=== === ============= ==== === === == == = == == === == == == == ==== === == == == == == == == = = == == == == == == == == ==== M U S I C T H E O R Y O N L I N E A Publication of the Society for Music Theory Copyright (c) 1993 Society for Music Theory +-------------------------------------------------------------+ | Volume 0, Number 3 June, 1993 ISSN: 1067-3040 | +-------------------------------------------------------------+ General Editor Lee Rothfarb Co-Editors David Butler Justin London Elizabeth West Marvin David Neumeyer Gregory Proctor Reviews Editor Claire Boge Consulting Editors Bo Alphonce Thomas Mathiesen Jonathan Bernard Ann McNamee John Clough Benito Rivera Nicholas Cook John Rothgeb Allen Forte Arvid Vollsnes Marianne Kielian-Gilbert Robert Wason Stephen Hinton Gary Wittlich Editorial Assistants Natalie Boisvert Cynthia Gonzales All queries to: mto-editor@husc.harvard.edu * * CONTENTS * * AUTHOR AND TITLE FILENAMES 1. Target Article Mark Lindley and Ronald Turner-Smith, An Algebraic Approach to Mathematical Models of Scales mto.93.0.3.lindley.art mto.93.0.3.lindley.fig (full listing of GIFs in the table of contents) 2. Discussion Threads Robert Judd mto.93.0.3.judd.tlk Joel Lester mto.93.0.3.lester.tlk Justin London mto.93.0.3.london.tlk Richard Parncutt mto.93.0.3.parncutt.tlk Stephen Smoliar mto.93.0.3.smoliar.tlk 3. Reviews (none) 4. Announcements mto.93.0.3.ann Feminist Theory and Music 5. Employment mto.93.0.3.job 6. New Dissertations James Boyd (Univ. of Michigan) mto.93.0.3.dis 7. Communications (none) 8. Copyright Statement +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+ 1. Target Article AUTHOR: Lindley, Mark and Turner-Smith, Ronald TITLE: An Algebraic Approach to Mathematical Models of Scales KEYWORDS: group, halfgroup, harmony, dodecaphony, semitones, temperament, Landini, Matteo da Perugia, Dufay, Schutz, Louis Couperin Mark Lindley P.O. Box 1125 Cambridge MA 02238-1125 USA Ronald Turner-Smith 19 Risingholme Road Harrow, Middx HA3 7EP England ABSTRACT: Although mathematical models of the scale have always been characteristic of Western music theory, in the last 200 years they have not been very much improved (although some interesting properties of scales have been defined in recent years). This article describes our effort in a new book to contribute to this part of music theory by using some appropriate concepts of modern algebra. ACCOMPANYING FILES: mto.93.0.3.lindley.fig (Figures 1-10) mto.93.0.3.lindley1.gif (10K) mto.93.0.3.lindley2.gif (16K) mto.93.0.3.lindley3.gif (6.5K) mto.93.0.3.lindley4a.gif (12.6K) mto.93.0.3.lindley4b.gif (23.1K) mto.93.0.3.lindley4c.gif (12.4K) mto.93.0.3.lindley4d.gif (22.7K) mto.93.0.3.lindley5a.gif (8.8K) mto.93.0.3.lindley5b.gif (13.8K) mto.93.0.3.lindley5c.gif (13.1K) mto.93.0.3.lindley5d.gif (12K) mto.93.0.3.lindley5e.gif (12.5K) mto.93.0.3.lindley5f.gif (6.6K) mto.93.0.3.lindley5g.gif (13.3K) mto.93.0.3.lindley6.gif (23.2K) mto.93.0.3.lindley7.gif (20.1K) mto.93.0.3.lindley8.gif (30K) mto.93.0.3.lindley9.gif (45.5K) mto.93.0.3.lindley10.gif (10.3K) [1] Our sequence of mathematical constructions is listed in Fig 1.(1) Our point of departure is the pitch continuum, which is not a mathematical construction but an intuition, since pitch is subjective. (Pitches as we hear them cannot be measured, but only judged.) Our first construction is a positive-number line for the sound-wave frequencies, which can be measured and which almost wholly determine the pitches. ----------------------------------------------------- 1. All the Figs are in the file mto.93.0.3.lindley.fig, which is available through the MTO FileServer, mto-serv. Consult the MTO Guide (information.txt) on how to retrieve files with mto-serv. ----------------------------------------------------- [2] It is the differences between pitches that interest us musically. Experience teaches us that there is a logarithmic relation between these subjectively judged differences and the differences among the corresponding pitch frequencies. So our second construction is a number line for the logarithms of the frequencies. We use logarithms to the base 2; this choice is due to another aspect of musical experience: the musical interval between two notes whose frequencies are in the ratio 2:1 is an octave, and notes one or more octaves apart from each other are intuitively heard as manifestations of the same note on different levels. Normally a scale repeats itself in each octave (there are exceptions, but they can be treated as special cases), so musicians today speak of pitch classes (see Fig 2) - that is, of equivalence classes of notes that are one or more octaves apart from each other - and of "pitch-class intervals" or "pitch-class relations", which are the analogous equivalence classes of musical intervals. With logarithms to the base 2 it is very easy to define the addition of logarithms-mod-1 (which we call "flogs") in such a way that the sum will be musically valid when the flogs for two pitch-class relations are added. [3] For a simple example, let us see how the flog for a 5th with a frequency ratio of 3:2, when added to itself, yields a flog for a whole- tone. It is readily reckoned that, to three decimal places, log 3 to the base 2 = 1.585, and hence log (3/2) to the base 2 = 0.585. Now instead of adding 0.585 + 0.585 and getting 1.170 for a major 9th (which differs by 1 from the log for a whole-tone), we reckon in terms of flogs and write .585 + .585 = .170. This is illustrated in GIF 1.(2) ----------------------------------------------------- 2. Each GIF is in its own data-file: mto.93.0.3.lindley1.gif, mto.93.0.3.lindley2.gif, etc. Some GIFs have separate parts, each in a separate file, e.g. mto.93.0.3.lindley4a.gif, ...lindley4b.gif, etc. (see the table of contents for MTO 0.3 for a complete listing. The command: "send mto.93.0.3.lindley*.gif" will retrieve all GIFs for the present essay from mto-serv. ----------------------------------------------------- [4] According to Max Weber,(3) there are two rational ways to construct a system of tones: by means of harmonic relations or else by dividing the octave into equal parts. From this hypothesis can be derived two types of generators for our pitch-class relations (see Fig 1e): either equal- division (whereby 1/n-octave is taken as a flog), or else harmonic (which will be described below). We are content to consider these two types in our book, but would admit any other valid type that could be adequately defined. ----------------------------------------------------- 3. Max Weber, *Die rationalen und soziologischen Grundlagen der Musik*, ed. Theodor Kroyer (Munich 1921). (The English translation published in 1958 is, alas, so inadequate that it quite misrepresents Weber's thinking.) ----------------------------------------------------- [5] Our next construction is the algebraic pair: (set, Abelian group), to which such generators give rise. The groups operate on the sets; the operation is addition. A positive number in a group means "add so much," but in a set means an amount that is *per se* so much. (There is of course no such thing as a "negative" or an "absolute-zero" pitch.) [6] Although music theorists have always represented notes as points, we provide, for the sake of greater validity, that each note in the scale will occupy a small neighborhood on the number line, in order to give every note some leeway for things like vibrato, inexact intonation etc. (We define the musical intervals as between the centres of these neighborhoods.) Thus the elements of our pitch-class sets are equivalence classes of neighborhoods around points-mod-1 (see Fig 3). To reach the highest degree of validity, one ought to allow that the neighborhoods for different notes in the same system may differ in size, and that in certain cases (for example, when a violinist has a wide vibrato) the neighborhoods for notes adjacent in the scale may overlap, obliging one to treat the neighborhoods as fuzzy sets. For the sake of simplicity, however, our book uses a kind of a modelling in which the neighborhoods have definite borders, do not overlap, and are uniform in any one system. We also postulate that in every system the leeway is at least a couple of ten-thousandths of an octave. Thus we reach (at Fig 1h) our next construction: "ideal systems". Each ideal system has a set of non-overlapping neighborhoods (and is thus finite) and a subset of one of our groups. This subset, which we call an "embedded halfgroup", is an unusual, indeed novel, algebraic structure: it is associative (when the sums are defined) and has an identity element and inverses, but is not closed with regard to group multiplication (compounds of the operation). The place of halfgroups vis a vis semigroups, quasigroups etc. in halfalgebra is indicated in GIF 2.(4) ----------------------------------------------------- 4. All the terms in GIF 2 except "halfgroup" are defined in Richard Herbert Bruck, *A Survey of Binary Systems* (Springer, Hamburg 1958). On the basis of our cordial correspondence with Eytan Agmon we think that a 12-oriented theory of diatonicism (as described in his "A mathematical model of the diatonic system" in *Journal of Music Theory*, 33:1) can be improved upon, insofar as validity in regard to certain music is concerned (e.g. Gregorian chant), by an acceptance of the concept of halfgroups as applicable to music. ----------------------------------------------------- [7] We wonder if in some cases the size of the neighborhood might guarantee a set of pitch classes small enough to be musically useful. Music is not only an art of sonorities but also (among other things) a cognitive game, and most composers wish to "juggle" with a set of pitch classes that is big enough to enable them to sustain an interesting 5-, 10- or 20-minute-long game of this kind - for which only three or four pitch classes would be insufficient - but not so big that the listener cannot grasp the cognitive game intuitively - for which, say, 40 pitch classes would be too many. One needs an intermediate number, something like 7, 8, 10, 12, 15 or 20. Gregorian chant normally has 7 or 8; Giovanni Gabrieli and Schutz had 14; some other well-known Renaissance composers (Costeley, Bull) composed music for 19 pitch classes; Bach, Debussy and the Beatles had 12. In non-Western music, the sizes of the sets are comparable. Now we have noticed that in outdoor genres - for instance, marching-band music - the intonation is not very exact (that is, the neighborhoods for the notes are quite wide) and the number of pitch classes in a phrase is normally closer to 7 than to 12. Is there a cause-and-effect relation here, in that such bands are usually unable to project chromatic harmony because their intonation is so inexact? The question has not been investigated (as the concept of pitch-class-leeways is new); we raise it in order to show that just where we come to our unusual algebraic structure (the halfgroup) we find a music-theory question which calls for empirical treatment - namely, the possible relation between *(a)* the limiting of the set which is due to the pitch- class leeways and *(b)* the limiting which one would want in any case for the sake of a cognitively viable juggling of the pitch classes. [8] It was by means of a natural mapping that we went over from notes and musical intervals to pitch classes and pitch-class relations; so now we return by reversing the natural mapping and thereby pulling back the system to an "unbounded" scale (repeating itself indefinitely, octave after octave) from which limited scales, each with a highest and a lowest note, can readily be derived (Fig 1i-j). One could then go farther, to scales in which certain pitch classes are omitted in certain octaves, or to scales in which every interval has a little something extra added to it (and thus the frequency-ratio for the octave, for instance, is a little bigger than 2:1, as on the piano) and so on. We prefer, however, to concentrate our attention on systems. [9] Most systems of Western music have had harmonic generators. There is a series of such generators, which - so experience teaches us - can be derived mathematically from the following series of primes: 2, 3, 5, 7. Our adaptation of the traditional Roman numerals of music theory for these generators is shown in Fig 4. With the first generator alone (which we write with the Roman numeral "I"), one can make a kind of minimal music in which all the notes belong to the same pitch class; with the first two generators (I and V), one obtains the most familiar kind of Medieval harmony, in which the 5ths (and their compounds and inversions), but not any 3rds, are used as consonant intervals; with the first three generators (I, V and III) one obtains the triadic harmony of the Renaissance; and with all four (I, V, III and VII), one has certain aspects of later harmony. [10] The extra "smaller flogs" (positive or negative) referred to in Fig 4 are used to obtain one or more equations between the generators, and thereby more pitch-class relations with a pitch-class set of a given size. These small amounts, which we designate with the letter "t", are necessary for this, because no multiple of the flog of one prime number can be equal to another such flog. (It is well known that no power of one prime number can have another prime as a factor.) The use of such small extra amounts is traditionally called "tempering", and in practice, temperaments - systems with tempered consonances - have been normal since the Renaissance. [11] How small should these small amounts be? To answer this question we have to classify empirically certain intervallic magnitudes, or rather, certain ranges of magnitude. Some amounts that are too small to be used for melodic intervals between notes (or for pitch-class relations) are nonetheless big enough - in the form of deviations from, say, log 3 or log 5 - to disturb a generating harmonic pitch-class relation by making the resulting intervals sound out of tune. A rough classification is shown in Fig 5. This is only a first approximation; empirically there is no particular validity in a tenfold relation between the different ranges. For example, while the semitones in equal temperament are each 1/12 octave, a semitone between an untempered major 3rd (i.e. where t^III^ = 0) and an untempered minor 3rd would be only some 6% of an octave (this is reckoned in GIF 3), so it would be better to say that the range of magnitudes for melodic steps is "20ths" rather than "10ths" of an octave. Also, an amount which would very likely render a 5th sour can in certain cases serve for the tempering of a major 3rd. (Such is the case in equal temperament, where the major 3rds are tempered by a little more than 1% of an octave; whereas a 5th tempered by such an amount would be melodically and harmonically too ugly to use in many kinds of music.) Thus the concept of ranges or orders of intervallic magnitude needs to be refined empirically by means of psycho- acoustical probings and a reading of the old music treatises (which often discuss temperaments). In general, however, tempering is taken to mean the dividing up of such inconvenient magnitudes as are labeled "out-of- tune-ness" in Fig 5 into smaller, less noxious amounts to be distributed amongst a suitable chain of generating pitch-class relations. [12] In order to find the most feasible possibilities for tempering "two- dimensional" systems (systems with V and III as generators, but not VII), we ask the following question: If one multiplies flog 3 by 1, -1, 2, -2, 3, -3 and so on, then which are the multiples that approach successively closer to flog 5 or its inversion? In the last (full) column of Fig 6 we see the smallest flogs by which it is possible to temper V and III at once by distributing the various such differences evenly amongst the group of generators.(5) The first two such flogs - T^1^ and T^2^ - are too big (they would mar the pitch-class relations); T^3^ is good; T^4^ is nearly insignificant, hence very good; to make use of T^5^ would involve more than 45 pitch classes - too many for traditional composition. Thus the equations at the far right in Fig 6 represent the most likely possibilities. They are the most feasible equations between harmonic generators for a two-dimensional system. ----------------------------------------------------- 5. When we go from one n to the next we get a rather lower T^n^ because s^n^ is smaller while at the same time m^n^ is bigger. (However, since each T is an average of some t^V^s and a t^III^; one of those t's may be less than T if the other is more.) Some intermediate multiple might yield a *slightly* lower average t (if the quasi-s is not as much bigger as the multiple is smaller), but it is reasonable, once the multiple becomes bigger than, say, 15 or 20, to demand a distinctly lower average t in return for involving more pitch classes. ----------------------------------------------------- [13] Now we are ready to discuss our system tree (see Fig 7). We include those possibilities that have been significant in the history of Western composition and theory. There are harmonic and equal-division systems, according to the type of generator. Among harmonic systems we have those of one, two or three dimensions, according to the number of generators (apart from the identity element). Also among harmonic systems, we distinguish between coherent sytems (in which all the pitch classes make one chain of 5ths) and non-coherent systems - which have proven musically so awkward that no well-known composer has ever written music for such a system, even though many theorists since the 16th century have described non-coherent, two-dimensional systems without any tempered intervals. (Mostly they were theorists who did not understand the problem to which tempering is the solution.) [14] Among coherent systems, we have temperaments and (one-dimensional) untempered systems. Then we have regular temperaments (in which each kind of consonant interval is of uniform size), semi-regular temperaments (discussed below), and irregular temperaments, in which some 5ths are tempered a little more than others and hence the 3rds etc. also vary. Certain irregular temperaments have been quite important historically. [15] Among regular, two-dimensional temperaments, there are two main types (as we have seen in connection with Fig 6); and when both of their defining equations (4V = III; -8V = III) are true for the same system, then we have an intersection of the two types, which is so important historically that it has its own name, "equal temperament"; and this name can refer as well to an equivalent one-dimensional type (i.e. with a "circle" of twelve 5ths but no consonant 3rds) which may conceivably have played a role in the history of lute music in the 15th century, and also to a three-dimensional type which practically everyone would agree is to be found in the music of, say, Villa-Lobos (and which we believe is to be found in some earlier music as well: think of how Wagner will resolve an appoggiatura to a 7th-chord from which the harmony is then just as free to move as it would be, in 17th-century harmony, from a triad). [16] Apart from equal temperament, there is a spectrum of musically good possibilities for each of the two types MT and QP (see Fig 7h). Meantone temperaments were very important for Renaissance and early Baroque music. They usually put at the composer's disposition two or three flats and three or four sharps: if three flats and four sharps, then in all the 14 pitch classes (7 chromatic and 7 diatonic) mentioned above in connection with Gabrieli and Schutz. (See Fig 8. It is well known that some keyboard instruments had 14 keys per octave, i.e. with "split keys" for Eb/D# and for Ab/G#.) The chain of 5ths had a beginning and end, and this was very important for the scheme of Renaissance modes, and often important also for the planning of compositions. In GIF 4a, for example, we see how Schutz in one of his pieces timed the successive steps toward the edges of his chain of 5ths.(6) In an 18th-century composition one would normally find the richest harmony in the middle of the movement, not just before the end. And why? Because in the 18th century, the chain of 5ths was closed to make a circle; to modulate "far away" did not mean to approach a border; so towards the end of the piece one would merely return to the freely-chosen central pitch class, with no opportunity to draw upon the structural discipline of an impending fence. In many late 19th-century compositions, on the other hand, all the pitch classes are introduced already in the first few bars. ----------------------------------------------------- 6. GIF 4a shows a diagram from page 144 of our book and GIFs 4b, 4c, and 4d show some relevant excerpts from the piece to which the diagram refers, "Die so ihr den Herren fuerchtet" (SWV 364). Twenty diagrams of this kind (with relevant musical examples) are included in Lindley, "Heinrich Schutz: intonazione della scala e struttura tonale" (with a long abstract in English), in *Recercare*, vol. i (1990). ----------------------------------------------------- [17] A quasi-Pythagorean temperament with twelve pitch classes was of some importance in the first half of the 15th century. The 5ths were either untempered or else so little tempered that no one at all was aware of it. The five chromatic pitch classes were linked to Bb in the chain of 5ths - we know this from contemporary treatises(7) - and so there was a "wolf 5th" between B and F#, as F# was tuned so low that it made a sour 5th with B. By chance, however, all these rather low chromatic notes made remarkably euphonious 3rds (hardly tempered at all) with the diatonic notes, as indicated by the slanted lines in the diagram at the beginning of GIF 5a.(8) Now in this transitional period between one-dimensional and two-dimensional harmony, certain composers would sometimes use such a 3rd at the end of a section of a piece (as can be seen in the musical examples in GIFs 5b-5g),(9) but no one would make such use of a 3rd without a sharp. To understand this interesting moment in the history of harmony, one must appreciate properly the significance of the system; we will say more about this below. ----------------------------------------------------- 7. Lindley, "Pythagorean Intonation and the Rise of the Triad", Royal Musical Association *Research Chronicle* 16 (1980). 8. The diagram is on page 55 of our book. 9. These examples (GIFs 5b-5g) show the conclusions of sections from the following pieces: Landini, "O fanciula giulia;" Matteo da Perugia, "A qui fortune" and "Le grant desir;" a Kyrie for organ from the Faenza Codex; Dufay, "Mon chier amy;" and a prelude from the Buxheim Organ Book (no. 242). ----------------------------------------------------- [18] For some of the regular temperaments in our system tree, there is an *equivalent* equal-division system - that is, physically the same though differently conceived (see Fig 7i-j). In a harmonic system there exists consonance (since the harmonic generators are the most consonant pitch-class relations) and hence nearly always also its counterpart, dissonance; and there is an important distinction between diatonic semitones (between two notes with different letter-names) and chromatic semitones (between two notes with the same letter-name). In an equal- division system there is no consonance, and therefore no dissonance. If the generator is 1/12-octave and if all twelve of the ensuing pitch classes are in the system, then one composes "mit zwoelf nur auf einander bezogenen Toenen" (in Arnold Schoenberg's words) and there is no distinction between diatonic and chromatic semitones: they are all qualitatively as well as quantitatively alike. This aspect of Schoenberg's dodecaphonic music is just as important as his atonality (the fact that each movement or piece is not somehow centered on one privileged pitch class), which is often said to be its most basic technical characteristic. [19] With such a perspective on how various systems have affected the art of composition, one can appreciate better the technical significance of enharmonic modulations in Romantic music. In most enharmonic modulations a given semitone is used first as a chromatic semitone and then as a diatonic one, or *vice versa*. More and more in the course of the 19th century, the significance of enharmonic modulations lay not so much in their momentary effect as in the way they enabled composers to exploit the same physical scale in terms of two systems at once: harmonic and equal-division. Thus David Lewin's analytical sketch (reproduced in GIF 6) of a well-known phrase in the prelude to Wagner's *Parsifal*(10) includes not only Roman numerals for a traditional harmonic analysis, but also Arabic numerals to show how 3 + 3 + 1 = 7 semitones (adumbrating the salient "Zauber-motif" in *Parsifal*) lead from Ab to a cadence on Eb. ----------------------------------------------------- 10. David Lewin, *Generalized Musical Intervals and Transformations* (Yale, 1987), p. 161. The 3 + 3 + 1 diagram ends with a high Eb, but the tune really goes to the Eb an octave lower after gliding down, step by step, from high Ebb (a minor 3rd above Cb) to middle D (a diatonic semitone below Eb). One hears an implicit equation between the Ebb and the D, inferring that they are an octave apart, and this gives the passage its enharmonic character. ----------------------------------------------------- [20] It is possible to distinguish certain "families" of equal-division systems (see Fig 7j) equivalent to the various kinds of temperaments. Fig 9 includes formulas (derived in one of the appendices of our book) for their generators. GIF 7 reproduces a diagram by Isaac Newton(11) showing how an equal-division system with 1/53-octave as generator and with 15 pitch classes is equivalent to the harmonic system represented in Fig 10. The diatonic semitones, labeled "mi-fa" in GIF 7, amount to 5/53-octave; the chromatic semitones amount to 4/53. Newton's harmonic system is not coherent, but if he had provided for an additional pitch class at "4" in the diagram, it would have made at once a good Ab to his Eb (at "35") and a good G# to his C# (at "26"), and thus he would have had a coherent, quasi-Pythagorean system.(12) ----------------------------------------------------- 11. *GB-Cu* add. 4000, fol. 105*v*. (Reproduced on page 57 of our book.) 12. Helmholtz for his "Harmonium mit natuerlicher Stimmung" used a system of this latter kind with 24 pitch classes. See Hermann von Helmholtz, *On the Sensations of Tone*, tr. Alexander J. Ellis, 2nd ed. (London, 1885), 316-19, or for a more succinct account, the entry on "Just intonation" in *The new Grove Dictionary of Musical Instruments*. ----------------------------------------------------- [21] Among the irregular temperaments, the most important historically were those used in the late 17th and 18th centuries. For most composers of that time, the various keys had much more individual character than they do today, and many contemporary music theorists said that it was due to the irregular temperaments of the day. An irregular temperament based on a circle of twelve Vs can be described as a variant of equal temperament, so in GIF 8 the sizes of each semitone in three such 18th- century schemes (by J. G. Neidhardt, J. H. Lambert and Vallotti) are described as some percent of 1/12-octave. The numbers at the outer edges of those diagrams show the differences between semitones that are adjacent in the circle of 5ths (in the sense that E-F and F#-G are adjacent to B-C) and thereby show that in each of these competently designed schemes, the semitones vary quite gradually, with B-C and E-F being the largest and F-Gb and A#-B the smallest. There is an analogous pattern of gradually varied nuances among the 3rds and 6ths, with C-E-G being tempered least and Gb-Bb-Db-F most. [22] On the silent screen we cannot demonstrate the acoustical differences amongst the different keys in such a system. But we can describe how, in the first section of Louis Couperin's famous F#-minor Pavane (see GIF 9), the composer used the pitch class F = E# in a special way. E#, which is essential to the key of F#-minor, was in the French- style irregular temperament tuned so high in relation to C# that the resulting interval was acoustically rather harsh. In bar 2 (at the first asterisk in GIF 9) E# is avoided: contrapuntally, our little ancillary example in GIF 9 would sound so much more natural that Couperin's alternative resolution of the chord F#-C#-G#-A is obviously an artful evasion. A similar avoidance of E# at the end of bar 5 (at the second asterisk) precipitates a modulation to A-major in the next two bars. In bars 10, 11 and 16 (at the next three asterisks) E# does appear, but each time in so dissonant a context (notice the A's and B's) that the acoustic sourness of E# with C# merely gilds the lily, as it were. C#-E# is at last heard in a straightforward triad at the end of the section; but then in the next bar (not included in GIF 9) the composer reverts immediately to a C#-minor chord, as if to say, "Alas! E# is *too* sharp for a straightfoward triad; let us revert to E-natural." This is an extreme case in that the consonant status of the major 3rd (or 10th) was actually compromised by its heavy tempering.(13) A wealth of subtler nuances involving some of the other 3rds in this piece are just as telling when the music is heard in a stylistically appropriate tuning (matching the conceptual system). It would be far more intelligent, however, to demonstrate such nuances than to try to describe them *in absentia*. ----------------------------------------------------- 13. It may be worth repeating that this is a French style. According to Bach's concept of the chromatic scale, which is reflected in what we know about his tuning (Lindley, "Bach's Harpsichord Tuning", *The Musical Times*, vol. 126, December) as well as his music, harmony in "extreme" keys is less constrained. ----------------------------------------------------- [23] In early 15th-century music such as represented in GIF 6, the distinctly euphonious quality of a 3rd or 6th with a sharp in the quasi- Pythagorean temperament is sometimes especially salient because it occurs right after (or right before) a prominent harmonic 3rd or 6th that is tempered by an entire comma (that is, by nearly as much as C#-E# in Louis Couperin's pavane). We may therefore speak of a "semi-regular" temperament (Fig 7m), because while the 5ths are uniform, the composer has evidently found two sizes of consonant or virtually consonant major 3rd, major 6th etc. in the scale. This is a queer kind of system, destined to play only a brief (though important), transitional role in the history of harmony even though it is physically the same as a regular system. [24] To measure the difference between any two systems that are physically almost but not quite the same, we have devised a "margin of equivalence", and with it the concept of "quasi-systems" which have no generators (and thus no subset of a group of pitch-class relations) but only a set of pitch classes, whose neighborhoods are, however, unequal. To put it very briefly: if two systems have the same number of pitch classes, then the margin of equivalence is the smallest overlap - i.e. where the notes differ most when the two systems are aligned as well as possible (as illustrated in GIF 10). [25] We hope that our book in which these and some related ideas are elaborated upon(14) will prove of value to music theory and to the study of music history. Renaissance and Baroque theorists took only some limited steps away from Medieval models of scales (by accepting ratios involving 5 and 7 as prime factors, and then by accepting irrational ratios), and even today many music theorists more or less vaguely favor the ancient Pythagorean idea that "Music is sonorous number." Here our algebraic approach could be of value, not only with regard to irregular temperaments (where the pitch-class relations have to be represented as functions of the pitch classes and not as numbers in their own right), but also for the designing of experiments to investigate the various musical and psycho-acoustical phenomena that give rise to pitch-frequency leeways for the notes. (Instead of a general leeway u, one could distinguish u^1^, u^2^, u^3^....) Some refinement of concepts pertinent to music history may also be derived from our work, as music historians have generally either neglected most of the various kinds of system which we describe or else have mistakenly treated them as a negligible aspect of performance practice - that is, as unconscious and inconsequential variants of equal temperament insofar as composition is concerned. ----------------------------------------------------- 14. *Mathematical Models of Musical Scales* (Verlag fur Systematische Musikwissenschaft, Postfach 9026, DW-5300 Bonn, Germany). ----------------------------------------------------- +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+ 2. Discussion Threads AUTHOR: Robert Judd TITLE: Commentary on Justin London's MTO 0.2 article REFERENCE: mto.93.0.2.london.art File: mto.93.0.3.judd.tlk I was interested in Richard Parncutt's comments, although I came up with a different perception. > Regarding durational accent, the last of the six notes (A) will tend > to sound accented due to its relatively long effective duration. This seems crucially important to me. > If, as the notation of Justin's example implies, the > first 3 notes represent C major harmony and the last three F major, > then the harmonic accent will fall on the fourth note, F. Is there an implied problem here? if we think CDE FGA, we'll imply groups of three (A weak). > Perhaps the strongest effect of all in the example is the primacy > effect, according to which the first note is the downbeat, simply > beacuse it is the first. OK, but I didn't sense this. I.e. We start with no expectations, we hear a note, we ask "is it a downbeat or an upbeat?" We listen to confirm one or the other alternative. BUT for the last note A, we have heard quite a few pitches in a series, we come to expect the same, but hear silence instead, which thus accents the last note strongly. I seem to hear the A as stronger than the C, thus I arrange duples and triples to come up with A as downbeat. I end up getting either c alone as upbeat in duple meter (C DE FG A) or c and d as upbeats in triple meter (CD EFG A) (a possibility not mentioned by RP). Both of these of course imply the melody as part of a context. My preferred interpretation, C DE FG A, is found in "doh, a deer"! Re Stephen Smoliar's comments [see mto.93.0.3.smoliar.tlk], drawing Beethoven's first into the picture: yes, I agree, there is a tradition of the musical phrase that goes way back, making the conveyance of downbeat "natural" and easy. One might also invoke the intro to Beethoven's seventh (i) or the Stravinsky Symphony in C (i) (which probably relied on just that literate ear to pick up the Beeth. refs.). The chicken-egg question that the perception-oriented person is wondering about is: do we hear it that way because we're so used to hearing it that way (i.e. is it learned), or do we hear it that way because of an a priori, physically based, perceptual inclination, one that has little reliance on previous experience? Bob Judd robert_judd@zimmer.csufresno.edu ********************************* AUTHOR: Joel Lester TITLE: Commentary on Justin London's MTO 0.2 article REFERENCE: mto.93.0.2.london.art File: mto.93.0.3.lester.tlk Like many of his other illuminating writings on these topics, Justin London's "Loud Rests and Other Strange Metric Phenomena (or, Meter as Heard)" in MTO 0/2 cogently presents some of the problems that have held center stage in writings on rhythm and meter in recent years (and probably for centuries in one or another form). But London's proposed solution to some of these problems via the notion of a "dynamic model" in which listeners interact with acoustic signals from the music they are hearing lacks a crucial component. London posits that we can avoid some of the severest problems of various metric theories by adopting a participatory attitude -- as he says in paragraph 12, "let's tap our feet and count along." The question he fails to address is: How do we know when to begin to count? Consider his one-line tune from paragraph 6: c-d- e-f-g-a (r) d-e-f-g-a-bb (r) b-c-b-c-b-c (r) d-c-bb-a-g-f, "where the duration of each pitch is an 8th note, and each rest two eighth notes (rests are indicated by (r)), at a tempo of a quarter-note = 100," and where b stands for B-natural and bb for B-flat. I was stymied by this example for for some time before being able to resume reading London's article. At first, I assumed I was going to create this (as a performer, of course! -- I'll return shortly to the differences between meter for the performer and the listener) with no preconceived meter and let the tune create its meter for me as I went along (in effect, I was role- playing having a split personality and being a performer and listener simultaneously). At first, I found myself grouping eighths in pairs starting on a strong eighth. The recurring pattern of six eighth-notes plus two-eighths rest made it clear by the end of the second string of notes that I was dealing with a periodicity of four quarter-notes worth of music. (By the way, I did have to impose that metric decision on the music retroactively from the end of the second string of notes.) The resulting harmonies and nonharmonic tones that were implied by my beginning on a strong eighth led me to question whether this was a common-practice-period tonal melody at all. As I "tapped my feet and counted along" in this manner, I was hardly hearing the melody that London presents later in that paragraph. I was hearing his pitch-string, but not his melody in the sense of the pitch-string organized by meter (and, consequently, harmony; I agree with Heinrich Christoph Koch's two-century old assertion that changing the placement of the beat changes the harmonic structure, and hence the nature of a phrase itself). I was unsatisfied with this first rendition, not least because I didn't think that London's previously-stated apology for producing "banal" examples posited such a weird melody. So I tried beginning with an upbeat eighth, still retaining the implicit 4/4 structuring. Once again, I ended up with a melody structured quite differently from the way London structures the pitch-string at the end of his paragraph. How does London explain how we know when to begin counting? Essentially, he provides us with a score to the melody with the metrics indicated (near the end of paragraph 6). London implies that he has not provided a score, but instead has invoked our commonly-recognized metric pattern of "sol-la-ti-DO." But this begs a host of questions. I will leave aside how we know to draw upon this particular pattern, when other common scale-step patterns beginning with two rising whole tones and semitone are just as possible. I will restrict myself to two questions even more primitive than that one: How do we know that the melody begins with an upbeat? And even more deeply, what evidence is there that when this melody begins we know that the first note is "sol?" Without knowing that, we cannot know to invoke London's pattern. Musical scores are sets of instructions for performers. And composers of tonal music included metrics in their scores to preclude precisely the sorts of problems I had in trying to figure out how to perform London's melody. Performers must know where the accented beats are located, or else they will not know how to perform the music. Imagine trying to put into effect Leopold Mozart's instructions to begin measures with downbows if you didn't know where the downbeats were! But listeners have to figure out where the beats are *without* scores (an issue I have addressed in "Notated and Heard Meter," in *Perspectives of New Music* 24 [1986]: 116-129, as well as in my *Rhythms of Tonal Music* [Carbondale: Southern Illinois University Press, 1986]). It is no accident that composers are usually a lot more friendly to listeners than London was to his hypothetical physicist, generally providing accompaniments, durational differentiations, textural differentiations, dynamic differentiations, and the like at the beginnings of pieces to allow the listeners to establish the metric grid so that they can indeed "tap their feet and count along." In other words, composers provide a variety of criteria for differentiating events from one another. Whether one wants to call those criteria *accents* (as distinct from *metric* accents), as I have done in my book on rhythm, or use some other conceptualization or locution, these criteria are necessary. To fail to consider them is to fail to consider the differences between scores as instructions for performers and sounding music as an adequate source of information for listeners. A final note: since I have not yet been able to get my PC to read the GIF file with examples, I read London's melody cited above in the tablature notation that appears in paragraph 6 of his article, not in a staff notation that he may have provided in the GIF file. As a result, I could not draw upon my long experience in sightreading staff notation to survey quickly the entire melody in one fell swoop -- I was forced to create the melody note by note as it occurs. I believe that we theorists are so adept in reading musical notation that in addition to failing to appreciate fully how scores are instructions for performers and not for listeners, we also often fail to realize how easily vwe use visual cues to create a synoptic perception of a passage instead of a diachronic perception that is closer to what a listener receives. Joel Lester CUNY les@cunyvms1.gc.cuny.edu ********************************* AUTHOR: Justin London TITLE: Preliminary response to the MTO 0.2 article REFERENCE: mto.93.0.2.london.art Whilst there were (as of yet) no "formal" responses to my article on "Loud Rests and Other Strange Metric Phenomena" I was most gratified to see the number and variety of responses, comments, and discussion that appeared on the SMT-list. I was especially pleased that a number of music-psychologists joined in the debate, as well as mainstream theorists (what ever that means, these days). For volume 0.4 I will prepare a formal reply to several of the discussion threads which arose in response to "Loud Rests", but for the moment I would like to offer a few comments on some issues raised by Joel Lester, Rich Parncutt, and others about my "weird" third example. Example 3 is metrically ambigious in a deadpan performance, and was so chosen precisely for that reason (though I will admit to moving rather quickly from the physicist's first hearing to the metrically-indexed version in paragraph [6]). I was/am assuming that in this example did not have a deadpan performance, but rather a performance which contains subtle, yet highly conven- tionalized variations in timing and dynamics (what Sloboda, op. cit. below, has termed "expressive variations") of each note within the anacrusis; these varitaions act as cues for the meter.(1) On the basis of those cues we can hear the first three durations, leaving any tonal interpretation(s) aside for the moment, as inidicative of a "and-four-and-ONE" metric pattern. Lester is completely correct in reminding us that the score is a set of directions for the performer, which is not the same thing as a knowledge- representation of the listener's metric cognition and understanding. ================================== (1) The study of timing and dynamics (and their metric implications) is currently a hot topic in music-cognition circles. Aside from the work by Eric Clarke on "Categorical Rhythmic Perception" cited in "Loud Rests," other relevant studies include: Alf Gabrielsson, "Timing in Music Performance and its Relations to Music Experience." in *Generative Processes in Music,* ed. John Sloboda, Oxford: Clarendon Press,1988; John A. Sloboda, "The Communication of Musical Metre in Piano Performance," *Quarterly Journal of Experimental Psychology* 35A:377-96, 1983; and Neil P. Todd, "A Model of Expressive Timing in Music," *Music Perception* 3:33-58, 1985. ================================== And as result of following the directions given by the score, musicians will produce a sound structure that encodes the metric information--the bar lines are thus "recoverable" to the listener. Lester also is quite right in pointing out that when we first hear the melody we have no idea that the opening pitches are "sol-la- ti-do." The first note can be anything; the first two are a whole step, which could be placed in various diatonic conetxts. But when we have the first three tones (assuming a diatonic context, which of course, may be an incorrect assumption), which span two whole steps, we now a fairly circumscribed number of tonal possibilities: do-re-mi, fa-sol-la, or sol-la-ti. Of these three only the first and third are viable candiates for a diatonic beginning--as starting on fa is rather unlikely (examples, anyone?). But at this point (3 notes into the "piece"), we will also have some timing and dynamic information about the three notes; as indicated above, a performer faithful to the notation will give varying emphasis to these pitches, something like "tone-Tone-tone-(TONE)". If we perceive that the second note is longer/louder (just a bit) that the first or third, we have a good reason to hear the tones as "sol-La-ti-DO" (OK, I've included the fourth note now), since the rhytmic emphasis is on the 2nd and (esp.) fourth notes of the scale. I posit that this interpretation is more likely than one which places these tones in a "do-Re-mi-FA" context, for that would be a highly atypical metric placement for tonic. Indeed, what I believe Lester did in grouping the 8ths in pairst, starting on the strong 8th, was to mentally perform the opening motive as "DO-re-Mi-fa-SOL-la (rest)", with appropriate expressive variations for that metric context. Along with expressive variations, and the scale-step limits that accrue as the melodic line unfolds, there is another factor which facilitates metric recognition, and that is that listeners know a lot about musical beginnings. It seems reasonable to imagine that we have a rich store of opening templates or schemata (of varying levels of specificity) which appertain to different musical styles. We have a "so-La-ti-DO" pattern already in our heads, in other words, and so it isn't so much a task of building the scale-step representations and metric placement from first principles, as it were, as it is a task of matching the given sound structure to our repertoire of opening gestures. As a final remark re the "problems of example 3", in "Loud Rests" I point out that metric cognition involves two phases, one of recognition, as well as one of continuation. What should be noted here is the metric recognition phase, which is what all the fuss is about, _must_ be retrospective--how could it be otherwise, unless we have metrically clairvoyant listeners--but nonetheless we are able to achieve metric recognition _very_ fast. This rapidity is due to the rich number of cues the music provices via expressive varations as well as our practiced experience in responding to those cues. Well, this is perhaps a bit more than just an informal reply, so I will conclude with a promissory note. In volume 0.4 I hope to comment on Smoliar's discusion of Desain's work on Expectancy Space, and its relevance to a dynamic model of meter, as well as the discussion thread spun by Judd, Demske, and others re musical and cognitive universals. Justin London Carleton College jlondon@carleton.edu ********************************* AUTHOR: Richard Parncutt TITLE: Commentary on Justin London's MTO 0.2 article REFERENCE: mto.93.0.3.london.art File: mto.93.0.3.parncutt.tlk I'd like to return to the recent analyses of Justin London's rhythm example by Robert Judd and myself. JL's example 3 began with the 6 notes c d e f g a, where all notes had the same duration except the last (a) which was longer. The story so far: I looked at the phenomenal accent of each note and concluded that c had a "primacy" accent, a had a durational accent, and d, f, and a were candidates for harmonic accents. RJ concluded that the sequence could be parsed in either of two ways -- either duple note groups with (metric) accents on d, f, and a, or triple note-groups with accents on e and a. The main difference between these two interpretations is the role of the primacy accent on the first note, c. My perceptual analysis referred to surface features heard on a first listening, while RJ performed a retrospective analysis of possible meters, arrived at after many listenings. In Judd's analysis, the primacy accent on c seemed relatively unimportant "in retrospect" -- at least by comparison to the durational accent on a. The difference between these two analyses points to a fundamental difference between music theory and psychology, of the kind alluded to by Greg Sandell in his recent letter to the list. Music-theoretic analyses generally assume previous familiarity with and understanding of the music, and are often based on "isolating ... passages of music and playing them several times" (a quote from Greg Sandell, somewhat out of context -- the point I want to make is that the theorist hears or imagines the music many more times than does the average listener). Psychological or perceptual approaches often go to the other extreme, exploring spontaneous responses to unfamiliar music or sound sequences presented in the "constrained conditions of an experiment run in a lab." I believe that a balanced combination of these two approaches could lead to significant progress in music theory. The difference between the approaches of Robert Judd and myself also involved levels of analysis. My analysis was focussed on a relatively low or "primitive" level -- phenomenal accent. Judd's concentrated on the next level up, the level of rhythmic strata (Yeston) or pulse sensations (my preferred term). In a systematic approach to rhythm, it may be useful to regard phenomenal accent and pulse sensations as independent and distinct, by first analysing phenomenal accents, and only then considering the resultant pulse sensations. Which of RJ's two solutions (duple, triple) is more likely? The relative importance of the two parsings may depend simply on the number of phenomenal accents that coincide with pulse events. This idea favors the duple grouping, as it involves more matching events than the triple grouping. Another effect is that of tempo. Research in rhythm perception (summarised by Fraisse, 1982) has suggested that pulse sensations are confined to a restricted range of tempi centered on about 100 beats per second ("moderate tempo"), and that most perceived pulses lie between a half and twice that value, that is, between about 50 and 200 beats per minute. According to this theory, at slow tempi, the RJ's duple note groups will be closer to moderate tempo, and will probably be preferred for that reason. At fast tempi, the triple note groups are more likely. The long-term aim of my research in rhythm is to develop an algorithm that predicts perceptual properties of simple rhythms in notated or performed music by the systematic application of a minimum number of specific rules or principles. Principles may be either perceptually "primitive" or specific to western music. The validity of the rules or principles may be checked by comparing predictions of the model with corresponding experimental results. This approach differs from most other music theory, in which the validity of analytic rules or principles is primarily determined by the perception and intuition of theorists. Traditional music theory nevertheless remains the primary foundation of, and motivation for, the model -- as well as most other research in music perception. ___________________________________________________ Reference Fraisse, P. (1982). Rhythm and tempo. In Deutsch, D. (Ed.), The psychology of Music (pp. 149-180). New York: Academic. Richard Parncutt McGill University parncutt@sound.music.mcgill.ca **************************************** AUTHOR: Stephen Smoliar TITLE: Commentary on Justin London's MTO 0.2 article REFERENCE: mto.93.0.2.london.art File: mto.93.0.3.smoliar.tlk There is one minor nit I would like to dispense with quickly concerning the matter of style. I felt as if this document had rather a parade of straw men in it. Each paragraph let me to raise my eyebrows and say, "Yes, but what about . . . ?," only to find that the but-what-about was covered in the following paragraph! It would have been nice had Justin not led us down quite so many garden paths in order to make his point, but perhaps I just happen to feel that way because I am in the thick of this stuff right now. What I REALLY want to write about is a but-what-about stone which was left unturned by Justin's discussion. It's a pretty heavy stone, though: But what about the fact that there is already a researcher who has worked out a potentially interesting quantitative model which not only accounts for the dynamic nature of meter but may even provide a viable quantification of just how loud some of those rests are? The research in question is named Peter Desain, and I want to address his work because I have been hard a work reviewing a recent book, MUSIC, MIND AND MACHINE: STUDIES IN COMPUTER MUSIC, MUSIC COGNITION AND ARTIFICIAL INTELLIGENCE, which Desain wrote with his colleague Henkjan Honing. (As an aside, my original intention was to write this review for ARTIFICIAL INTELLIGENCE. However, it began to grow into something more like a paper than a review; so I ended up sending it to COMPUTER MUSIC JOURNAL. Until I read Justin's paper, it had not occurred to me that it might be suitable for MUSIC THEORY SPECTRUM. For now, however, I just want to summarize one particular aspect of the work reported in this book.) Desain's model is called an EXPECTANCY SPACE. It was actually introduced to comparatively evaluate systems concerned with the detection of metric beat in performances of music. For example, given a timetable of MIDI events from a keyboard performance, the system being evaluated should be capable to translating the real-time durations of events into the discrete symbols of music notation. An algorithm to solve this problem was first proposed by Christopher Longuet-Higgins in the Seventies, and Desain wanted to compare the performance of this algorithm with a system of his own design based on a neural network. The principle behind the expectancy space is similar to that of Meyer's expectations. Given a past history of duration events, the question is whether or not that history predisposed the model in favor of certain durations rather than others. For example, if the last six events have all been interpreted as the duration of an eighth note, the expectancy space gives what amounts to a high probability that the next note will also be an eighth note, somewhat lower probabilities that it will be a sixteenth or quarter note, and so on down to a very low probability that it will be a whole note. Note that the purpose of the algorithm is to reflect how the interpretation system actually performs, but that means that each interpretation system in turn may be viewed as reflecting a particular kind of listening behavior. Neither of the systems being compared present particularly convincing expectancy spaces. This is because they are both based on the rather trivial goal of trying to establish simple integer ratios between successive durations. Thus, everything is evaluated on a note-by-note basis without any attempt to hypothesize how notes are grouped into measures or any other higher-level construct. However, the expectancy space could be used to evaluate any other system which tries to take this sort of rhythmic dictation. What is important is that it treats such a system as a dynamic function processing data in real time and displays the relationship between specific data and the behavior of that function. The most important element of this technique is that it is quantitative. One is not dealing with highly subjective measurements which try to capture how strong an expectation it. At any moment in the course of a performance, the system gives a numerical weight of predisposition for the duration of the next event. If that next event does not happen, as would be the case with a rest, it would not be too far fetched to interpret that weight as the "loudness" of the rest. The only real problem with Desain's results to date is that you have to have a model implemented before you can evaluate it with an expectancy space. Thus, the main thing we learn from his report is that note-to-note relations do not give us a particularly effective model, particularly when they only take duration into account. If one were to try to develop more realistic expectancy spaces, one would first have to assemble a more comprehensive model, taking account not only the recognition that duration is organized at a higher level than individual notes but also the roles of other parameters of performance, such as the pitches of the notes being performed, their dynamics, and perhaps their articulation. Such a model may still be a ways in the future, but the expectancy space now obliges US to think much more seriously and quantitatively about how it could be implemented. Let me close with one final nit. In paragraph [14] Justin writes: "Meter is neither a parameter like pitch or timbre, nor is it a part of a nested measuring of durational patterns and/or periodicities. It is something that is heard and felt." Are not ALL aspects of musical sound elements that are "heard and felt?" Justin's acknowledgement of phenomenology is all very well and good, but I do not think he gives it sufficient attention. ANYTHING which is either a musical object or a parameter of a musical object is ultimately a construction of the interpreting mind. That is as true of the sonority of a minor triad in first inversion as it is of a ternary metric pattern. The real question concerns the nature of the operations of construction which are brought into play in the course of listening. Justin is quite right that they are dynamic for meter; but, most likely, they are dynamic for all other aspects as well. The dynamic nature is not the issue. More important will be how well we shall be able to describe that nature in quantitative terms. [second commentary] Joel Lester raises some interesting points in his response to the London article. I think it is particular important to recognize the score as a set of instructions for performance whose information content should not be confused with that of sounding music. My guess is that one could augment his list of cues through which the sounding music can guide how one tap's one's foot; but enumerating those cues is not as important as acknowledging that such cues are there to be "picked up" from the audible signal. However, no matter how rich our supply of cues may be, it is rarely foolproof. Ultimately, there really is no good answer to the question: How do we know when to begin counting? The only absolute answer is: We don't; we HYPOTHESIZE a count. If we then discover that our count really does not "fit," we update our hypothesis. It is this updating of a running hypothesis which makes the model "dynamic," in London's sense of the word. Unfortunately, his paper only began to scratch the surface of those dynamics (as did Desain's work, coming from a different direction). The biggest rub, however, has to do with the question of "goodness of fit:" How do we determine whether or not our current hypothesis should be abandoned. That, I think, is where the sorts of cues Joel enumerated enter the picture. If too many of those cues offer too much evidence against where the hypothesis says the downbeat is, then it is time to change hypotheses. As a final point I think it is probably important that most of the cues which tend to be invoked to assess the running hypothesis are SURFACE features. When one analyzes a score, one can find no end of "deep" structural features which offer evidence as to where the downbeat REALLY is. However, I content that those features are another part of the landscape of instructions for performance. Listeners tend not to read scores, just as listeners to natural language tend not to diagram the sentences they hear. Rather, they pick up on surface features and respond to them. Perhaps, then, the real ART of performance concerns how the deep features which are the result of careful analysis may be made available as surface features to the listening ear. I am getting a bit worried about the way in which we are all jumping on Example 3 in Justin's paper. What worries me the most is a methodological danger which I shall call "selective denial of context." It seems as if each interpretation chooses to bar certain experiential elements from the context in order to make its point, and I am not sure this is a terribly healthy way to go. For example, I have now read several accounts which basically have tried to abstract away from the way in which Example 3 is actually notated, as if any intelligent ear should be able to infer the notation from the listening experience. This strikes me as being akin to looking down the wrong end of a telescope. I prefer Lester's view of the score as a set of instructions for performance. Thus, in this case the "game" is not one of inferring where, and how hard, to tap your foot. The score tells you that already; and it is the responsibility of the performer to make sure you "get the message." Rather, the "game" is determining when you bring your foot down which particular emphasis on a rest; where, to some extent, the energy of your stomp may then me taken as a rough measure of the loudness of the rest. This is not a question of the listener resolving any ambiguities which are latent in the score. That's the performer's job. Rather, the question is how the performer endows the listener with a mental state based on expectancies which set his foot tapping in the first place. The reason I trotted out my Desain hobby horse at the beginning of this discussion was because his expectancy space provides a means by which such a mental state may be inferred from strings of perceived durations. Having said all that, let me now stir up the pot which a bit more context which has received little attention. Having now sung Example 3 to myself so many times that it is beginning to invade my dreams, I have discovered that it is beginning to co-mingle with some more concrete musical memories. For example, while the resemblance is not note perfect, it begins with a gesture which we all know and love from the last movement of Beethoven's first symphony. I feel that such a "family resemblance" is particularly important when considering the "responsibility" of the performer. What I mean is that, because this particular passage is so similar in both pitch and rhythm to a passage which is in so many listeners' memories, the performer really does not have to do very much to communicate this particular set of score instructions. Indeed, the memory may well be triggered before even that first rest has been reached, thus making it all the easier for a mind with a rich memory to control the tapping foot. Stephen W. Smoliar; Institute of Systems Science National University of Singapore; Heng Mui Keng Terrace Kent Ridge, SINGAPORE 0511 Internet: smoliar@iss.nus.sg FAX: +65-473-9897 ***************************************************** 3. Reviews (none) ***************************************************** 4. Announcements FEMINIST THEORY AND MUSIC II: A CONTINUING DIALOGUE The Eastman School of Music, Rochester, New York, will host a conference on Feminist Theory and Music, to be held on June 17-20, 1993. Scholars from the fields of musicology, ethnomusicology and music theory as well as other disciplines, will join composers and performers for three and one-half days of paper sessions, lecture recitals, study sessions and workshops that focus on the insights of feminist theory as they relate to all subdisciplines of music. For registration materials, write to the University Conference and Events Office, University of Rochester, Rochester, NY 14627-0041; phone 716-275-4111 or 4171; FAX 617-275-8531. Email contact: Gretchen Wheelock (gwlk@troi.cc.rochester.edu) ***************************************************** 5. Employment NOTICE OF VACANCY IN MUSIC THEORY At McGILL UNIVERSITY Position: one non-tenure-track appointment at the rank of Faculty Lecturer; one-year contract renewable to a maximum of two subsequent years Qualifications: Ph.D. in music theory completed or well underway; proven excellence as a teacher will be an important criterion in hiring Nature of Duties: teaching of core undergraduate theory and musicianship (ear training) at any one of three levels; maximum of twelve classroom hours per week plus coordination duties for multi- sectional courses Salary Offered: current base salary: $33,000.00 Appointment Date: September 1, 1993 Closing Date: June 15, 1993 Applicants should submit a curriculum vitae and arrange to have three letters of reference sent to: Professor Bo Alphonce Chairman, Department of Theory Faculty of Music McGill University 555 Sherbrooke Street West Montreal, Qc H3A 1E3 In accordance with Canadian Immigration requirements, this advertisement is directed in the first instance to Canadian citizens and permanent residents. +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+ 6. New Dissertations AUTHOR: Boyd, James W. WORKING TITLE: Mahler and Directional Tonality INSTITUTION: University of Michigan School of Music Department of Music Theory Earl V Moore Bldg 1100 Baits Dr. Ann Arbor, MI 48109-2085 BEGUN: January 1992 COMPLETION: September 1993 ABSTRACT: Carl Dahlhaus's view that the modern era of music began in 1889 with, among other works, Mahler's Symphony No. 1, can be taken as a challenge to explore the origins of this work in detail. There is no better place to begin this exploration than the preceding orchestral song cycle, *Lieder eines fahrenden Gesellen*, from which the symphony borrows both body and soul. The tonal plans of the two works, however, contrast sharply. It is my purpose to determine if the directional tonal scheme of the song cycle is a phenomenon determined by the poetry, and, if so, what impact this might have had upon the symphony. The linear contrapuntal aspects of these works are historically significant, since they come on the heels of Wagner's *Parsifal*, but do not necessarily follow in its footsteps. KEYWORDS: Directional Tonality, Schenker, Tonal Function, Linear Analysis, Recomposition, Text-based, Absolute Music, Lieder, Symphony TOC: Chapter One - Defining and Interpreting Tonality Chapter Two - Analytical Approaches to "Directional" Tonality Chapter Three - Application to Mahler: *Das klagende Lied* and *Lieder eines fahrenden Gesellen* Chapter Four - A Comparison of Tonal Relationships in the *Lieder eines fahrenden Gesellen* and the First Symphony. Contact: James Boyd 29 Winona Ave. Lawrence, KS 66046 913-865-2330 e-mail: mboyd@kuhub.cc.ukans.edu +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+ 7. Communications (none) +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+ 8. 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