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PROPORTION AND SYMMETRY AS MUTUAL ANTAGONISTS IN TUNING: SOME QUARTER-TONE RESOURCES
Published online by Cambridge University Press: 28 August 2024
Abstract
Quarter-tones have the dubious honour of being the microtonal default in Western art music, yet they have been of little recent interest to those most involved with extended intonation. Other microtonal equal divisions have appealed as pragmatic approximations of consonant just-intonation intervals, something that quarter-tones do not offer. This article proposes that quarter-tones can be valued in a different way, for their ability to generate symmetrical harmonic resources that divide the fourth and fifth as the tritone does the octave. These resources are offered as examples of a broader aesthetic of symmetry, which is contrasted with an aesthetic of proportion. These antagonistic principles are explored through the case of the ever problematic tritone, illustrating how proportion and symmetry are best understood using the symbolic resources of just intonation and equal temperament respectively. Drawing on the work of Robert Hasegawa, Georg Friedrich Haas and Ivan Wyschnegradsky, the article argues for a hybrid approach that embraces both just intonation and equal temperament.
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- RESEARCH ARTICLE
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- Copyright © The Author(s), 2024. Published by Cambridge University Press
References
1 This hegemony has never resulted in a homogenous tuning practice, as performers actualise tuning systems in diverse ways. Throughout this article, I am interested in the conceptual and symbolic resources of tuning systems rather than their performed results. The distinction is discussed in Pedro Laranjeira Finisterra's ‘(Un)Equal Tunings: Exploring Multiple Levels of Resolution between Equal Tunings and Intonational Practices in Composition’ (Ph.D. commentary, Guildhall School of Music and Drama, 2024).
2 Gann, Kyle, The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician (Urbana: University of Illinois Press, 2019), pp. 102–103CrossRefGoogle Scholar.
3 Wyschnegradsky, Ivan, Kaplan, Noah and Kaplan, Rosalie, Manual of Quarter-Tone Harmony (New York: Underwolf Editions, 2017)Google Scholar; Madrid, Alejandro L., In Search of Julián Carrillo and Sonido 13 (Oxford and New York: Oxford University Press, 2015)CrossRefGoogle Scholar; Suzette Mary Battan, ‘Alois Hába's “Neue Harmonielehre des diatonischen, chromatischen, Viertel-, Drittel-, Sechstel und Zwölftel-Tonsystems”’ (Ph.D. dissertation, Eastman School of Music, University of Rochester, 1980).
4 Gann, The Arithmetic of Listening, p. 205.
5 Franck Jedrzejewski, ‘Generalized Diatonic Scales’, Journal of Mathematics and Music, 2, no. 1 (2008), pp. 21–36; Alain Louvier, ‘Recherche et classification des modes dans les tempéraments égaux’, Musurgia, 4, no. 3 (1997), pp. 119–31.
6 Just intervals are those expressed through frequency ratios. For example, a 5:4 ratio is the interval between the fourth and fifth harmonic of a shared fundamental – that is, a just major third. The simplicity of these ratios is derived from the size of their prime factors and the size of the numbers in the ratio.
7 Marc Sabat and Robin Hayward, Towards an Expanded Definition of Consonance: Tuneable Intervals on Horn, Tuba and Trombone (Berlin, Germany: Plainsound Music Edition, 2006), p. 4.
8 Robert Hasegawa, ‘Parcours de l'oeuvre: Georg Friedrich Haas’, 2014, B.R.A.H.M.S. ircam, http://brahms.ircam.fr/georg-friedrich-haas#parcours (accessed 17 October 2023); Ezra Sims, ‘Yet Another 72-Noter’, Computer Music Journal, 12, no. 4 (1988), p. 28; Julia Werntz, ‘Adding Pitches: Some New Thoughts, Ten Years after Perspectives of New Music's “Forum: Microtonality Today”’, Perspectives of New Music, 39, no. 2 (2001), pp. 159–210.
9 James Tenney, From Scratch: Writing in Music Theory, eds Larry Polansky, Lauren Pratt (Oxford: Oxford University Press, 2015), pp. 306–307.
10 Ibid., p. 306.
11 Ivor Darreg, ‘The Place of QUARTERTONES in Today's Xenharmonics’, Tonalsoft, n.d., www.tonalsoft.com/sonic-arts/darreg/dar8.htm (accessed 22 February 2024).
12 Sims, ‘Yet Another 72-Noter’, p. 28.
13 Robert Hasegawa, ‘Clashing Harmonic Systems in Haas's Blumenstück and in vain’, Music Theory Spectrum, 37, no. 2 (2015), pp. 205, 209.
14 Hasegawa, ‘Clashing Harmonic Systems in Haas's Blumenstück and in vain’.
15 Ibid., p. 222.
16 Gann, The Arithmetic of Listening, p. 40.
17 Used here non-logarithmically to illustrate the distinction between logarithmic and linear scales.
18 See, for example, Chiyoko Szlavnics, ‘OPENING EARS: The Intimacy of the Detail of Sound’, Filigrane, 4 (2006); Ben Johnston, ‘Maximum Clarity’ and Other Writings on Music, ed. Bob Gilmore (Urbana: University of Illinois Press, 2006).
19 Olivier Messiaen, The Technique of My Musical Language (Paris: A. Leduc, 1956), pp. 87–94.
20 Dmitri Tymoczko, A Geometry of Music Harmony and Counterpoint in the Extended Common Practice (New York: Oxford University Press, 2011).
21 Ibid., p. 53. Nearly even chords might also be chords that fill out the octave as evenly as possible within a given intonation. For example, an even sevenfold division of the octave is not possible within 12TET, but the diatonic scale is a nearly even equivalent, being as close to even as is possible within 12TET.
22 Tymoczko, A Geometry of Music Harmony and Counterpoint in the Extended Common Practice, p. 97.
23 Ernő Lendvai, Béla Bartók: An Analysis of His Music (London: Kahn & Averill, 2007).
24 One note, in this case C, must be duplicated because of the odd number of intervals in a diatonic scale.
25 Harry Partch, Genesis of a Music: An Account of a Creative Work, Its Roots and Its Fulfillments, 2nd edn, (New York: Da Capo Press, 1974), pp. 188–90.
26 Robert Hasegawa, ‘Composing with Hybrid Microtonalities’, Živá hudba, 11 (2020), pp. 112–26.
27 Hasegawa, ‘Clashing Harmonic Systems in Haas's Blumenstück and in vain’, p. 206.
28 Ibid., p. 220.
29 Ibid., p. 205.
30 Tenney, From Scratch, pp. 378–79.
31 Michael Bruschi, for example, argues that listeners’ encultured expectations govern pitch perception, meaning that microtonal intervals are heard relative to 12TET for Western listeners; see Michael Bruschi, ‘Hearing the Tonality in Microtonality’ (Ph.D. dissertation, Yale University, 2021), p. 129.
32 In practice, I have found subdividing intervals a major third or smaller to be less interesting. The small ambit makes it hard to discern self-similar patterns and close voice-leading.
33 I draw the subminor stack's unusual name from Philip Tagg, whose work analysing quartal harmony has been vital for my understanding of subminor harmony; see Tagg, Philip, Everyday Tonality II: Towards a Tonal Theory of What Most People Hear (Larchmont: MMMSP, 2017), p. 292Google Scholar.
34 Beaulieu, Marc, ‘Cyclical Structures and Linear Voice-Leading in the Music of Ivan Wyschnegradsky’, Ex-Tempore, 5, no. 2 (1991)Google Scholar, www.ex-tempore.org/beaulieu/BEAULIEU.htm (accessed 18 October 2023).
35 Hasegawa, ‘Clashing Harmonic Systems in Haas's Blumenstück and in vain’, p. 210.
36 Wyschnegradsky et al., Manual of Quarter-Tone Harmony.
37 Louvier, ‘Recherche et classification des modes dans les tempéraments égaux’; Young, Gayle, ‘The Pitch Organization of Harmonium for James Tenney’, Perspectives of New Music, 26, no. 2 (1988), p. 204CrossRefGoogle Scholar.
38 Ives, Charles, Essays before a Sonata, and Other Writings, ed. Boatwright, Howard (New York: Norton, 1962), p. 115Google Scholar.
39 A similar set of relationships will be found in any equal-tempered scale that can equally subdivide the perfect fourth and fifth, such as 36TET. Joe Bates, Wound Honey, on Visions, terra invisus. Bandcamp, 2024.
40 Joe Bates, Wound Honey (2023).
41 In the subminor circle shown, the generator interval of a subminor third is inverted, becoming a subminor seventh. This does not change the structure of these but makes them easier to visualise.