¹ Cardew quoted from the Tractatus in correspondence in the spring of 1959: ‘“The world is everything that is the case” […] Read Wittgenstein and confuse yourself.’ See Cornelius Cardew, letter to Earle Brown dated ‘The last Sunday in April’ (26 April 1959), postmarked 27 April 1959. Earle Brown Collection, Paul Sacher Foundation, Basle. The first mention of Wittgenstein in Cardew’s journal appears in an entry dated 24 October 1959. See Tilbury, John, Cornelius Cardew (1936–1981): A Life Unfinished (Matching Tye: Copula, 2008)Google Scholar, 78, 118. For ‘deep impression’, see Cardew’s journal, entry dated 18 November 1966, quoted ibid., 227–8.

² Cardew, Cornelius, Stockhausen Serves Imperialism, and Other Articles: With Commentary and Notes (London: Latimer, 1974), 122Google Scholar.

³ For ‘impeccable’, see Smalley, Roger, ‘A Beautiful Score’, Musical Times, 109 (1968), 462 CrossRefGoogle Scholar, and Nyman, Michael, Experimental Music: Cage and Beyond, 2nd edn (Cambridge: Cambridge University Press, 1999), 117 Google Scholar.

⁴ Cardew’s journal, entry dated 18 November 1966, quoted in Tilbury, Cornelius Cardew, 227–8.

⁵ See Dennis, Brian, ‘Cardew’s “Treatise” (Mainly the Visual Aspects)’, Tempo, new ser., 177 (1991), 10–16 CrossRefGoogle Scholar (pp. 11, 15); Anderson, Virginia, ‘“Well, It’s a Vertebrate …”: Performer Choice in Cardew’s Treatise ’, Journal of Musicological Research, 25 (2006), 291–317 (p. 295)CrossRefGoogle Scholar; and Tilbury, Cornelius Cardew, 236–7.

⁶ See Parsons, Michael, ‘Introduction’, in Cornelius Cardew: A Reader, ed. Prévost, Edwin (Matching Tye: Copula, 2006), ix–xviii Google Scholar (p. xi), and Tilbury, Cornelius Cardew, 238–9.

⁷ Wittgenstein, Ludwig, ‘Logisch-philosophische Abhandlung’, Annalen der Naturphilosophie, 19 (1921), 185–262 Google Scholar; Tractatus logico-philosophicus, trans. Charles (C. K.) Ogden (London: Kegan Paul, Trench & Trübner, 1922).

⁸ For possible links with Wittgenstein’s later philosophy, see Tilbury, Cornelius Cardew, 240–4, 248, 279 nn. 34 and 37. For links with the overall course of his philosophy, see ibid., 241; Dennis, ‘Cardew’s “Treatise”’, 16; and Parsons, ‘Introduction’, in Cornelius Cardew, ed. Prévost, xi.

⁹ The distance between Wittgenstein’s later ideas and his earlier opinions is disputed. For a topography of the debate, see Stern, David G., ‘How Many Wittgensteins?’, Publications of the Austrian Ludwig Wittgenstein Society, new ser., 2 (2006), 205–29Google Scholar.

¹⁰ Cardew, Cornelius, Treatise Handbook (London: Edition Peters, 1971)Google Scholar, i.

¹¹ Ibid.

¹² Cornelius Cardew, letter to Earle Brown, 1 February 1963. Earle Brown Collection, Paul Sacher Foundation, Basle.

¹³ Cardew, Cornelius, ‘Notation – Interpretation, etc.’, Tempo, new ser., 58 (1961), 21–33 CrossRefGoogle Scholar (p. 25); Treatise Handbook, x. In the case of Treatise, he also envisaged non-musical interpretations (ibid., xi).

¹⁴ Cardew, Treatise Handbook, x.

¹⁵ Tilbury, Cornelius Cardew, 62, 72, 88–9; Tony Harris, The Legacy of Cornelius Cardew (Farnham: Ashgate, 2013), 30.

¹⁶ Iddon, Martin, New Music at Darmstadt: Nono, Stockhausen, Cage, and Boulez (Cambridge: Cambridge University Press, 2013)CrossRefGoogle Scholar, 286, 295.

¹⁷ Elsewhere, I have distinguished between a substantive and a procedural concept of indeterminacy. See Cline, David, ‘Two Concepts of Indeterminacy in Music’, Musical Quarterly, 102 (2019), 82–110 CrossRefGoogle Scholar. The substantive concept is used throughout this article.

¹⁸ Tilbury, Cornelius Cardew, 62, 88. Only a year later, Cage would describe Cardew as among those ‘who imbibe American actions’. See John Cage, letter to Peter Yates, 28 December 1959. The Selected Letters of John Cage, ed. Kuhn, Laura (Middletown, CT: Wesleyan University Press, 2016), 210–13Google Scholar (p. 211). In a letter to Cage written soon after completing Treatise, Cardew stated, ‘Without your work it [Treatise] w[oul]d never have been written.’ See Cornelius Cardew, letter to John Cage, 13 March 1967. John Cage Collection, Northwestern University Music Library, Evanston, IL.

¹⁹ Cage, John, ‘Experimental Music’ (1958), Silence: Lectures and Writings, 5th edn (London: Marion Boyars, 1999), 7–12 Google Scholar (p. 10). See also Cage, ‘History of Experimental Music in the United States’ (1959), ibid., 67–75 (p. 71), and ‘Composition as Process: 1. Changes’ (1961), ibid., 18–34 (pp. 22–3).

²⁰ Brown attributed his interest in the performer’s point of view to various factors, including his early experiences as a jazz trumpeter. See Brown, Earle, ‘On December 1952 ’, American Music, 26 (2008), 1–12 Google Scholar (p. 6).

²¹ Cage’s prefatory note with the notation labelled ‘AR’ in ‘Solo for Piano’ from Concert for Piano and Orchestra (1958) reads: ‘Play in any way that is suggested by the drawing.’

²² Cardew, ‘Notation – Interpretation, etc.’, 23, 25, and ‘Piano (Three Hands) – Morton Feldman’, Accent, 4 (1962), 31–5.

²³ Cardew worked as Stockhausen’s assistant in 1958–60.

²⁴ The works discussed by Stockhausen were Sylvano Bussotti’s Pièces de chair II, Cage’s Concert for Piano and Orchestra, Cardew’s Two Books of Study for Pianists (1958) and February Piece I (1959), Mauricio Kagel’s Transición II (1959) and Stockhausen’s own Zyklus (1959).

²⁵ Cardew, ‘Notation – Interpretation, etc.’, 23. These comments appear in a note dated October 1959.

²⁶ See Gutkin, David, ‘Drastic or Plastic: Threads from Karlheinz Stockhausen’s “Musik und Graphik”, 1959’, Perspectives of New Music, 50 (2012), 255–305 Google Scholar (pp. 267–73), and Iddon, New Music at Darmstadt, 242–4.

²⁷ Cardew, ‘Notation – Interpretation, etc.’, 21–3, 30.

²⁸ Ibid., 23 (emphasis original).

²⁹ An annotation on p. 4 implies that the middle line of the lowest stave is a B, whereas the stave itself is preceded by an alto clef.

³⁰ Cardew, ‘Notation – Interpretation, etc.’, 23.

³¹ Cardew, Treatise Handbook, x. Cardew completed a one-year, part-time course in typography at the London College of Printing in the summer of 1962 and commenced the first of several stints at Aldus Books in October of that year. See Tilbury, Cornelius Cardew, 129–30.

³² Cardew, ‘Notation – Interpretation, etc.’, 23, 29–30.

³³ Ibid., 29.

³⁴ Ibid., 30.

³⁵ ‘A’ and ‘A’ are two tokens of the same letter type. See Peirce, Charles (C. S.), ‘Prolegomena to an Apology for Pragmaticism’, Monist, 16 (1906), 492–546 CrossRefGoogle Scholar (pp. 505–6).

³⁶ Cardew, ‘Notation – Interpretation, etc.’, 23 (emphasis original).

³⁷ Cardew’s criticisms of Five Piano Pieces for David Tudor assume that Bussotti intended to produce a serviceable notation, but one commentator has suggested that his aim was ‘fundamentally magical and egocentric’. According to this view, Bussotti used indeterminacy in an attempt to ‘avoid the mediation of “composition”, to achieve a direct communication of the composer’s being to and through the interpreter’. See Ulman, Erik, ‘The Music of Sylvano Bussotti’, Perspectives of New Music, 34 (1996), 186–201 CrossRefGoogle Scholar (p. 192; emphasis added).

³⁸ Cardew, Treatise Handbook, i, v.

³⁹ Anthony Quinton, ‘Contemporary British Philosophy’, A Critical History of Western Philosophy, ed. O’Connor, Daniel (D. J.) (London: Collier-Macmillan, 1964), 530–56CrossRefGoogle Scholar (p. 540).

⁴⁰ Cardew, ‘Notation – Interpretation, etc.’, 32–3 (emphasis original).

⁴¹ Cardew, Treatise Handbook, i.

⁴² Cornelius Cardew, letter to John Cage dated ‘3 1 67’ (probably 1 March 1967). John Cage Collection, Northwestern University Music Library, Evanston, IL.

⁴³ The reasons why this idea came to nothing are unclear. Cage agreed to provide an introduction, but insisted that he would need to see the finished score in order to do so (John Cage, letter to Cornelius Cardew, 8 March 1967. John Cage Collection, Northwestern University Music Library, Evanston, IL.) Cardew sent a copy of the score, but, in an accompanying note, added: ‘let’s forget the idea of an introduction’ (Cornelius Cardew, letter to John Cage, 13 March 1967. John Cage Collection, Northwestern University Music Library, Evanston, IL.)

⁴⁴ Cornelius Cardew, letter to Hans Swarsenski (Hinrichsen Edition), 29 October 1967. Peters Edition, London. I am grateful to Andrew Kemp for providing me with a transcript.

⁴⁵ Cardew, Treatise Handbook, iv (emphasis original).

⁴⁶ Ibid., iii.

⁴⁷ Ludwig Wittgenstein, Tractatus logico-philosophicus, trans. David (D. F.) Pears and Brian (B. F.) McGuinness (London: Routledge & Kegan Paul, 1961), 2.12. Unless otherwise indicated, all subsequent references to the Tractatus are to this translation and cite Wittgenstein’s paragraph numbers.

⁴⁸ Bogen, James, Wittgenstein’s Philosophy of Language (London: Routledge & Kegan Paul, 1972), 15 Google Scholar.

⁴⁹ Wittgenstein, Tractatus logico-philosophicus, 2.1514.

⁵⁰ Ibid., 2.15.

⁵¹ Ibid., 2.221.

⁵² Frege, Gottlob, ‘On Sense and Reference’ (1892), Translations from the Philosophical Writings of Gottlob Frege, ed. Geach, Peter T. and Black, Max (Oxford: Basil Blackwell, 1970), 56–78 Google Scholar (p. 57).

⁵³ Wittgenstein, Tractatus logico-philosophicus, 2.21–2.222.

⁵⁴ Ibid., 2.

⁵⁵ Ibid., 1.1–1.2. The debate about the meaning of Wittgenstein’s terms Tatsache (‘fact’), Sachverhalt (initially translated as ‘atomic fact’, subsequently as ‘state of affairs’) and Sachlage (initially translated as ‘state of affairs’, subsequently as ‘situation’) commenced before the Tractatus was first published and continues to this day. For the earlier translations, see Wittgenstein, Tractatus logico-philosophicus, trans. Ogden (see above, n. 7). My presentation reflects the later translations by Pears and McGuinness (see above, n. 47), which portray Sachverhalt and Sachlage as possible combinations of objects. For the history of the dispute, see Plourde, Jimmy, ‘States of Affairs, Facts and Situations in Wittgenstein’s Tractatus ’, Philosophia, 44 (2016), 181–203 CrossRefGoogle Scholar.

⁵⁶ Wittgenstein, Tractatus logico-philosophicus, 2.14–2.141.

⁵⁷ Bogen, Wittgenstein’s Philosophy of Language, 18–19; Dummett, Michael, ‘Frege and Wittgenstein’, Perspectives on the Philosophy of Wittgenstein, ed. Block, Irving (Oxford: Basil Blackwell, 1981), 31–42 (pp. 37–8)Google Scholar.

⁵⁸ See, for example, Shwayder, David (D. S.), ‘Critical Notice’ of Erik Stenius, Wittgenstein’s ‘Tractatus’: A Critical Exposition of its Main Lines of Thought (Oxford: Basil Blackwell, 1960)Google Scholar, Mind, 72 (1963), 275–88 (p. 281), and Bogen, Wittgenstein’s Philosophy of Language, 18.

⁵⁹ Wittgenstein, Tractatus logico-philosophicus, 2.161.

⁶⁰ Ibid., 2.15.

⁶¹ Wittgenstein, Ludwig, Notebooks, 1914–1916, ed. von Wright, Georg (G. H.) and Anscombe, Elizabeth (G. E. M.), trans. Anscombe, 2nd edn (Oxford: Basil Blackwell, 1979)Google Scholar, 7e. The Tractatus was compiled from a selective reworking of material that Wittgenstein extracted from several notebooks. See von Wright, ‘Historical Introduction: The Origin of Wittgenstein’s Tractatus’, in Wittgenstein, Prototractatus: An Early Version of ‘Tractatus logico-philosophicus’, ed. Brian (B. F.) McGuinness, Tauno Nyberg and von Wright, trans. David (D. F.) Pears and McGuinness (London: Routledge & Kegan Paul, 1971), 1–34 (pp. 3–7). The importance of the three notebooks that survive for an understanding of the Tractatus is generally recognized. See, for example, von Wright, ‘Ludwig Wittgenstein: A Biographical Sketch’, Philosophical Review, 64 (1955), 527–45 (p. 534), and Stern, David G., Wittgenstein on Mind and Language (New York: Oxford University Press, 1995)CrossRefGoogle Scholar, 6. For a dissenting voice, see Shwayder, David (D. S.), ‘Gegenstände and Other Matters: Observations Occasioned by a New Commentary on the Tractatus ’, Inquiry, 7 (1964), 387–413 CrossRefGoogle Scholar (pp. 412–13). The notebooks were first published in 1961 – that is, two years before Cardew began work on Treatise.

⁶² Wittgenstein, Tractatus logico-philosophicus, 2.15.

⁶³ Ibid., 2.151.

⁶⁴ Ibid., 2.171.

⁶⁵ Wittgenstein’s explanations are often unclear and occasionally inconsistent, as Frank Ramsey was among the first to point out in an otherwise deeply appreciative review published in 1923. See his ‘Critical Notice’ of Ludwig Wittgenstein, Tractatus logico-philosophicus, trans. Ogden (above, n. 7), Mind, 32 (1923), 465–78 (pp. 467–73, 476).

⁶⁶ Frascolla, Pasquale, Understanding Wittgenstein’s Tractatus (London: Routledge, 2007), 8–45 CrossRefGoogle Scholar.

⁶⁷ See above, note 60.

⁶⁸ Griffin, James, Wittgenstein’s Logical Atomism (Oxford: Clarendon Press, 1964), 88–95 Google Scholar; Malcolm, Norman, Nothing Is Hidden: Wittgenstein’s Criticism of his Early Thought (Oxford: Basil Blackwell, 1986)Google Scholar, 4, 63.

⁶⁹ Warnock, Geoffrey (G. J.), English Philosophy Since 1900 (London: Oxford University Press, 1958), 65 Google Scholar; Anscombe, Elizabeth (G. E. M.), An Introduction to Wittgenstein’s Tractatus (London: Hutchinson University Library, 1959), 67 Google Scholar.

⁷⁰ Stenius, Wittgenstein’s ‘Tractatus’ (see above, n. 58). In addition to Shwayder’s ‘Critical Notice’, two of the many reviews of this book were important essays in their own right: Bergmann, Gustav, ‘Stenius on the Tractatus: A Special Review’, Theoria, 29 (1963), 176–204 Google Scholar, and Jarvis, Judith, ‘Professor Stenius on the Tractatus ’, Journal of Philosophy, 58 (1969), 584–96CrossRefGoogle Scholar.

⁷¹ Wittgenstein, Tractatus logico-philosophicus, 2.18.

⁷² Stenius, Wittgenstein’s ‘Tractatus’, 91–8. Anscombe had previously used this term once in An Introduction to Wittgenstein’s Tractatus (p. 67).

⁷³ For the differences between Stenius’s conception of isomorphism and the mathematical concept, see Stenius, Wittgenstein’s ‘Tractatus’, 93.

⁷⁴ Ibid.

⁷⁵ Stenius barely mentioned this passage, but its importance for his view was highlighted by Pasquale Frascolla, who also cited Tractatus 3.21 (‘The configuration of objects in a situation corresponds to the configuration of simple signs in the propositional sign’). See Frascolla, Understanding Wittgenstein’s Tractatus, 32, 225 n. 19.

⁷⁶ Cardew, Treatise Handbook, xix.

⁷⁷ Wittgenstein, Tractatus logico-philosophicus, 4.011.

⁷⁸ Tilbury (Cornelius Cardew, 238–9) misreads it in this way. Both he (ibid., 239) and Michael Parsons (‘Introduction’, in Cornelius Cardew, ed. Prévost, xi) mistakenly posit a special connection between the picture theory and serialism, whereas Wittgenstein’s words indicate that he regarded all conventionally notated musical scores as pictures.

⁷⁹ See Cole, Hugo, Sounds and Signs: Aspects of Musical Notation (London: Oxford University Press, 1974), 9 Google Scholar.

⁸⁰ Cardew, Treatise Handbook, iv.

⁸¹ Ibid., iii.

⁸² Ibid.

⁸³ Some years after completing Treatise, Cardew would attack this idea, which he came to regard as a ‘disease’. See Cardew, Stockhausen Serves Imperialism, 79–86.

⁸⁴ Cardew, Treatise Handbook, iii.

⁸⁵ Cardew, Treatise Handbook, vi.

⁸⁶ Ibid.

⁸⁷ For criticism of other aspects of Stenius’s views, see Bogen, Wittgenstein’s Philosophy of Language, 20–8.

⁸⁸ Wittgenstein, Tractatus logico-philosophicus, 4.01, 4.021.

⁸⁹ Ibid., 3.141.

⁹⁰ Ibid., 2.131, 3.22, 4.0312. ‘Go proxy for’ is a standard alternative translation of vertreten, which is rendered by Pears and McGuinness as ‘be the representative of’.

⁹¹ I have slipped from talking of pictures as presenting situations to talking of propositions as asserting them. Wittgenstein seems to have made an analogous slip, although not all commentators agree. A different view, defended by Shwayder (‘Critical Notice’, 278–87) and Schwyzer, Hubert (‘Wittgenstein’s Picture Theory of Language’, Inquiry, 5 (1962), 46–64 CrossRefGoogle Scholar), is that Wittgenstein regarded all picturing as inherently assertive. This reading makes Wittgenstein’s views on non-linguistic picturing less credible, but Shwayder argued that these were only an adjunct to his main concern, which was with linguistic representation. Both authors conceded that several remarks in the Tractatus, including those about music, argue against this interpretation. Wittgenstein’s general discomfort with this aspect of the analogy between propositions and pictures is evident in his notebooks. See Wittgenstein, Notebooks, 1914–1916, ed. von Wright and Anscombe, 33e.

⁹² Wittgenstein, Tractatus logico-philosophicus, 3.1432 (emphasis original).

⁹³ Copi, Irving M., ‘Objects, Properties, and Relations in the “Tractatus”’, Mind, 67 (1958), 145–65 (pp. 152–60)CrossRefGoogle Scholar; Ishiguro, Hidé, ‘Subjects, Predicates, Isomorphic Representation, and Language Games’, Essays in Honour of Jaakko Hintikka, ed. Saarinen, Esa, Hilpinen, Risto, Niiniluoto, Ilkka and Provence Hintikka, Merrill B. (Dordrecht: D. Reidel, 1979), 351–64 (pp. 351–61)CrossRefGoogle Scholar.

⁹⁴ Bertrand Russell, ‘Introduction’, in Wittgenstein, Tractatus logico-philosophicus, trans. Ogden, 7–23 (p. 7), and in Wittgenstein, Tractatus logico-philosophicus, trans. Pears and McGuinness, ix–xxii (p. ix). Ramsey disputed Russell’s views on this point in his ‘Critical Notice’ of the Tractatus (see above, n. 65), p. 465. For compelling evidence that the theory was meant to apply to sentences of ordinary language, see Wittgenstein, Tractatus logico-philosophicus, 4.011, 5.5563.

⁹⁵ Bertrand Russell, ‘On Denoting’, Mind, 14 (1905), 479–93; Wittgenstein, Tractatus logico-philosophicus, 4.0031.

⁹⁶ Wittgenstein, Tractatus logico-philosophicus, 4.002.

⁹⁷ For ‘completely analysed’, see ibid., 3.201.

⁹⁸ Ibid., 4.21.

⁹⁹ Ibid., 5.

¹⁰⁰ For Wittgenstein’s treatment of the former, see ibid., 5.52–5.5262. For his views on the latter, see ibid., 5.54–5.542.

¹⁰¹ Ibid., 3.01. This is the line taken in Black, Max, A Companion to Wittgenstein’s ‘Tractatus’ (Cambridge: Cambridge University Press, 1964), 97 Google Scholar, and in Hintikka, Merrill B. and Hintikka, Jaakko, Investigating Wittgenstein (Oxford: Basil Blackwell, 1986), 102 Google Scholar. For a dissenting voice, see Copi, ‘Objects, Properties, and Relations’, 149.

¹⁰² Wittgenstein, Tractatus logico-philosophicus, 3.202. For ‘simple’ as unanalysable, not physically unstructured, see ibid., 3.26–3.261, and Bogen, Wittgenstein’s Philosophy of Language, 56–7.

¹⁰³ For this argument, see Wittgenstein, Tractatus logico-philosophicus, 2.02–2.0212. For another argument for the existence of simples, see ibid., 3.23.

¹⁰⁴ Ibid., 2.027–2.0271.

¹⁰⁵ Malcolm, Norman, Ludwig Wittgenstein: A Memoir, rev. edn (London: Oxford University Press, 1966), 86 Google Scholar. See also Anthony Kenny, The Legacy of Wittgenstein (Oxford: Basil Blackwell, 1984), 16.

¹⁰⁶ Ramsey is usually credited as the original source of this interpretation. See Frank Ramsey, ‘Universals’, Mind, 34 (1925), 401–17 (pp. 408, 416). Supporters of this view disagree about which particulars he had in mind. For objects as ‘bare’ particulars, see Copi, ‘Objects, Properties, and Relations’, 160–5. For objects as ‘material points’, see Griffin, Wittgenstein’s Logical Atomism, 152–3.

¹⁰⁷ Those who have challenged the venerable tradition include Stenius, Wittgenstein’s ‘Tractatus’, 63; Peter (P. M. S.) Hacker, Insight and Illusion: Themes in the Philosophy of Wittgenstein, rev. edn (Oxford: Clarendon Press, 1986), 70; Hintikka and Hintikka, Investigating Wittgenstein, 30–44; and Frascolla, Understanding Wittgenstein’s Tractatus, 71–84.

¹⁰⁸ Wittgenstein, Tractatus logico-philosophicus, 2.023, 2.0271; Shwayder, ‘Critical Notice’, 276.

¹⁰⁹ Wittgenstein, Tractatus logico-philosophicus, 4.22.

¹¹⁰ Ibid., 4.21, 2.03. Both formulations imply that an elementary proposition contains at least two names, but this conclusion is disputed. For a defence of the contrary view and the significance of the issue, see Sellars, Wilfrid, ‘Naming and Saying’, Philosophy of Science, 29 (1962), 7–26 CrossRefGoogle Scholar (pp. 12–14, 17–22).

¹¹¹ Wittgenstein, Tractatus logico-philosophicus, 4.021.

¹¹² Ibid., 4.024.

¹¹³ Ibid., 2.061–2.062, 5.134, 6.3751.

¹¹⁴ Ibid., 5.122.

¹¹⁵ Wittgenstein, Ludwig, ‘Some Remarks on Logical Form’, Proceedings of the Aristotelian Society, supp. vol. 9 (1929), 162–71CrossRefGoogle Scholar; Waismann, Friedrich, ‘Shorthand Transcript of Wittgenstein’s Talks and Conversation between December 1929 and September 1931’, in Wittgenstein, Philosophical Remarks, ed. Rhees, Rush, trans. Hargreaves, Raymond and White, Roger (Oxford: Basil Blackwell, 1975), 315–46Google Scholar (p. 317).

¹¹⁶ Carl Ginet argued that Wittgenstein’s insistence on the mutual independence of propositions was not only inconsistent with other aspects of the picture theory, but also inessential to it. See Ginet, ‘An Incoherence in the Tractatus’, Canadian Journal of Philosophy, 3 (1973), 143–51.

¹¹⁷ Wittgenstein, Tractatus logico-philosophicus, 4.016, 4.02–4.021, 4.024, 4.027, 4.03, 4.0311, 4.06. See also Peter (P. M. S.) Hacker, ‘The Rise and Fall of the Picture Theory’, Perspectives on the Philosophy of Wittgenstein, ed. Block, 85–109 (pp. 87–92).

¹¹⁸ Cardew, Treatise Handbook, iv (emphasis original).

¹¹⁹ For the various ways of reading a page, see Hall, John, ‘Reading (Il)legible Pages’, Performance Research, 9 (2004), 15–23 CrossRefGoogle Scholar.

¹²⁰ Tilbury suggested that the extended length of some groupings was a direct response to a perceived weakness of Autumn 60 and Octet ’61 for Jasper Johns, which was that these earlier works tended to invite a discontinuous, event-based interpretation. See Tilbury, Cornelius Cardew, 250.

¹²¹ For ‘life-line’, see Cardew, Treatise Handbook, x.

¹²² Ibid., ix.

¹²³ Ibid., xiii.

¹²⁴ Cardew’s description of the June 1964 performance states that ‘pp 57–60 and 75–9 were played as two separate sections’ (ibid., ix). His commentary on the September 1965 performance includes: ‘Page 74 was coordinated in detail as a piece on its own’ (ibid., x).

¹²⁵ Ibid., xiii.

¹²⁶ Quoted in Harris, The Legacy of Cornelius Cardew, 46.

¹²⁷ Cardew, ‘Notation – Interpretation, etc.’, 33.

¹²⁸ Cardew, Treatise Handbook, xi–xii.

¹²⁹ This key is not entirely mechanical, not only because each note ‘may be articulated in any way, not necessarily as a single held duration corresponding to the length of the line’, but also because it specifies two alternative intervallic progressions for each type of intersection (‘minor or major third down’, for example). See Cardew, Treatise Handbook, xi.

¹³⁰ Ibid., xii.

¹³¹ Ibid., xi.

¹³² Ibid.

¹³³ Ibid., xii.

¹³⁴ Cardew, Treatise Handbook, xii.

¹³⁵ Ibid., xiii.

¹³⁶ The score of Volo solo forms part of the Treatise Handbook, which also includes a set of explanatory notes for Bun no. 2. The score of Bun no. 2 is supplied with the handbook, but as a separate item.

¹³⁷ Tilbury, Cornelius Cardew, 257.

¹³⁸ The title Volo solo implies an unaccompanied soloist, but the explanatory notes state that it ‘may be performed by a number of virtuoso performers simultaneously, in which case the performers should begin each group together and finish independently’.

¹³⁹ This quotation is taken from the explanatory notes published with the score.

¹⁴⁰ Cardew, Treatise Handbook, i.

¹⁴¹ Notations, ed. Cage, John and Knowles, Alison (New York: Something Else Press, 1969)Google Scholar, unpaginated.

¹⁴² The disparity occurs in the first system on page 12. In the published edition, the penultimate event consists of a single note (a′′), but in the pre-publication version, the corresponding event is a dyad (c/a′′).

¹⁴³ It is initially unclear whether this assessment can be made consistent with Cardew’s comment – in a 1966 programme note included in the Treatise Handbook (p. x) – that ‘about 80 pages exist’.

¹⁴⁴ Tilbury, Cornelius Cardew, 263.

¹⁴⁵ Tilbury commissioned Volo solo in 1963 and gave its first performance in 1965 (ibid., 266). Consequently, he may have been referring to a pre-publication copy of the score when counting pitches.

¹⁴⁶ Ibid., 263.

¹⁴⁷ Cardew, Treatise Handbook, xiv (emphasis added).

¹⁴⁸ ‘Cornelius Cardew’ (interview), International Times, 2–15 February 1968, 8.

¹⁴⁹ In both sources, the first occurrence of e′′′ on page 1 is preceded by a redundant natural sign. Context suggests that e♭′′′ was intended, and I have counted it as such. Elsewhere, Cardew’s use of accidentals occasionally creates ambiguities, which I have sought to disentangle.

¹⁵⁰ Cardew, Treatise Handbook, i.

¹⁵¹ There are six differences between the sources. Five strings differ because an interruption present in the published edition is absent in the handwritten manuscript. A sixth string differs because the final repetition present in the handwritten manuscript is missing in the published edition.

¹⁵² Cardew, Treatise Handbook, i.

¹⁵³ This description appears in the explanatory notes that accompany the score.

¹⁵⁴ ‘Cornelius Cardew’ (interview), 8.

¹⁵⁵ Tilbury, Cornelius Cardew, 263.

¹⁵⁶ For aggregate completions, see Zatkalik, Miloš, ‘Reconsidering Teleological Aspects of Nontonal Music’, Music Theory and its Methods: Structures, Challenges, Directions, ed. Collins, Denis (Frankfurt am Main: PL Academic Research, 2013), 265–300 Google Scholar.

¹⁵⁷ For Cage’s use of rhythmic structures, see Cline, David, The Graph Music of Morton Feldman (Cambridge: Cambridge University Press, 2016), 175–82CrossRefGoogle Scholar. For ‘micro-macrocosmic’, see Cage, John, ‘A Composer’s Confessions’ (1992), John Cage: Writer – Previously Uncollected Pieces, ed. Kostelanetz, Richard (New York: Limelight Editions, 1993), 27–44 Google Scholar (p. 35).

¹⁵⁸ Wittgenstein, Tractatus logico-philosophicus, 4.014.

¹⁵⁹ Ibid., 4.0141.

¹⁶⁰ The imprecision that I have in mind includes vagueness (andante, mf, legato, etc.) and other forms of inexactness (free ornamentation, improvised cadenzas, figured bass, etc.).

¹⁶¹ Ibid., 3.323–3.325, 4.002.

¹⁶² For the notion of a ‘criterion of identity’, as the term is used here, as a constitutive principle applicable to a specific concept with countable instances, see Frege, Gottlob, The Foundations of Arithmetic, trans. Austin, John (J. L.), 2nd edn (Oxford: Basil Blackwell, 1953)Google Scholar, §§62–9.

¹⁶³ Wittgenstein, Tractatus logico-philosophicus, 4.04.

¹⁶⁴ Rosenberg, Jay F., ‘Wittgenstein’s Theory of Language as a Picture’, American Philosophical Quarterly, 5 (1968), 18–30 Google Scholar (p. 24).

¹⁶⁵ Hertz, Heinrich, The Principles of Mechanics Presented in a New Form, trans. Jones, Daniel (D. E.) and Walley, John (J. T.) (London: Macmillan & Co., 1899)Google Scholar.

¹⁶⁶ For discussion of its influence on the early Wittgenstein, see Stern, Wittgenstein on Mind and Language, 36–7. For a more extreme assessment, which maintains that the picture theory ‘comes almost in its entirety’ from Hertz, see Griffin, Wittgenstein’s Logical Atomism, 5, 99, 100–2, 108–10, 149–51.

¹⁶⁷ Hertz, The Principles of Mechanics, 175–7.

¹⁶⁸ Wittgenstein, Tractatus logico-philosophicus, 2.202–2.203.

¹⁶⁹ Ibid., 2.0233.

¹⁷⁰ Ibid., 2.0123.

¹⁷¹ Ibid., 2.18.

¹⁷² Ibid., 4.12.

¹⁷³ Wittgenstein, Tractatus logico-philosophicus, 4.12.

¹⁷⁴ Ibid., 4.121–4.1212.

¹⁷⁵ Stern, Wittgenstein on Mind and Language, 44, 49.

¹⁷⁶ Ibid., 47–50.

¹⁷⁷ Wittgenstein, Tractatus logico-philosophicus, 4.462.

¹⁷⁸ Ibid.

¹⁷⁹ Ibid., 6.12–6.1201.

¹⁸⁰ Ibid., 6.12, 6.124.

¹⁸¹ Ibid., 6.124.

¹⁸² Stern, Wittgenstein on Mind and Language, 69–70.

¹⁸³ Wittgenstein, Tractatus logico-philosophicus, 6.54.

¹⁸⁴ Cardew, Treatise Handbook, i.

¹⁸⁵ Russell, ‘Introduction’, in Wittgenstein, Tractatus logico-philosophicus, xx–xxii.

¹⁸⁶ Other notable examples of this standard reading include Anscombe, An Introduction to Wittgenstein’s Tractatus, 162, and Hacker, Insight and Illusion, 18–19.

¹⁸⁷ Diamond, Cora, ‘Throwing Away the Ladder’, Philosophy, 63 (1988), 5–27 CrossRefGoogle Scholar; Conant, James, ‘Must We Show What We Cannot Say’, The Senses of Stanley Cavell, ed. Fleming, Richard and Payne, Michael (Lewisburg: Bucknell University Press, 1989), 242–83Google Scholar. A watershed in the elaboration of their non-standard interpretation was the publication of The New Wittgenstein, ed. Alice Crary and Rupert Read (London: Routledge, 2000). For the ensuing debate, see Beyond the Tractatus Wars: The New Wittgenstein Debate, ed. Rupert Read and Matthew A. Lavery (London: Routledge, 2011).

¹⁸⁸ Cardew, Treatise Handbook, xi. Lou Gare, Eddie Prévost and Keith Rowe founded AMM in 1965. Lawrence Sheaff joined shortly thereafter. Cardew became a member in January 1966. See Tilbury, Cornelius Cardew, 284–5.

¹⁸⁹ Cardew, Treatise Handbook, x.

¹⁹⁰ For his interest in jazz, see Cornelius Cardew, ‘Rome Letter: Nuova consonanza’, Financial Times, 25 May 1965, 28; Tilbury, Cornelius Cardew, 283; and Olewnick, Brian, Keith Rowe: The Room Extended (Brooklyn, NY: Powerhouse Books, 2018)Google Scholar, 101.

¹⁹¹ Cardew, ‘Notation – Interpretation, etc.’, 26.

¹⁹² Cornelius Cardew, letter to Earle Brown, undated (late 1959). Earle Brown Collection, Paul Sacher Foundation, Basle.

¹⁹³ Cornelius Cardew, ‘Autumn ’60 – A Lecture Given to the Heretics Society, Cambridge’, transcribed from a notebook entry dated 5 September 1962, Cornelius Cardew: A Reader, ed. Prévost, 49–54 (p. 54).

¹⁹⁴ Years later, Cardew would attribute his previous resistance to improvisation to other factors. See Cardew, Treatise Handbook, xi, and Olewnick, Keith Rowe, 134.

¹⁹⁵ Cornelius Cardew, letter to Morton Feldman, 20 April 1961. Morton Feldman Collection, Paul Sacher Foundation, Basle (emphasis added). Feldman misunderstood Cardew’s reference to ‘versions’. See Cline, The Graph Music of Morton Feldman, 249.

¹⁹⁶ For Cage’s views on improvisation, see Feisst, Sabine M., ‘John Cage and Improvisation: An Unresolved Relationship’, Musical Improvisation: Art, Education, and Society, ed. Solis, Gabriel and Nettl, Bruno (Urbana, IL: University of Illinois Press, 2009), 38–51 Google Scholar, and Kim, Rebecca Y., ‘John Cage in Separate Togetherness with Jazz’, Contemporary Music Review, 31 (2012), 63–89 CrossRefGoogle Scholar.

¹⁹⁷ For this aspect of Tudor’s approach, see John Holzaepfel, ‘David Tudor and the Performance of American Experimental Music, 1950–1959’ (Ph.D. dissertation, City University of New York, 1994); Holzaepfel, ‘Painting by Numbers: The Intersections of Morton Feldman and David Tudor’, The New York Schools of Music and Visual Arts: John Cage, Morton Feldman, Edgard Varèse, Willem de Kooning, Jasper Johns, Robert Rauschenberg, ed. Steven Johnson (New York: Routledge, 2002), 159–72; Iddon, Martin, John Cage and David Tudor: Correspondence on Interpretation and Performance (Cambridge: Cambridge University Press, 2013)CrossRefGoogle Scholar; and Cline, The Graph Music of Morton Feldman, 264–91.

¹⁹⁸ Cardew, Treatise Handbook, xi (see also p. xix).

¹⁹⁹ Quoted in Harris, The Legacy of Cornelius Cardew, 46.

²⁰⁰ Cardew, Treatise Handbook, xix.

²⁰¹ Ibid., xvii–xviii.

²⁰² See, for example, Brendel, Alfred, ‘A Case for Live Recordings’ (1984), Music Sounded Out (London: Robson, 1990)Google Scholar, 200–7, and Fabian, Dorottya, ‘Classical Sound Recordings and Live Performances: Artistic and Analytical Perspectives’, Recorded Music: Philosophical and Critical Reflections, ed. Doğantan-Dack, Mine (London: Middlesex University Press, 2008), 232–60 (pp. 243–5)Google Scholar.

²⁰³ Cornelius Cardew, ‘Programme Notes’, The Macnaghten Concerts: 40th Anniversary Season 1971–72, issued in connection with the first performance of Paragraph 5 from Cardew’s The Great Learning (1970), 21 January 1972, unpaginated. For other conceptions of improvisation, see Nettl, Bruno, ‘Thoughts on Improvisation: A Comparative Approach’, Musical Quarterly, 60 (1974), 1–19 CrossRefGoogle Scholar.

²⁰⁴ The ‘complexity, as well as the difficulty of making any single logical code of “translation” from graphics to sound […], can thoroughly inhibit a musician such as myself from attempting to “realize” even a small section of the piece’ (Dennis, ‘Cardew’s “Treatise”’, 15).

²⁰⁵ Tilbury, Cornelius Cardew, 249.

²⁰⁶ Ibid., 238.

²⁰⁷ Cardew, Treatise Handbook, i.

²⁰⁸ Some years later, Cardew described Treatise as ‘an attempt to […] encourage improvisation amongst avant garde musicians’. See Cardew, Stockhausen Serves Imperialism, 79.

²⁰⁹ Cardew, Treatise Handbook, vii.

²¹⁰ Ibid., x.

²¹¹ Prévost, Edwin, ‘The Almost Unintended Consequences of Indeterminacy’, Minute Particulars: Meanings in Music-Making in the Wake of Hierarchical Realignments and Other Essays (Matching Tye: Copula, 2004), 32–9Google Scholar (p. 34).

²¹² Cardew, Treatise Handbook, xix (emphasis added). Philip Alperson argued that it makes ‘no sense’ to characterize an improvisation as an interpretation. See Alperson, ‘On Musical Improvisation’, Journal of Aesthetics and Art Criticism, 43 (1984), 17–29 (p. 26). For the contrary view, that every improvisation is also an interpretation, see Benson, Bruce Ellis, The Improvisation of Musical Dialogue: A Phenomenology of Music (Cambridge: Cambridge University Press, 2003), 143–6CrossRefGoogle Scholar.

²¹³ Cardew, Treatise Handbook, vii (emphasis original).

²¹⁴ Dennis, ‘Cardew’s “Treatise”’, 11, 15; Wittgenstein, Tractatus logico-philosophicus, 7.

²¹⁵ Wittgenstein, Tractatus logico-philosophicus, 6.42–6.421, 6.432, 6.51–6.521. For a list of topics on which silence is deemed necessary, see Hacker, Peter (P. M. S), ‘Was He Trying to Whistle It?’ (2000), Wittgenstein: Connections and Controversies (Oxford: Clarendon Press, 2001), 98–140 Google Scholar (pp. 98–100).

²¹⁶ Dennis, ‘Cardew’s “Treatise”’, 15.

²¹⁷ Ibid.

²¹⁸ Cardew, Treatise Handbook, iv. In a subsequent entry Cardew explained, ‘The score was drawn on a grid’ (ibid., v).

²¹⁹ Wittgenstein, Tractatus logico-philosophicus, 6.54.

²²⁰ For this view of the Tractatus as circular, see Friedlander, Eli, Signs of Sense: Reading Wittgenstein’s Tractatus (Cambridge, MA: Harvard University Press, 2001), 22–3Google Scholar.

²²¹ Dennis, ‘Cardew’s “Treatise”’, 11.

²²² Ludwig Wittgenstein, letter to Ludwig von Ficker, undated (1919). Quoted in von Wright, ‘Historical Introduction’, in Wittgenstein, Prototractatus, ed. McGuinness, Nyberg and von Wright, 15–16 (p. 16).

²²³ Wittgenstein, Tractatus logico-philosophicus, 6.522.

²²⁴ Cardew, Treatise Handbook, v.

²²⁵ Dennis, ‘Cardew’s “Treatise”’, 11.

²²⁶ Cardew, ‘Notation – Interpretation, etc.’, 23.

²²⁷ This principle is certainly false, as there is no inconsistency involved in imagining languages in which typography or handwriting is semantically significant.

²²⁸ ‘Only in the last two pages did the tension finally relax (at least in me). From amongst the various sustaining sounds music seemed to be issuing gently from some unspecifiable source’ (Cardew, Treatise Handbook, xiii). See also Andrew Porter, ‘Treatise’, Financial Times, 10 April 1967, 16.