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
This PTG archive page provided courtesy of Moy Piano Service, LLC