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OTHER HARMONY: BEYOND TONAL and ATONAL by Tom JoHnSon

(Publisher: Editions 73 Two–Eighteen Press; illuminating coherence, regarding aspects practice. Bartók and unnamed others are Second Printing edition (Nov 18, 2014)) of tonality and atonality–and their differ- for Johnson kindred spirits whose com- by MARK ZUCKERMAN ences–as matters of style based on a wide positional process is unencumbered by and deep knowledge of relevant musical parochialism–exemplified by tonality and While it may be tempting to sample com- literature. atonality, whatever they may be. posers’ published recipes for their source Johnson’s approach is entirely differ- e bulk of the book presents pitch material, these may not be good fare for ent. He introduces the book by describing materials generated by a number of differ- those on low–salt diets. Such is the case a debate he observed raging in the late ent means drawn from Johnson’s personal with Tom Johnson’s Other Harmony, 1960s between two opposing camps of experience as a composer supplemented though the amount of salt needed may composers: tonalists (as represented by by his research at the French National depend on your background and outlook. Ned Rorem) and atonalists (e.g., Charles Library. e latter he admits was some- e book’s subtitle, beyond tonal and Wuorinen). So what matters is not so what cursory–reluctant as he was to atonal, is an intriguing promise. ere are much what tonality and atonality have to “become a musicologist” at his stage of few musical issues more intensely examined than “tonality” and “atonality.” Among This is an age–old problem typical of the scientific community’s the myriad approaches, perhaps the most attempts to explain musical phenomena, many of which may be interesting regards each as a framework to based on scientific truths but musical fallacies. apprehend a piece of music rather than an inherent property of the music itself. From offer as the conflict between them: tonal- life–largely just refreshing his acquain- this perspective, as eloquently articulated ity vs. atonality. In other words, the subject tance with books he had encountered in by John Rahn in his classic Basic Atonal isn’t experiential, theoretical, or musico- the past, most of which, unsurprisingly, eory, tonality and atonality are filters logical, but political–and in the fashion of are unconventional and obscure. employed by a listener who experiences current politics, Johnson casts a plague on ere are chapters on the work of a piece of music as “tonal” or “atonal” both their houses: musicians: Josef Matthias Hauer, a con- depending on the filter. How satisfactory …Nobody seemed to really temporary of Schoenberg’s who wrote the experience turns out depends on how notice that all this time Bartok music using tropes, an adaptation of well the filter integrates what’s being heard and others were writing fine medieval practice. ere are several Rus- and how well the listener uses it. music without taking a posi- sians: Nicolas Slonimsky, best known as e fundamental objective of listen- tion on one side or the other. I a musical lexicographer, a supporter of ing is achieving coherence, from which think we finally need a new har- avant–garde music, and a humorist, who all other experiential aspects–visceral, mony book that goes beyond theorized about pentatonic collections; emotional, and intellectual–flow. Music tonal and atonal and considers Nicolas Obuhow, a composer of the first th emerges through coherence; incoherence all the Other Harmony that half of the 20 Century who worked is noise. Music theories set forth principles has dominated and continues with chords containing the entire chro- of coherence used in constructing concep- to dominate music. matic; and Joseph Schillinger, creator of tual models for music, either descriptive Never mind that Bartók died a good 20 the “Schillinger System” of composition (i.e., analysis as listening aid–Rahn’s “fil- years before this debate purportedly took whose best known acolyte was George ter”) or prescriptive (compositional aid). place and that nowhere does Other Har- Gershwin. Schillinger, like Hauer, devel- Musicology provides another approach to mony attempt to explain Bartók’s harmonic oped an updated version of a medieval

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practice (isorhythm) as well as a catalogue cal” for musicians and “too musical” for is is not to deny that mathematics of chords constructed systematically by mathematicians, a too generous assessment can have valuable applications in music combining intervals without regard to a in keeping with Johnson’s that Euler was a theory, especially in creating models for tonal reference. (with the emphasis given pioneer of Other harmony too radical for his musical dimensions like pitch and making to Russians, it’s curious that Alexander time because he threatened “sacred rules.” their properties apparent. Milton Babbitt’s Scriabin isn’t included, considering the ere are two problems with this view: the fundamental articles on 12–tone music comparative value of his music and his first is that Euler’s harmonics have to be (Some Aspects of Twelve–Tone Composition, wider influence.) e best known composer detuned to fit on the equal–tempered chro- Twelve–Tone Invariants and Compositional in this collection is Olivier Messiaen, who matic scale that was coming into vogue at the Determinants, and Set Structure as a Com- developed a system of extracting chords time and is still in use today. is detuning positional Determinant) demonstrate this. from modes whose intervallic construc- compromises the neat ratios that purport- Mathematics can also be used to inspire the tion is such that their content is preserved edly generate the notes and are put forth as structuring of musical materials, such as when transposed. Examples of these are the basis for the theory’s musical rationale. using Fibonacci series to control rhythmic or the “whole–tone scale” (Messiaen’s Mode More important, Euler relies on an abstrac- temporal dimensions. 1) and “octatonic scale” (Messiaen’s Mode tion of music that fails as a model of musical ese are both a far cry from the kind 2); transposing the former by any even coherence. is is an age–old problem typi- of numerology behind Johnson’s chapter chromatic interval and the latter by any cal of the scientific community’s attempts to Heights and Sums, where we are asked to multiple of 3 semitones retains the original explain musical phenomena, many of which add the pitch class numbers (not the inter- content. Although Johnson uses Messiaen may be based on scientific truths but musical vals) of the members of a chord to determine as an example of Other, this property of fallacies. J.K. Randall’s paper ree Lectures its “height” and to construct collections of his modes is basic atonal theory. to Scientists presents cogent analyses of this chords with the same height. e collection Before we get to these musicians, how- issue. with a “C–major triad” (0,4,7 = 11) would ever, we hear about a mathematician. e Unfortunately, this problem is the pre- also contain C#–D#–G (1,3,7 = 11), C–F– th 18 Century pioneer Leonhard Euler dominant flavor of Other Harmony, which F# (0,5,6 = 11) and so on. is is certainly is known for many wide–ranging con- holds mathematics in high regard but low “other” harmony (i.e., not tonal and not tributions, including the invention of insight. e brief chapter on tonality makes commonly atonal), but without superimpos- mathematical notational conventions and passing references to the seminal music theo- ing another means of organization, how does ground–breaking work in number theory, rists Jean–Philippe Rameau and Heinrich this means of generating chords relate them graph theory, logic, physics, astronomy, Schenker before leaving them far behind, musically and suggest a model for musical and applied mathematics. Most of these instead extolling the virtues of modern coherence? are still considered fundamental (some graphical “visualizations” of the tonal system A passage from the beginning of the chap- even bear his name). His application of that “give us a much clearer view of what ter Sums Modulo N illustrates in a nutshell mathematics to music theory–creating Bach and Beethoven and Brahms were really many of the ingredients of Other Harmony chords consisting of overtones generated doing…”–although there are no examples that are hard to swallow. Johnson introduces by combining prime numbers–is not one offered to demonstrate the legitimacy of the application of modular arithmetic by of these. We might well ask why Bach, this rather remarkable claim. Graphs play partitioning the intervals in the chromatic Handel, Haydn, or Mozart (to name a an outsized role in the balance of the book, collection modulo 2: five “even” intervals (0 few contemporaneous composers) did not which demonstrates a variety of methods to mod 2) and six “odd” intervals (1 mod 2) become Eulerites, but their failure to do generate notes, some algorithmic and oth- and observes: so apparently didn’t impede the quality of ers using mathematics largely as gimmickry. At least since Fux, music their output nor earn them the reputation e graphs mostly display networks of com- theorists have tried to agree on for being stodgy. mon tone relationships, some of which are how to divide the intervals into ere are some who speculate that the the product of the generating methodology a consonant half and a disso- reason Euler’s theory of harmony didn’t gain but many are merely a way of organizing the nant half, and they never really traction was because it was “too mathemati- material after the fact. agree, because it’s all a matter of

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opinion, but we can divide the dissonance function in schemes of musical of harmonic function and progression. In intervals in a completely ratio- propulsion when they have any meaning other words, both are concerned with mat- nal way by simply separating at all. Even in theories of what might be ters of musical coherence. the even from the odd…and called “quasi–tonal” music, as presented In Other Harmony, we are asked to accept this separation is easily found in Vincent Persichetti’s Twentieth–Century a much lower standard for music theory– in musical practice… e odd Harmony: Creative Aspects and Practice, “dis- succession as opposed to progression, no intervals are not always disso- sonance” and “consonance” are subsidiary notion of function, coherence seemingly nant, but somehow we hear that to patterns of “tension” and “relaxation”–in taken for granted–represented as progress. the two notes do not belong to other words, musical context. Admittedly, as Persichetti writes, “Only Using modular arith- when theory and technique are combined once again we are expected to metic to partition the with imagination and talent do works of conflate science and music chromatic is simply importance result.” But what are the rela- a roundabout way of tive contributions of theory, technique, the same whole tone scale. ey identifying cycles produced by intervals imagination, and talent? inking of their are not coming from one world, that are integral divisors of 12 (the size of work as recipes: for Piston and Persichetti, but are somehow juxtapositions the chromatic collection). e cycle of 2 harmony theory is a source of nourish- of two worlds. ere is some- semitones (“mod 2”) is the “whole tone ment, whereas for Johnson, harmony, thing rather rigid and logical scale”; the chromatic contains 2 of these, apparently, is spice. Without some main and structured about the odd each with 6 notes. Likewise, the cycle of ingredient, this produces a pretty thin intervals and something softer, 3 semitones (“mod 3”) is the “diminished stew, however flavorful (if you can tolerate smoother, and more melliflu- seventh chord”, 3 instances of 4 notes each; the salt). ous about the sound of the even 4 semitones (“mod 4”), the “augmented In memory of Steven R. Gerber, intervals, those that come from triad” (4 instances of 3 notes); 6 semitones composer and friend. the whole tone scale. (“mod 6”), the “tritone” (6 instances of 2 Although eschewing research required to notes). e cycles of the remaining inter- “become a musicologist,” Johnson, here as vals (1, 5, 7, and 11) reproduce the entire elsewhere in the book, shows no compunc- chromatic, since they are not integral divi- tion about making sweeping statements sors of 12; their cycles are, respectively, with seeming (but undeserved) musico- the transformations of identity, “cycle of logical authority. at aside, once again fourths,” “cycle of fifths”, and inversion. music theory is cast as a matter of politics, All this is fundamental to atonal theory based on “opinion” rather than a coherent and to 12–tone theory in particular. It framework. Once again we are expected to provides perhaps a more “rational” way to conflate science and music: assessing the divide the intervals than merely “separat- “consonance” or “dissonance” of an inter- ing the even from the odd” and it doesn’t val in isolation is an acoustic judgment, rely on vague evasions like “somehow” or not a musical one, which depends on con- “something.” text. In classic tonal terms, for example, the What should we expect of a theory of interval of 3 semitones can be “consonant” harmony (“other” or otherwise)? We can (minor third) or “dissonant” (augmented start with classic models. Walter Piston’s second); they “sound” the same in isolation Harmony, first published in 1941, is an but function differently in context. Func- excellent example, as is the Persichetti, tion is the crucial difference between the published 20 years later. Both are con- musical and acoustic notions of these over- cerned with how harmony is realized in loaded terms; in music, consonance and musical passages. Both develop notions

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