Greetings,
In last week's dust-up inre temperament, there were several references
made to the orchestral use of Just intonation. However, there is a variety
of definitions available for this term, and they are not necessarily all
inclusive.
The following is a reprint from Margo Schulter's posting to the 'tuning"
list, it gives a fairly broad overview of the different ways of defining JI.
Regards,
Ed Foote
--------------------------------------------
What is Just Intonation?:
A Definition in Musical Context
--------------------------------------------
Hello, everyone, and one of the great pleasures of the Tuning List is
an opportunity to share in dialogue that can encourage an exploration
of new concepts and approaches to familiar terms. "Just Intonation,"
or JI for short, may be such a term.
Especially for newcomers, the world of JI may be difficult to survey
as a whole. Often the term is associated with assumptions which apply
to specific styles of music, and may not apply to JI systems
appropriate to other eras or styles or world musical traditions.
Here I would like to suggest a definition of JI which takes into
account not only the objective qualities of a tuning system, but also
the qualities of the music for which its "justness" is being judged.
In other words, the same tuning may be an instance of "JI" when
applied to one piece of music, but not an instance of "JI" when
applied to another.
Too often, it is tempting to define a JI system as "a system that does
intonational justice to my favorite music, or at least a significant
subset of it." Rather than simply repudiate this human proclivity, I
would like here to bring it into good service: to define "JI" as a
meeting between a tuning system with certain characteristics and a
style of music to which the tuning system can "do justice."
It is my pleasure warmly to dedicate this article to Johnny Reinhard,
whose views on intonational issues in the works of Charles Ives
(1874-1954) have led me to the conclusion that what is JI for Perotin
may be a deliberate and artful departure from JI for Ives. Possibly
one might speak of an outlook of "intonational relativity," where the
justness of a tuning depends on the music's frame of reference. In any
case, while the flaws of this article are of course my own responsi-
bility, I would like to thank Johnny Reinhard for his thoughtful
contributions to this list and his bold enthusiasm for many kinds of
music.
----------------------------------------------
1. What is Classic JI?: A threefold definition
----------------------------------------------
In defining a JI system, I would like to focus on three requirements,
two being reasonably objective and intrinsic to the tuning system
itself, the third being contextual, and depending upon the specific
musical composition or style to which the system might be applied.
After giving the following formal definition, I would like to explain
each of the three points of the definition in a more informal and
conversational manner:
A tuning system is a classic just intonation system
in the context of a specific musical composition or
style if:
(1) All intervals in the system are defined as
integer ratios;
(2) The system provides a complete set of
low-integer ratios up to an odd factor or
"odd-limit" of at least 3; and
(3) This odd-limit is high enough to provide
pure or low-integer ratios for all stable
concords in the given musical context.
To anticipate a curious question for some readers and an ardent cause
for others, I would explain for the moment that in addition to what I
term "classic JI" systems meeting all three requirements, there are
"adaptive JI" systems.[1] The latter systems meet points (2) and (3),
but typically include some intervals based on irrational rather than
integer ratios (see Section 3).
Returning to more conventional or "classic" JI, let's consider each of
the three points.
--------------------------------------------------------------
1.1. Classic JI systems define all intervals as integer ratios
--------------------------------------------------------------
Our first requirement simply says that JI systems, as opposed to other
systems, derive all their intervals from integer ratios. In other
words, JI systems and tempered systems with at least some irrational
ratios are here treated as mutually exclusive categories. This test
tells us that neither 12-tone equal temperament (12-tet) nor 1/4-comma
meantone (despite its pure 5:4 major thirds) is a JI system, because
both include irrational ratios.[2]
Within this constraint of "all intervals as integer ratios," a JI
system may be built in various ways. One elegant method is simply to
build from the powers of 3:2 (the pure fifth) or 4:3 (the pure
fourth), plus 2:1 (the pure octave). Thus we obtain a pure major
second (9:8) from two pure fifths minus an octave, and an intriguing
major sixth (27:16) and major third (81:64) from three or four pure
fifths respectively, and so on. Since 3 is the highest prime factor in
this system, it is called a "3-prime-limit" JI system.
A "5-prime-limit" system would add intervals such as pure major thirds
(5:4) and minor thirds (6:5), and also, for example, a minor seventh
at 9:5. Such a system could also have what might be less obvious
intervals, for example a variety of minor second at 135:128, as the
prime factors 2, 3 and 5 interact in various ways.
A "7-prime-limit" system would add yet more intervals, both obvious
favorites for advocates of this system such as the 7:4 minor seventh,
and more intricate ones such as the neutral third (between minor and
major) of 49:40. Here the prime factors 2, 3, 5, and 7 are free to
interact.
Current JI systems may have prime limits as high as 17 or 19; beyond
this point, there is some debate as to whether yet higher primes
really have a tangible musical identity.
While this first requirement is significant in what it excludes
(temperament in its various forms), it is also significant in what it
permits. Both the variety of JI systems, from 3-limit to 19-limit or
higher, and the variety of musics based on these systems, are awesome.
Sometimes people speak categorically of a "just scale" or "just
intonational ideal." In such cases, one must ask, "Which one?" As
Admiral Grace Hopper remarked of software standards, we might remark
of JI standards: the nice thing about them is that there are so many
to choose from.
---------------------------------------------------------------------
1.2. Classic JI systems provide pure concords to at least 3-odd-limit
---------------------------------------------------------------------
Our second requirement focuses on the expectation that a JI system
will provide ideally pure or simple ratios for the stable concords in
a given music. This ideal is sometimes described as "beatlessness."
For example, in a style of music using fifths and fourths as the
richest stable concords, ratios for these intervals of 3:2 and 4:3
respectively should make them sound ideally smooth and blending.
In another style favoring stable sonorities built from not only these
intervals but also major and minor thirds, tuning these thirds at 5:4
and 6:5 should likewise produce an ideally smooth and blending
quality.
Still another style might prefer richly stable sonorities of 4:5:6:7,
6:7:9, or 12:14:18:21, and here the availability of pure or simple
integer ratios such as 7:4, 7:6, and 9:7 should again optimize the
blend or concord.
In order that a JI system may fulfill such expectations, it must
provide a complete set of intervals with low-integer ratios up to an
"odd-limit" of 3 or greater. That is, any JI system must at least
provide those pure intervals with an odd factor of 3 or less: namely
octaves, fifths, and fourths. Optionally, a JI system may have an
odd-limit of 5, 7, or higher.
A 3-limit system, for example, includes pure or just octaves (2:1),
fifths (3:2), and fourths (4:3). A 5-limit system additionally
includes pure major thirds (5:4) and minor thirds (6:5). A 7-limit
system additionally includes 7-based versions of the major second
(8:7), minor third (7:6), diminished fifth (7:5), and minor seventh
(7:4). A 9-limit, 11-limit, or higher system will add yet other
"small-integer" ratios (e.g. 9:7, 11:9, 13:8, 17:12, 19:16).
Note that this second requirement excludes some systems based on
entirely on integer ratios which nevertheless do not meet a prime
objective of JI: optimizing the tuning of stable concords to make them
as blending as possible.
For example, in 1766, Johann Philipp Kirnberger proposed a method for
approximating 12-tet by finding a series of pure fifths and thirds
together forming a fifth of 16384:10935, an interval repeated 11 times
to complete the system. Here all intervals are based on (very large!)
integer ratios, and the system provides pure 2:1 octaves, but not pure
3:2 fifths or 4:3 fourths. Thus Kirnberger's "quasi-12-tet" is not a
JI system in our sense.
--------------------------------------------------------
1.3. Classic JI systems in context: Pure stable concords
--------------------------------------------------------
To this point, our criteria for a JI system have been fairly
objective: does the system base all intervals on integer ratios, and
provide pure intervals up to an odd-limit of 3 or higher?
Our third criterion, however, depends not only on the tuning system
but on the music to which it is applied.
In order to realize the ideal of just intonation _in a specific
musical context_, the system must have an odd-limit high enough to
provide pure intervals for all stable concords of the composition or
style in question.
For example, a 3-limit system can beautifully realize the just
intonation ideal of "beatless stable concords" for the music of
Perotin, a Gothic composer of the era around 1200. However, it would
not do intonational justice to the music of the 16th-century composer
Orlandi di Lasso (1532-1594), which calls for a 5-limit system in
order to make all stable concords pure or beatless.
Nor can a 5-limit system realize the ideal of just intonation when
applied to music calling for 7:4 minor sevenths, or 11:9 neutral
thirds; and so on.
Thus at its conceptual lowest common denominator, "JI in action"
occurs when a tuning system meeting our first two objective criteria
actually provides a full set of pure stable concords for a given
musical composition, improvisation, or style.
-----------------------------------
1.4. Concord, discord, and contrast
-----------------------------------
While JI systems as here defined share in common the trait of
providing pure ratios for all stable concords of a given music,
such musics need not be limited to stable concords alone. Both
"dual-purpose" sonorities regarded as relatively blending but
unstable, and yet more tense "discords" felt urgently to seek
resolution, can provide creative conflict and contrast in the
unfolding of a composition or improvisation.
One serious misunderstanding about JI systems holds that such systems
must seek pure or ideally concordant ratios for _all_ intervals --
including those which may be intriguing dual-purpose sonorities or
outright discords in a given musical style. In fact, a contrast
between "beatfulness" in unstable sonorities and purity or beatless-
ness in stable ones, like conflict in a novel or drama, can add to
musical interest. Some JI systems make the most of this contrast when
applied to an appropriate musical style.
For example, in the music of Guillaume de Machaut (1300-1377), both
4:6:9 (e.g. G3-D4-A4) and 64:81:96 (e.g. G3-B3-D4) might be termed
"dual-purpose" combinations; they are relatively blending, but
unstable, always calling for further music. (Here I use a notation
with C4 as middle C, with higher numbers showing higher octaves.) As
it happens, 3-limit JI makes the first sonority ideally pure, and the
second rather complex and "beatful." Both the energetic fusion of the
first sonority and the pleasant edginess of the second add to the
flavor of the music.
Likewise, in the music of Monteverdi (1567-1643), a 5-limit tuning of
10:15:18 for a sonority such as D3-A3-C4 lends the rather tense
quality of the 9:5 minor seventh to this stylistically discordant and
engaging combination. A theorist of the time, defending Monteverdi's
boldness with such sonorities, describes them as a mixture of "the
sweet and the strong": both the pure fifth (3:2) and minor third
(6:5), and the edgy minor seventh, might fit this ideal.
While many JI systems invite such contrasts, they are an optional
feature; it is quite possible to compose music where _all_ intervals
are regarded as stable concords, and have pure ratios included within
an applicable tuning's odd-limit.
Thus JI musicians and advocates may differ not only in their choice of
odd-limits and musical styles, but in their preferences regarding
levels of contrast between concords, dual-purpose sonorities, and
discords. Such varying tastes lend emphasis to the point that an ideal
JI system for one kind of music may do great intonational injustice to
another.
--------------------------------------------------------
2. Just tunings and motivations: intonational relativity
--------------------------------------------------------
A curious consequence of our definition is that the same tuning may
constitute a JI system when applied to one kind of music, but a non-JI
system (quite possibly favored precisely as such) when applied to
another kind of music.
For example, a 3-limit JI tuning can indeed realize just intonation in
a musical sense for the Gothic polyphony of Perotin or Machaut. As
discussed above, it not only provides pure ratios for all stable
concords, but fits the intricate spectrum of concords, dual-purpose
sonorities, and discords.
When the same tuning is applied to the music of Charles Ives[3], where
thirds and sixths seem to serve as stable concords, the result is
something quite in contrast to JI in a musical sense. Major and minor
thirds, for example, have rather complex and "beatful" ratios of 81:64
and 32:27 where their musical role would call in a JI interpretation
for the ideally pure ratios of 5:4 and 6:5.
>From a musical point of view, the 3-limit tuning may in fact have
almost opposite meanings for these two styles. In a Gothic setting,
where thirds are unstable, their beatfulness lends point to the
listener's secure expectation of a resolution sooner or later to
stably concordant sonorities with ideally pure octaves, fifths, and
fourths. In a setting such as the music of Ives, where thirds are
treated as stable, their beatfulness instead may suggest a pervasive
sense of restlessness and inconclusiveness, an "Unanswered Question."
Additionally, the use of a 3-limit tuning in Ives might be, as the
composer himself suggests, motivated by a desire to emphasize the
melodic or horizontal dimension. Such a tuning provides narrow or
compact diatonic semitones: Db is closer to C, for example, while C#
is closer to D. Both this enhanced "melodic pull" of the semitones,
and the pervasive vertical tension when a 3-limit tuning is applied to
a music where thirds and sixths are stable concords -- or in other
terms, are left unresolved -- may accentuate the discrete layers of
the texture, the distinctness of the melodic lines.
Of course, the melodic factor of compact semitones may be just as
important (and attractive) in Gothic music as in Ives; but here, this
factor pulls in tandem with vertical resolutions from unstable
sonorities to stable and purely intoned ones. Thus we have true JI in
a musical sense, with the two dimensions in equilibrium. In Ives, the
same tuning produces an artful disequilibrium.
If we define a JI system in terms not only of the tuning itself but of
its interaction with a given musical style, then the same tuning can
indeed be JI for Machaut and artful non-JI for Ives.
----------------------
3. Adaptive JI systems
----------------------
While classic JI systems realize _all_ intervals as integer ratios,
"adaptive JI" systems include some tempered intervals based on
irrational ratios, but nevertheless meet the second and third prongs
of our test. That is, they provide a set of intervals based on
low-integer ratios up to 3-odd-limit or higher; and when used for
applicable musical styles, they provide such pure intervals for all
stable concords.
In 1555, Nicola Vicentino describes such a system for his
_archicembalo_ or "superharpsichord" with 36 notes per octave. While
his first tuning features a temperament dividing the octave into 31
more or less equal parts, his second tuning is an adaptive 5-limit JI
system when applied to the 16th-century music for which it was
conceived, where all stable concords fall within a 5-odd-limit.
In this system, the 19 notes of the instrument's first manual (Gb-A#)
are tuned in a meantone temperament[4] identical to his first scheme,
likely a 1/4-comma tuning with pure 5:4 major thirds or something very
close (by the mid-17th century, his tuning was being interpreted as
31-tet). The remaining 17 notes of the second manual, however, are
tuned in pure fifths with these. Thus if one plays a pure 5:4 major
third on the first manual (e.g. C3-E3) -- assuming a 1/4-comma tuning
for this manual -- and adds the version of the note G4 on the second
manual, the result is a pure 5-limit sonority having also a 3:2 fifth
and 6:5 minor third.
Vicentino's adaptive JI involves a distinction between a basic gamut
of _melodic_ intervals based on a tempered meantone, and an available
set of _vertical_ intervals providing pure ratios for all the stable
concords of Renaissance music.
Adaptive JI systems, then and now, may be especially attractive for
tunings needing to juggle two or more odd prime factors, in
Vicentino's 5-limit case both 3 and 5; and in more recent cases, often
also larger primes such as 7, 11, 13, etc. If one builds such a system
on integer ratios only, then complications such as awkward melodic
shifts may result.
For 3-limit JI, the classic approach of deriving all intervals from
integer ratios based on the powers of the primes 2 and 3 seems to
avoid most of these complications, so there is less motivation for an
adaptive approach.
-------------
4. Conclusion
-------------
In this article, I have attempted mainly to suggest that JI involves a
meeting between a tuning system and a musical context in which the
system can provide pure ratios for all stable concords.
This outlook of "intonational relativity" suggests that what is often
called the "out-of-tuneness" of an interval in a given JI tuning when
applied to a musical context demanding a higher odd-limit might rather
be called "out-of-styleness."
For example, in a musical context with stable concords calling for a
5:4 major third or a 7:4 minor seventh, a 7-limit enthusiast might
speak of a 3-limit major third (81:64) or 5-limit minor seventh (9:5)
as being respectively about 22 cents and 49 cents "out of tune."[5] In
appropriate musical settings where the 3-limit or 5-limit can include
all stable concords, however, these intervals could be regarded as
perfectly in tune: they were never advertised to be beatless or
stable, and the musical context may make them ideal exactly as they
are.
If the question is stated as one of contextual "out-of-styleness"
rather than inherent "out-of-tuneness," then a difference in musical
contexts may be clarified. The flaw lies not in the interval itself,
but in a collision between the interval and the musical context.
While leaving many "loose ends" indeed loose[6], I have attempted to
suggest an approach which may introduce newcomers to the variety of JI
systems and the relativistic nature of JI as a meeting between such a
tuning system and a musical setting where it can realize the ideal of
pure stable concords. If this approach ultimately promotes more
concord among people applying different JI systems to a variety of
musics, it will have served its purpose.
-----
Notes
-----
1. Being aware of the dangers of attributing any innovation in tuning
theory to a specific author when it may be found to go back centuries
earlier, I would to thank Paul Erlich for bringing the term and
concept of "adaptive JI" to my attention in a series of Tuning List
posts. The concept, at least, would seem to go back at least to
Vicentino's second archicembalo tuning of 1555 (see Section 3).
2. In a meantone temperament, the most popular keyboard tuning from
around the later 15th to later 17th centuries, fifths are slightly
narrowed or tempered in order to provide pure or near-pure major
thirds. In 1/4-comma tuning, they are narrowed by 1/4 of a syntonic
comma (81:80, ~21.51 cents), the difference between the active thirds
and sixths built from pure fifths in a 3-limit tuning and the pure
5-limit versions of these intervals. Since a major third is built from
a chain of four fifths (e.g. F-C-G-A), a 1/4-comma tuning makes such
thirds one comma narrower than the 3-limit 81:64, i.e. a pure 5:4.
A cent is equal to 1/1200 octave, so that the equal semitones of
12-tet are each 100 cents.
3. As already mentioned in the dedication at the beginning of this
article, I am most deeply indebted to Johnny Reinhard, and especially
to his discussions of intonation in Ives.
4. On meantone, see n. 2.
5. Since there are 100 cents in a 12-tet semitone (see n. 2), these
are substantial differences -- about 2/9 and 1/2 of a semitone
respectively.
6. Such loose ends might include, for example, JI systems based on one
limit which are extended far enough to approximate closely the pure
ratios of a higher limit, and are then applied to music with stable
concords calling for such a higher limit. If such an extended system
is applied to musical styles with all stable concords within its usual
limit, then there is no definitional problem: it meets the test for
classic (or adaptive) JI while offering some extra intervals for
"special effects."
Most respectfully,
Margo Schulter
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