1. A useful comparison may be made between Nono's re-ordering procedures, which I shall discuss here, and order-relations as explored by other composers: Milton Babbitt, "Set Structure as a Compositional Determinant," Journal of Music Theory 5 (1961): 72-94, "Twelve-Tone Rhythmic Structure and the Electronic Medium," Perspectives of New Music 1.1 (1962): 49-79, and "Since Schoenberg," Perspectives of New Music 12.1 (Fall-Winter 1973): 3-28; Philip N. Batstone, "Multiple Order Functions in Twelve-Tone Music," Parts 1, 2, Perspectives of New Music 10.2 (Spring-Summer 1972): 60-71 and Perspectives of New Music 11.1 (Fall-Winter 1972): 92-111; Dave Headlam, "The Derivation of Rows in Lulu," Perspectives of New Music 24.1 (Autumn-Winter 1985): 198-233; David Kowalski, "The Construction and Use of Self-Deriving Arrays," Perspectives of New Music 25.1-2 (1987): 286-361; David Lewin, "On Certain Techniques of Re-Ordering in Serial Music," Journal of Music Theory 10.2 (Winter 1966): 276-287 and "On Partial Ordering," Perspectives of New Music 14.2 and 15.1 (Spring-Summer/Fall-Winter 1976): 252-59; Andrew Mead, "Some Implications of the Pitch Class/Order Number Isomorphism Inherent in the Twelve-Tone System: Part One," Perspectives of New Music 26.2 (Summer 1988): 96-163; Robert Morris, "On the Generation of Multiple-Order-Function Twelve-Tone Rows," Journal of Music Theory 21.2 (Autumn 1977): 238-62 and "Review: John Rahn, Basic Atonal Theory," Music Theory Spectrum 4 (1982): 135-54; Daniel Starr, "Sets, Invariance, and Partitions," Journal of Music Theory 22 (1978): 1-42 and "Derivation and Polyphony," Perspectives of New Music 23.1 (Fall-Winter 1984): 180-257; Peter Westergaard, "Toward a Twelve-Tone Polyphony," Perspectives of New Music 4.2 (Spring-Summer 1966): 90-112.
2. Under Bruno Maderna's guidance, Nono and Maderna transcribed many polyphonic works and studied historical treatises. A list of intended transcriptions can be found among the holdings in the Archivio Luigi Nono in Venice, ALN M02.01.06/8-35. Notes on treatises can be found in ALN M02.01.03/1-49.
3. Nono's preoccupation with canonic symmetry is discussed in Erika Schaller, Klang und Zahl: Luigi Nono--Serielles Komponieren zwischen 1955 und 1959 (Ruhr-U. Bochum: Ph.D. dissertation, 1996). Canons and canonic devices populate Nono's choral works, as discussed in Jeannie Ma. Guerrero, "Tintoretto, Nono, and Expanses of Silence," paper presented at the Dublin International Conference on Music Analysis (University College Dublin, Ireland), 2005.
4. An extended discussion of multidimensional evolution in Nono's choral music may be found in Jeannie Ma. Guerrero, "Text-Setting Techniques in Luigi Nono's Choral Works (1956-1960)" (Harvard University: Ph.D. dissertation, 2003).
5. Jeannie Ma. Guerrero, "Multidimensional Counterpoint and Social Subversion in Luigi Nono's Choral Works," Theory & Practice 28 (2003), 1-26.
6. Luigi Nono, "Un autobiografica dell'autore raccontata da Enzo Restagno (1987)," in Angela Ida De Benedictis and Veniero Rizzardi, eds., Luigi Nono: Scritti e colloqui II (Lucca: Casa Ricordi, 2001), 489. All translations are the author's unless otherwise noted.
7. Luigi Nono introduces the term in his handwritten notes for Guai ai gelidi mostri (1983), in Angela Ida De Benedictis and Veniero Rizzardi, eds., Luigi Nono: Scritti e colloqui I (Lucca: Casa Ricordi, 2001), 491-492. The term appears a year later in "Verso Prometeo. Frammenti di diari," in Angela Ida De Benedictis and Veniero Rizzardi, eds., Luigi Nono: Scritti e colloqui I (Lucca: Casa Ricordi, 2001), 385-396.
8. Remarks about Sospeso's novel textures can be found in Luigi Nono, Interview by Hansjörg Pauli, in Jürg Stenzl, ed., Luigi Nono: Texte, Studien über seine Musik (Zürich: Atlantis Musikbuch-Verlag, 1975), 200: "Ich wollte eine horizontale melodische Konstruktion, die sämtliche Register ergreift; ein Schweben von Laut zu Laut, von Silbe zu Silbe: eine Linie, die manchmal aus der Abfolge von Einzel-Tönen oder Einzel-Tonhölen entsteht und manchmal sich verdickt zu Klängen." An English translation by Angela Davies appears in Christoph Flamm, Preface to Luigi Nono's Il canto sospeso (London: Eulenberg, 1995), ix: "I wanted an horizontal melodic construction encompassing all registers; floating from sound to sound, from syllable to syllable: a line which sometimes consists of a succession of individual tones or pitches, and sometimes thickens into chords."
9. For an examination of the various generative complexes used throughout Sospeso, see Kathryn Bailey, "'Work in Progress': Analysing Nono's Il canto sospeso." Music Analysis 11.2-3 (July-October 1992): 279-334.
10. Lettere di condonnati a morte della resistenza europea (Turin: Giulio Einaudi, 1954) was also reprinted in German as Und die Flamme soll euch nicht versengen (Z�rich: Steinberg-Verlag, 1955).
11. Another socially engaged composition by Schoenberg, the Ode to Napolean, provided the musical basis for Nono's first published work, Variazioni canoniche sulla serie dell'op. 41 di Arnold Schoenberg (1950). His first published choral work, the first of the three Epitaffios auf Federico Garc�a Lorca (1952-1954), sets poems by Federico Garc�a Lorca and Pablo Neruda that pertain to the Spanish Civil War (1936-1939). Like Survivor, the choir, soprano, and baritone soloists deliver texts as Sprechstimmen against a sparse texture in the orchestra. A traditional Spanish melody sung in unison concludes the third movement, also calling Survivor to mind. La Victoire de Guernica (1954) for mixed choir and orchestra also takes the Spanish Civil War as its subject, setting Paul Éluard's verses and Picasso's painting of the same title. While short textual phrases undergo polyphonic treatment that somewhat confuses their apprehension, longer textual phrases are delivered as Sprechstimmen, thus continuing the link to Schoenberg's sound-world. Like the first Epitaffio, as well as Survivor, the piece ends with the choir singing in unison.
12. Luigi Nono, Il canto sospeso (Edition: Ernst Eulenberg, 1995), 90-91.
13. The innovative choral textures of No. 9, in particular, persist in Nono's choral writing for the remainder of his career. Their precise production through serial mechanisms receives more extensive discussion in Guerrero, "Text-Setting Techniques."
14. Karlheinz Stockhausen, "Sprache und Musik II," Darmstädter Beiträge zur Neuen Musik 1 (1958). Reprinted in Texte zu eigenen Werken, zur Kunst Anderer, Aktuelles II (1952-1962) (Köln: DuMont Schauberg, 1964), 157-166.
15. Luigi Nono, "Text-Musik-Gesang (1960)," in Jürg Stenzl, ed., Luigi Nono: Texte, Studien über seine Musik (Z�rich: Atlantis Musikbuch-Verlag, 1975), 41-60.
16. Nono's commentary focuses predominantly on the work's textual innovations, as in "Text-Musik-Gesang," and letters between Nono and Eigel Kruttge revolve around the performance implications for the choir. Excerpts from these letters may be found in Flamm, v-vii.
17. Massimo Mila, "Nonos Weg-Zum 'Canto Sospeso'," in Jürg Stenzl, ed., Luigi Nono: Texte, Studien über seine Musik (Z�rich: Atlantis Musikbuch-Verlag, 1975), 380-393.
18. Bailey, "Work in Progress."
19. I diverge from Bailey and others in matters of terminology. The twelve-tone series that generates Sospeso belongs to the class of "All-Interval Series," as discussed by Daniel Starr and Robert Morris in "The Structure of All-Interval Series," Journal of Music Theory 18.2 (Autumn 1974): 364-389. I shall refer to this series in my discussion by a more specific name, "All-Interval Wedge," to distinguish it from other all-interval series. Nono uses an AIW beginning on pc-0 in the choral works La terra e la compagna (1957) and Cori di Didone (1958), as well as two instrumental works, Incontri (1955) and Canti per 13 (1955). The AIW appears throughout Sospeso, beginning on pc-9 instead.
20. Ivanka Stoianova, "Testo-musica-senso. 'Il canto sospeso,'" in Restagno, Enzo, ed. Nono (Turin: Edizioni di Torino, 1987), 126-142. Stoianova argues that Nono's processes are akin to the 'exploration of intertextual space' discussed in works of literary criticism, notably by Mikhail Bakhtin, Ferdinand de Saussure, and Claude Levi-Strauss.
21. Wolfgang Motz, Konstruktion und Ausdruck: Analytische Betrachtungen zu �Il Canto sospeso� (1955/56) von Luigi Nono, Saarbrücken: PFAU-Verlag, 1996; and "Konstruktive Strenge und kompositorische Freiheit im ersten Satz des Canto Sospeso," in Gianmario Borio, Giovanni Morelli, and Veniero Rizzardi, eds., La nuova ricerca sull'opera di Luigi Nono (Florence: Olschki, 1999), 53-66.
22. Schaller, Klang und Zahl.
23. Nicolaus A. Huber, "Luigi Nono: Il canto sospeso VIa, b," Musik Konzept 20 (1981): 58-79.
24. Gianmario Borio, "Sull'interazione fra lo studio degli schizzi e l'analisi dell'opera," in Gianmario Borio, Giovanni Morelli, and Veniero Rizzardi, eds., La nuova ricerca sull'opera di Luigi Nono (Florence: Leo Olschki, 1999), 1-21.
25. Nono, "Un autobiografia," 507. The Schoenberg reference cites a letter dated 27 July 1932 regarding String Quartet No. 3, Op. 30, and printed in Italian in Arnold Schönberg, Letture, ed. E. Stein and L. Rognoni, transl. L. M. Rubino (Florence: La Nuova Italia, 1969), 179-180.
26. Nono, Interview by Hansjörg Pauli, 23-24.
27. Elizabeth West Marvin employs the term and concept of "beat-division" in her examination of duration contours in "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varèse." Music Theory Spectrum 13.1 (Spring 1991): 61-78.
28. The duration-factor 5 leads a particular series based on the integer succession 1 to 12, as I shall show later.
29. The timpani's pc-9 at m. 3 has a notated duration-factor of 9 rather than 10. I shall discuss this note more extensively later.
30. Again, duration factors are limited to the integers 1 through 12, so numbers larger than 12 arise secondarily.
31. The note appears in an earlier draft (ALN 14.13) and does not undergo further alteration in the published version. A similar truncation occurs in the bass clarinet at m. 497 (No. 8), where a serial duration-factor of 6 is truncated to 5 (a difference of 1/8 a quarter note) within a decrescendo. Another timpani note is truncated at m. 498 from duration-factor 7 to 6 (a difference of 1/10 a quarter note), this time serving a crescendo--here, concern for the timpani's resonance still seems to motivate the shortening. There are in total thirteen alterations throughout No. 8 of which I am currently aware, without ruling out others that may be in the remaining movements.
32. "Cosmetic" permutation tables in dynamics, as she calls them, are not created by any self-replicating process that generates tables in other dimensions. As a smoking gun, the dynamics table for No. 9 appears to have been filled in one column at a time (thus without regard for the base series), whereas other tables are filled one rank at a time through the recursive re-ordering of the base series.
33. According to unpublished work by Borio and Veniero Rizzardi, cited by Borio in "Sull'interazione," Nono and Bruno Maderna developed the technique in or around 1951. Nono first used it in his Composizione per orchestra and Maderna in his Improvvisazione No. 1. Borio also illustrates the technique's implementation for the fourth movement of Il canto sospeso, but without the level of detail I provide here. Christoph Neidh�fer also provides a high level of detail concerning the technique's use in several of Maderna's compositions from the 1950s in "Bruno Maderna's Serial Arrays," paper presented at the 2004 meetings of the Music Theory Society of New York State (Rochester, NY) and the Society for Music Theory (Seattle, WA).
34. I shall use italicized numbers to denote ordinal positions in the hope of minimizing confusion between ordinal positions and values contained by these positions. I enumerate ordinal positions beginning with 1, as opposed to 0, to ease comparison with Bailey's "permutation tables," which employ the numbering from 1 to 12. More importantly, the enumeration at 1 matches that used by David Lewin in his explication of the multiplicative operation Mn in "On Certain Techniques of Re-Ordering in Serial Music," Journal of Music Theory 10.2 (Winter 1966), 279. The rearrangement described here corresponds to Lewin's M7. Dave Headlam shows Berg's use of essentially the same rearrangement to derive the Acrobat's row in "The Derivation of Rows in Lulu," 214; and The Music of Alban Berg (New Haven: Yale University Press, 1996), 306. Headlam categorizes the rearrangement as an "extended rop-cycle," where a "rop" is a "referential order position/pc pair" ("Derivation of Rows," 203).
35. The particular permutations shown by Table 2 correspond to the Mallalieu complex described by Andrew Mead in "Some Implications of the Pitch-Class/Order-Number Isomorphism Inherent in the Twelve-Tone System Part Two: The Mallalieu Complex: Its Extensions and Related Rows," Perspectives of New Music 27.1 (Winter 1989): 180-233. The shuffled ordinal positions are shown below:
The reordered ordinal positions in the Mallalieu Complex arise by applying the twelve Mn ordinal-position operations (where n runs from 1 to 12) to the Mallalieu Row (0 1 4 2 9 5 e 3 8 t 7 6) and then ordering the resulting pitch-class transpositions by ascending index number (a unique property of the Row is that each M operation produces its own pitch-class transposition of the Row). The Mallalieu Row itself has received attention from Lewin, "Certain Techniques of Re-Ordering," as well as Starr and Morris, "The Structure of All-Interval Series." Interestingly, the same results can be achieved by recursively applying M7 to each successively transformed row, which seems to describe Nono's procedures more precisely. Without recourse to a proper mathematics arsenal, I shall attempt to explain the presence of the invariant columns as follows:
Within the Mallalieu complex, columns contain rotations of an invariant numeric string, 2x mod 13, where x=0 to 11; horizontal ranks produce the strings 2xy mod 13, where y=1 to 12. That is, there is a 2-cycle in the second rank, a 4-cycle in the third, and so on until the last rank, which features a 2048-cycle. The vertical invariance arises because of the particular, and unique, way in which 2x and its multiples map out in mod 13, a prime-number modulus. While the prime-number modulus ensures that each of the twelve arithmetic series (i.e., the number cycles 1 to 12) yield all twelve factors without replication (incidentally, the prime-number modulus 7 assures the same for diatonic sets; John Clough makes this observation in "Diatonic Interval Sets and Transformational Structures," Perspectives of New Music 18.1-2 (Autumn 1979-Summer 1980), 473.), it does not guarantee the same results for all exponential series. In fact, only four exponential series (2x, 6x, 7x, and 11x) produce a twelve-member invariant string mod 13; it is the twelve-member string that guarantees the presence of invariant columns throughout the complex. The remaining exponential series produce shorter invariant strings, and thus when multiplied generate two or more distinct, invariant strings that are doled out into separate columns. The mappings change, as might be expected, with different prime-number moduli. I offer one last notable observation: 7x (which is generated by recursive M2) yields the retrograde of the vertical string produced by 2x (which is generated by recursive M7).
While the complex shown above stems from M7, there are rearrangement schemes used elsewhere in Sospeso that cannot be produced through a simple multiplicative operation, as happens for No. 8 (to be discussed later). Bailey's Figure 1b ("Work in Progress," 282) reflects the order-position transformations for No. 8:
Two distinct operations recursively produce the collections above. Lewin's M2 operates on ordinal positions 1-6, and (order-inversion) I12(M2) operates on ordinal positions 7-12. The resulting invariant columns resemble those of the Mallalieu Complex except for two columns that bear the values 5 and 10 exclusively. Also, the 2x series reverses direction and breaks into fragments, as do the cycles in the horizontal ranks. A heterogeneously-derived complex such as this is not remarked upon in the Mallalieu literature and, admittedly, merits further investigation and formalization.
36. The diagram replicates information from Figura 1 in Borio, "Sull'interazione," 16.
37. While Borio calls ranks in the arrangements by the term "Tondistanzreihen," I prefer Motz's more generic term "Positionreihe" and call the arrangements "Position-Grids" to reflect the interlocking between ranks and columns.
38. These time-points fall at Column 12 of Position-Grid IV and throughout Position-Grid V (mm. 260-263).
39. The orchestration omits the piccolo, oboe, bassoon, timpani, Glockenspiel, celesta, and harp.
40. The placement of canonic devices within the choral works exhibits symbolic consistency, as discussed in Guerrero, "Tintoretto, Nono,"
41. Motz, "Konstruktive Strenge."
42. Again, noting the one exception in the timpani, whose duration-factor is 9 rather than 10.
43. The boxes correspond to the first three grids of the system. Further, Box 8 represents an overlap between the end of the first duration-factor series and the beginning of the second (i.e., 10 8 6, etc.).
44. The elements form the basis for the content of the entire cantata, so their presence might have some affinity with a direct statement of the AIW or an unimpeded application of duration factors.
45. The table coincides with the upper-left block of Figure 27 in Bailey, "Work in Progress," 318. Bailey does not include the tenth collection. The permutation of ordinal positions is discussed above in Note 35.
46. As discussed in Note 35, two distinct order-number multiplications shuffle ordinal positions in No. 8. M2 operates on ordinal-positions 1 through 6, while I12(M2) operates on ordinal-positions 7 through 12.
47. The table adds beat-division information to Motz's Beispiel 49 ("Konstrucktion und Ausdruck," 126) and Bailey's Figure 27 ("Work in Progress," 318), which show only duration factors.
48. The overrides account for the high degree of distortion in Bailey's Figure 27 towards the end of No. 8.
49. A similar multidimensional silence, in the guise of a retrograde canon, concludes Nono's 1960 choral piece Sarà dolce tacere which is discussed extensively in Guerrero, "Multidimensional counterpoint."
50. In Guerrero, "Tintoretto, Nono," canons are shown to be consistently associated with silence and death in the choral works. It would appear that No. 4, and perhaps earlier canonic structures, belonged to a larger current in Nono's compositional output.
51. Nono, Il canto sospeso, 89.
End of footnotes