Abstract
In The Continuum, Hermann Weyl notes (Weyl, The Continuum: A Critical Examination of the Foundations of Analysis. Dover Books, Mineola, 1994):
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Notes
- 1.
In twelve-tone music, the underlying structural unit is an ordering of the twelve notes in the scale (the “tone row”), which is acted on by a permutation group (the “tone group,” generated by inversion, transposition and retrograde).
- 2.
The row is subdivided into sets {C ♯, E♭, D}, {B, F ♯, F, E}, {B♭, A♭, A}, {G, C}. The overall form is P 0 → I 9 → RI 9 → I 4 → RI 4 → P 0 → Coda. I 9 and RI 9 are the only row operations that preserve two blocks ({C ♯, E♭, D} and {B♭, A♭, A}) from the initial partition, and I 4, RI 4 are the only operations that preserve a single block ({B♭, A♭, A}) common to I 9 and RI 9. This method of dividing the row into pitch sets and looking for invariance under row operations is characteristic of late Webern, though it was never employed in such a logically reductive manner.
- 3.
Schoenberg, who developed the twelve-tone method in the 1920s, sometimes describes twelve-tone music as an undirected space of relations, untethered from the tonal notion of a root. In Style and Idea, he writes “All that happens at any point of this musical space has more than a local effect. It functions not only in its own plane, but also in all other directions and planes, and is not without influence even at remote points…there is no absolute down, no right or left, forward or backward. Every musical configuration, every movement of tones has to be comprehended primarily as a mutual relation of sounds” [15, p. 109].
- 4.
These are collected in An Anthology, a classic document of the early 1960s New York avant-garde edited by Young.
- 5.
The initial Harvard performance was organized by Henry Flynt, and carried out by Young and Robert Morris. See Curtis [3, p. 89].
- 6.
For instance, an octave is assigned the frequency ratio 2:1, a fifth 3:2, a fourth 4:3. Less familiar intervals such as 28:27, 49:48, and 64:63, which are normally heard only as overtones of a fundamental, also play an important role in Young’s music.
- 7.
Young draws a number of interesting conclusions from this view, for instance the impossibility of tuning an equal-tempered tritone, whose frequency ratio is \(\sqrt{2}: 1\).
- 8.
In fact, there have been five different versions of Trio. The 1958 version, three just intonation versions (1984, 2001, 2005), and most recently (2015) a three hour tuned version based on Young’s original sketches for the piece.
- 9.
Although distinct from Conceptual Art of the later 1960s, it is interesting to note Flynt’s connection to the genre. Flynt is part of the artistic circle of Robert Morris and Walter de Maria at this time, and each contribute to An Anthology (Flynt’s contribution is Concept Art). Flynt notes “there was a milieu which may have consisted only of Young, Morris, myself, and one or two others, which was never chronicled in art history” [5, p. 2].
- 10.
For instance, Flynt’s Each Point On This Line Is A Composition (1961) appears to be a specifically foundational interpretation of Young’s Composition #9 1960, in which a line is printed on a notecard.
- 11.
Flynt later expresses this system in more familiar logical notation, stipulating “an associated ratio is a sentence,” an “axiom is the first sentence one sees,” and “sentence A implies sentence B if the associated ratio of B is the next smallest ratio of all sentences you see” [6, p. 24].
- 12.
Flynt believes this new framework simplifies issues surrounding the continuity of perception in Illusion-Ratios, and subsequently the cardinality of its language.
- 13.
Translation from Hesseling [14, p. 140].
- 14.
The then current offshoot of twelve-tone music, where rhythm, duration, timbre, etc. are acted on by the permutation group.
- 15.
In these pieces, rationally tuned sine tones gradually go in and out of phase. Although Young refers to “tuning as a function of time” as one of his key theoretical constructs, it is interesting to note this philosophy may in part have stemmed from technological limitations of the time. In works such as Dream House (1969) Young envisions sustaining tuned intervals for weeks or longer by electronic means. However, the realization of such works proved difficult due to the instability of commercially available oscillators of the time. EMS had recently purchased phase-locked oscillators, which Young was interested in testing.
- 16.
More generally, choice sequences can be understood in terms of spreads. A spread consists of a spread law Λ M , which is a lawlike characteristic function on \(\mathbb{N}^{<\mathbb{N}}\), and a complementary law Γ M which assigns a mathematical object to each finite sequence 〈a 1, a 2, . . a n 〉 such that Λ M (〈a 1, …, a n 〉) = 1. See Hesseling [14, p. 65].
- 17.
Although the title is suggestive of a finite process, Hennix refers to the piece as infinitary. She envisions it to be composed of three infinitely sustained sine tones [9, p. 2].
- 18.
For instance, a finite sequence like (1,0,0,1,0) would indicate which harmonics were detected as present or absent, based on some enumeration of all harmonics of the fundamental. Hennix has more recently remarked that she associates the intuitive continuum with the spectrum of the tambura, the basic acoustical reference for Young’s music since the 1970s.
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Gerhardt, S. (2017). Minimalism and Foundations. In: Kossak, R., Ording, P. (eds) Simplicity: Ideals of Practice in Mathematics and the Arts. Mathematics, Culture, and the Arts. Springer, Cham. https://doi.org/10.1007/978-3-319-53385-8_17
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