Colleges and universities are known worldwide as centers for formal research as well as for informal thought about “what is” and “what is not.” They don’t always fulfill this mission with precision, but the search persists. Unfortunately, although inconsistent with empirically based research, popular topics can stay alive for decades.

Scholarly research into choice topics often extends over decades. Even when its value may be questionable, its completion (and publication) is a major source for academic success. As the old saying goes, “publish or perish.” But the values of research should not be judged exclusively by pop pressures of the day. There is a limit to how extensive and deep research should be into the music of John Lennon; there is a fence beyond which research into the remnants of Guillaume de Machaut need not step.

Some research in music can’t escape charges of “pointless” or “excess baggage.” One common cause for such conditions hasn’t received the attention it deserves: it is a tendency over the years for academic scholars to be seduced by formalized systems, approaches to creating or talking about music that provide an aura of incorruptible factuality. Any framework that promises a direct path to creativity or analytical power is welcomed into our abstract world of tones-in-time. I’ll begin my discussion with one that was a respected part of the music scene back in the 20th century.

 The Schillinger System

It was never a pedagogical tool in the curricula of many colleges and universities, but Joseph Schillinger’s method of composition was a hot subject for composers and theorists in the 1940s. Acclaimed as an “objective” path for putting notes together, it was created during the years when the prevailing avant-garde sought mathematical means for explaining things and designing objects: it was “The Bauhaus Age.” Many agreed with renowned music scholar Nicholas Slonimsky¹ that Schillinger did for music what Mendeleyev did for Chemistry.²

 For some, Schillinger’s value was evident. As a practical approach to composition, a few of its elements enjoyed a decade of fame among a wide range of notables, from Henry Cowell to George Gershwin and Glenn Miller.³ In fact, once his system became known, Schillinger himself enjoyed a posh life in New York City, but his work’s notoriety began to fade within the same decade. The system nonetheless is still regarded as a useful tool in a small part of the academic world. Schillinger’s goal was monumental yet simple: it was a “…way to arrive at a musical decision” without “…vagueness and haphazard speculation.”⁴

In 1945 an ardent Schillinger student, Lawrence Berg, opened the “Schillinger House” in Boston. It later became the Berklee College of Music, where the system was a vital part of the school’s curriculum until the 1960s. It is unlikely that a student in any American college or university today can sign up for a course in “The Schillinger System,” yet recent dissertations have been written about it at the University of Chicago (1997), and at the City University of London (1996); a two-volume “how to” edition was published in 2006 by Rose Books. As a system it survives because some, like Professor Jeremy Arden, who teaches it at Morley College in London and the University of Herefordshire, are convinced its virtues outweigh its sins. He reminds the world that the system “…produced many magnificent insights and original ideas which far outweigh the occasional flaws.”⁵

What is the Schillinger system? It is a complex, obtuse compilation of instructions/guides aimed mainly at the hopeful composer. Its basic instructions in The Schillinger System of Musical Composition, consumes 1640 pages, the two volumes divided into twelve separate “books.” Beginning with “Theory of Rhythms,” it covers pitch scales, “Variations by Means of Geometrical Projection,” melody, and harmony. Just trying to follow explanations of basic rhythmic components, pulse to meter, through what are called “attack groups” and “master signatures” (meter), makes a harrowing experience for the reader.

Schillinger’s system lays a numerical groundwork for music. Noting that since a uniform series of durations of fours (say of quarter notes) can be ”interfered with” by a series of the value 3, a universe of rhythms can be created (2 ÷1, 3 ÷ 2, 4 ÷ 3, et al.). The same process of recurring coefficients is later applied to such things as scale, melody, harmony, texture (i.e., “harmonic density”). Applying what are called “coefficients of recurrence” can produce a melody. Applied to two or more simultaneous melodies it can yield counterpoint, with special attention to their implied harmonic contents, among other things. The composer must recognize that there are 24 potential variations of any “group” consisting of four parts: (ABCD), (ADBC), (ACBD), (ABDC), (4=1X2X3X4=24). Also, three types of harmonic progression are specially identified in another section as Diatonic, Diatonic Symmetric, and Symmetric. From these the composer must choose an appropriate structure to fit the varied forms of  “tension,” both constant and variable, desired for a particular context. American composer Henry Cowell was an early convert to Schillinger’s guides. His preface to Schillinger’s first volume happily notes that “…it is a compositional procedure in which numeric calculations precede compositional decisions about rhythms, chords and melodies.”

The system has drawn repeated criticism over the years, perhaps the most direct and knowledgeable that of John Backus, a physicist who also enjoyed a musical background. Published in the Journal of Music Theory,⁶ he found only errors and false inferences in the method: terms are misused; terms are used undefined; familiar terms are used in unfamiliar ways. Capping these flaws, the writing is obtuse, even beyond the mix of math and music. Backus concludes that what Schillinger put together is a fetching though bewildering and flawed numerology. His methods “...have no scientific or mathematical foundation, and to claim they have is to perpetrate a fraud on a defenseless musical public.”

The Schenker System

A deeper and broader notoriety is enjoyed worldwide by a system put together before Schillinger’s: Heinrich Schenker’s approach to musical analysis. It’s a prime example of this lure in academe for systemization. As early as 1935, theorist Adele Katz extolled the system’s virtues in The Musical Quarterly,⁷ noting that “He has been the first modern theorist to find out how the imagination of a composer works, and to differentiate between the raw material itself and the consummate art that turns this raw material into a great masterpiece.” For many music scholars it became—and for some it remains—the touchstone for discussions of tonal music, boasting even a website and a scholarly journal. Graduate courses in “Schenkerian Analysis” abound; when not a full course, the topic looms large in classroom discussions of analysis.

In one way Schenker’s system is the direct opposite of Schillinger’s. Rather than begin with the tiniest bits from which wholes emerge, it begins with wholes and progresses to tiniest bits. But the two systems are alike in that both deal with the sonic art from details to totalities.

For those unfamiliar with this venerable regimen, a brief résumé might help. As mentioned earlier, its basic goal is the representation of pitch structure of any tonal work or any section within. Its reduction formats are limited to a fundamental bass (Bassbrechung) whose potential notes are limited mainly to I and V, although IV and III are potentials within the I—V—I format. Then an upper line (Urlinie) reveals the basis of melodic content, its representation confined to downward motion within one of three possible ranges: tonic-to-tonic octave, V down to I, or III down to I. Together, the Bassbrechung and Urline form the basic pitch representation of any work, Its Ursatz. Example 1 can help one understand fundamentals.⁸

Example 1: Schenker’s  Ursatz,  Bach Prelude in C major, WTC

 

 

 

Such extreme reductions often lead to puzzlements. For instance, if one examines Bach’s score, the high A’s in measure 5 act as an interior “arrival point.” By the end of the Prelude they turn out to have been the highest pitches of the whole. They are accorded no significance in such a thorough reduction as Schenker’s Ursatz (although given passing attention in a first level of analysis, called the Takte). For Schenkerian advocates this reductive pathway is the modern answer to analysis, a replacement for outworn figured bass operations of the past. Its value is magical, as Katz claimed in her 1935 paean, its supreme value lying in “...its ability to reveal what was in the mind of the composer himself.”⁹ And that’s quite an order to fill!

Over the past several decades there have been thorough and robust reactions to the system that find it wanting in large and small ways. Most provocative is the thorough discussion by Eugene Narmour in Beyond Schenkerism; it remains the most penetrating negation I know. He points out that Schenker himself commits such systemic sins as to add “root” tones that are not present in the actual music. And further, Schenker's occasional demand that certain chords operate as undefined (thus unanalyzable) structures “…is to allow favored relations to predetermine structural features of certain individual works of art—the real world. And it is this sort of fallacious apriorism that prevents Schenkerism from being accepted as a valid axiomatic system.”¹⁰

Other serious critiques have surfaced over the years. David Temperly’s exhaustive review of Schenkerian links with perception¹¹ finds some parts of the system justified, others doubtful. It is his judgment we first must question that “…listeners spontaneously generate reductions of music as they listen,” and the “potential for the theory to inform us about musical motion also seems doubtful, or at least unproven.”¹²

Aside from theoretical critiques, a few basic and straightforward observations are pertinent to Schenkerian analysis and weaken its lure. For example, any person who knows music of that 18th-19th century era will recognize at least three simple facts: (1) Any music of that era will possess tonality; (2) it will possess a V—I chordal relationship within its parts, especially its closing moments; (3) its melodic content will teem with raw materials for a 1—1, 5—1, or 3—1 relation to fill out an Urlinie. Choices for filling in all of these requirements are left to the analyst, but they will derive from a feast of available choices. Bach’s C-major Prelude is a perfect object for demonstrating the Schenkerian system: it has the right nuts and bolts in the right places.

This is not to say that all musics of our tonal past readily fill the bill. Works can project a sense of strong musical unity and deliver an effective aural message but still defy squashing into the Schenkerian mold. A prime example—and by no means unique--is Chopin’s Prelude in E minor, op.28. Aurally, it’s a brief, simple and readily comprehended work, but fulfilling the Schenkerian plan demands some crafty decisions. Its sin? A proper Urlinie is difficult to fulfill. Only if one accepts a brief G3 neighboring tone at the beginning of m. 12 (halfway through the piece), or a climatic G4 in m. 16 (an octave too high) can one identify the 3 of what would fulfill the work’s 5-4-3-2-1 Urlinie.

Example 2: Chopin E minor Prelude, mm. 1-2, 12-13, 16-17

 

 

 

 

 

 

 

Dedicated Schenkerians nonetheless insist upon living with the system’s flaws; they are viewed as negligible when recognized at all. Yet a close look at many analyses in the Schenker mode reveal notes in the requisite Urlinie chosen more for their fulfillment of systemic needs than revelation of structural essentials. A passing sixteenth note of no melodic eminence is chosen because its pitch is needed to complete the Urlinie recipe. When this happens the system has become more important than the object analyzed.

But there is a positive side. Anyone who diligently applies the system to any music learns a lot about that music. And yet, let us note that anyone who diligently searches through the guts of any composition without utilizing Schenker’s guidelines also will learn a great deal about that music. The system is not required for penetrating and understanding basic structure. 

Finally, Schenker’s system can leave us with “facts” that are irrelevant to music as heard. That embedded B—A—G Urline within Mendelssohn’s Song Without Words, Op. 62, No. 1, has no bearing on the work as I listen to or perform it; a proper Bassbrechung and Urline would be required only to fulfill the demands of the system, not to enhance the aural reality of the piece. I find this true with much music of the 18th and 19th centuries that make complete aural sense.

The Schoenberg System

As systems go, Schenker’s takes a back seat to the most highly touted systemization of musical pitch of the past century. As a path for composition and analysis it continues to claim the attention of musicians, even though it squirms around in an unsubstantiated bed of truth. It’s the literature that began with Schoenberg’s 12-tone method early in the last century.

Despite extensive psychological studies in aural perception and cognition—as well as a gradual dismissal of the music itself¹³—academic research keeps alive this “system” and its auxiliary manifestations. The overarching attitude seems to be that systemic truth and sonic relevance are separable qualities. Tenured positions in the world’s leading universities continue to be held by professors who excel in tedious analyses of its musical produce; it remains one of those mumpsimus beliefs that ignore root conditions of cognitive and perceptual reality, conditions as much a part of “human nature” as digestion, meiosis, mitosis, or enzyme secretion.

Serialist enthusiasm is well documented. In 1961 an extensive bibliography of its literature, along with that of electronic music, was compiled by Ann Basart and published by the University of California Press. Its over 800 titles attest to the widespread interest in the two subjects, and its opening page reflects the bravura of the times: “Twelve-tone technique, once thought to be the private and unintelligible musical language of a small group of composers, is today one of the most important influences of European and American music.”

I have written at length about the shortcomings of the system as compositional and analytical processes.¹⁴ Anyone wishing to delve more deeply into claims that analyses of 12-tone music are discussions of music as heard should refer to those two books. My discussion here will be brief; I wish to note--only to the degree necessary—that the “method” cannot be supported as relevant to human auditory experience.

From Schoenberg to Allen Forte, Milton Babbitt, Harold Lewin, and Andrew Mead,¹⁵ its credibility as an aurally relevant subject has depended upon answers to a single question: can a 12-tone row establish itself as a perceived unity, so that its parts or its permutations (Inversion, Transposition, Retrograde, Retrograde Transposition) are recognized as its progeny?

Schoenberg himself laid the keel when he explained the perceptual basis for his system of composing. In his words, “...every unit of a piece being a derivative of the tonal relations in a basic set of twelve tones, the Grundgestalt, is coherent because of the permanent reference to the basic set.”¹⁶

Later concurrences with Schoenberg’s claim are not hard to find. In 1979 theorist Robert Morris argued unquestioned credibility for the functional reality of all 12-tone structures.¹⁷ Since permutations of a row contain the same intervals, “...it is therefore rational to assume that compositions will derive sonic and syntactic unity from the use of sets selected from the same SC” [Set-Class].¹⁸

“Rational” though it may be, aurally related and logically related are not the same; negative answers provided by empirical psychology over the years have denied aural realization. Serious research began as early as 1959, when psychologist Robert Francés initiated controlled empirical studies. His conclusions were simple and direct: “Serial unity lies more on the conceptual than on the perceptual level.”¹⁹ Four decades later found no change when extensive studies by Niccola Dibben concluded that atonal works projected “no underlying structure from the musical surface.”²⁰ Confirming stories were numerous: W.J. Dowling (1972), Diana Deutsch (1975-1987), Carol Krumhansl--also with Sandell and Sargent—(1979-1987), Don Gibson, Jr. (1986-1988), Nicola Dibben (1999), Zoltan Dienes and Christopher Longuet-Higgins (2004).

Let us get straight the verdicts of all juries: a 12-tone row does not have the power of imposing a memorable pitch unity on a listener’s mind, so that its whole or its parts, however permuted (I, T, R, RT), provide cohesion for its parent artifact. Accumulated over four decades, those results sounded a death knoll for the theorizing that has ensued from the day of the system’s birth. As I have observed,²¹ these heavily touted set-to-notes relationships cannot be questioned: their transformation into a world of set-to-tones is another matter. Let us remember Stephen Pinker’s comment about denials of human nature: “…this is the mentality of a cult, in which fantastic beliefs are flaunted as proof of one’s piety.”²² Studied carefully and thought through as sonic events, fastidious analyses of 12-tone music prove for the most part to be irrelevant—unless one’s goal is nothing more than following the composer’s path from predetermined note sets to notes on the page.

Schoenberg’s System has left us with more than a way for composing; it also has left us with an analytical process whose aura of impressive knowledge surpasses even that of Schenker’s. Unfortunately, scholars who possess the know-how to expose the limitations of its claims neglect to take the time. But there have been exceptions. In 1996 Thomas Regelski posted a valuable warning:²³

In seeking the “exact observation and strict correlation of data” more suitable to the inanimate subject matter of the physical sciences, many music researchers have disintegrated the subject matter of music—its musical integrity, its human interest—to the degree that their results are irrelevant and of no theoretical interest or pragmatic use.

It would be hard to find a more revered champion of the dodecaphonic cause than the late Milton Babbitt. His treasury of dodecaphonic analyses remains as a classic remnant that supports Regelski’s claim of irrelevance, of pragmatic waste. He was brilliant musically and mathematically. He was not a man to spare words; in one classic example he devotes a sentence of 324 words to a model of an analysis of the opening 10-measures of Schoenberg’s 4th String Quartet, Mvt. III.²⁴ Within this careful dissection are revealed—presumably—ways in which note relations from the work’s basic set “control” what is heard. I shall deal with only one of Babbitt’s claims, his closing words²⁵. They tell us how, within two measures, the first violin pitches tie together parts of the basic set. In his words “...functional ‘orchestration’ (the six-note unit of the first violin in 620-21 combines with the six-notes of 622-23 to form a set); et cetera.” This tells us that Vln. I’s six notes of ms. 620-21 “combine” with those of 622-23 to “complete” the established set of 12 notes.

Example 3:  Set, Schoenberg Quartet IV

 

 

 

 

 

 

 

 

 

Very interesting. The notes are there, but this claim of “combining” and “completion” has little relevance to sonic reality; neither pattern fulfillment nor row completion occur for a listener. Babbitt’s discoveries are classic models of the dodecaphonic oeuvre. Thought through as aural events, such explanations prove to be irrelevant.

In Babbitt’s footsteps, more than a decade later, Bryan Alegant and Andrew Mead used the composed cadenza25 of Schoenberg’s Violin Concerto as a source for set-to-music explanations. Their analysis is not easy to follow.

Example 1(a) illustrates the procedure of hexachordal inversional combinatoriality with the “home” rows of the Violin Concerto, P9 and I2. The rows form a combinatorial array containing two aggregates, each aggregate comprising six dyads sharing the same order position in the rows: {2, 9} {1, A}, {3, 8} {B, 0}, {4, 7} and {6, 5}. (The dyads are the cycles of index number 11, or B.) The array can be partitioned in various ways to generate “row harmonies.” One harmonization combines the non-overlapping dyads of the rows into tetrachords, with the first one containing the elements at order positions 0 and 1, the second having the elements at order positions 2 and 3, and the remaining tetrachords combining the elements at order positions 4 and 5, 6 and 7, 8 and 9, and A and R. Example 1(b) shows that the resulting tetrachords belong to set classes 4-1 [0123], 4-7 [0145], and 4-17 [0347]; note that, in this array, tetrachords of the same set class are related by T6. We call this set of six tetrachords the dyadic complex, and we label its tetrachords [1] through [6].

Example 4 shows the set of the Concerto followed by the Cadenza’s opening. 

Ex. 4. Set, Schoenberg Vln. Concerto

I must confess that l have not fully matched the Alegant/Meade claims of set divisions and aggregations to the opening of the cadenza. It is not hard to detect visually that permutations of the original set flourish within the example shown here. But their fastidious breakdown does nothing to enhance my aural comprehension of the cadenza, nor does it strengthen my sense of unity of the concerto’s pitch content. 

Does this reporting of analytical irrelevancies mean that 12-tone music is itself flawed? Certainly not, although a robust claim of its basic shortcomings was published by Diana Raffman in 2003.²⁶ Academic tenures have been gained by such reporting; dodecaphony remains as a solid element in higher education. But our concern here is that of music descriptions, not esthetic evaluations. When it comes to descriptions, one contribution of Schoenberg’s system is little appreciated today: it is that pitch is not the only attribute from which musical structure is molded. Rhythm, texture, loudness, timbre, and contour play vital roles in feeding the unity/variety mix that shape our art of sound. They actually reign supreme in some 12-tone music, but to date it is the note system that prevails as a descriptive base.

Put together with Schillinger and Schenker, it provides another chance to hesitate when faced with yet another systemization that claims deep and thorough musical insight; all three have enjoyed unique lives of fame in the musical world. Their acceptance at face value, over the years, reveals the desire of musicians to possess some form of objective knowledge in their abstract sonic world. Music lacks the lexical powers of poetry; it cannot match the representational simplicities of painting or sculpture. In terms of objectivity it is weak.  But let us stand guard when told some new system of musical creation and/or description has been developed. It may be seductive. It also may be as flawed as those of our past.

Notes

¹ As Slonimsky told me in a personal conversation back in the early 1950s.

²Mendeleyev created the periodic law of chemistry in which elements are presented in order of increasing atomic number. The standard form of the table consists of a grid of elements laid out in 18 columns and 7 rows, with a double row of elements below that.

³That Gershwin used the system in composing remains an argued point. It has long been claimed that Miller’s Moonlight Serenade was composed using it.

⁴Schillinger, Joseph. A Memoir. NYC: Da Capo Press, 1976. P. 1356

⁵Ibid., 231.

⁶Backus was awarded the Acoustical Society of America’s Silver Medal in 1986.

⁷Vol. 23. 

⁸Direct sources are Free Composition (Der Frei Satz). Five Graphic Analyses (Funf Urlinie Tafeln). Other analyses by Schenker are those of C.P.E. Bach’s music in Kalavierwerke, Neue Kritische Ausgabe von Heinrich Schenker.

⁹Katz, Loc. Cit, 311.

¹⁰Ibid., p. 15.

¹¹“Composition, Perception and Schenkerian Theory.”

¹²Ibid., 163.

¹³Russell Platt  p. 22) implies the greater story when he observes that Schoenberg’s Verklärte Nacht “. . . would become his only truly popular work.” (Even it is rarely played today.)

¹⁴Schoenberg’s Error and Metamusic Versus the Sound of Music.

¹⁵As exemplary works: Schoenberg, Style and Idea; Allen Forte, “Context and Continuity in Atonal Works: A Set Theoretic Approach.” David Lewin, "A Theory of Segmental Association in Twelve-tone Music;" Milton Babbitt, The Collected Essays of Milton Babbitt; Andrew Mead (with Bryan Alegant), “Having the Last Word.”

¹⁶Style and Idea, 91.

¹⁷“A Similarity Index for Pitch-Class Sets." 

¹⁸Ibid., 445.

¹⁹Francés, Robert, “Recherches Expérimantales sur le Perception des Structures Musicales.” 191.

²⁰In “The Cognitive Reality of Hierarchic Structure in Tonal and Atonal Music.” 

²¹Especially in Metamusic Versus the Sound of Music.

²²How the Mind Works, 308.

²³“Scientism in Experimental Music Research," 12.

²⁴The Collected Essays of Milton Babbitt.

²⁵“Having the Last Word.”

²⁶“Is Twelve-Tone Music Artistically Defective?” 

Bibliography

Backus, John. “Pseudo-Science in Music, Journal of Music Theory IV (1960), 221-232.

Basart, Ann. Serial Music: A Classified Bibliography of Writings on Twelve-tone and Electronic Music. Berkely, CA: University of California Press, 1961.

DeLannoy, Christian. "Detection and Discrimination of Dodecaphonic Series,” Interface (1972), 13-27.

Deutsch, Diana, “The Processing of Structured and Unstructured Tonal Sequences, Perception and Psychophysics 28 (1980), 381-389; ”The Processing of Pitch Combinations”, Psychology of Music 2nd ed., 349-411, Academic Press, 1999;

Dibben, Nicola, “The Cognitive Reality of Hierarchic Structure in Tonal and Atonal Music,” Music Perception XII (1994), 1-25; “The Perception of Structural Stability in Atonal Music: the Influence of Salience, Stability, Horizontal Motion, Pitch Commonality, and Dissonance,” Music Perception 16 (1999), 265-204.

Dowling, W.J., “Recognitions of Melodic Transformations: Inversion, Retrograde, and Retrograde Inversion,” Perception and Psychophysics 12 (1972), 417-435;

Forte, Allen, “Context and Continuity in Atonal Works: A Set Theoretic Approach.”  Perspectives of New Music I (1963), 72-82.

Francés, Robert, “Recherches Expérimantales sur le Perception des Structures Musicales.” The Perception of Music (trans. Dowling), 191.

Gibson, Don, “The Aural Perception of Non-Traditional Chords in Selected Theoretical Relationships: A Computer-Generated Experiment,” Research in Music Education 34 (1986), 5-24; “The Aural Perception of Similarity in Non-Traditional Chords Related by Octave Equivalence,” Journal of Research in Music Education 36 (1988), 4-17; “The Effects of Pitch and Pitch-Class Content on the Aural Perception of Dissimilarity in Complementary Hexachords,” Psychomusicology 12 (1993), 58-72.

Katz, Adele, “Heinrich Schenker’s Method of Analysis,” Musical Quarterly 21 (1935), 311-329.

Krumhansl, Carol L., Gregory J. Sandell, and Desmond D. Sargent. “The Perception of Tone Hierarchies and Mirror Forms in Twelve-Tone Serial Music,” Music Perception 5 (1887), 31-38.  “The Cognitive Reality of Hierarchic Structure in Tonal and Atonal Music," Music Perception 12 (1994). 1-25

Lewin, A Theory of Segmental Association in Twelve-tone Music,” Perspectives of New Music I (1962), 89-101.

Mead, Andrew and B. Allegant, “Having the Last Word: Schoenberg and the Ultimate Cadenza.” Music Theory Spectrum 34 (2012), 107-136.

Morris, Robert, “A Similarity Index for Pitch-Class Sets," Perspectives of New Music 18 (1979), 445 – 460.

Narmour, Eugene, Beyond Schenkerism, Chicago: University of Chicago Press, 1980.

“Composition, Perception and Schenkerian Theory,” Music Theory Spectrum 32 (2011), 146-168

Pinker, Steven, How the Mind Works. NYC, W.W. Norton, 1997.

Platt, Russell, The New Yorker, July 8, 2013, p. 22.

Quist, Ned, “Toward a Reconstruction of the Legacy of Joseph Schillinger, MLA Notes 58/4 (June 2002), 765-786.

Raffman, Diana, “Is Twelve-Tone Music Artistically Defective?” Midwest Studies in Philosophy XXVII (2003), 69-87.  Regelski, Thomas, “Scientism in Experimental Music Research,” Philosophy of Music Education Review IV (1996), 12.

Schenker, Heinrich, Free Composition (Trans. of Der Frei Satz by Ernst Oster). NYC: Longmans, , 1979.  Five Graphic Analyses (Funf Urlinie Tafeln); Kalavierwerke, (Intro. Felix Salzer), NYC: Dover Books, 1969.

Schillinger, Joseph. A Memoir. NYC: Da Capo Press, 1976. Schillinger System of Musical Composition, Vol. I and II, (compiled by Lyle Dowling and Arnold Shaw), NYC: Rose Books, 2005.

Schoenberg, Arnold. Style and Idea (Ed. Leonard Stein). London: Faber and Faber, 1975.

Temperly, David. “Composition, Perception and Schenkerian Theory,” Music Theory Spectrum 32 (2011), pp. 146-168.

The Collected Essays of Milton Babbitt (Ed. Stephen Peles), Princeton: Princeton Univ. Press, 2003.

Thomson, William, Metamusic Versus the Sound of Music: A Critique of Serialism. Lewiston: Mellen Books, 2010. Schoenberg’s Error, Philadelphia: Univ. of Pennsylvania Press, 1991.