Yahoo Tuning Groups Ultimate Backup tuning Re: JI definitions and types -- a three-dimensional approach (PRI)

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Just intonation definitions and typologies:
A three-dimensional approach (PRI)
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Attempts to define "just intonation" (JI) may say as much about the
musical outlooks and philosophies of the interlocutors as about the
concept of JI itself. Part of the richness and complexity of this
dialogue may lie in the fact that typically "JI" may actually imply a
combination of musical traits and associations, leaving different
people free to emphasize different aspects.

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1. The 16th-century ideal type for "JI": the PRI model
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From an historical point of view, the development in Western European
theory of a concept of "pure" or "perfect" intonation distinct from
the newly standardized process of _participatio_ or temperament seems
to be a development of the later 15th and 16th centuries. A system of
pure intervals and ratios featuring stable sonorities based on the
prime factors of 2, 3, and 5 (2-3-5 JI) stands in contrast both to
earlier Gothic systems of Pythagorean tuning (2-3 JI) and to regular
temperaments.

By the time of Fogliano (1529), Zarlino (1558), or Bottrigari (1594),
three distinctive aspects of such a system based on the syntonic
diatonic of Ptolemy have gained recognition:

1. Purity (P). A fixed-pitch instrument in a JI tuning
presents consonant intervals in their most "perfect"
or "sonorous" form.

2. Rationality (R). A JI tuning is based on integer
ratios, especially those of Zarlino's _senario_
(the natural numbers 1-6), in contrast to the
irrational ratios of temperament.

3. Intricacy/Incommensurability (I). A JI tuning,
unlike either the older Pythagorean tuning or
a meantone temperament, is not regular, but
involves such intervals as the syntonic comma
resulting from what has been termed the
"incommensurate" nature of prime factors
such as 3 and 5.

All three themes emerge in 16th-century discussions of just tuning and
temperament. Zarlino, for example, describes JI keyboards as
"perfectly tuned in accord with the perfection of the harmonic
numbers," and as including "minute intervals" such as the syntonic
comma of 81:80 (~21.51 cents).[1]

He therefore focuses on a "perfectly tuned" instrument, our purity
property (P); the "perfection of the harmonic numbers," our
rationality property (R); and the presence of "minute intervals," our
intricacy or incommensurability property (I).

In his dialogue, Bottrigari likewise has his expositor explain that a
listener accustomed to tempered keyboard instruments has heard only
the diapason or octave in its "true form," and that a tuning based on
the "true plurality of strings" must include diatonic notes (e.g. two
versions of D, Bb) not found on conventional instruments.[2]

The continued use of such terms as "the Just scale" (i.e. the syntonic
diatonic with its pure 2-3-5 ratios), and of such categorical
distinctions as "Pythagorean/meantone/just," suggests that these
16th-century concepts may still present a kind of ideal type for "JI,"
albeit an ideal type subject both to extension and to broadening.[3]

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2. Variations on PRI: counting to 7
-----------------------------------

Starting from the classic 16th-century model of 2-3-5 JI with its pure
tunings of all stable sonorities (P), rational ratios for all
intervals (R), and intricate syntonic commas (I), we can extend or
vary this PRI model by adding or substituting other primes.

For example, admitting the prime factor of 7, we have such additional
PRI possibilities as these:

2-3-5-7 JI For example, a tetradic system based on the
the stable 4:5:6:7 or 12:14:18:21.

2-3-7 JI For example, a neo-Gothic system with stable
2-3 sonorities (e.g. 2:3:4) and unstable
2-3-7 sonorities (e.g. 14:18:21:24).

In such systems, all stable sonorities (and in the neo-Gothic example
also many unstable sonorities) would be represented by pure intervals;
all intervals derived from integer ratios; and intricacy would prevail
in the form of the syntonic comma (81:80) and/or septimal comma
(64:63, ~27.26 cents).

Another facet of intricacy is the element in these systems of notably
unequal whole-tones: 9:8 and 10:9 in classic 2-3-5 JI; 9:8 and 8:7 in
2-3-7 JI; and all three sizes in 2-3-5-7 JI.

While some of these systems may involve interval aesthetics quite
distinct from those of 16th-century music, I suspect that most people
would readily accept them as instances of "JI" tunings; the PRI model
is extended or altered, but not compromised in its conceptual traits.

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3. Some broadened interpretations of "JI" -- PR, RI, PI
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If the PRI model offers an "ideal type" for JI, what happens if we
consider less characteristic "JI" systems featuring the intersection
of two of our three properties, but not all three? Let us consider
some of the possibilities.

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3.1. Purity and Rationality -- Pythagorean tuning or 2-3 JI
-----------------------------------------------------------

In the musical setting of Continental Western Europe during the Gothic
era of the 13th and 14th centuries, Pythagorean tuning or 2-3 JI may
present an ideal case of a PR system:

1. (P)urity. All stable intervals and sonorities (2:3:4,
3:4:6) are available in pure form.

2. (R)ationality. All intervals are derived from integer
ratios, specifically the powers of 2 and 3.

3. (I)ntricacy. As a regular tuning, Pythagorean
does _not_ feature the kind of intricacy present in
2-3-5, 2-3-5-7, or 2-3-7 systems involving the
syntonic and/or septimal commas; all regular
whole-tones have a single size of 9:8.

Thus we have P and R, but not I. Such a "PR" or 2-3 system seems
instinctively to fit the usual aural and mathematical concepts of JI,
although its regularity and comparative simplicity may give it a
radically different feeling for the player of a fixed-pitch
instrument than the intricacy of a 2-3-5, 2-3-5-7, or 2-3-7 system.

More specifically, we might say that the Gothic 2-3 and Renaissance
2-3-5 systems share a property of "Pstable" (with "stable" to be read
as a subscript) -- the Purity property holds for at least the subset
of intervals which may appear in stable sonorities.

Other JI systems might fit such additional versions of the P property
as "Pcompatible" or "Pcompat" for short, for example a neo-Gothic 3-7
system where all usual diatonic intervals deemed in Gothic theory to
be stable or "unstable but compatible" (notably excluding m2, M7, and
A4 or d5) are available within a 9-odd-limit.

In a Pythagorean PR system, as in more intricate PRI systems uniting
three or more prime factors, we have an intersection of two basic
premises which seem to characterize "JI" for many people, albeit with
differing degrees of emphasis on one premise or the other:

1. A JI system should provide the purest possible
intervals for stable sonorities -- or some
equivalent set in musics not based on such
concepts as stability/instability; and

2. A JI system should be based exclusively on
rational ratios, as opposed to the irrational
ratios of temperament.

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3.2. Rationality and Intricacy: Complex RI systems
--------------------------------------------------

While the element of "purity" is associated in the classic
16th-century model with integer ratios, this association (in usual
harmonic timbres) more specifically suggests _small_ integer ratios
with an audible "locking-in" of partials. Modern theorists such as
Paul Erlich and Dave Keenan have suggested that for interval ratios
a:b, the limit of such "purity" may be reached at around a*b=99 or
a*b=104 (for example, respectively 11:9 and 13:8).

Traditional PRI and PR systems, of course, typically include _some_
complex ratios going beyond this "limit of aural purity," and it is
also possible to design some nontraditional PRI system featuring such
complex ratios.

For example, suppose we decide on a 2-3-7-11 scheme featuring stable
2:1 octaves, 2:3 fifths, and 4:3 fourths (as in Pythagorean 2-3 JI),
plus unstable major thirds at 14:11 and minor thirds at 33:28. Such a
scheme fulfills PR (i.e. Pstable, R), and additionally introduces the
intricacy factor of the 896:891 comma, ~9.69 cents (the difference,
for example, between 14:11 and the usual Pythagorean 81:64).

However, what about a system where all intervals are derived from
integer ratios, but there is no set of pure intervals corresponding to
that of a PRI or PR system?

If the system is defined _only_ by the R property, then it might
approximate any familiar (or unfamiliar) temperament as closely as
desired. One might ask, "What is distinctively 'JI' about this tuning
from a musical perspective?"

However, such a non-P tuning might combine Rationality with Intricacy
-- for example, by featuring various kinds of commas or schismas
shaping its complex ratios into a pattern somewhat analogous to that
of a PRI system.

Such an RI system would then seem to have qualities distinct from
either a usual temperament with its less intricate structure, or a PRI
or PR system of JI with its set of pure intervals.

As it happens, the initials RI could also stand for "Rational
Intonation," a term sometimes applied also to PR or PRI systems where
complex sonorities play a central role along with pure ones (for
example, the unstable cadential sonorities of Gothic or neo-Gothic
music).

Here it should be noted that radical RI systems, as described for
example by Jacky Ligon, might focus especially on _melodic_ aspects of
integer-based music. The ancient Greek interest in superparticular
melodic intervals, for example semitones of 16:15 or 28:27, may
provide a time-honored precedent for this type of JI approach, with a
focus on _emmeles_ or "intervals apt for melodic progressions."

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3.3. Purity and Intricacy (PI) -- Two possible interpretations
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A focus on the "just" elements of Purity and Intricacy (PI) might
suggest a Setharean approach based on irrational ratios fitting the
spectrum of a given timbre, especially a notably inharmonic one where
such ratios might in fact maximize the traditional JI value of ideal
sensory consonance.

A different interpretation of PI, actually including the R property
also, might follow along the lines of an artist such as LaMonte Young:
using "intricately" complicated sonorities based on integer ratios in
order to _make_ certain complex integer ratios _contextually_ "pure,"
i.e. featuring aurally distinct meshings of partials.

To borrow a distinction from medieval theory, such complex intervals
are "pure" _per accidens_ ("by circumstance") rather than _per se_
("in themselves") -- or, one might say, they participate in sonorities
which have such a property of purity.

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4. "Adaptive JI": PRstable-vert
-------------------------------

While the term "adaptive JI" can apply to various kinds of keyboard
arrangements (e.g. neo-Gothic 3-7 JI realized by two Pythagorean
manuals tuned a septimal comma apart as an "adaptive" modification of
a regular Pythagorean tuning with the manuals a Pythagorean comma
apart), I would like here to address an "adaptive" model of a kind
which might be applicable to either fixed or continuous sets of
pitches.

This variety of "adaptive JI" might be summed up in terms of PRI about
as follows:

1. (P)urity. All stable sonorities are available in forms
with pure intervals, satisfying this JI property.

2. (R)ationality. All _vertical_ intervals of _stable_
sonorities are based on integer ratios, but melodic
steps and unstable sonorities typically or always
feature irrational ratios, as in tempered systems.

3. (I)ntricacy. Like traditional (e.g. meantone)
temperaments, these "adaptive JI" systems are
designed in part precisely to avoid such
intricacies as the syntonic or septimal comma.

Thus while such schemes share a P property (e.g. Pstable for
Vicentino's adaptive JI system of 1555 with two manuals in 1/4-comma
meantone tuned 1/4 syntonic comma apart) like that of classic PRI or
PR systems, they exhibit a _partial_ R property often applying
specifically to the subset of sonorities covered by the P property,
and more specifically to the _vertical_ intervals of such sonorities.

For example, in Vicentino's system, while a vertical fifth of 3:2 in a
sonority of 4:5:6 or 10:12:15 is pure, a melodic fifth is typically
tempered. Also, unstable vertical sonorities such as a suspended c-f-g
or C3-F3-G3 (simplest ratio 6:8:9), or a suspended or suspension-like
c-g-a or C3-G3-A3 (simplest 2-3-5 ratio 10:15:18), may involve
tempered vertical intervals.

In such a system, then, R as well as P applies to a subset of stable
sonorities, and specifically to the _vertical_ intervals of such
sonorities.

Possibly a notation like PRstable-vert might express such a system.

One might say that such "adaptive JI" systems are designed to preserve
the P property while "tempering" the I property -- and a logical
result of this compromise is the partial presence of R in a mixed
environment of rational and irrational ratios.

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5. Conclusion
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Starting with the classic 2-3-5 JI model of the 16th century, I have
sought to suggest that we may explore various permutations of the PRI
properties which together make this model an "ideal type" for what
people often imply by the term JI: pure vertical intervals, pervasive
use of integer ratios, and intricate implementations on fixed-pitch
instruments.

Other PRI schemes such as 2-3-5-7 or 2-3-7 JI also fit these musical
implications of "just intonation," albeit with often quite different
musical specifics than those of Zarlino.

A Gothic or neo-Gothic PR scheme of 2-3 JI (Pythagorean intonation)
also fits the properties of Purity (for stable sonorities) and
Rationality (for all intervals), while at the same time constituting a
regular tuning with less "Intricacy."

An RI scheme based on complex rational ratios intricately ordered
(possibly with an emphasis on melodic aspects), or a PI scheme based
on Setharean spectra realized by intricate irrational ratios for
maximal "purity" or sensory consonance, or a contingent (P)RI scheme
where complex ratios have a "pure" effect in a setting of intricately
structured sonorities (e.g. LaMonte Young), are other interpretations
of "JI" which may have musical qualities distinctive from those of
usual tempered systems.

"Adaptive" JI systems in the sense of PRstable-vert or the like
resemble conventional PR or PRI systems in seeking purity, and regular
temperaments in seeking to disperse or minimize intricacy, especially
the kind of aural "intricacy" represented by syntonic or septimal
comma shifts/drifts.

Since an exclusively integer-based tuning might approximate any
regular temperament as closely as desired, and a regular temperament
of arbitrarily large size might likewise approximate pure integer
ratios as accurately as desired, a definition of "JI" in terms of
rationality (R) or purity (P) alone may not be especially
distinctive.

We might speak, for example, of an R-based "quasi-temperament" or an
irrationally-based "quasi-JI" system: Kirnberger's approximation of
12-tone equal temperament (12-tET) by a ratio for the fourth of
10935:8192, for example; or 118-tET used to emulate a 2-3-5 JI system.

One way to approach such quasi-temperaments and quasi-JI systems is to
say that the fine conceptual structure of the tuning falls in one
category (Kirnberger's scheme as "R-based JI," 118-tET as an
irrational or logarithmic division of the octave), but its intended
and audible musical application in another (Kirnberger's scheme as
"virtual 12-tET," 118-tET as "virtual 2-3-5 JI").[4]

An advantage of the PRI approach is that it permits us to take a
"three-dimensional" approach in explaining, for example, that while
Kirnberger's scheme is R-based, it differs from more characteristic JI
systems in being both non-P (no pure intervals) and non-I (no
intricacy of the kind associated with PRI systems, but rather a
regular structure for all practical purposes identical to 12-tET).

We can thus distinguish Kirnberger's "virtual 12-tET" not only from
PRI or PR systems of JI, but from a non-P system of the RI variety
with an intricate "emmelic" structure of integer ratios featuring
commas, schismas, or the like.

This last example brings us to a very vital point made by Bill Alves,
whose scholarship and graciousness alike served to impress it upon me
in an earlier dialogue. Any model of "JI" based on the Western
European evolution of this concept over the past 500 years or so is
necessarily a culturally-specific one, and if used as a "universal"
standard may distort and deprecate the nature of integer-based musics
in other world cultures.

Complex "emmelic" JI systems focusing on melodic structure rather than
vertical "purity" -- in the terms of our typology, RI systems --
represent a different approach than that of Zarlino, as he himself
acknowledges in distinguishing between "ancient" and "modern" styles.
In his view, the "ancient" Greek style based essentially on one
melodic line permits the artful use of the chromatic and enharmonic
genera as well as the diatonic; but only the diatonic fits "the modern
style, with its multiplicity of parts, always seeking the perfect
harmony."[5]

Thus the PRI approach, itself taking 16th-century Western European
practice and theory as a starting point, might serve merely as one
possible three-dimensional model for integer-based systems of music,
inviting other models based on the specific structures and concepts of
a variety of world musics.

-----
Notes
-----

1. Gioseffo Zarlino, _The Art of Counterpoint: Part Three
of Le Istitutioni harmoniche, 1558_, trans. Guy A. Marco and
Claude Palisca (W. W. Norton, 1976), ISBN 0-393-00833-9, Chapter 17,
at p. 35. Interestingly, Zarlino suggests that voices "do tend to
approximate more closely those instruments that are tuned to the
Pythagorean steps, where these minute intervals are not present, than
those that are perfectly tuned in accord with the perfection of the
harmonic numbers." At the same time, he asserts that voices "seek the
perfection of intervals." One way to reconcile these two statements
would be to posit a technique of "adaptive JI" of the kind described
in Section 4 below; and scholars such as Paul Erlich have proposed
such an adaptive model for polyphonic vocal intonation.

2. Ercole Bottrigari, _Il Desiderio, 1594_, trans. Carol MacClintock,
Musical Studies and Documents 9 (American Institute of Musicology,
1963), pp. 41-42. "...[Y]our ears have never heard any consonances
other than those of the Diapason in its true form; and are accustomed
to hear those only as tempered, defective, and altered on all the
stable and stable-alterable instruments."

3. Here I am much indebted to Joe Monzo for the insight that the
concept of "perfect" intonation or JI seems to emerge in Western
European theory along with the concept of temperament, both concepts
addressing the intonation of Renaissance sonorities combining
incommensurable ratios of 3 and 5.

4. For the example of 118-tET as a "virtual 2-3-5 JI system," I am
indebted to Dave Keenan.

5. Zarlino, n. 1 above, Chapter 74, p. 272. Here Zarlino is
specifically rejecting the adaptation of all three Greek genera to
modern polyphonic practice as advocated by Nicola Vicentino, with
prominent use of the chromatic semitone and the enharmonic diesis of
approximately 1/5-tone (e.g. the diesis G#-Ab in 1/4-comma meantone at
128:125 or ~41.06 cents). While this judgment may be most debatable,
Zarlino's recognition that 16th-century Western European polyphony is
not necessarily a universal standard for music or intonation deserves
appreciation.

Most respectfully,

Margo Schulter
mschulter@value.net

--- "M. Schulter" wrote:
> -----------------------------------------------
> Just intonation definitions and typologies:
> A three-dimensional approach (PRI)
> -----------------------------------------------
>
> Attempts to define "just intonation" (JI) may say as much about the
> musical outlooks and philosophies of the interlocutors as about the
> concept of JI itself. Part of the richness and complexity of this
> dialogue may lie in the fact that typically "JI" may actually imply
a
> combination of musical traits and associations, leaving different
> people free to emphasize different aspects.
>

Margo,

To comment here, I must begin by saying that I stand in humble awe of
the deeply insightful content of this paper, and find that you
suggest a fresh framework for discussion of the many important faces
of Rational Music practice. The brilliance of these categorizations,
is that it allows one to clearly see and identify the use of these
facets of Rational Intonation, which may be at work within a given
composer's style (as we know, the contemporary meaning of JI seems to
have leapt off the pages of the OED, and went running off on a
mission of its own!). This could only come from someone in possession
of many profound years of experience of active practice and
performance with tunings. The voice of experience delivering the
profound truths of "taking the theory to the music", has such a
refreshing and clear sound to my ears.

> --------------------------------------------------
> 3.2. Rationality and Intricacy: Complex RI systems
> --------------------------------------------------
>
> Here it should be noted that radical RI systems, as described for
> example by Jacky Ligon, might focus especially on _melodic_ aspects
of
> integer-based music. The ancient Greek interest in superparticular
> melodic intervals, for example semitones of 16:15 or 28:27, may
> provide a time-honored precedent for this type of JI approach, with
a
> focus on _emmeles_ or "intervals apt for melodic progressions."
>

Thanks for introducing me to this term: "emmeles". Now I know what to
call it!

One of the most prolific and experienced microtonal composers I know,
basically told me that the most important and primal thing about
music is also the most difficult to categorize and qualify in
mathematical terms - "MELODY". And if one thinks about it, they will
see it's true. Melody being the very soul of music (and "Intricacy"
for that matter), is something that can't be explained fully or
easily by means of harmonic theory. Obviously, what looks good on
paper may not always work in a musical setting - and moreover, what
works melodically, may be lacking in harmonic properties and visa
versa. It has taken many years for my ears to climb to the
appreciation of higher primes and complexities, and the mother-load
of pure gold that I've found is in the melodic realm.

Something interesting, is that I demonstrated to my guests this
weekend, everything from Pythagorean, to O-U Hybrids, to maddeningly
high prime scales, and we all agreed that it is important to grasp
the concept that RI structures can generally be optimized for melody
or harmony - this is my native tongue. Some high prime scales I
played on our new TX802 timbre, to show nakedly what the harmonic
attributes of the tuning were (some didn't reach the Keenanian 50%!),
but when they would be used melodically on a real flute sample
(oxymoron of the day!), it was wonderful! I think it's important to
embrace both of the poles, from "P" to "I".

> -------------
> 5. Conclusion
> -------------
>
> Complex "emmelic" JI systems focusing on melodic structure rather
than
> vertical "purity" -- in the terms of our typology, RI systems --
> represent a different approach than that of Zarlino, as he himself
> acknowledges in distinguishing between "ancient" and "modern"
styles.
> In his view, the "ancient" Greek style based essentially on one
> melodic line permits the artful use of the chromatic and enharmonic
> genera as well as the diatonic; but only the diatonic fits "the
modern
> style, with its multiplicity of parts, always seeking the perfect
> harmony."[5]

It's more than a little fascinating to realize the antiquity of the
knowledge of how to navigate the higher primes and complexities. No
doubt we come culturally from the harmonic side of things being
emphasized. Being culturally conditioned to harmony should not become
a barrier to appreciation of "I" melody.

Thanks,

Jacky Ligon

P.S. I should add - and also with deepest possible insight, my friend
pointed out that you never hear anyone humming or whistling chord
progressions. Something from the Xen Master!!!

Re: JI definitions and types -- a three-dimensional approach (PRI)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗ligonj@northstate.net

🔗Joseph Pehrson <pehrson@pubmedia.com>