system also makes available pure ratios combining these
intervals with major seconds or ninths at 9:8 or 9:4, and minor
sevenths at 16:9, for example 4:6:9 or 6:8:9. These sonorities occur
in traditions as diverse as Japanese koto music, Laotian music for the
khene (often known as a "mouth organ"), and the vocal polyphony of
Georgia or Gothic Europe.

Of course, a 2-3 JI or Pythagorean system also makes available a
host of intervals with more complex ratios, which may represent
various degrees of stylistic concord or discord depending on the
musical language and also, at least to some degree, on the timbre.

Another possibility, evidenced both in the sruti system of India and
in Renaissance Europe is a 2-3-5 JI system featuring major thirds at
5:4 and minor thirds at 6:5. Possibly because of the contrast between
"just" and "tempered" systems which had become clear in 16th-century
Western Europe, the term "JI" often tends to imply specifically this
kind of system, although the more general concept of "low integer
ratios" or "audible locking-in of partials" covers a much wider
ground.

An alternative approach is a 2-3-7 JI system -- or some might say a
2-3-7-9 system (listing all relevant odd factors) -- featuring pure
sonorities such as 12:14:18:21 or 14:18:21:24. This is typical of one
main approach to neo-Gothic music, where these sonorities provide a
contrast with stable concords of 2:3:4 -- while also having a rather
concordant effect themselves.

Such an approach might be somewhat analogous to what may be a more
familiar type of 2-3-5-7 JI system, where 4:5:6:7 may serve as an
unstable but rather concordant sonority resolving to 4:5:6 or the
like.

From what I take as a Keenan/Erlich type of perspective, ratios of 11
may still participate in recognizably just dyads (e.g. 11:7, 11:8,
11:9), but nearing the limit of recognizability for _simple
intervals_. Thus something like 17:14 would not be considered in
itself a "just" interval, although it might be used in some larger
sonority considered recognizably just such as 10:12:14:17.

In contrast, the use of a sonority such as 14:17:21 in a neo-Gothic
setting may represent RI rather than a Keenan type of JI: here 17:14
is regarded simply as a very interesting flavor of minor (or
supraminor) third, rather than necessarily reflecting any kind of
unifying structure of partials.

From an harmonic entropy perspective, the question might still be
raised: could the total effect of 14:17:21, influenced possibly by the
"anchoring" outer fifth or the difference tones, represent some kind
of "local minimum" or "valley"?

From a neo-Gothic perspective, the main point is simply that 14:17:21
is an intriguing sonority with beautiful resolutions, and that the
integer ratio provides an apt emblem and landmark for this general
type or flavor of sonority (somewhere around 0-336-702 cents).

From a Keenan perspective, the label "just" would imply a valley where
intonational effects are sensitive to small changes in tuning: would
changing the sonority to 0-331-702 cents or 0-340-702 cents audibly
affect its "purity"?

If not, then a Keenan approach, if I understand it correctly, would
regard 14:17:21 as a variety of "tempering by ratio" -- that is,
choosing a given region which is not sharply defined by a distinct
"valley," Erlich's "entropy minimum."

This kind of approach might apply not only to JI or RI systems, but to
tempered systems -- in fact, Erlich's main interest, and also an
important one for Keenan.

For example, let us consider some resources of 36-tET for a neo-Gothic
style, realized as a "24-out-of-36" system with two 12-note manuals,
each with a 12-tET set, at 1/6-tone apart. (With a full 36-note
tuning, all intervals would be available from all steps.)

Either manual will have quite close approximations of such 2-3 JI
sonorities as 2:3:4, 6:8:9, 4:6:9, etc.

Additionally, either manual will include sonorities of 0-300-700
cents, which has been noted as a nice approximation of 16:19:24, and
could also be taken as not far from the Pythagorean 54:64:81 (around
0-294-702 cents).

Combining notes from the two manuals makes available a favorite kind
of neo-Gothic sonority closely approximating another type of "JI,"
based on factors of 2-3-7-9: for example 14:18:21:24 (here
0-433-700-933 cents) or 12:14:18:21 (here 0-267-700-967 cents).

Additionally, 36-tET has excellent renditions of the more complex
14:17:21 type (here 0-333-700 cents), which might be considered RI as
opposed to JI in Keenan's interpretation.

Thus from one perspective, 36-tET has an assortment of low-integer JI
sonorities, as well as the more "RI-like" approximation of 14:17:21.

From a stylistic perspective specifically seeking 4:5:6 or 4:5:6:7,
however, 36-tET would not be considered a noteworthy "JI-like" tuning,
because the closest approximation to a pure 5:4 is the "not especially
close" 400-cent major third.

This example may indicate how, within a general framework of "JI based
on recognizably pure intervals," the same tuning may be considered
either an outstanding "optimization" (36-tET as an approximation of
2-3-7-9 JI), or not so outstanding (36-tET as an approximation of
2-3-5 or 2-3-5-7 JI). For the latter type of system, of course, going
to 72-tET neatly adds accurate approximations of 5, as well as other
types of intervals.

Maybe I should conclude this section with a curious aside on
"rootedness," and the possible "rootless cosmopolitanism" of many
neo-Gothic sonorities involving thirds, treated as often relatively
concordant but unstable.

Such forms as 6:7:9, 7:9:12, 12:14:18:21, 14:18:21:24, 14:17:21, and
28:33:42 (around 0-284-702 cents) -- the latter two on the RI side --
share the property of not having Erlich's "rootedness," with the
lowest note as an octave or power of two in relation to a fundamental.
Compare 4:5:6, 4:5:6:7, or also 16:19:24.

Mark Lindley, writing of the transition in the 15th century from
Pythagorean intonation to meantone on keyboards, has suggested that
the stylistic difference pointing toward the choice of meantone may be
not so much the use of lots of thirds and sixths, but their use in
sonorities intended to have more "solidity" than Pythagorean tuning
would permit.

In contrast, as I have found with Pythagorean tuning, the "floating"
quality of a texture like early 15th-century fauxbourdon with a series
of sixth sonorities eventually moving to a cadence -- but in a kind of
prolonged "free fall" of mild coloristic instability -- can be very
pleasing with complex ratios, or with simple but not "rooted" ones
such as 7:9:12.

A similar kind of effect might explain some of the appeal of "gently
Sethareanized" neo-Gothic tunings like 17-tET: a sonority like
0-424-918 cents, or 0-424-706 cents, is "smoothed" but still
intriguingly complex, in contrast to a "rooted" form.

-------------------------------
3. An open xenharmonic universe
-------------------------------

Having suggested the interplay of smoothness/roughness and
simplicity/complexity in permutations shaped by the factors of tuning
and timbre, and emphasized the variety of JI systems and outlooks
(with some mention of temperament also), I might conclude by offering
a few observations on "smoothed plateaux" as one very fertile and
creative choice, but not _the_ one choice of merit.

For example, while a meantone "well-timbrement" is a choice opened by
a Setharean approach which I find most beautiful and engaging, it
does not make traditional meantone timbres (or their more or less
close counterparts on synthesizer) less beautiful or relevant in the
21st century.

For music in Renaissance and Xeno-Renaissance styles, whether
conventional or highly experimental, traditional timbres bring out the
"justness" of the pure 5:4 major thirds -- and also the strident
quality of a 32:25 diminished fourth or the like, lending a unique
color to the music which Sethareanizing somewhat mutes or
"pastelizes."

SImilarly, for meantone music mixing Xeno-Renaissance and neo-Gothic
elements, the cadential action and excitement provided by a sonority
like 0-427-931 cents (e.g. F#3-Bb3-Eb4 resolving to E3-B3-E4) with its
usual "beatful" quality can be most exhilarating.

Pastelizing favors circulation, or more generally "interchangeability
with modal color," and also adds much room for creative ambiguity, but
without the sharper definitions and contrasts of a more traditional
timbre.

On a 24-note instrument with two 12-note manuals, either choice
permits such options as Renaissance/Manneristic fifthtone progressions
of a kind favored by Vicentino and Colonna, as well as neo-Gothic
progressions (often mixing notes from the manuals) treating diminished
fourths and sevenths as cadential major thirds and sixths, for
example.

Pastelizing to the point of well-timbrement opens up full
transposibility within the framework of a Xeno-Renaissance style, and
also bring us into a universe of "modal color" with musical geometries
and permutations possibly not yet adumbrated, let alone explored in
practice.

Of course, it is precisely practice which often reveals such patterns.
Thus while a bit of arithmetic leads me to propose a "well-timbred"
24-note meantone system including 288 "modalities" -- the 12 modes of
Glareanus (1547) or Zarlino (1558) transposed to each of the 24 steps
of the system -- actual musicmaking is the way to find out about its
realms of sonority and motion.

The mixing of neo-Gothic styles and progressions with Xeno-Renaissance
ones adds to these possibilities -- without, of course, making
dedicated neo-Gothic tunings obsolete, or other kinds of meantone
timbres offering more colorful contrasts for this kind of stylistic
mixture.

My main point in this discussion may be that the concept of a
continuum does not imply a uniform or featureless terrain, but rather
a diverse choice of terrains and textures offering new choices without
invalidating more familiar ones, which may be appreciated for their
uniqueness all the more.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗Paul Erlich <paul@stretch-music.com>

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

> Here I would like especially to address two basic points. The first
is
> that the "sensory consonance" of Sethares -- a factor or dimension I
> might term smoothness/roughness -- can interact with Erlich's
harmonic
> entropy or what I'll call simplicity/complexity in assorted
> permutations.

I absolutely agree. For example, in the tuning of the CEGAD chord,
which I've been discussing with Bob Wendell, it's the
smoothness/roughness consideration which is the dominant factor in
determining an "optimal" tuning for the chord. For other chords, such
as a half-diminished seventh chord, the reverse seems to be true.
>
> In approaching the fruitful and often astonishing universe of
> timbre/tuning interactions, I would like to suggest a two-
dimensional
> model in which the sensory consonance of Bill Sethares and the
> harmonic entropy of Paul Erlich or Dave Keenan may complement each
> other, with myriad permutations and choices open for musicians.

Again, I've always emphasized that this needs to be a two-dimensional
model just as you spell out. However, there may be some
misunderstanding about the "choice" aspect of this. All choices
should always be open for musicians. I don't see what a model of
concordance/discordance has to do with this.
>
> Here I will address "sensory consonance" first, because its musical
> salience and power is so dramatically tangible. Very simply, as
> demonstrated by traditional gamelan ensembles as well as by
composers
> of new musics such as Ivor Darreg or Bill Sethares, the factor of
> timbre can radically alter perceptions of consonance/dissonance or
> stability/instability.
>
> For me, an early and effective demonstration of this sonorous
reality
> was with 17-tET, a tuning where the regular major third has a size
of
> 6/17 octave or ~423.53 cents. In the synthesizer timbres I was
using,
> this interval seemed a bit too "strange" or "tense" to serve
routinely
> as a Gothic-style major third, which in a 13th-14th century kind of
> European style should be active and unstable, but _relatively_
> concordant.
>
> At some point, however, I happened to try 17-tET using a more gentle
> timbre on the Yahama TX-802 called "Puff Pipes" (voice A56), and
found
> this same interval and tuning absolutely delightful for a 14th-
century
> style in the manner of Guillaume de Machaut. Now those major thirds
> had a quality of "partial concord," at once rather blending and
> intriguing, like Pythagorean major thirds at 81:64 (~407.82 cents)
can
> have in assorted somewhat brighter timbres.
>
> Recently I have been relishing a more dramatic application of
sensory
> consonance: a "well-timbred" 1/4-comma meantone in which a
diminished
> sixth at around 738 cents is musically interchangeable with a
regular
> fifth at around 697 cents; and likewise a diminished fourth at 32:25
> (around 427 cents) with a usual major third at a pure 5:4 (around
386
> cents).
>
> As it happens, the "Piccolo" timbre of the TX-802 (voice A27) can
give
> 1/4-comma meantone this quality of a "well-timbrement," with
sensorily
> concordant forms of the usual 16th-century consonances (at least to
my
> ears) available from every step of a 12-note tuning. Two such
> circulating 12-note keyboards a diesis apart (128:125, ~41.06 cents)
> make available a gamut of "modal color," with many opportunities for
> creative musical ambiguity.
>
> For example, an interval of 32:25 might represent now a remote
> transposition of a usual Renaissance major third; now a
> Renaissance-style diminished fourth with a resolution typical of
that
> era; and now a bold neo-Gothic major third inviting a 14th-century
> kind of resolution to some variety of stable fifth.
>
> For more on meantone well-timbrement, and some sound files most
helpfully
> and skillfully produced by Jacky Ligon, please see the latest issue
of
> _The Microtonal Activist_ or TMA, volume 2, an edition featuring
articles
> on topics ranging from the related theme of tunings/timbres with
stretched
> or compressed intervals (Jacky and Jeff Scott) to Joseph Pehrson's
lecture
> at Moscow Conservatory's Theremin Center (transcription by Anton
Rovner)
> to Robert Walker's invitation to compose for the spontaneous
musician:
>
> <http://www.geocities.com/jacky_ligon/TMA_archive_index.htm>
>
> Having illustrated both a "moderate" application of Sethareanization
> with 17-tET in a neo-Gothic setting, and a dramatic application with
> 1/4-comma meantone,

Margo, the interrelationship between tuning and timbre, as regards
concordance and discordance, is undeniable, and thank you for the
detailed examples. However, do you have any evidence that the effects
you're mentioning are actually in accord with Sethares' model? Have
you worked out the mathematics? I would think, due to the
mathematical nature of Sethares' writings, that "Sethareanization"
would have to refer to some operation justified on the basis of
mathematical calculations using Sethares' model, if not the explicit
derivation of timbre from tuning or vice versa that is Sethares'
original contribution in this area.

> I would now like to turn to a harmonic entropy
> kind of approach as one kind of strategy for putting these phenomena
> in a larger perspective.
>
> While Setharean "sensory consonance" often focuses on
the "smoothness"
> of an interval or sonority, the harmonic entropy of Erlich or Keenan
> focuses on what I might term the "simplicity" of an interval.

Actually, for intervals (i.e., dyads) with harmonic timbres, the two
models are in close agreement, making it hard to tell exactly what is
going on. It is for chords of three or more notes that the two models
begin to differ significantly in their concordance/discordance
rankings.

> For example, as we have seen, a 17-tET major third at around 423.5
> cents may be either "rough" or "smooth" depending on the timbre;
from
> a viewpoint of harmonic entropy, it is in any event "complex,"
located
> in a "plateau" region between the two simple ratios of 5:4 (~386.31
> cents) or 9:7 (~435.08 cents).
>
> Similarly, the 737.64-cent diminished sixth of 1/4-comma meantone is
> historically famed as a "Wolf fifth" with its "howling" beats in a
> typical Renaissance or Baroque timbre -- the essence
of "roughness" --
> but can become intriguingly smooth when "well-timbred." Again,
> however, from a perspective of harmonic entropy, it remains a
complex
> or plateau interval, in contrast to the usual meantone fifth at
about
> 696.58 cents located in the "valley" around the simple ratio of 3:2
> (~701.955 cents).

A truly "Sethareanized" version of an interval, such as the 423.5
cent and 737.64 cent intervals you mention, would use inharmonic
timbres to acheive smoothness. A harmonic entropy calculation on
these intervals would then have to take into account all the partials
(since they are no longer just multiples of the fundamental
frequencies). Such a calculation is not likely to find these
intervals at local maxima of harmonic entropy, as it does for
harmonic partials.
> >
>
> What harmonic entropy may tell us is that while a smooth valley and
a
> smooth plateau may be in some ways analogous or even musically
> interchangeable, they are not musical identical.

I'm not following you here, though I'd like to. Can you elaborate?
>
> In a circulating meantone well-timbrement with 12 notes per octave
in
> 1/4-comma, for example, the difference of some 41 cents in the size
of
> the diatonic and chromatic semitones will lend a different effect
than
> in an unequal well-temperament such as Werckmeister where these
sizes
> are not so disparate, let alone 12-tET.

This is a very important point, and very true, and something I often
bring up. I'm not sure what it has to do with the issue at hand,
though.
>
> From what I take as a Keenan/Erlich type of perspective, ratios of
11
> may still participate in recognizably just dyads (e.g. 11:7, 11:8,
> 11:9), but nearing the limit of recognizability for _simple
> intervals_. Thus something like 17:14 would not be considered in
> itself a "just" interval, although it might be used in some larger
> sonority considered recognizably just such as 10:12:14:17.

I think you've captured the Keenan/Erlich perspective very well,
though personally I'd never drag the word "just" into this . . . I'd
substitute "concordant" instead in the paragraph above. I have
always, and will continue to, allow 17:14 to be considered a "just"
interval -- the simplest reason would be that one could tune
10:12:14:17 easily by ear on say, a harmonium, and then proceed to
use the bare interval 17:14 in one's music. That's enough to make
it "just" for me (going by the culture in which I live and the
literature to which I've been exposed, rather than by a dictionary
entry).
>
> From an harmonic entropy perspective, the question might still be
> raised: could the total effect of 14:17:21, influenced possibly by
the
> "anchoring" outer fifth or the difference tones, represent some kind
> of "local minimum" or "valley"?

I'm almost certain that, at least if the chord is played in a
somewhat high register (say three to four octaves above middle C),
such a local minimum could be audibly recognized, while a Sethares-
based approach to discordance would fail to predict one, if harmonic
timbres are used.
>
> From a neo-Gothic perspective, the main point is simply that
14:17:21
> is an intriguing sonority with beautiful resolutions, and that the
> integer ratio provides an apt emblem and landmark for this general
> type or flavor of sonority (somewhere around 0-336-702 cents).

Certainly, though if I were in the position of lending advice to a
novice at microtonality, and if 14:17:21 were not being perceived as
a point of stability relative to its immediate neighbors on the
triadic plane, I would advise that it would probably prove more
useful to think of the sonority as a just fifth split into a 336-cent
interval and a 366-cent interval . . . mainly because addition and
subtraction (as with cents) are so much easier than multiplication
and division (as with ratios) as a way of understanding musical
intervals -- for example, one can easily see which intervals are
larger and which are smaller; one can plan resolutions and other
progressions without having to be constrained by cumbersome, and
often limiting, ratio calculations; and one can immediately see when
familiar, concordant intervals are approximated in the course of
combining other intervals -- something which the multiple-digit
ratios which arise in the course of JI arithmetic actually obscure.
>
> Such forms as 6:7:9, 7:9:12, 12:14:18:21, 14:18:21:24, 14:17:21, and
> 28:33:42 (around 0-284-702 cents) -- the latter two on the RI side -
-
> share the property of not having Erlich's "rootedness," with the
> lowest note as an octave or power of two in relation to a
fundamental.

Though I'd add that some modicum of "rootedness" does manage to
impart a degree of stability to these chords, due to the lowest note
participating as the "2" in a 2:3 relationship with one other note.
>
> In contrast, as I have found with Pythagorean tuning, the "floating"
> quality of a texture like early 15th-century fauxbourdon with a
series
> of sixth sonorities eventually moving to a cadence -- but in a kind
of
> prolonged "free fall" of mild coloristic instability -- can be very
> pleasing with complex ratios, or with simple but not "rooted" ones
> such as 7:9:12.

What would an example of a "rooted" version of a "sixth sonority"?
>
> Pastelizing favors circulation, or more
generally "interchangeability
> with modal color," and also adds much room for creative ambiguity,
but
> without the sharper definitions and contrasts of a more traditional
> timbre.

I would agree, and would offer my own observation
that "pastelization" (an expression I like much better
than "Sethareanization" for referring to the use of timbres which
mute the differences in concordance between simple-integer ratios
[other than octaves] and intervals far from them -- thanks Margo) is
very useful when trying to work with 9-tET, which has powerful
melodic and modulatory resources but rather harsh harmonies in many
conventional timbres.
> >
> My main point in this discussion may be that the concept of a
> continuum does not imply a uniform or featureless terrain, but
rather
> a diverse choice of terrains and textures offering new choices
without
> invalidating more familiar ones, which may be appreciated for their
> uniqueness all the more.

I heartily affirm this point, Margo. Thanks, as always, for the
eloquent and elaborate exposition of your extra-dodecaphonic
experiments and entertainments -- edifying as ever!

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗mschulter <MSCHULTER@VALUE.NET>

Hello, there, Paul Erlich, and thanks for a most generous and
thoughtful response to my article:

</tuning/topicId_26941.html#26978>

> I absolutely agree. For example, in the tuning of the CEGAD chord,
> which I've been discussing with Bob Wendell, it's the
> smoothness/roughness consideration which is the dominant factor in
> determining an "optimal" tuning for the chord. For other chords,
> such as a half-diminished seventh chord, the reverse seems to be
> true.

Yes, the smoothness/roughness and simplicity/complexity dimensions can
have some interesting tradeoffs, and a discussion you've had with
Joseph Pehrson maybe helps me in responding to some of what follows.

> Again, I've always emphasized that this needs to be a
> two-dimensional model just as you spell out. However, there may be
> some misunderstanding about the "choice" aspect of this. All choices
> should always be open for musicians. I don't see what a model of
> concordance/discordance has to do with this.

First, thanks for emphasizing a very important point: these models are
designed, maybe, in good part precisely to illuminate some of the
choices available which might not otherwise be recognized. In that
way, while all choices are always open, a good theory can help by
revealing what some of those choices, maybe less familiar ones, are --
or what some possibly yet unrealized choices might be.

> Margo, the interrelationship between tuning and timbre, as regards
> concordance and discordance, is undeniable, and thank you for the
> detailed examples. However, do you have any evidence that the
> effects you're mentioning are actually in accord with Sethares'
> model? Have you worked out the mathematics?

Please let me admit that at least at this point, you or Bill Sethares
might be a better judge of that than I: when I understand more about
the model, maybe I could attempt a comment.

At this point, I might sum up my understanding of the data about as
follows. As described in the TMA article, Jacky Ligon very helpfully
did a spectrum analysis on the TX-802 "Piccolo" timbre (preset A27),
and found partials at an octave-reduced 428 cents and 744 cents. These
looked very interesting to me for optimizing intervals like the
427-cent diminished fourth or 738-cent diminished sixth (or "Wolf
fifth") of 1/4-comma meantone.

So I understand that the partials seem to approximate the vertical
intervals, but of course either your model or that of Sethares might
address all kinds of questions. For example, why is it that the usual
meantone intervals (e.g. 386-cent major thirds, 697-cent fifths) also
sound consonant in this timbre? In some Sethareanized timbres, the
"familiar" ratios can sound quite dissonant.

Of course, to me, it's a thrilling musical experience, whatever the
exact model -- but that shouldn't stop me from asking questions and
learning from you and Bill Sethares and others who have considered the
fine points here.

> I would think, due to the mathematical nature of Sethares' writings,
> that "Sethareanization" would have to refer to some operation
> justified on the basis of mathematical calculations using Sethares'
> model, if not the explicit derivation of timbre from tuning or vice
> versa that is Sethares' original contribution in this area.

Of course this is a very reasonable point -- just as Einsteinian
special relativity has definite and known mathematics, which I've
taken pleasure in applying to calculate the time dilation effect of my
travel at around 1.5 meters per second or whatever, possibly a comment
on the practical relevance of the some of the math I post here.

I must admit that for me, "Sethareanization" has much a more
colloquial meaning, basically: "Combining tuning and timbre and
happening to arrive at a notable effect, especially if it involves
making some interval strikingly more consonant or less dissonant than
might be expected."

With the 738-cent fifth, I'd say additionally that the tuning/timbre
combination did apparently involve a rough fit between partials and
vertical intervals.

However, I realize that we could cite everyone from gamelan builders
and players to Sabaneev, Darreg, Gamer, and Carlos for the idea of a
tuning/timbre interaction, so the question comes up of what is
_distinctively_ "Setharean," and the math seems one obvious answer.

In my kind of usage, "Sethareanization" is often a byword simply for
finding a timbre that happens to match a tuning, or to have a nice
effect with a specific piece in that tuning.

>> While Setharean "sensory consonance" often focuses on the
>> "smoothness" of an interval or sonority, the harmonic entropy of >
>> Erlich or Keenan focuses on what I might term the "simplicity" of
>> an interval.

> Actually, for intervals (i.e., dyads) with harmonic timbres, the two
> models are in close agreement, making it hard to tell exactly what
> is going on. It is for chords of three or more notes that the two
> models begin to differ significantly in their
> concordance/discordance rankings.

This is an interesting comparison, and one I might better understand
when I better grasp the two models. I tend to associate the Sethares
model with smoothness, the critical band, and so forth; and harmonic
entropy with the probability of recognition of an interval as
representing a given ratio type.

> A truly "Sethareanized" version of an interval, such as the 423.5
> cent and 737.64 cent intervals you mention, would use inharmonic
> timbres to acheive smoothness. A harmonic entropy calculation on
> these intervals would then have to take into account all the
> partials (since they are no longer just multiples of the fundamental
> frequencies). Such a calculation is not likely to find these
> intervals at local maxima of harmonic entropy, as it does for
> harmonic partials.

This is really interesting. With the timbre Jacky analyzed as having
a partial around 744 cents, it would be interesting to see what the
minima and maxima would look like (also taking the other partials into
account, as you say).

>> What harmonic entropy may tell us is that while a smooth valley and
>> a smooth plateau may be in some ways analogous or even musically
>> interchangeable, they are not musical identical.

> I'm not following you here, though I'd like to. Can you elaborate?

Reading a further dialogue between you and Joe Pehrson has maybe
helped me to appreciate one possible difference, although I should be
careful to seek your guidance on whether harmonic entropy theory would
lead to kind of distinction I'm about to discuss.

The point you and Joe were discussing was that an inharmonic or a
"pastelized" timbre differs from a harmonic timbre in other ways than
just happening to have different minima or maxima of local consonance,
or a different degree of smoothness or consonance for a given
interval.

What the two of you were discussing is the idea that pitch definition
itself can be more vague or imprecise in an inharmonic timbre: the
tuning, or scale, may not be so sharply defined. Maybe it's a bit of
"Impressionistic" pastelization (a very visual metaphor); the colors
are more gentle, but the outlines of the scene itself are a bit
blurred or indistinct.

To borrow my metaphor, we're in a garden-like park that reminds us of
a valley, but thousands of meters up on a plateau where there are
mists or clouds that make the scene different than in the valley.

>> In a circulating meantone well-timbrement with 12 notes per octave
>> in 1/4-comma, for example, the difference of some 41 cents in the
>> size of the diatonic and chromatic semitones will lend a different
>> effect than in an unequal well-temperament such as Werckmeister
>> where these sizes are not so disparate, let alone 12-tET.

> This is a very important point, and very true, and something I often
> bring up. I'm not sure what it has to do with the issue at hand,
> though.

Actually, though maybe I wasn't sure just where I was going when I
wrote this, a question does now occur to me maybe related more
directly to the issues we're discussing. Might the melodic contrasts
between those step sizes in a circulating tuning/timbre, for example,
make up for the somewhat indistinct pitch definition that could occur
because of the inharmonic timbre?

We seem very much in agreement that the melodic aspect shouldn't get
neglected, and one issue here might be how timbral choices mainly
directed to vertical consonance might affect the definition of melodic
lines.

>> From what I take as a Keenan/Erlich type of perspective, ratios of
>> 11 may still participate in recognizably just dyads (e.g. 11:7,
>> 11:8, 11:9), but nearing the limit of recognizability for _simple
>> intervals_. Thus something like 17:14 would not be considered in
>> itself a "just" interval, although it might be used in some larger
>> sonority considered recognizably just such as 10:12:14:17.

> I think you've captured the Keenan/Erlich perspective very well,
> though personally I'd never drag the word "just" into this . . . I'd
> substitute "concordant" instead in the paragraph above. I have
> always, and will continue to, allow 17:14 to be considered a "just"
> interval -- the simplest reason would be that one could tune
> 10:12:14:17 easily by ear on say, a harmonium, and then proceed to
> use the bare interval 17:14 in one's music. That's enough to make it
> "just" for me (going by the culture in which I live and the
> literature to which I've been exposed, rather than by a dictionary
> entry).

Thanks for this clarification, and I know that usages can vary here
even among theorists with generally quite similar views. Personally I
would tend to speak in terms of "simple" or "complex" intervals: it's
interesting to consider how flexible a term like "consonance" can be,
and I know that I find it natural to use in at least three or four
different ways.

>> From an harmonic entropy perspective, the question might still be
>> raised: could the total effect of 14:17:21, influenced possibly by
>> the "anchoring" outer fifth or the difference tones, represent some
>> kind of "local minimum" or "valley"?

> I'm almost certain that, at least if the chord is played in a
> somewhat high register (say three to four octaves above middle C),
> such a local minimum could be audibly recognized, while a Sethares-
> based approach to discordance would fail to predict one, if harmonic
> timbres are used.

That's really interesting, and I'd be really curious about the results
of this kind of experiment. Are there specific elements of a triadic
harmonic entropy that would predict the minimum? One idea that occurs
to me is difference tones, for example, which Heinz Bohlen has focused
on in writing about consonance issues.

By the way, I'd agree that for a newcomer to microtonality, and often
more generally, something like your explanation of a splitting of the
fifth into intervals around 336 and 366 cents might be more flexible
and descriptive than 14:17:21. In fact, the idea is a region of
supraminor and submajor thirds which often vary a few cents in either
direction, depending on the tuning.

The one tuning I use with a pure 14:17:21 is a "Pythagorean
enharmonic," where the ratio defines the tuning scheme. Typically,
however, it could range from 0-331-703 to 0-341-705, for example, more
of a zone, or a mood, than a specific defined ratio.

>> Such forms as 6:7:9, 7:9:12, 12:14:18:21, 14:18:21:24, 14:17:21,
>> and 28:33:42 (around 0-284-702 cents) -- the latter two on the RI
>> side -- share the property of not having Erlich's "rootedness,"
>> with the lowest note as an octave or power of two in relation to a
>> fundamental.

> Though I'd add that some modicum of "rootedness" does manage to
> impart a degree of stability to these chords, due to the lowest note
> participating as the "2" in a 2:3 relationship with one other note.

Right, and this is the "anchoring" of the 2:3 that I've read about in
20th-century harmony books. I wonder if that's a possible term to
distinguish it from the "rootedness" based on the lowest note as an
octave or 2^n of the fundamental?

>> In contrast, as I have found with Pythagorean tuning, the "floating"
>> quality of a texture like early 15th-century fauxbourdon with a
>> series of sixth sonorities eventually moving to a cadence -- but in
>> a kind of prolonged "free fall" of mild coloristic instability --
>> can be very pleasing with complex ratios, or with simple but not
>> "rooted" ones such as 7:9:12.

> What would an example of a "rooted" version of a "sixth sonority"?

With 7:9:12, or for that matter 12:15:20 in a later Renaissance
setting, I'd agree with what I take as your implicit point that such
sonorities aren't "rooted" by either the n^2 kind of concept or the
anchoring effect of a fifth above the lowest note. In the early
15th-century setting -- and to a degree in the later setting also
(meantone or adaptive JI) -- the cadential role of a sixth expanding
to an octave may also have a connection with the kind of effect I
describe.

Curiously, some other kinds of sixth sonorities can be "rooted" or at
least "anchored." For example, 14:18:21:24 (0-435-702-933 cents) has a
fifth above the lowest note. These would involve the fifth and sixth
above the lowest note.

For simple ratios of major sixths that harmonic entropy theory would
consider "recognizable" as simple dyads, I notice that the lowest
voice of the sonority will have some prime factor other than 2, e.g
3:5, or 7:12.

>> Pastelizing favors circulation, or more generally
>> "interchangeability with modal color," and also adds much room for
>> creative ambiguity, but without the sharper definitions and
>> contrasts of a more traditional timbre.

> I would agree, and would offer my own observation that
> "pastelization" (an expression I like much better than
> "Sethareanization" for referring to the use of timbres which mute
> the differences in concordance between simple-integer ratios [other
> than octaves] and intervals far from them -- thanks Margo) is very
> useful when trying to work with 9-tET, which has powerful melodic
> and modulatory resources but rather harsh harmonies in many
> conventional timbres.

Maybe "pastelization" could refer to a general muting of contrasts, as
you say, with the implication that usual concords remain concordant;
in some tunings of Bill Sethares, such concords with simple ratios may
actually take on a dissonant quality, and then we'd want some other
term.

By the way, in a cadential context in 20-tET, I use the term "pastel"
to describe a lessened contrast between sizes of melodic steps; maybe
the general idea would muting either vertical or melodic contrasts, so
that the two usages could be related.

>> My main point in this discussion may be that the concept of a
>> continuum does not imply a uniform or featureless terrain, but
>> rather a diverse choice of terrains and textures offering new
>> choices without invalidating more familiar ones, which may be
>> appreciated for their uniqueness all the more.

> I heartily affirm this point, Margo. Thanks, as always, for the
> eloquent and elaborate exposition of your extra-dodecaphonic
> experiments and entertainments -- edifying as ever!

Thanks also for your comments, and patient explanations, not only
clarifying concepts but providing incitements to new music.

Most appreciatively,

Margo

🔗Paul Erlich <paul@stretch-music.com>

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

> At this point, I might sum up my understanding of the data about as
> follows. As described in the TMA article, Jacky Ligon very helpfully
> did a spectrum analysis on the TX-802 "Piccolo" timbre (preset A27),
> and found partials at an octave-reduced 428 cents and 744 cents.
These
> looked very interesting to me for optimizing intervals like the
> 427-cent diminished fourth or 738-cent diminished sixth (or "Wolf
> fifth") of 1/4-comma meantone.

Fascinating? Just curious . . . was this spectrum analysis done on
the results of playing many different pitches in the "Piccolo"
timbre, or just one? How many octaves above the fundamental were
these partials? Were they true partials or simply local maxima of
some broad-band noise "humps"? Did they die out quickly, or were they
sustained as long as the key was held down? The reason I ask is that
a true, acoustic, sustained piccolo sound, should be characterized by
an approximate harmonic series, plus noise. The reason I
say "approximate" is that, since the physics of turbulent air is not
so well understood, I don't feel confident asserting that an exact
harmonic series results (as it does in reed, brass, bowed string, and
human voice sustained sounds).
>
> So I understand that the partials seem to approximate the vertical
> intervals, but of course either your model or that of Sethares might
> address all kinds of questions. For example, why is it that the
usual
> meantone intervals (e.g. 386-cent major thirds, 697-cent fifths)
also
> sound consonant in this timbre?

The timbre must be primarily harmonic or nearly so.
>
> With the 738-cent fifth, I'd say additionally that the tuning/timbre
> combination did apparently involve a rough fit between partials and
> vertical intervals.

If that was in fact the case (subject to the questions I asked
above), then yes, Sethareanization, or better "Carlosization", would
be an apt description.
> >
> > Actually, for intervals (i.e., dyads) with harmonic timbres, the
two
> > models are in close agreement, making it hard to tell exactly what
> > is going on. It is for chords of three or more notes that the two
> > models begin to differ significantly in their
> > concordance/discordance rankings.
>
> This is an interesting comparison, and one I might better understand
> when I better grasp the two models. I tend to associate the Sethares
> model with smoothness, the critical band, and so forth; and harmonic
> entropy with the probability of recognition of an interval as
> representing a given ratio type.

This is correct, and, since harmonic partials are in simple-ratio
relationships, is why the two models tend to agree for dyads with
harmonic timbres.
>
> > A truly "Sethareanized" version of an interval, such as the 423.5
> > cent and 737.64 cent intervals you mention, would use inharmonic
> > timbres to acheive smoothness. A harmonic entropy calculation on
> > these intervals would then have to take into account all the
> > partials (since they are no longer just multiples of the
fundamental
> > frequencies). Such a calculation is not likely to find these
> > intervals at local maxima of harmonic entropy, as it does for
> > harmonic partials.
>
> This is really interesting. With the timbre Jacky analyzed as having
> a partial around 744 cents, it would be interesting to see what the
> minima and maxima would look like (also taking the other partials
into
> account, as you say).

At present a harmonic entropy calculation with a large number of
inharmonic frequencies is beyond my computational capabilities :(

> >> What harmonic entropy may tell us is that while a smooth valley
and
> >> a smooth plateau may be in some ways analogous or even musically
> >> interchangeable, they are not musical identical.
>
> > I'm not following you here, though I'd like to. Can you elaborate?
>
> Reading a further dialogue between you and Joe Pehrson has maybe
> helped me to appreciate one possible difference, although I should
be
> careful to seek your guidance on whether harmonic entropy theory
would
> lead to kind of distinction I'm about to discuss.
>
> The point you and Joe were discussing was that an inharmonic or a
> "pastelized" timbre differs from a harmonic timbre in other ways
than
> just happening to have different minima or maxima of local
consonance,
> or a different degree of smoothness or consonance for a given
> interval.
>
> What the two of you were discussing is the idea that pitch
definition
> itself can be more vague or imprecise in an inharmonic timbre: the
> tuning, or scale, may not be so sharply defined. Maybe it's a bit of
> "Impressionistic" pastelization (a very visual metaphor); the colors
> are more gentle, but the outlines of the scene itself are a bit
> blurred or indistinct.
>
> To borrow my metaphor, we're in a garden-like park that reminds us
of
> a valley, but thousands of meters up on a plateau where there are
> mists or clouds that make the scene different than in the valley.

If you're asking me whether harmonic entropy theory has anything to
say about the "pastelization" of pitch itself, I would say yes,
absolutely. Take a look at the very recent discussions between Carl
and myself on harmonic_entropy@yahoogroups.com, and join in if you
have questions or comments.
>
> Actually, though maybe I wasn't sure just where I was going when I
> wrote this, a question does now occur to me maybe related more
> directly to the issues we're discussing. Might the melodic contrasts
> between those step sizes in a circulating tuning/timbre, for
example,
> make up for the somewhat indistinct pitch definition that could
occur
> because of the inharmonic timbre?

Perhaps one would tend to hear slightly unequal steps as equal when
pitch definition was indistinct, so perhaps added inequality in step
sizes is sometimes aesthetically appropriate when "pastelizing" . . .
is that what you're suggesting? Note that this wouldn't speak against
using ETs with pastelized timbres, if you were planning to base you
melodies on MOS or other unequally-spaced _subsets_ of these ETs.
>
> That's really interesting, and I'd be really curious about the
results
> of this kind of experiment. Are there specific elements of a triadic
> harmonic entropy that would predict the minimum?

The groundwork for calculating triadic harmonic entropy has been laid
out on harmonic_entropy@yahoogroups.com -- what remains is to perform
the actual (vast) calculation.

> One idea that occurs
> to me is difference tones, for example, which Heinz Bohlen has
focused
> on in writing about consonance issues.

I understand Heinz Bohlen's views but I think he ignores the
importance of harmonic entropy (which tends to agree with a
difference-tone-based analysis) and critical-band roughness (which
often disagrees with it) for consonance and dissonance, especially in
cases where difference tones are inaudible -- perhaps partly as a
result of the particular tunings and timbres he has concentrated on.

>
> >> Such forms as 6:7:9, 7:9:12, 12:14:18:21, 14:18:21:24, 14:17:21,
> >> and 28:33:42 (around 0-284-702 cents) -- the latter two on the RI
> >> side -- share the property of not having Erlich's "rootedness,"
> >> with the lowest note as an octave or power of two in relation to
a
> >> fundamental.
>
> > Though I'd add that some modicum of "rootedness" does manage to
> > impart a degree of stability to these chords, due to the lowest
note
> > participating as the "2" in a 2:3 relationship with one other
note.
>
> Right, and this is the "anchoring" of the 2:3 that I've read about
in
> 20th-century harmony books. I wonder if that's a possible term to
> distinguish it from the "rootedness" based on the lowest note as an
> octave or 2^n of the fundamental?

Whether one distinguishes it or not, I think it's clear that the
brain seems to "try out" all subsets of the tones it hears, looking
for a clear root, willing to treat the additional tones as "noise" or
as a separate stimulus with its own root. One of the inspirations for
harmonic entropy was Parncutt's approach in his book _Harmony: A
Psychoacoustical Approach_. In the book he characterizes chords by
two perceptual qualities: their salience (clear relation to a root)
and multiplicity (the apparant number of tones sounding). Harmonic
entropy for chords of more than two tones, as has been discussed,
would have to consider ratio-interpretations for all possible
subsets, as well as for the chord as a whole, and even a subset that
exhibits "rootedness" should effectively decrease the multiplicity of
a chord (sorry, my food-deprived brain can't make more sense right
now).

Anyhow, I welcome your terminological distinction, and would suggest
that sometimes a 5:4, and perhaps even a 7:4, can serve
the "anchoring" function, and not just 3:2.