While it may seem probable that basing an entire musical composition solely on a single interval cycle may cause problems with invariance of musical color, it seems not to be the case. The potential to generate multiple cyclic-interval relationships is inherent in each individual cycle because the place-ment of more than two pitches in the time and space of a musical context will generate pitch relations between the fi rst two notes and all the ones
that may follow. If, for example, we unfold the three pitches of any tritone cycle [C–F#–C], then tritone relationships will be created between the F#
and the two Cs, and an octave relationship will be created between the two Cs. While there are many musical works based exclusively on single inter-val cycles, the pieces that seem to off er the clearest examples of basic cyclic relationships tend to be didactic works like études or character pieces. The reason for this is that these kinds of works are explicitly formulated to showcase the colors generated by cyclic sonorities and, in the case of études, designed to teach new technical skills necessary for the execution of new cyclic musical textures.
Though not nearly as extensive and comprehensive as Bartók’s Mik-rokosmos, Wallingford Riegger’s (1885–1961) series of children’s pieces explores the unique sounds generated by the cycles. To this end, he gave his set of miniatures simple, self-explanatory titles such as The Aug-mented Triad and The Tritone. The Major Second, as its title suggests, consists solely of two pairs of whole-tone dyads [Db–Eb/D–E and Gb–
Ab/G–A] that represent and juxtapose dyads from the two whole-tone or interval-2/10 cycles. The complete pitch-class content of the miniature is unfolded in the opening four measures. The opening whole-tone 1 dyad7 [Db–Eb] of the left hand is immediately contrasted by the whole-tone 0 dyad [D–E] of the right hand (mm. 1–2). Together, these two dyads gener-ate a chromatic tetrachord [Db–D–Eb–E] that suggest that the chromatic (interval-1/11) cycle is being equally subdivided into its two whole-tone partitions.8 The process is confi rmed when the second set of dyads [Gb–
Ab and G–A] are subjected to the same treatment (mm. 3–4), generating a transposition of the opening chromatic tetrachord [Gb–G–Ab–A]. This time, however, the tone 0 dyad is in the left hand while the whole-tone 1 dyad is in the right hand.
The switching of the two whole-tone cycles between the two hands gen-erates a second relationship between the whole-tone dyads and the perfect fi fth (7/5) cycle that is expressed linearly by the individual right-hand and left-hand parts. In the right hand, the D–E dyad is followed by the G–A dyad. Together they generate a four-note segment of the perfect fi fth cycle [G–D–A–E] in juxtaposition with another four-note segment of the same cycle [Gb–Db–Ab–Eb], generated linearly by the left-hand dyad pair [Db–
Eb and Gb–Ab]. The pitch relations generated in this small piece encom-pass the relationship between the two extremes of the cyclic realm, the chromatic, represented by the interval-1/11 cycle, and the diatonic, repre-sented by the interval-7/5 cycle. The chromatic (1/11) and the perfect fi fth/
fourth (7/5 or 5/7) interval cycles are the only two cycles that generate all twelve pitch classes and therefore express the same pitch content. The only diff erence between the two cycles is that of sequential order. If we lay out the complete perfect fi fth cycle and partition it in exactly the same way that we did the chromatic cycle, we will obtain the same two whole-tone cycles from the partitioning of the chromatic cycle.
C–D–E–F#–G#–A#
C–G–D–A–E–B–F#–C#–G#–D#–A#–E#
G–A–B–C#–D#–E#
The whole-tone cycles can therefore be seen as the gateway between dia-tonic and chromatic spheres of music and it is these cycle that create the binding relationship between the semitone and perfect fi fth cycles. This is made even clearer if the two cycles are aligned with one another.
Interval-1/11 cycle C—C#—D—D#—E—F—F#—G — G#—A—A#—B Interval-2/10 cycle C D E F# G# A#
Interval-7/5 cycle C—G — D—A — E—B—F#—C#—G#—D#–A#—E#
The second of Ligeti’s piano études, Cordes à vide (1981), is based exclu-sively on the interval-5/7 cycle. The closest meaningful English translation of the French title is “Hollow Chords.” The meaning of this title is directly refl ected by the continuous stream of perfect fi fths outlined by both upper and lower parts that generate a purely intervallic texture that has no clear melody and an accompaniment that is missing the rich harmonic sonority characteris-tic of complete chords. The basic compositional technique used in this piece is the linking of segments of the perfect fi fth (7/5) cycle in loops or spirals that are themselves linked to one another by either perfect fi fth or semitone.
The upper part consists of paired musical fi gures that either unfold the perfect fi fths in strict cyclic order (fi gure (a)], or that interlock the perfect fi fths at the semitone [fi gure (b)]. In the opening measures (mm. 1–2), the fi rst fi gure unfolds a descending three-note segment of the perfect fi fth cycle [A–D–G]. This fi gure is chromatically linked to the second fi gure, which consists of two chromatically adjacent perfect fi fths [Bb–Eb/Cb–Fb]. The chromatic linking of the two fi gures occurs between the fi rst fi fth of the fi rst fi gure [A–D] and the fi rst fi fth [Eb–Bb] of the second fi gure (Figure 2.1).
The procedure is repeated (mm. 2–3) with a restatement of the fi rst fi g-ure [A–D–G] that is linked to a transposed version of the second fi gg-ure [C#–F#/D–G/Eb–Ab] via its second fi fth [D–G]. This pairing occurs two more times at diff erent transpositional levels (mm. 4–7). The accompani-ment of the opening measures (mm. 1–4) consists of three interval-5 cyclic segments that are linked by perfect fi fths that generate a closed cyclic loop that begins and ends on the same pitch class [Bb–Eb–Ab–Db–Gb–Cb → B–E–A–D–G–C → F–Bb] unfolding the complete interval-7/5 cycle. Since the moto perpetuo quality of the texture lacks any of the traditional struc-tural markers, the completion of the cycle in the fourth measure is sig-nifi cant because it heralds the end of a traditional four-measure phrase in much the same way that a cadential pattern would in a tonal work.9
↑
↑
↑ ↑ ↑ ↑ ↑ ↑
↑ ↑ ↑ ↑ ↑ ↑
The accompaniment to the second phrase (mm. 4–7) is made up of two intertwining interval-5 cyclic strands that coil around one another like a DNA double helix. Ligeti achieves this by subdividing each cyclic strand in an alternating pattern.
Strand 1: C#–F#–B–E–A–D G–C–F–Bb–Eb Strand 2: D#–G#–C#–F#–B E–A–D–G–C–F
The upper part of the third phrase (mm. 7–8) commences with an inverted version of the original fi gure (a) that outlines a four-note cyclic segment [Db–Ab–Eb–Bb]. This is linked by fi fth to a second (a) fi gure [F#–B–E].
These two (a) fi gures are followed by three (b) fi gures, the second and third of which articulate a new ordering of the chromatically adjacent fi fths.
The fi rst of these two fi gures outlines a series of chromatically ascending fi fths [F–C, Gb–Db, and G–D] while the second fi gure outlines a descend-ing series [E–B, Eb–Bb, and D–A]. If these two series were to be aligned against one another, they would describe an inversional symmetrical pro-gression in which every note is equidistant from an axis of symmetry.
D G Db Gb C F
D/D Axis of symmetry B
E Bb
Eb A
D
Figure 2.1 Ligeti, Cordes à vide (mm. 1–4). Ligeti Études Pour Piano, Book I.
Copyright © 1986 Schott Music GmbH & Co. KG, Mainz—Germany. All Rights reserved. Used by permission of European American Music Distributors LLC, sole U.S. and Canadian agent for Univeral Edition A.G, Wien.
In the context of this piece, this particular disposition of the fi fths describes a new way in which cyclic elements can be arranged, while also making evident how one set of symmetrical pitch relations (those of the cycle itself) can generate secondary pitch relationships in the form of axes of symmetry. These axes can be used as the basis for the generation of both cyclic and modal pitch collections or as a new means of tonal progres-sion. The concept of axial symmetry, for example, is used in many 20th century works as a means of establishing a new kind of tonal centricity where the axes operate in much the same way as do traditional tonal cen-ters in traditional tonal works.10 The closing of the opening section (mm.
11–13) is marked by the appearance of the fi rst dyad chords that become the characteristic feature of the second section of the piece. Noticeably, both parts end the fi rst section by unfolding the same fi fth that begins the piece, D–A in the upper part and Ebb–Bbb in the lower part. This sym-bolizes not only the end of the section but the closing of the interval-7/5 cyclic loop as a formal marker.
At the beginning of the second section (m. 14) the D–A dyad, now at the center of the texture is cyclically extended in opposite directions by both the upper [D–A–E] and lower [A–D–G] parts creating a cyclic link between the formal sections.11 This shows that the cycle plays a role in the formal layout of the work. This principle is confi rmed by the link that exists between the second section and the modifi ed recapitulation (m. 26), where the last dyad of the former section [E#–B#] is cyclically extended by the fi rst dyad of the upper part [F–Bb] of the recapitulation.
The second formal section of the piece (mm. 14–26) is based on varied forms of the original (a) and (b) fi gures which are now often combined, often blurring their individual identity. The (a) fi gure often appears as ver-tical perfect fi fth dyads rather than arpeggiated fi fths, while the (b) fi g-ure is gradually extended. Alternating (a) fi gg-ures partition the chromatic spectrum into “white note” and “black note” cyclic collections. The fi rst (a) fi gures (mm. 14–15) outline a white-note segment of the perfect fi fth cycle [C–G–D–A–E], followed immediately by a new (a) fi gure that out-lines a black note segment [Gb–Db–Ab–Eb–Bb]. Chromatic completion by the fused fi gural pairs becomes more frequent and occurs in progres-sively shorter time frames. The fusion of fi gures achieved by interlocking cyclic dyads becomes one of the most prominent aspects of the section.
The (a) and (b) fi gures (mm. 15–16) are now strictly linked by perfect fi fths [C–G–D + Gb–Db–Ab–Eb–Bb and F–C + D–A–E–B–F#–C#] to the point that the chromatically adjacent fi fths unfold all twelve tones (mm. 15–16), creating yet another complete loop. 12
This piece clearly illustrates the binding relationship between the perfect fi fth and chromatic cycles, the one being a diff erent ordering of the other.
Both the Ligeti and Riegger case studies also illustrate that in music based purely on cyclic pitch relations, the diff erent dispositions of a single cycle within a musical texture automatically generates other cycles. The specifi c
disposition of the whole-tone dyads in the Riegger example generates both perfect fi fth and chromatic cyclic segments. Within a cyclic musical con-text, an individual cycle is but a single expression of the entire system of interval cycles. In a certain sense, the entire system of interval cycles is inherently present in any single cycle.
The formation of multiple cycles and resultant intercyclic relationships that occur in the works discussed thus far were generated by single inter-val cycles. There are, however, a large number of works in which cyclic relationships are created by the concurrent use of more than one cycle.
This does not mean that the cyclic relationships in these works are any diff erent from those found in the former category of pieces. What it does mean is that diff erent intervallic properties of one cycle are simultaneously expressed by another present cycle. It also means that the number of simul-taneous cyclic relationships may be greater because of the presence of a larger number of cycles.