Finding Slonimsky’s Thesaurus of Scales and Melodic Patterns in Two Concerti by John Adams

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John Adams has candidly acknowledged the significance of Nicolas Slonimsky’s Thesaurus of Scalesand Melodic Patterns (1947) in his own compositional output. Recent studies affirm that Adams’sworks incorporate patterns from this book of cyclically related musical synonyms, yet there is littleunderstanding of the properties and organizational principles that make up the Thesaurus and, mostimportantly, how Adams integrates these ideas in his music. This study provides a closer look at theThesaurus and examines two representative movements by the composer, drawn from the ViolinConcerto (1993) and the piano concerto titled Century Rolls (1996).

Keywords: John Adams, minimalism, Nicolas Slonimsky, Thesaurus of Scales and Melodic Patterns,Violin Concerto, Century Rolls

Ever since its publication in 1947, The Thesaurus ofScales and Melodic Patterns by Nicolas Slonimsky hasbeen a widely used source of musical composition and

improvisation. The organizing elements of this work remain anenigma to most casual Thesaurus practitioners today, but itsinfluence on numerous musicians from vastly different back-grounds is undeniable. In the classical world, important figuressuch as Henry Cowell, Leonard Bernstein, Virgil Thomson,Howard Hanson, Arthur Honegger, and Arnold Schoenberghave written testimonials that reveal, at least, a passing familiar-ity with the Thesaurus.1 In the jazz community, too, the Thesaurushas gained enormous respect. Several reports on John Coltraneassert that he devoted time studying this book and, in fact,scholars have convincingly demonstrated how part of “GiantSteps” is directly derived from Slonimsky’s work.2 Another greatjazz musician who has been influenced by Slonimsky is DaveBrubeck, whose 2003 album Park Avenue South draws directlyfrom the Thesaurus.3 In the rock genre, musicians including FrankZappa and guitarists Steve Vai and, more recently, Bucketheadhave candidly acknowledged use of Slonimsky’s patterns.4

Another recent musician who has been forthright on hisadoption of the materials found in the Thesaurus is the com-poser John Adams. His fascination not only for Slonimsky’swork but also for his charismatic personality stems from theirclose friendship. Adams describes Slonimsky as a “character ofmind-boggling abilities” and a “coiner who never tired of

minting his own.”5 The kind of impact Slonimsky had onAdams carried over to his music. Following Slonimsky’s deathin 1995, Adams composed an orchestral work entitled Slonim-sky’s Earbox (1996), which “memorializes [Slonimsky’s] witand hyper-energetic activity, but [it] also acknowledges [Adams’s]great debt to his Thesaurus.”6 One of Slonimsky’s “mind-boggling abilities” was his attempt to form scales and melodicpatterns “in such a way as to cover every kind of combination.”7

In the words of Schoenberg, this was “an admirable feat ofmental gymnastics.”8

Compositional materials and techniques from the Thesauruscan be observed in Adams’s works from the 1990s onwards.9

Yet although Adams has openly acknowledged borrowing fromthis source, he has not specified the nature or desire of his influ-ence. In this study, I explore the manner in which he incorporatesSlonimsky’s patterns and scales, as well as other relevant issues:how does Adams’s employment of these patterns interact within

I wish to acknowledge Michael Buchler, Christina Fuhrmann, and theanonymous reviewers of this journal for their helpful suggestions on earlierdrafts of this article.

These accounts are included in the back cover of the 1975 Charles Scrib-ner’s Sons edition.

Porter (1998, 149); Bair (2003). Brubeck’s song Elegy, for instance, consistently employs one of the melodic

patterns Slonimsky introduces in the preface to the Thesaurus. See Kostelanetz (1997).

Adams (1999, 65). Ibid. Slonimsky (1995, 37). The notion that Slonimsky created an exhaustive

book of patterns is misleading. Sanchez-Behar’s dissertation elaborates thispoint further (2008, 28).

Schoenberg (1949). Sanchez-Behar (2008, 1, 3). Adams’s works ranging from 1992 to the

present are composed in a new style: “[Adams’s] career as a composer maybe divided into four periods: (1) 1970–77, confined to Adams’s initialexperimentation with minimalism, emulating Reichian tape techniques;(2) 1977–87, marked by gradually changing harmonies in a minimalistand post-minimalist style, concluding with the opera Nixon in China;(3) 1987–92, a transitional period characterized by Adams’s discontentwith and reassessment of his compositional style, during which he beginsto show a discernable preoccupation for contrapuntal writing in The Death

of Klinghoffer (1989–91); and (4) 1992–the present, a conscious break fromharmonic structures in favor of a contrapuntal style, which comes intofruition with the Chamber Symphony (1992), a work modeled afterSchoenberg’s Kammersymphonie, Op. 9” (2).

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a surrounding musical passage, how do patterns relate to oneanother and interact within a musical work, and what is themanner or substance of his application? In other words, does theinclusion of patterns affect merely the surface structure, or does itbear implications for the larger components of a work?

To understand the properties of these musical scales and mel-odic patterns, and as a precursor to the discussion of Adams’suse of them, I will begin with an exploration of Slonimsky’sThesaurus. Subsequently, my study will consider various ways inwhich Adams utilizes Slonimsky’s melodic patterns: (1) quotingthem in their entirety, (2) gradually mutating them throughvarious kinds of pitch modifications, and (3) paraphrasing themto create unique patterns that resemble those from Slonimsky’sThesaurus. Observing the manner in which melodic patternsfunction and interact with their surrounding context will helpelucidate the kinds of conditions necessary for a seamless incor-poration of Slonimsky’s materials and techniques. This finalportion of the study will illuminate the types of harmonies thatSlonimsky suggests for his patterns and how Adams re-createsthem in his instrumental works.

The bulk of material found in the Thesaurus contains overone thousand melodic patterns arranged into chapters accordingto interval cycles (referred to as principal tones) that divide oneor more octaves into equal parts. Slonimsky devotes the mostattention to patterns whose principal tones divide a singleoctave into various equal parts, which he labels Semitone,Whole tone, Sesquitone, Ditone, and Tritone Progressions.Slonimsky’s prefix “sesqui” signifies the addition of a semitoneto any given interval; thus, a sesquitone corresponds to a minorthird. The pitches that divide the octave into equal distances arein turn ornamented through the insertion of notes below (infra-polation), between (interpolation), and/or above (ultrapolation)these given pitches. Slonimsky’s title “Thesaurus” suggests thata composer can look for musical “synonyms” according to theirdivision of the octave, type(s) of inserted pitches, as well as theircardinality.10 One can observe, for instance, patterns such asTritone Progressions (or cycles) that contain an interpolation ofone pitch (thus a pattern length of four pitches), two pitches(which yields patterns of six pitches), and so on.

Slonimsky’s Progression in Example 1 shows how his orna-mentations can be inserted into an interval cycle. Slonimskyclassifies Pattern 10 as a Tritone Progression formed throughthe insertion of two interpolated notes. The melodic pattern inthis illustration is shown in ascending and descending form, afeature that is characteristic of nearly all melodic patterns

from the Thesaurus. Drawing on Stephen Heinemann’s nota-tion for pitch-class set multiplication, the ascending form ofSlonimsky’s Pattern 10 can be represented as 0–2–5 ⊗<06> = 0–2–5–6–8–e, where ordered segment 0–2–5, referredto as the multiplicand, is transposed using a tritone or interval6-cycle <06>, which is the multiplier.11 The series that com-prises the union of these two operators, called the product,results in pcs 0–2–5–6–8–e, which Slonimsky notates as C–D–F–F♯–G♯–B. Heinemann accurately notes that despite hispitch-class approach to multiplication, Slonimsky’s patterns arerealized in pitch space.12 For instance, patterns that revolvearound an interval 3-cycle <0369> can be generated with princi-pal tones using minor thirds, such as the chapter “SesquitoneProgression: Equal Division of One Octave into Four Parts,” orwith major sixths, as in “Sesquiquadritone Progression: EqualDivision of Three Octaves into Four Parts.”

The types of patterns found in Slonimsky’sThesaurus containinteresting musical properties. The main core of Slonimsky’sbook consists of patterns that are transpositionally and inver-sionally symmetrical.13 Some of the patterns meet EdwardGollin’s14 criteria for what he calls multi-aggregate cycles,which consist of repeated patterns of two or more distinctintervals that complete the aggregate more than one timebefore returning to their point of origin.15 Several theoristshave developed methods for predicting when and where pcs areduplicated in patterns such as those found in the Thesaurus.16

. Interpolation of two notes in Slonimsky’s Pattern 10. THESAURUS OF SCALES ANDMELODIC PATTERNS ByNicolas Slonimsky Copyright © 1947 (Renewed) Schirmer Trade Books, a division of Music Sales Corporation International Copyright

Secured. All Rights Reserved. Reprinted by Permission.

Sanchez-Behar (2014, 58).

Heinemann (1998). Heinemann’s theorems for multiplication could not be expressed as ele-

gantly in pitch space without further refinement of his system. A handful of scholars have discussed the properties of musical patterns like

those found in Slonimsky’s work: Brown (2003), Cohn (1988), Gollin(2007), Heinemann (1998), and Lambert (1990).

Gollin (2007). Multi-aggregate cycles are, in fact, quite a common occurrence in the The-

saurus, particularly in the chapters titled “Diatessaron Progression,” “Dia-pente Progression,” and “Sesquiquinquetone Progression,” which generatepatterns based on cycles 1 and 5. Diatessaron Patterns 826–829, forinstance, illustrate dual-aggregate cycles.

Heinemann (1998) discusses how pitch-class duplications can be predictedby comparing the respective interval-class vectors of a multiplicand andmultiplier. The number of repeated pcs in the product is derived by multi-plying each interval class from the vectors compared and adding their total,with the exception of ic6, which is transpositionally symmetrical and thusyields twice as many duplications. If one interval class other than ic6 isshared between operands, one note will be repeated in the product, result-ing in a total cardinality of |A| × |B| – 1 (this method of finding set cardi-nality is severely limited in most other instances). If there are no intervalclasses in common, then there are no repeated pitch classes in theirproduct. For example, the interval-class vectors from the operand sets in

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If one considered Slonimsky’s melodic patterns as unorderedcollections of pitch classes, then Richard Cohn’s musicalproperties detailed for transpositional combination (e.g., thecommutative and associate properties) would be fully applica-ble. Aside from the theoretical properties inherent in thesepatterns, a broad assertion about their pitch-class content canbe made: a compilation of the melodic patterns reveals thatnearly all consist of either the octatonic collection (set-class8–28), the enneatonic collection (set-class 9–12), varioussubsets of these two collections, including enneatonic subsetssuch as the hexatonic collection (set-class 6–20) and thewhole-tone collection (set-class 6–35), and last, twelve-tonerows and other patterns that complete the chromatic aggre-gate.17 The pervasiveness of the octatonic and enneatoniccollections is naturally a by-product of transpositional combi-nation through interval cycles that maximize specific intervals,which in turn minimize others; the octatonic has the highestic3 representation of all octachords, while the enneatonic doesthe same for ic4.

The patterns within each interval cycle are organized in asystematic manner. Consider Example 2, which illustrates theinterpolation of two notes within a 6-cycle. Here, the multipli-cands begin with the most compact trichord 0–1–2 and proceedin the following order: 0–1–3, 0–1–4, 0–1–5, 0–2–3, 0–2–5,0–3–4, 0–3–5, 0–4–5. From this arrangement of patterns, onecan surmise that Slonimsky did not consider inversionally orrotationally related sets as equivalent. Given that any of Slonim-sky’s patterns “can be transposed to any tonal center accordingto a composer’s requirements,”18 0–1–2 ⊗ <06>, 0–1–5 ⊗<06>, and 0–4–5 ⊗ <06> are related by rotation. The proximityof patterns that are inversionally or rotationally equivalent is aby-product of Slonimsky’s order, and for this reason Slonimskydraws close association of this musical relationship through hissynonyms. The remaining 6-cycle patterns in Example 2belong to the same set class, Forte’s 6–30 [013679], and arerelated by rotation and inversion: 0–1–3 ⊗ <06>, 0–2–5 ⊗<06>, and 0–3–4 ⊗ <06> are inversionally related to 0–1–4 ⊗<06>, 0–2–3 ⊗ <06>, and 0–3–5 ⊗ <06>. Slonimsky offers no

explanation for why the linear segment 0–2–4 ⊗ <06>, knownas the whole-tone collection, does not appear in this list,though it can be found elsewhere in the Thesaurus (Patterns 36and 569).19 Based on this and other comparable instances, itseems evident that Slonimsky tried to avoid the duplication ofthis pattern, as well as other “modes of limited transposition”such as the octatonic scale, which can be included within a3- and a 6-cycle.

Adams’s instrumental works incorporate patterns from theThesaurus through various means. One of these ways entails theuse of exact quotations that stem from Slonimsky’s work.When present, these complete representations assume domi-nance over an extended passage of music. Adams referencesSlonimsky’s complete Pattern 425 in the third movement ofhis piano concerto Century Rolls (1996) in Example 3. LikeSlonimsky’s Thesaurus, Adams’s passage features the pattern inprime and retrograde forms. Slonimsky begins all of his patternson pc 0, but here Adams transposes this pattern to begin onpc 6 (or G♭) to accommodate the surrounding musical space.Slonimsky classifies this pattern as having an ultrapolation ofthree notes within a 3-cycle <0369>. In Heinemann’s notation,Slonimsky’s pattern consists of an ascending 0–4–t–6 ⊗<0369>, and the retrograde of its expansion, or 0–3–7–1 ⊗<0963>. Here, the tetrachord 4–25 [0268], which can bethought of as the familiar Mm7(♭5) (or less likely thought of as aroot-position French augmented-sixth chord), is transposedaround a cycle of minor thirds. Adams renotates Slonimsky’spattern so that each triplet group comprises an ascending ordescending major third in the prime form, but because pc 6 isnot repeated before beginning the retrograde, the pattern trans-forms into a series of alternating ascending minor thirds anddescending tritones (mainly augmented fourths). The resultyields the notes from an octatonic collection, which is one ofAdams’s preferred collections in his recent instrumental

Pattern 402 = 0–5–4 ⊗ <0369> do not share interval classes; thus, theiroperation yields twelve pitch classes, resulting in a derived twelve-toneseries, which is generated by either one of the recurring operands. In Hei-nemann’s words, “two operand sets A and B with no interval classes incommon will produce a set with a cardinality equal to |A| × |B|” (79). Theexact order position of an initial duplication can also be predicted (Lambert[1990]). Lambert details a method for finding the position of the firstduplication in what he refers to as a combination cycle, which equates toSlonimsky’s pattern interpolation of one note within an interval cycle.

The enneatonic is also known as the “Tcherepnin” scale in Russia becausethe composer Alexander Tcherepnin (1899–1977) had used it more exten-sively than any other composer. It is discussed in a number of writings byJohn Schuster-Craig and more recently in a dissertation by Kimberly AnneVeenstra (2009). Incidentally, Olivier Messiaen classifies this collection ashis third mode of limited transpositions in Technique de mon langage

musical (1944) and refers to it as “the best of all modes [of limited transpo-sition]” (Messiaen [1994, 64]).

Slonimsky (1975, vi).

One of the forerunners to Slonimsky’s Thesaurus, titled The Schillinger

System of Music Composition, likewise lists a series of patterns and symmetri-cal scales. In the section on the interpolation of two notes within a 6-cycle,the author Joseph Schillinger includes the whole-tone collection andarranges the patterns in a slightly different order: 0–1–2 ⊗ <06>, 0–1–3,0–2–3, 0–1–4, 0–3–4, 0–1–5, 0–4–5, 0–2–4, 0–2–5, and 0–3–5 (vol. I[148–54]). Schillinger’s monumental treatise on musical composition andhis theoretical concepts includes two volumes and extends over 1,500pages. His work is more highly driven by mathematical logic and its signifi-cance on music. The thrust of his ideas revolves around permutated musicalstructures that create a sense of continuity. The Schillinger System was highlyinfluential during its day; notable musicians such as George Gershwin,Glenn Miller, and Benny Goodman have been known to study underSchillinger’s tutelage. An even earlier attempt at organizing symmetricalscales and patterns, which predates Schillinger’s work, is Alois Hába’sNeue Harmonielehre (1927). However, Hába’s organizing principle com-bines patterns that solely bear inversional symmetry to others such as thoselisted in Example 2. Ultimately, Slonimsky’s Thesaurus has proven to bethe most successful book of scales and patterns, in part due to its user-friendly approach that attempts to exhaust all patterns with little jargonmostly confined to the book’s preface, rather than present few patterns asgerminating ideas for further exploration and creation of other patterns.

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works, such as Slonimsky’s Earbox (1996). This pattern has aneven distribution of pitch repetitions and therefore projectsthe octatonic collection twice before returning to its point oforigin.20 Aside from Adams’s own transposition and the addedrhythmic swing effect, his excerpt is identical to Slonimsky’spattern.

Adams presents complete replications of Slonimsky’s pat-terns without disguise, thereby directly acknowledging thepoint of origin or source for inspiration and, subsequently, sub-jects them to a series of pc modifications, such as those found in

Example 4. The example shows a straightforward illustration ofthis process in the opening measures of Adams’s Violin Con-certo, first movement (1993). In this incipit, Adams featuresSlonimsky’s Pattern 10, transposed to begin on E.21 ImaginingE as pc0, Slonimsky’s Pattern 10 can be notated as: ascending0–2–5 ⊗ <06>; descending 0–e–8 ⊗ <06>. The set classformed by these pcs consists of the hexachord collection 6–30[013679], a subset of the octatonic collection. The ascendingpattern first maintains the register of the original, and shortlyafter Adams transposes various pitches down an octave to

. Interpolation of two notes in a C6-cycle.

. Slonimsky’s Pattern 425 in Adams’s Century Rolls, third movement. THESAURUS OF SCALES ANDMELODICPATTERNS By Nicolas Slonimsky Copyright © 1947 (Renewed) Schirmer Trade Books, a division of Music Sales CorporationInternational Copyright Secured. All Rights Reserved. Reprinted by Permission. Century Rolls by John Adams © Copyright by Hendon

Music, Inc., a Boosey & Hawkes company. Reprinted by permission.

Slonimsky’s Pattern 425 reveals a similar characteristic to what Gollindescribes as a multi-aggregate cycle, though with the octatonic collection;hence, it produces what I call a multi-octatonic cycle.

This musical depiction compiles the highest textures from the violin I,violin II, and viola parts. Occasionally chromatic notes are enharmonicallyrespelled in the score transcribed for violin and piano.

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prevent a continuous melodic rise. After commencing the con-certo with a transposed replica of Slonimsky’s Pattern 10,Adams begins a process of gradual transformation through theremoval of pcs. Each asterisk in the illustration signals omittedpcs from Slonimsky’s ordering.22 The omitted notes from theascending patterns (B♭, C, D♯, E, F♯) are generally arrangedaccording to the same structure as Pattern 10, which producesthe intervals <+2+3+1>. Thus, the absence of these expectednotes reinforces the intervallic structure that defines the initialpattern. Like Slonimsky, Adams also incorporates the retro-grade form of the pattern (starting on m. 10), albeit in modifiedform. Omitted notes in Adams’s retrograde form are far morefrequent and occasionally outline A–C–E, or an A-minor triad.In the modification that I refer to as a “pitch-class interchange,”Adams swaps the ordering of two notes. The influence of thispattern on the Violin Concerto can be traced throughout theentire movement to a greater or lesser degree, and its effectbears a direct impact on the surface and structure of the entiremovement.

Another compositional technique that displays influence ofSlonimsky’s Thesaurus involves paraphrasing a pattern’s contin-uous melodic ascent and descent (prime and retrograde),guided by a recurring interval to form a cycle. The genesis ofAdams’s newly composed patterns can be traced directly toSlonimsky’s work because they appear after a transparent repro-duction of one of Slonimsky’s patterns has been introduced. In

this manner, a traceable pattern is used as a starting point;Adams’s own twist can be interpreted as variations on a pattern.Example 5 illustrates how Adams reworks Pattern 425 in “HailBop” from Century Rolls, given in Example 3 (mm. 128–31).The recurring tetrachord from Pattern 425—by coincidence amember of 4–25 [0268]—has been altered, yet Adams’s newpatterns resemble Slonimsky’s in their contour; moreover, boththe prime and retrograde forms appear in the score. Adams’svariations alternate between two pitch-class collections, withthe exception of the [026] trichord in m. 134, which is never-theless a subset of the [0258] tetrachord heard in the followingmeasure. Each variation utilizes two pitch-class collections: var-iation 1 contains [0258] or its subset [026], as well as [0146];variation 2 includes [0147] and [0135]; variation 3 employs[036] and [0156]. Adams’s newly composed patterns formthree different hexachordal collections: 6–21 [023468] in mm.134–35, 6-Z40 [012358] in mm. 144–47, and 6-Z28 [013569]in mm. 152–55. It is intriguing to discover that none of thehexachords are subsets of Slonimsky’s octatonic pattern. Bydeparting from the parent collection, Adams highlights aprocess of development in his own variations.23

The employment of two set classes to create a new kind ofpattern, as Adams develops in Example 5, is not foreign to the

. Slonimsky’s Pattern 10 in Adams’s Violin Concerto, first movement. THESAURUS OF SCALES AND MELODICPATTERNS By Nicolas Slonimsky Copyright © 1947 (Renewed) Schirmer Trade Books, a division of Music Sales CorporationInternational Copyright Secured. All Rights Reserved. Reprinted by Permission. Violin Concerto by John Adams © Copyright by Hendon

Music, Inc., a Boosey & Hawkes company. Reprinted by permission.

In the context of minimal music, Warburton (1988) refers to this gradualremoval of notes from a pattern as a block reductive process.

Timothy A. Johnson’s (2005) presentation “Diatonic Transformations inthe Music of John Adams” confirms a similar point on contrasting seem-ingly unrelated pitch-class collections. Namely, Adams signals new formalsections with pitch collections that have fewer common tones than thosemaintained within a former section.

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treatment of a small number of patterns in the Thesaurus,such as those titled “Non-Symmetric Interpolation” (Patterns49–52), and “Bitonal Arpeggios” (Patterns 1191–1213). However,one of the primary organizing principles of the Thesaurus concernsdirected intervals, rather than the collection of unordered pcs toform set classes. Therefore, the directed intervals that generate thealternating set classes will remain the same (or at times be inverted)when a set class is produced. Yet Adams’s variations depart fromthe kind of linear regularity seen in Patterns 49–52 and 1191–1213.Like Slonimsky’s Pattern 425, Adams’s variations revolve around a3-cycle <0369>, except that Adams distorts one of the principalcyclic tones with each variation (the only hexachordal collection thatallows for a complete 3-cycle is found in variation 3, but Adamschooses to deviate away from Slonimsky’s pattern of transposition).

Although one cannot make a direct one-to-one correlationbetween Adams’s unique variations and patterns from theThesaurus, each of the variations contains embedded portionsfrom patterns that stem from the same chapter in which onefinds Pattern 425, entitled “Sesquitone Progression.” Pattern425 features the ultrapolation of three notes within a 3-cycle,while traces from Patterns 528, 533, and 540 found in the varia-tions all feature the infra-inter-ultrapolation of three notes, alsowithin a 3-cycle.24 Considering that there are 177 patterns

. Slonimsky’s Pattern 425 reworked in “Hail Bop” from Century Rolls. THESAURUS OF SCALES ANDMELODICPATTERNS By Nicolas Slonimsky Copyright © 1947 (Renewed) Schirmer Trade Books, a division of Music Sales CorporationInternational Copyright Secured. All Rights Reserved. Reprinted by Permission. Century Rolls by John Adams © Copyright by Hendon

Music, Inc., a Boosey & Hawkes company. Reprinted by permission.

These three patterns contain the following pcs shown using Heinemann’snotation for multiplication: Pattern 528 = 0–t–1–9 ⊗ <0369>; Pattern533 = 0–t–2–9 ⊗ <0369>; Pattern 540 = 0–9–2–6 ⊗ <0369>. To trace theresemblance, one must examine Slonimsky’s patterns with directed

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found in the chapter “Sesquitone Progression” and 12 subchap-ters that list the kinds of embellishments added to the principaltones, whether or not one could attribute a direct influence ormarginalize it as merely coincidental, it appears that Adams’schoice of patterns from just one subchapter is germane to hismusical treatment of Slonimsky’s patterns. Given that Adamssets into motion different kinds of patterns that are related bya binding interval cycle, one could assert through inductivereasoning that his conception of a musical synonym is in thisinstance broader than that suggested by Slonimsky throughhis division of cyclically related patterns by processes of inter-, infra-, or ultra-polation. As shown in Example 2, the criteriafor Slonimsky’s arrangement are threefold: the cyclic compo-sition of a pattern; its cardinality; and the type of embellish-ment(s) chosen to realize the pattern. Adams, on the otherhand, retains a hexachordal cardinality here, yet relates patternsmore broadly as long as they fit within the same pattern length.

The transformation of Adams’s patterns at work inExample 5 can be closely examined using the concept of stepclasses in modulo-6 space.25 Christoph Neidhöfer defines astep-class interval as the directed distance in steps between twonotes of a given collection.26 The step-class sequence <+2 +3–2>serves as the driving factor that generates Adams’s variations.By arranging the pitch-class content of the first variation inascending order (0, 2, 6, 8, 9, t), the resulting step-class intervalscan be derived: <+3 +3 -2 -2 +2 +3 -2 -1 +1 +3 -2 -1 +2 +3 -2>.Each trichord or tetrachord illustrated in Example 5 is indicatedby the underlined, directed step-class intervals, while the con-nection between them, which highlights the connectionbetween collections within the 3-cycle, is shown by the remain-ing integers. Any deviation from the generating step-classsequence <+2 +3 -2> normally results from the interactionbetween trichords and tetrachords in this and other variations.The second variation is derived in the same manner <+2 +3 -2-1 +2 +3 -2 -2 +2 +3 -2 -1 +2 +3 -2>, as is the third variation<+3 +2 -2 -1 +2 +3 -2 -2 +2 -2 +3 -1 +2 +3 -2>. The end resultof Adams’s method for composing variations reveals the mal-leable nature of Slonimsky’s patterns as they transform into newkinds of patterns that draw from the original in their contourand emphasis on a specific interval cycle, yet incorporate recur-ring embedded step-class intervals at each variation.

The method in which Adams integrates and transformsSlonimsky’s patterns does not appear to occur simply at thesurface level but rather at the very core, retaining motivic andstructural relationships that bind a whole work. In the thirdmovement from Century Rolls, the octatonic collection comesto the fore subsequent to Slonimsky’s pattern and Adams’sunique molding of Pattern 425 that ensues in mm. 126–57.The movement continues with a particularly lucid manipula-tion of the octatonic collection represented through scales andother motivic material reminiscent of Adams’s earlier han-dling of this collection. Moreover, the musical setting preced-ing m. 126 associates motivic interrelationships throughoctatonic subsets as well as a rhythmic character that augurs thesyncopated figures found subsequently. In the first movement ofthe Violin Concerto, Adams’s opening, stemming directly fromSlonimsky’s Pattern 10, virtually shapes and molds the entiremovement. Adams describes the first movement, which bearssome resemblance to a concerto in its use and placement of acadenza for the soloist: “The large organism is a picture of thesmallest cellular structure. For example, in the first movement,those rising waves of triads become basic genetic material forthe entire movement. They make their effect felt everywhere,even in the cadenza.”27 Adams candidly acknowledges that thesource of the smallest cellular structure derives from the Thesau-rus.28 The opening pattern is subjected to various transpositionsjuxtaposed to create parallel second-inversion major triads.

Formal development in the Violin Concerto, arising fromAdams’s microcosmic treatment of Pattern 10, shapes thelarger gestalt. Adams’s more extensive approach to implement-ing Slonimsky’s patterns in the Violin Concerto departs fromthe occasional use found in Century Rolls. Close examination ofAdams’s works appears to pinpoint the Violin Concerto as thecomposer’s initial and deliberate adoption of this resource.In this regard, the first movement can be understood as anexperimental study of sorts, as one musical realization derivedfrom Slonimsky’s raw materials and molded by Adams. In theViolin Concerto, Pattern 10 develops in a variety of ways, someof which I will illustrate; moreover, Pattern 10 interacts withother closely related patterns. Pattern 9 = 0–2–3 ⊗ <06> andPattern 10 = 0–2–5⊗ <06> are the most prominent ones; they aretypically established in a contrasting fashion. These two patternsfeature the interpolation of one note in a 6-cycle; thus, Adamsheeds Slonimsky’s recommendation for finding related synonyms.

Example 6 illustrates various other patterns from the firstmovement of the Violin Concerto. In mm. 11–12, the violinsolo features a complete octatonic collection (OCT01), which isclassified as Pattern 392, a 3-cycle Progression appearing as thefirst pattern under the section entitled Sesquitone Progression.Although Patterns 10 and 392 appear in different sections inSlonimsky’s book, Pattern 10 shares a subset relationship to theparent octatonic collection. In mm. 18–20, the violin solo playsSlonimsky’s Pattern 11 = 0–3–4 ⊗ <06> (as marked in an

intervals in modulo 12 space. Slonimsky’s Pattern 528, for instance, gener-ates the following repeating pc series four times before returning to its start-ing point: <-2+3+8-6>. Adams’s variation 1 rotates the pattern to beginwith the second directed interval <+3+8-6-2>, while the other two varia-tions also rotate Slonimsky’s pattern by one note.

The significance of step-class intervallic analysis in Adams’s instrumentalworks from the 1990s onwards has been raised by Sanchez-Behar (2008),and the earliest trace of Adams’s initial workings in step classes has beendocumented in China Gates (1977), which Adams and scholars alike regardas his first opus (Sanchez-Behar [2014]). Closer examination of Adams’sworks reveals that the notion of interval transposition in diatonic space or amodulo-space of smaller cardinality is pervasive as a means of motivicdevelopment.

For a discussion and application of the term “step class,” refer to Neidhöfer(2005) and Santa (1999).

Adams (1996, 91). Ibid., 99.

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earlier example, the asterisk symbol highlights the notes that aresporadically omitted from the pattern). Pattern 11 can also bethought of as a rotation of Pattern 10, and naturally, it alsoforms a subset of the octatonic collection. Measure 45 illustratesone instance in the concerto of Slonimsky’s Pattern 3 = 0–4 ⊗<06>, a subset of Pattern 10. In mm. 302–5, the violin solo fea-tures the retrograde form of Pattern 9, written as 0–9–8⊗ <06>.

Close examination of the Violin Concerto reveals that Adams’snotion of musical “synonyms” at times is as Slonimsky prescribes,and at times departs from the method for resemblance set up inthe Thesaurus. Adams does not restrict his choice of patterns to asingle cardinality, but rather relates patterns through set/subsetrelations. Considering Slonimsky’s “synonyms” in Example 2(Patterns 5–13), only those patterns that formed subsets to theoctatonic collection—such as 9, 10, and 11—appear prominentlyin the first movement of the Violin Concerto. The melodic pat-terns utilized by Adams in the Violin Concerto share a commonproperty: maximal saturation of ic6 given their cardinality. The tet-rachord 4–25 [0268] formed by Pattern 3 has two tritones, thehexachord 6–30 [013679] formed by Patterns 9, 10, 11 has threetritones, and the octatonic collection has four tritones.

A summary of my findings thus far shows that, after present-ing a pattern from the Thesaurus and gradually altering itthrough various pitch and rhythmic modifications, Adamscreates his own patterns that are nevertheless influenced by the

Thesaurus. In other instances, he gradually introduces variouspatterns from the Thesaurus that interact with one another andare related in pitch content. In the Violin Concerto, Adamshandles the interaction of patterns in a variety of ways. An over-view of the compositional techniques from the Violin Concertothat are coupled with Slonimsky’s patterns will help elucidatehis synthesis of raw materials and their manipulation throughhis unique minimalist treatment.

Transposition is the most pervasive operation Adams appliesto Slonimsky’s patterns. The first approach for integratingtransposition involves the use of a single pattern. Example 7illustrates the various transpositions of Pattern 10 and theiremployment in the solo violin part in mm. 27–30; these mea-sures are emblematic of Adams’s method for transpositionthroughout the concerto. Due to the transpositional symmetryof Pattern 10, there are six possible transpositions; the remain-ing six patterns form rotational equivalents to the ones shown inthe chart below. The product of T6, for instance, yields a rota-tion of T0 with pcs 6–8–e–0–2–5. Adams employs all transposi-tions of Pattern 10, giving rise to a highly chromatic passage.29

. Other Melodic Patterns in Adams’s Violin Concerto, first movement. Violin Concerto by John Adams © Copyright by HendonMusic, Inc., a Boosey & Hawkes company. Reprinted by permission.

In an interview with Adams, Rebecca Jemian and Anne Marie de Zeeuw(1996) noted that in the opening measures of the Violin Concerto “thesolo instrument seems to have free access to the twelve tones” (98). Themelodic line expressed in the opening has an improvisational quality that is

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The chromatic aggregate results from the combination of pcsfrom at least three Pattern 10 transpositions. Transpositions ofthis kind tend to overlap with one another, and the higher theinvariance between transpositions, the longer the overlappingspan tends to be: the Pattern 10 interval-class vector <224223>reveals four common notes between T0–T3, and T4–T1. After abrief passage of double stops leading up to m. 27, Adams rein-troduces Pattern 10 in a straightforward manner that adheres toSlonimsky’s own succession of intervals <+2+3+1>, and soonafterwards, patterns are modified either through rotation, spora-dic omission of one or more pcs or rejection of the scalar orderestablished by Slonimsky. The transposition of a single patterncan also be associated with certain formal functions withinthe structure of this movement. In the return of the primaryviolin solo theme (starting in m. 184), Adams uses a transposi-tion of Pattern 10 (see the violin solo in mm. 181–87) that pro-gresses from T0 to T1. The only pcs shared between these twotranspositions are those that formed the parent 6-cycle. Thisrelationship seems meaningful in light of Johnson’s (2005)findings that demonstrate how Adams often develops formalsections by maintaining a high degree of common tones and,conversely, highlights the arrival of subsequent sections withpitch collections that retain a limited number of shared notes.30

The second approach to transposition entails the interactionbetween two patterns from the Thesaurus. One of the mostvisible procedures in the Violin Concerto concerns the interplayof Pattern 9 = 0–2–3 ⊗ <06> and Pattern 10 = 0–2–5 ⊗ <06>while keeping the level of transposition between the two intact.This technique can be found in virtually all instruments atvarious times, including the violin solo and other instrumentparts, and it is particularly noticeable in the Synthesizer 1 line(for a transparent example, turn to the Synthesizer 1 in mm.1–22). Utilizing the same level of transposition between Pat-terns 9 and 10 enables a high retention of common tones.Adams’s penchant for utilizing this technique with greatestclarity at formal beginnings—rather than as concluding or tran-sitional gestures—bears parallels to the methods for elaborationand development considered part and parcel of the minimaliststyle. He stretches the presentation of each pattern for a sub-stantial period and gradually subjects them to a type of fragmen-tation that expedites the alternation of patterns.

The third and final technique is central to the types of trans-formations encountered in the Violin Concerto; it consolidatestransposition with step-class intervals drawn from Patterns 9and 10. Example 8 illustrates the Synthesizer 1 texture frommm. 49–55, doubled by the lowest pitches found in the dove-tailed string section. Subsequent to his manipulation of Patterns9 and 10 in the opening measures of the concerto, Adams con-cocts a melodic rise that highlights not only the interactionbetween patterns but, at times, also their unification throughpatterns that periodically utilize only invariant pcs. An examina-tion of traditional ordered pitch-intervals would reveal unequal

. Transposition of Pattern 10 in Adams’s Violin Concerto. Violin Concerto by John Adams © Copyright by Hendon Music, Inc.,a Boosey & Hawkes company. Reprinted by permission.

unlike the unyielding rise of Slonimsky’s pattern; however, the aim of suchchromaticism is not aggregate completion, but rather a transpositionaltreatment of Slonimsky’s patterns. Slonimsky’s manual in the preface tothe Thesaurus is compatible with Adams’s free approach: “Fragments ofthe scales and patterns in the THESAURUS may be used as motives andthemes” (iv).

Johnson’s model, called the common-tone index (CTI), tallies pc relation-ships retained between chords, sonorities, and fields. Johnson describes theterm “sonority” as all sounding pitches, whereas “field” consists of the

diatonic collections implied by musical context. For additional informationon Johnson’s analytical model, please refer to his 1991 dissertation.

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distances to generate each transposition, but step-class intervalsappropriately illustrate intervallic consistency <+2 +1 +2> mea-sured in scale steps according to Slonimsky’s patterns. Adamsagain creates variety by exhausting all transpositions fromPattern 9 (and nearly all from Pattern 10). Moreover, the rota-tion of the borrowed patterns is frequent; in other words, hebegins a pattern with any desired note from that collection.Considering the transpositional level between contiguous pat-terns, Adams focuses on those that are one or two levels apart.This passage can be perceived as creating a closing gesturethrough an ascending figure reminiscent of the opening butfragmented to half its duration. The texture that follows thefragmentation of this motivic material “achieves a shifting ofgears in [the] music” through a gradual accelerando and shortlyafter a metric modulation.31

The types of harmonic structures that accompany Adams’sborrowed patterns at times parallel Slonimsky’s approach toharmonizing his own patterns. In several appendices from the

Thesaurus, Slonimsky suggests one or more possible harmoniza-tions for many of his patterns (Patterns 1–568).32 Slonimskyprimarily recommends the use of “master chords” to harmonizethe linear patterns. These master chords consist of Mm7 chordswithout a chordal fifth. In 3-cycle patterns, Slonimsky suggestsas little as one harmonization, and as many as six possibleharmonizations. These harmonizations differ in their transposi-tional levels, at times blending well with the melodic patternsby using the same pcs, and at times clashing strongly with con-trasting notes. Example 9 shows the only harmonization sug-gested for Slonimsky’s Pattern 425, first introduced inExample 3 to interpret a passage from the third movement ofCentury Rolls. Slonimsky prescribes the use of cyclically revolv-ing master chords at T3 for all 3-cycle patterns in ascending anddescending form. The bass of each root-position chord is atritone apart from the melodic notes that fall on each beat.

. Step-class transposition from Patterns 9 and 10. Violin Concerto by John Adams © Copyright by Hendon Music, Inc., a Boosey& Hawkes company. Reprinted by permission.

. Slonimsky’s suggested harmonization of Pattern 425. THESAURUS OF SCALES ANDMELODIC PATTERNS ByNicolas Slonimsky Copyright ©1947 (Renewed) Schirmer Trade Books, a division of Music Sales Corporation International Copyright

Secured. All Rights Reserved. Reprinted by Permission.

Adams (1996).

Yet his book is arranged according to linear patterns, which suggests thatharmony was a secondary consideration—indeed, most of the harmoniza-tions are shown in the appendices.

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Together the harmonization and the melody notes on the beatform the whole-tone subset 4–25 [0268]. While the verticalaspect of this pattern forms the 4–25 collection, the linearunfolding of Pattern 425 segmented into tetrachords also gen-erates the same collection.

When Adams borrows Slonimsky’s patterns, he either usesSlonimsky’s prescribed harmonization when first presentingthe melodic pattern in its untouched state, or he conceives newharmonies following an initial introduction to a melodicpattern. In “Hail Bop” from Century Rolls, for example, Adamsharmonizes the ascending form of Pattern 425 in the mannersuggested by Slonimsky, though with the addition of an addedperfect fifth above the bass (this harmonization begins inm. 126). Once Adams begins to develop Slonimsky’s patterns,the accompanying harmonic structures also begin to take theirown form. Example 10 revisits variation 1 from Example 5,where I described how the melodic pattern retains the samepitch contour as Pattern 425 (in mm. 134–41). The harmonicaccompaniment in the piano in Example 10 is generated by thenotes expressed melodically. In the Thesaurus, Slonimskyderives a related technique in his chapter “Autochordal Harmo-nization,” where new scales are harmonized “with the aid ofchords formed by the notes of the scale itself.”33 Slonimsky uti-lizes root-position major and minor triads for each note of amelodic pattern. Adams’s approach differs from Slonimsky inthat each melodic segment circled in the example is representedharmonically on each offbeat (I have only circled the first threemeasures, but this process applies to the entire passage). WhenAdams’s melodic segments form a trichord in the excerpt from“Hail Bop,” the respective harmony will contain three distinctnotes, whereas when a segment consists of a tetrachord, itsharmony will yield four notes. Unlike Slonimsky’s prearrangedharmonizations, Adams’s harmonies are not drawn from arecurring master chord subjected to T3 operations; instead, thewoodwinds and piano illustrate linear and vertical expressions ofthe same process. As a direct result, his harmonic structuresdiversify their content through various dyads, triads, and tetra-chords that form subsets of the parent collection 6–21 [023468]formed by the melodic pattern.

Another formula Adams employs for the harmonization ofmelodic patterns concerns the use of root-position major triads.The opening measures of the Violin Concerto, shown inExample 11, illustrate Adams’s harmonization of Slonimsky’sPattern 10 discussed earlier in Examples 1, 2, and 4. In aninterview, Adams recalled how he “derived the rising figure as adiatonic figure which [he] then submitted to modal transposi-tions.”34 The result in Example 11 and throughout the move-ment is a linear succession of Pattern 10 in three different andsimultaneous transpositions, vertically forming second-inver-sion major triads.35 Adams’s triadic approach to the

harmonization of melodic patterns has a precedent in the The-saurus. Slonimsky derives another formula for his scales andpatterns that strictly engages root-position major triads. Eachnote of a pattern is harmonized with a new triad, and notesfrom the melodic pattern consist of either a root, third, or fifth.This technique diverges from “Authochordal Harmonization”in that the harmonies do not stem directly from the melody;rather, major triads are matched with the melody’s root, third,or fifth, while remaining notes can lie outside the collection ofthe melodic pattern itself. Concerning Adams’s triadic approach,the composer states that we perceive the linear aspect of thesemeasures more strongly than the vertical structures: “A majortriad is a very pleasant and sonically user-friendly interval. Butwhen they’re perfectly parallel and follow lines that are essentiallyatonal, it creates a very interesting effect where I think the atonal-ity dominates over the tonality of the vertical arrangement.”36

This statement is significant in light of Adams’s composi-tional approach to instrumental works from the 1990s onwards,which receive greater emphasis on counterpoint and polyphonythan his earlier works written in the 1970s and 80s. The ViolinConcerto, Century Rolls, and other works from the 1990sachieve “a successful integration of the older minimalist tech-niques (repetitive motifs, steady background pulse and stableharmonic areas) [to] the more complex, more actively contra-puntal language of the post-Klinghoffer pieces.”37 In this light,Adams’s greater interest in contrapuntal writing during thisperiod offers a plausible explanation for his adoption of thelinear patterns found in the Thesaurus.

I have presented an overview of Adams’s method of integrat-ing and molding patterns and accompanying harmonies fromthe Thesaurus in individual movements from two concerti. Thispractice naturally extends to other movements, as well as instru-mental works from the 1990s onwards, including Slonimsky’sEarbox and Scratchband (both composed in 1996). In thesecond movement from the Violin Concerto, a Chaconne thatquotes Pachelbel’s celebrated Canon in D, Adams employs theoctatonic collection as a preamble to Slonimsky’s melodic pat-terns and scales. The octatonic collection serves as a primaryorganizational strategy that helps generate melodic subsets andstructure the work. During the opening measures, the secondsynthesizer chords outline a major-triad harmonization arounda complete 3-cycle, exhausting all the notes from an octatoniccollection. The solo violin begins with a free counterpoint thatfrequently employs notes from the same collection, but withadded embellishments. Adams gradually morphs the tonallyderived ostinato into the whole-tone subset 4–24 [0248] inmm. 62–71 and subsequently into the octatonic subset 5–19[01367] in mm. 119–32 while keeping the original contourintact.38 This movement of the Violin Concerto also makes per-vasive use of numerous heptatonic scales, which are also

Slonimsky (1975, vi). Adams (1996, 99). Another instance of this technique can be observed in Adams’s American

Berserk, composed in 2001.

Adams (1996, 99). Adams (www.earbox.com/orchestra/slonimsky-s-earbox). Whole-tone and octatonic collections are well represented in the Thesaurus,

comprising over 200 patterns.

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. Adams’s harmonization of Pattern 425 in “Hail Bop.” Century Rolls by John Adams © Copyright by Hendon Music, Inc., a Boosey & Hawkes company.Reprinted by permission.

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referenced in Slonimsky’s Thesaurus, including Patterns 1038–1040, and 1048.39

The orchestral work Slonimsky’s Earbox features octatoni-cism prominently, and its transpositionally symmetrical melodicpatterns can frequently be traced to 3-cycle and 6-cycle patternsfrom the Thesaurus. The opening loosely models Stravinsky’ssymphonic poem Le chant du rossignol, a work that has beenexamined by scholars for its pentatonic-diatonic-octatonicinteraction and appeals to Adams for its “brilliant eruption ofcolors, shapes, and sounds.”40 The opening scalar passagessuggest a melodic-minor scale that leads to an octatonic collec-tion in m. 5. He borrows Slonimsky’s Pattern 6 in the secondclarinets in mm. 6–7, an octatonic subset containing 0–1–3 ⊗<06> = 0–1–3–6–7–9–0, then superimposes its two other rota-tions, Pattern 10, or 0–2–5 ⊗ <06>, and Pattern 11, or 0–3–4⊗ <06>, in the piccolo and first clarinets. Slonimsky’s pat-terns and various other octatonic subsets appear in prime and/or retrograde form. Throughout the work, Adams employspatterns from Slonimsky’s Tritone Progression (6-cycle) orSesquitone Progression (3-cycle), though he also balancesthe symmetrical nature of these patterns with nonsymmetricalones whose exact origin in Slonimskian antecedents is at timesdifficult to pinpoint.

There is little doubt that Slonimsky’s Thesaurus informs oneimportant facet of Adams’s compositional technique. Yet it isthe manner in which Adams employs Slonimsky’s melodic pat-terns that is most telling. Initially Adams incorporates a singlepattern or a variety of musical synonyms according to theirintervallic cycle and cardinality, while in subsequent statementsthe patterns serve as a model for creating newly composed pat-terns. This method for employing and reworking Slonimsky’smelodic patterns is particularly notable in the third movementfrom Century Rolls. Likewise, the first movement of the ViolinConcerto presents another study of Adams’s procedure for

employing and developing Slonimsky’s patterns. From theonset, Slonimsky’s Pattern 10 assumes a role akin to a Grundg-estalt, by which other patterns revolve around and transformthrough transpositional operations into myriad shapes andforms. In reading Slonimsky’s own words for composing withthe Thesaurus, I cannot help but feel a sense that Slonimskynever meant for these patterns to be used with any degree ofrigidity. Slonimsky encouraged Thesaurus users to transpose hispatterns to suit a composition and explained that “fragments ofthe scales and patterns in the THESAURUS may be used asmotives and themes. The rhythmical elaboration is left to theimagination of the composer.”41 Exemplifying this attitude, Slo-nimsky ventured composing fifty-one short piano pieces, titledMinitudes (1972–76), which occasionally showcase patterns fromthe Thesaurus in an unrestricted manner. Adams’s handling ofthese patterns, first presenting the germinating idea and thengradually molding the basic shape into his own, exemplifies thekind of flexible approach that is implicit in Slonimsky’s writingand music devoted to or influenced by his Thesaurus.

Adams, John. 1999. Liner notes to John Adams Earbox.Nonesuch Compact Disc 79453–2.

———. “Slonimsky’s Earbox.” http://www.earbox.com/orchestra/slonimsky-s-earbox (accessed 6 December 2013).

Adams, John, Rebecca Jemian, and Anne Marie de Zeeuw.1996. “An Interview with John Adams.” Perspectives of NewMusic 34 (2): 88–104.

Bair, Jeff. 2003. “Cyclic Patterns in John Coltrane’s MelodicVocabulary as Influenced by Nicolas Slonimsky’s Thesaurusof Scales and Melodic Patterns: An Analysis of SelectedImprovisations.” Ph.D. diss., University of North Texas.

Brown, Stephen C. 2003. “Dual Interval Space in Twentieth-Century Music.”Music Theory Spectrum 25 (1): 35–57.

. Adams’s harmonization of Pattern 10 in the Violin Concerto. Violin Concerto by John Adams © Copyright by HendonMusic, Inc., a Boosey & Hawkes company. Reprinted by permission.

I would like to acknowledge Michael Scott Cuthbert (and his studentAnna Rose Lawrence) for drawing my attention to these scales and theoctatonic progression in the synthesizer incipit.

Adams (www.earbox.com/orchestra/slonimsky-s-earbox). Slonimsky (1975, iv).

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Cohn, Richard. 1988. “Inversional Symmetry and Transposi-tional Combination in Bartók.” Music Theory Spectrum 10(1): 19–42.

Gollin, Edward. 2007. “Multi-Aggregate Cycles and Multi-Aggregate Serial Techniques in the Music of Béla Bartók.”Music Theory Spectrum 29 (2): 143–76.

Hába, Alois. 1927. Neue Harmonielehre. Leipzig: Fr. Kistner &C. F. W. Siegel.

Heinemann, Stephen. 1998. “Pitch-Class Set Multiplicationin Theory and Practice.” Music Theory Spectrum 20 (1):72–96.

Jemian, Rebecca, and Anne Marie de Zeeuw. 1996. “An Inter-view with John Adams.” Perspectives of New Music 34 (2):88–104.

Johnson, Timothy A. 1991. “Harmony in the Music of JohnAdams: From Phrygian Gates to Nixon in China.” Ph.D.diss., State University of New York at Buffalo.

———. 2005. “Diatonic Transformations in the Music of JohnAdams.” Paper presented at the Music Theory Society ofNew York State, 9–10 April, New York.

Kostelanetz, Richard. 1997. The Frank Zappa Companion: FourDecades of Commentary. New York: Schirmer Books.

Lambert, J. Philip. 1990. “Interval Cycles as CompositionalResources in the Music of Charles Ives.” Music TheorySpectrum 12 (1): 43–82.

Messiaen, Olivier. 1944. Technique de mon langage musical.Paris: Ed. musicales Alphonse Leduc.

———. 1994. Musique et couleur. Trans. E. Thomas Glasow.Portland: Amadeus Press.

Neidhöfer, Christoph. 2005. “ATheory of Harmony and VoiceLeading for the Music of Olivier Messiaen.” Music TheorySpectrum 27 (1): 1–34.

Porter, Lewis. 1998. John Coltrane: His Life and Music.Ann Arbor: University of Michigan Press.

Sanchez-Behar, Alexander. 2008. “Counterpoint and Polyph-ony in Recent Instrumental Works of John Adams.” Ph.D.diss., Florida State University.

———. 2014. “Symmetry in the Music of John Adams.”Tempo 68 (268): 46–60.

Santa, Matthew. 1999. “Defining Modular Transformations.”Music Theory Spectrum 21 (2): 200–29.

Schillinger, Joseph. 1946. The Schillinger System of MusicComposition. Vols. I and II. New York: Carl Fischer.

Schoenberg, Arnold. February 8, 1949. Arnold Schoenberg toNicolas Slonimsky. Letter. From Library of Congress, NicolasSlonimsky Collection, Performing Arts Reading Room, box158, folder 4.

Slonimsky, Nicolas. 1975. Thesaurus of Scales and MelodicPatterns. New York: Charles Scribner’s Sons.

———. 1995. “Why I Wrote the Thesaurus of Scales andMelodic Patterns.” Piano Stylist and Jazz Workshop January/February: 37.

Veenstra, Kimberly Anne. 2009. “The Nine-Step Scale ofAlexander Tcherepnin: Its Conception, Its Properties, andIts Use.” Ph.D. diss., Ohio State University.

Warburton, Dan. 1988. “AWorking Terminology for MinimalMusic.” Intégral 2: 135–59.

Music Theory Spectrum, Vol. 37, Issue 2, pp. 175-88, ISSN 0195-6167,electronic ISSN 1533-8339. © The Author 2015. Published by OxfordUniversity Press on behalf of The Society for Music Theory. All rightsreserved. For permissions, please e-mail: journals.permissions@oup.com.DOI: 10.1093/mts/mtv013

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