Abstract
Following a lead found in Bartók’s Music for String Instruments, Percussion, and Celesta (Sect. 6.1), a “note-reception system” is constructed in Sect. 6.2, as follows. Relative to a referential note selected from a set of notes termed “core,” a received pitch is paired with the note a primary interval away. Section 6.3 then explores the idea of self-communicating the core in a potentially endless feedback cycle, such that the “received message” of one cycle becomes the “transmitted message” of the next. “Stability” is reached if at some iteration and onwards the received message always equals the original transmitted message, namely the core. It is proven that if stability is reached at all, it is reached at the very first iteration. In Sect. 6.4 “diatonic note” is defined as a transmitted note the reflexive image of which (at some iteration) is constant relative to every core element. It is shown that every note (at some iteration) is diatonic if and only if the system is stable, and thus the core itself is a set of diatonic notes. It is shown further that a diatonic core consists of exactly ⌊a/2⌋ + 1 elements that may be ordered quintically. Finally, Sect. 6.5 revisits interesting properties of diatonic systems as previously studied by Balzano, Agmon, Clough and Douthett, and Carey and Clampitt.
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- 1.
Bartók’s self-analysis may still be found in Boosey and Hawkes’s study score. See Hunkemöller (1983) for an interesting perspective on Bartók’s analysis, including its comparison to two other analyses of the work, also by the composer.
- 2.
Exceptions do occur in the fragmentary C
and G entries of mm. 34–37, and, most notably, the movement’s climax, mm. 45–56, supposedly “in D.” - 3.
On Bartók’s notational practices see Gillies (1982, 1983, 1989, and 1993); see also Leichtentritt (1929), von der Nüll (1930), and Brukman (2008). Gillies cites Bartók’s ethnomusicological activities as possible influences on his notation. For relevant passages in Bartók’s “Harvard Lectures,” see Gillies (1989, pp. 288–290). Gillies (1989, pp. 26, 133–138) specifically addresses the “enharmonically consistent” notation of the fugue subject of Music for String Instruments, evoking the idea of “modulation” to account for the single D
/E
inconsistency. See footnote 5. - 4.
The large-scale structure of Bartók’s movement, where the subject is successively transposed by ±1,…, ±6 perfect fifths relative to the first entry, reflects the finding that the usual primary intervals together with the non-primary A4/d5 form a mirror-symmetrical connected segment of an intervallic “line of fifths.” Interestingly, Hindemith ’s (1937) “Series 1” consists of the same set of thirteen intervals relative to the “scalar root.” See for example his Fig. 65 (p. 96).
- 5.
In choosing between D
and E Bartók seems to be guided by local primary-intervallic relations. Thus, Bartók writes E in the context of D and C, whereas in the context of C and E he writes D. For Gillies
(1989, p. 135), “the change to a D notation in the third phrase is the sign that a modulation has taken place to an in-filled fourth B–E…” - 6.
Cf. Clough and Douthett’s (1991, p. 100) Definition 1.7: “A set of pcs is maximally even… if it has the following property: the spectrum of each dlen is either a single integer or two consecutive integers.” Clough and Douthett’s Definition 1.7 is closely related to “Myhill’s Property ,” according to which “every generic interval appears in exactly two specific sizes” (Clough and Myerson 1985, p. 250). For further discussion see Sect. 8.3.
- 7.
See Carey and Clampitt (1989), p. 196.
and G
inconsistency. See footnote 5.References
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Agmon, E. (2013). Receiving Notes. In: The Languages of Western Tonality. Computational Music Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39587-1_6
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