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ABSTRACT
This study contributes to the investigation of formal similarities between music and language by examining commonalities between reversing operations found in language games and pitch patterns found in serial music. Our approach provides a new framework for analysing transformations in serial compositions, especially those involving pitch–class relationships between invariant hexachords. Although transformations in serial repertoire have been argued to be complex and difficult to aurally comprehend, we demonstrate that the structure of these transformations parallels language games commonly found across languages. Reference to language game patterning illustrates systematic relationships between hexachords which cannot always be related using conventional serial operators.
Acknowledgements
We wish to thank audiences at the Manchester Phonology Meeting, the Annual Conference of the Texas Society for Music Theory, and the College Music Society National Conference as well as Jonah Katz for helpful feedback and discussion. We would also like to thank Editor Alan Marsden and two anonymous reviewers for detailed comments on this work.
Notes
1 In the cases where discrete hexachords are not relatable through Tn/TnI, they are derived from the same series (or row) class. Throughout this paper, order numbers will be italicised, ordered sets will be written within angle brackets (<>), and unordered sets within curly brackets ({}). Pitch-classes will be written as pitch-class integers, numbered chromatically from 0 to 11 (i.e. C=0, C#=1, D=2, except that pitch classes 10 and 11 will be represented respectively by the letters t and e). In some cases A (order position 0–5) and B (order positions 6–e) are used to represent the order of discrete hexachords with a serial row. All of the terms used throughout this paper are from Straus (Citation2016).
2 The advantages of using the contextual transformations outlined in this paper, independent of language game comparisons, are discussed in Argentino (Citation2019).
3 For a succinct introduction to other methodologies for exploring serial music see Robert Morris (Citation2007, especially pp. 96–97). See Ernst Krenek (Citation1940) as well as James Houser (Citation1977) for information on serial rotation and its usage in Krenek’s compositions.
4 In particular, the notion of compositional versus perceptual organisation of serial music was also outlined in Lerdahl and Jackendoff’s GTTM (Citation1983, p. 299). Interestingly, the authors discuss the analysis of serial music from the perspective of smaller constituents (sets). For specific commentary on contemporary music, including serial organisation, see Lerdahl and Jackendoff (Citation1983, pp. 296–301).
5 This example is intended as an analogy to illustrate the role of length and memory in manipulation of strings of elements. Of course, the manipulations of text do not necessarily correspond precisely to the musical operations, nor do they correspond to language game operations discussed later which operate over speech sounds and not over written characters.
6 Although see Nevins and Endress (Citation2007) and Nevins (Citation2010) for an experimental study comparing the learning of language games to the learning of manipulations in pitch order. Their results will be discussed in section 4.
7 Conventional English spelling is used here to represent the sequence of speech sounds in the word ‘avocado’. English spelling in general is not a good representation of pronunciation as there are inconsistencies in spelling across words and multiple sounds may be represented with a single symbol or vice versa. Each speech sound in the word ‘avocado’ happens to be spelled with a single character, however. Examples from languages other than English will be shown as transcribed in the International Phonetic Alphabet or in the orthographies of the languages, depending on what is available in the source documents.
8 There are a number of prosodic constituents above the word, such as the intonational phrase and the utterance. Most language games operate at the level of the word, however, and we will not be concerned with constituents above the word here.
9 The International Phonetic Alphabet symbol ‘t∫’ represents the sound that occurs initially in English words like ‘chip’. Although this sound is written with two characters, both in phonetic transcription and in common English spelling, it is a single speech sound and not a sequence.
10 This game type is referred to as ‘transposition’ in the language game literature. We will use the term ‘edge-anchored movement’ to avoid confusion with the meaning of ‘transposition’ in music analysis.
11 The source of the Luchazi data reported in Bagemihl (Citation1989) is Trevor and White (Citation1955). Word-by-word translations are not given in Trevor and White (Citation1955) so there is no English meaning provided for the forms here.
12 The symbol ‘nd’ represents a prenasalized stop. This is similar to a sequence of ‘n’ and ‘d’ but functions as a single segment. The second syllable of the Luchazi form ‘kundzivo’ has three segments, ‘nd’, ‘z’, and ‘i’.
13 See McCarthy (Citation1981); Clements and Keyser (Citation1983); Hayes (Citation1986) for various proposals concerning the structure of the timing tier.
14 The operations in Figure are capable of accounting for all reversing games illustrated with the exception of false reversal. See Bagemihl (Citation1989) for discussion of particular issues that arise in the analysis of false reversals.
15 The text in Figure is taken directly from Nevins (Citation2010, p. 223) with the exceptions that the term ‘inversion’ has been changed to ‘reversal’ and ‘transpose’ has been changed to ‘exchange’ to avoid confusion with the distinct meanings of the terms ‘inversion’ and ‘transpose’ in musical terminology. In addition, one operation that has not been considered here, doppel, has been omitted.
16 This transformation (i.e. constituent reversal) is defined as a contextual transformation entitled semitone dyad rotation (i.e. SDR) in Argentino (Citation2019).
17 For other examples of ‘constituent reversals’ in Modern Psalm see bars 51–55 (bass clarinet and clarinets); bars 57–58 (bass clarinet and violin/violas); bar 62 (clarinet/horn and violas); bars 74–76 (violas/altos); bars 75–77 (violas/altos and basses); bars 82–83 (between bass clarinet and b-flat clarinet, e-flat clarinet/oboe).
Other modified forms of reversals also occur throughout Modern Psalm. For instance, <9,1,t,2,5,6> and <2,t,1,9,6,5> can be understood as a dual transformation involving retrograde (segmental reversal) and constituent reversal. Reversing the order of the pitch classes parallels segmental reversal (<9,1,t,2,5,6> becomes <6,5,2,t,1,9>) and constituent reversal (<6,5,2,t,1,9> becomes <2,t,1,9,6,5>). The combination of game forms that parallel conventional serial operators will be addressed later in the article.
18 We realise that there are shortcuts for figuring out TnI relations between related pitch groups or transforming a pitch collection through TnI. For instance, inversionally related elements between related pitch-class collections will sum to the index number.
20 For instance, since <2,t> occurs ordered in the first hexachord of Figure , and unordered in the second hexachord of Figure , we will assume an ordering of <2,t> for both dyad pairs. This logic follows for the other two ordered dyad pairs. <8,4> and <3,9> – which occur ordered in the second hexachord of Figure – will be the assumed ordering for the parallel unordered dyads that occur within the first hexachord of Figure .
21 These reversing transformations at the dyad level occur at the moment that ‘ADONAI’ is sung by the choir, reflecting the symmetrical perfection of God.
22 Argentino (Citation2019) defines edge-anchored movement as a contextual transformation entitled dyad shift (or DS).
24 Figure appears as Example 4 in Argentino (Citation2019). Interchange is defined by Argentino as an LDR transformation (i.e. last dyad retrograde).
25 Listeners were told that they were hearing a Martian ceremony where one Martian gives an utterance (the regular ‘Martian’ form) and the other Martian must reply appropriately (the game form) (Nevins & Endress, Citation2007, p. 4).
26 This transformation is termed RSWAP by Argentino (Citation2019), who provides additional discussion of similar examples.
27 Serial combinatoriality can be understood as row pairs related through edge-anchored hexachord pairs.