¹ Cohn’s book is one of three monographs to appear in quick succession, each of which stakes out its territory in different but complementary ways. The others are Tymoczko, Dmitri, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (New York and Oxford: Oxford University Press, 2011)Google Scholar and Rings, Steven, Tonality and Transformation (New York and Oxford: Oxford University Press, 2011)CrossRefGoogle Scholar. All three belong to the Oxford Studies in Music Theory series.

² Klumpenhouwer, Henry, ‘History and Structure in Richard Cohn’s Audacious Euphony ’, Intégral 25 (2011): 159–175 Google Scholar; Smith, Kenneth, ‘The Transformational Energetics of the Tonal Universe: Cohn, Rings and Tymoczko’, Music Analysis 33/2 (2014): 214–256 CrossRefGoogle Scholar, which considers all three books together; Scott Murphy, Review of Cohn, Richard, Audacious Euphony: Chromaticism and the Triad’s Second Nature (New York and Oxford: Oxford University Press, 2012)CrossRefGoogle Scholar, Journal of Music Theory 58/1 (2014): 79–101; Jason Yust, Review of Cohn, Richard, Audacious Euphony: Chromaticism and the Triad’s Second Nature (New York and Oxford: Oxford University Press, 2012)CrossRefGoogle Scholar, Music Theory Online 18/3 (2012) (accessed 28 December 2015); Steven D. Matthews, Review of Cohn, Richard, Audacious Euphony: Chromaticism and the Triad’s Second Nature (New York and Oxford: Oxford University Press, 2012)CrossRefGoogle Scholar, Notes 70/1 (2013): 113–16.

³ Distance here means voice-leading proximity, in contrast to its traditional measurement along the cycle of fifths.

⁴ Set 6-20, contains two augmented triads a semitone apart – e.g., C, E, G♯, and C♯, F, A – and serves as a construct for explaining chromatic successions in nineteenth-century music resisting illumination by Schenkerian and other methodologies. The set-class terminology originates in Forte’s, Allen The Structure of Atonal Music (New Haven, CT: Yale University Press, 1977)Google Scholar, more recently disseminated via Straus’s, Joseph N. Introduction to Post-Tonal Theory (Englewood Cliffs, NJ: Prentice-Hall, 1990)Google Scholar.

⁵ In neo-Riemannian theory, parsimony is synonymous with the law of the shortest way, that is, the minimal voice-leading work required to pass from one triad to another.

⁶ Cube dance represents the six triads of a hexatonic system and its two flanking augmented triads as the eight corners of a cube; conjunction of such representations for each hexatonic system produces the cube dance. See Douthett, Jack and Steinbach, Peter, ‘Parsimonious Graphs: A Study in Parsimony, Contextual Transformation, and Modes of Limited Transposition’, Journal of Music Theory 42/2 (1998): 241–264 CrossRefGoogle Scholar and Cohn, Richard, ‘Weitzmann Regions, My Cycles, and Douthett’s Dancing Cubes’, Music Theory Spectrum 22/1 (2000): 89–103 CrossRefGoogle Scholar.

⁷ The approachability of Cohn’s text has been variously noted, as for example in Smith, ‘The Transformational Energetics of the Tonal Universe’, p. 215 and Klumpenhouwer, ‘History and Structure’, p. 160.

⁸ Weitzmann, Carl Friedrich, Der übermässige Dreiklang (Berlin: T. Trautweinschen, 1853)Google Scholar. See also: Saslaw, Janna, trans., ‘Two Monographs by Carl Friedrich Weitzmann, I: The Augmented Triad’, Theory and Practice 29 (2004): 133–228 Google Scholar. Cohn’s historicism is interestingly appraised by Klumpenhouwer, who situates it in relation to recent conflicting impulses to subordinate theory to history on the one hand, or to make the history of theory yield new ideas on the other; see especially ‘History and Structure’, 170–73.

⁹ Throughout this essay, I represent major triads by capital letters (C, E, G, etc.) and minor triads with lower-case letters (c, e, g, etc.). Neo-Riemannian theory often applies + and – symbols to the same end (C⁺, E⁺, G⁺; C⁻, E⁻, G⁻).

¹⁰ Murphy’s review focuses more closely on the concept of average voice-leading work as foundational to Cohn’s system, and the theoretical implications flowing from it.

¹¹ See Cohn, Richard, ‘Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions’, Music Analysis 15/1 (1996): 9–40 CrossRefGoogle Scholar, and especially 17.

¹² These issues do not arise in relation to the comparable modelling in Chapter 7 of Tristan-genus chords as minimal perturbations of diminished sevenths in octatonic space, because the combination of adjacent octatonic systems yields the total chromatic.

¹³ More attention has been paid to the surrogacy of subsets of the octatonic than to issues of hexatonic/nonatonic identity. For a recent address on such issues, see Taruskin, Richard, ‘Catching Up with Rimsky-Korsakov’, Music Theory Spectrum 33/2 (2011): 169–185 CrossRefGoogle Scholar, and the various responses in the same issue from Kofi Agawu, Robert O. Gjerdingen, Marianne Kielian-Gilbert, Lynne Rogers, Dmitri Tymoczko, Pieter van den Toorn, Arnold Whittall and Lawrence M. Zbikowski.

¹⁴ The Tonnetz is one of the core concepts of neo-Riemannian theory; its various formulations are too diverse to enumerate here. For recent consideration, see Edward Gollin, ‘From Matrix to Map: Tonbestimmung, the Tonnetz and Riemann’s Combinatorial Conception of Interval’, and Cohn, Richard, ‘Tonal Pitch Space and the (Neo-)Riemannian Tonnetz’, in Gollin and Alexander Rehding, eds., The Oxford Handbook of Neo-Riemannian Music Theories (New York and Oxford: Oxford University Press, 2011): 271–293 Google Scholar and 322–49. Its most commonly cited source in Riemann’s, Hugo work is in ‘Ideen zu einer “Lehre von den Tonvorstellungen”’, Jahrbuch der Bibliothek Peters 21–22 (1914–15): 1–26 Google Scholar.

¹⁵ Cohn likens this reversal to the linguistic concept of code switching, which accounts for the cognitive processes guiding bilingualism. He has in mind specifically Gardner-Chloros, Penelope, Code-Switching (Cambridge: Cambridge University Press, 2009)CrossRefGoogle Scholar.

¹⁶ Cohn’s analysis of this passage is considered by Smith (‘The Transformational Energetics of the Tonal Universe’, p. 226), together with the first movement of Schubert’s D. 960 and the third movement of D. 944. Smith regards a lack of engagement with the issue of melodic kinetics as problematic in Cohn’s work, and advocates a general re-engagement with concepts of harmonic function as a possible remedy.

¹⁷ This issue is considered extensively by Murphy, who places a novel spin on some of Cohn’s ideas. See p. 89, where Murphy assesses triadic syntax post-1800 as applying the pre-1800 notion of tonic triadic proximity to triads in general.

¹⁸ Cohn references McCreless, Patrick, ‘An Evolutionary Perspective on Nineteenth-Century Semitonal Relations’, in William Kinderman and Harald Krebs, eds, The Second Practice of Nineteenth-Century Tonality (Lincoln: University of Nebraska Press, 1998): 87–113 Google Scholar.

¹⁹ See for example Jan, Stephen, ‘The Evolution of a Memeplex in Late Mozart: Replicated Structures in Pamina’s ‘Ach Ich fühl’s’, Journal of the Royal Musical Association 128/1 (2003): 30–70 CrossRefGoogle Scholar. The pre-adaptation of the triad, and the associated issues of ‘over-determination’, are a central focus for Murphy, who subjects them in his review to penetrating scrutiny; see particularly 85–6.

²⁰ I have in mind Foucault, Michel, The Order of Things: An Archaeology of the Human Sciences (London: Tavistock, 1974)Google Scholar and The Archaeology of Knowledge (London: Tavistock, 1972).

²¹ As in Dahlhaus, Carl, Foundations of Music History, trans. J. Bradford Robinson (Cambridge: Cambridge University Press, 1983): 149 CrossRefGoogle Scholar.

²² For a more extended consideration of the appropriateness of competing geometrical models, see Brower, Candace, ‘Paradoxes of Pitch Space’, Music Analysis 27/1 (2008): 51–106 CrossRefGoogle Scholar. For a defence of musical geometry, see Tymoczko, A Geometry of Music, 19–21.

²³ I think of Lerdahl, Fred, Tonal Pitch Space (New York and Oxford: Oxford University Press 2001)Google Scholar and Hepokoski, James and Darcy, Warren, Elements of Sonata Theory: Norms, Types, and Deformations in the Late-Eighteenth-Century Sonata (New York and Oxford: Oxford University Press 2006)CrossRefGoogle Scholar.

²⁴ See Spitzer, Michael, ‘The Metaphor of Musical Space’, Musicae Scientiae 7/1 (2003): 101–120 CrossRefGoogle Scholar.