¹ Slonimsky, Nicolas, Thesaurus of Scales and Melodic Patterns (New York: Charles Scribner's Sons, 1947).Google Scholar

² Brower, Candace investigates these types of symmetries in her ‘Paradoxes of Pitch Space’, Music Analysis, 27, no. 1 (2008), pp. 51–106.CrossRefGoogle Scholar

³ Various authors have described symmetry in this manner. See Weyl, Hermann, Symmetry (Princeton: Princeton University Press, 1952)CrossRefGoogle Scholar and Darvas, György, Symmetry: Cultural-Historical and Ontological Aspects of Science-Arts Relations (Basel: Birkhäuser, 2007)Google Scholar.

⁴ The application of a hyperboloid stems from Daniel J. McConnell's analysis of China Gates: ‘John Adams's Perpetual Motion Machine’, unpublished paper presented at the Society for Music Theory Annual Meeting (Boston, 2005).

⁵ My designation of modes, which concurs with Daniel J. McConnell's analysis, is based on a low pedal note that signals the opening of each section. This interpretation also accords with Timothy A. Johnson's preference rules specifically designed for analyzing modes and chords in Adams's music. See his ‘Harmonic Vocabulary in the Music of John Adams: A Hierarchical Approach’, Journal of Music Theory, 37, no. 1 (1993), pp. 117–56CrossRefGoogle Scholar, esp. p. 130.

⁶ John Adams, ‘China Gates’, http://www.earbox.com/piano-solo-or-duet/china-gates (accessed 2 October 2013).

⁷ Adams, ‘China Gates’.

⁸ While McConnell's geometric depiction is compelling, and his modal designation of formal sections is accurate, his analysis overlooks other symmetries that interact with the hyperboloid structure. For instance, McConnell's shadings in the lower cone do not reflect those of the upper cone. Furthermore, McConnell's dark- and light-shaded regions are represented on different rows, while my own hyperboloid reinterprets these regions on the same horizontal plane to associate another aspect of this work's symmetry I will soon detail.

⁹ György Darvas, Symmetry, p. 4.

¹⁰ Darvas, Symmetry, pp. 4–5.

¹¹ Gretchen Horlacher derived the term reiterating fragment to describe a repetitive pattern that is similar to an ostinato, except that its iterations can be modified or offset by rest. See her ‘The Rhythms of Reiteration: Formal Development in Stravinsky's Ostinati’, Music Theory Spectrum, 14, no. 2 (1992), pp. 171–87CrossRefGoogle Scholar, esp. p. 180.

¹² CHINA GATES, by John Adams. © 1983 by Associated Music Publishers, Inc. (BMI). International Copyright Secured. All Rights Reserved. Used by Permission.

¹³ McConnell states that formal sections are related by retrograde-inversion, but he fails to acknowledge step-class transformations.

¹⁴ For a detailed discussion of step-class intervals in analytical literature, see: Neidhöfer, Christoph, ‘A Theory of Harmony and Voice Leading for the Music of Olivier Messiaen’, Music Theory Spectrum, 27, no. 1 (2005), pp. 1–34CrossRefGoogle Scholar and Santa, Matthew, ‘Defining Modular Transformations’, Music Theory Spectrum, 21, no. 2 (1999), pp. 200–229.CrossRefGoogle Scholar

¹⁵ Bernard, Jonathan, ‘Space and Symmetry in Bartók’, Journal of Music Theory, 30, no. 2 (1986), p. 192.CrossRefGoogle Scholar

¹⁶ John Adams, ‘Phrygian Gates’, http://www.earbox.com/piano-solo-or-duet/phrygian-gates (accessed 2 October 2013).

¹⁷ John Adams, ‘Phrygian Gates’.

¹⁸ John Adams, liner notes to Phrygian Gates and Shaker Loops (1750 Arch Records S-1784, 1980).

¹⁹ Pellegrino, Catherine Ann, ‘Aspects of Closure in the Music of John Adams’, Perspectives of New Music, 40, no.1 (2002), pp. 147–175Google Scholar, here 150–51. The late K. Robert Schwarz also discusses the third movement and draws his analysis from an interview with Adams. See ‘Process vs. Intuition in the Recent Works of Steve Reich and John Adams’, American Music 8, no. 3 (1990)Google Scholar, p. 257.

²⁰ Pellegrino, ‘Aspects of Closure in the Music of John Adams’, p. 150.

²¹ This hyperboloid resembles McConnell's depiction of China Gates. Pellegrino also recognises a palindrome in the fourth movement without introducing any geometric representations.

²² This shift from process-driven works to a more intuitive conception of music is examined by K. Robert Schwarz. See Schwarz, ‘Process vs. Intuition in the Recent Works of Steve Reich and John Adams’, American Music, 8, no. 3 (1990), pp. 245–73CrossRefGoogle Scholar. In an interview with Schwarz, Adams said ‘I've stopped worrying about whether intuiting a structure is right or not; as far as I can tell, most nineteenth-century composers wrote on intuitive levels’ (‘Process vs. Intuition’, p. 247). There are elements in Adams's work that can be interpreted as efforts to break free from the early minimalist aesthetic that, through a detachment of the composer's voice, generated self-mechanised processes and eventually led to a more personal, intuitive style. According to Schwarz: ‘Not only does Adams exploit this modal conflict to create contrasts in melodic patterns, textural density, rhythmic figuration, and dynamics, but he does so with a directionalised motion that sweeps toward climaxes—a motion far removed from the stasis of minimalism. Such a subjective approach works to loosen the bonds of musical process and heighten the role of intuition’ (p. 258). The sheer size of Phrygian Gates also poses a challenge to maintaining audible symmetrical structures; a primary concern of the early minimalist style was creating gradual, perceptible processes.

²³ Pellegrino's explanation of the core seems to miss the mark, in my opinion. Rather than acknowledging a certain structure to the modal ordering, she states that ‘in m. 923, Adams abandons key signatures and uses accidentals to generate the pitches needed for the modes’ (‘Aspects of Closure’, p. 152). Furthermore, her explanation of the axis point does not explain Adams's preference for using D♯/E♭ over G♯/A♭: ‘these measures clearly demonstrate that the focus of the movement is in the alternation between G♯ Lydian and A♭ Phrygian, and the enharmonic equivalence between D♯ and E♭. There is no other reason why Adams would have notated this pitch in two different ways, other than to make this point. The enharmonic equivalence between D♯ and E♭ indicates that there is an underlying conceptual justification for this unusual notation’ (p. 153).

²⁴ See Timothy A. Johnson, ‘Harmony in the Music of John Adams: From Phrygian Gates to Nixon in China’ (PhD diss., State University of New York at Buffalo, 1991).

²⁵ Adams, John, Hallelujah Junction: Composing an American Life (New York: Farrar, Straus and Giroux, 2008)Google Scholar, p. 149.

²⁶ John Adams, Hallelujah Junction, p. 149.

²⁷ Debra Lee Traficante concurs with the importance of the musical phrases intertwined between pianos: ‘Of greatest melodic interest in the entire work is the introduction of a gospel-style melody found in the pianos ... The confidently stated gospel-style melody assists in providing a terraced build-up to the only non-vocable text, “For I have seen the promised land”’ (‘An Analysis of John Adams’ Grand Pianola Music’, DMA diss., University of Oklahoma, 2010), pp. 108–9).

²⁸ According to Weyl, near symmetries maintain some components of symmetry, but introduce at least one asymmetrical feature. See Weyl, Symmetry, p. 9–11.

²⁹ Brower, Candace, ‘Memory and the Perception of Rhythm’, Music Theory Spectrum 15, no. 1 (1993), pp. 19–35CrossRefGoogle Scholar, here p. 28.

³⁰ Fink, Robert, ‘(Post-)minimalisms 1970–2000: the Search for a New Mainstream’, in The Cambridge History of Twentieth-Century Music, ed. Cook, Nicholas and Pople, Anthony (Cambridge: Cambridge University Press, 2004)Google Scholar, p. 542.

³¹ The employment of Neo-Riemannian connections in this work reveals a keen similarity to Adams's opera Nixon in China (1985–87). In Act 1 Scene 2, recurring L-transformations are prominent when Mao Tse-tung calls on his ancestors and makes a declaration that the world has come. See Johnson, Timothy A., John Adams's Nixon in China: Musical Analysis, Historical and Political Perspectives (Farnham: Ashgate, 2011), pp. 174–77.Google Scholar

³² Weyl, Symmetry, p. 16.

³³ Adams, John, Jemian, Rebecca and de Zeeuw, Anne Marie, ‘An Interview with John Adams’, Perspectives of New Music, 34, no. 2 (1996), pp. 98–9.CrossRefGoogle Scholar

³⁴ A more thorough examination of Slonimsky's Thesaurus appears in Alexander Sanchez-Behar, ‘Counterpoint and Polyphony in John Adams's Recent Instrumental Works’, PhD diss., (Florida State University, 2008).

³⁵ Of course, this is not the only reason Adams might have resorted to the Thesaurus. Adams and Slonimsky shared a close friendship for many years.

³⁶ For more information on the enneatonic collection, refer to Kimberly Anne Veenstra, ‘The Nine-Step Scale of Alexander Tcherepnin: Its Conception, Its Properties, and Its Use’ (PhD diss., Ohio State University, 2009).

³⁷ Examples from Nicholas Slonimsky, Thesaurus of Scales and Melodic Patterns © 1947 (Renewed) Schirmer Trade Books, a division of Music Sales Corporation. International Copyright Secured. All Rights Reserved. Used by Permission. Violin Concerto and Century Rolls by John Adams © Hendon Music, Inc., a Boosey & Hawkes company. Reprinted by permission.

³⁸ Heinemann, Stephen, ‘Pitch-Class Set Multiplication in Theory and Practice’, Music Theory Spectrum, 20, no. 1 (1998), pp. 72–96.CrossRefGoogle Scholar Heinemann's multiplication signified by ⊗, transposes the underlined multiplicand series by a cyclic multiplier to yield its union, known as the product. In Pattern 11, for instance, the multiplicand 0-3-4 is transposed to pc 6 giving a product of 0-3-4-6-9-10.

³⁹ For general information on interval cycles, see Straus, Joseph N., Introduction to Post-Tonal Theory (Upper Saddle River: Pearson, 2005).Google Scholar

⁴⁰ Weyl, Symmetry; Darvas, Symmetry, p. 4.

⁴¹ Adams, Jemian and Zeeuw, ‘An Interview with John Adams’, p. 91.

⁴² Weyl, Symmetry, p. 13.

⁴³ Steve Reich discusses symmetry in this manner in ‘The Canon’, http://www.studio360.org/story/106790-the-canon (accessed 13 October 2013).

⁴⁴ Adams in John Adams, Hail Bop! A Portrait of John Adams, produced by James Wills and John Kelleher, 98 min, (Kultur International Films, DVD, 2006).