MODAL COMPLETION OF THE CYCLES

Both simple and compound interval cycles can be created by combining or interlocking two modes, where the individual modes may be from either the same or diff erent modal families. One of the most prevalent examples of this principle occurs when two modes interlock to generate the complete inter-val-1/11 cycle or chromatic scale. As a general principle, the completion of this cycle indicates that it is the source pitch collection of the unordered twelve-tone continuum and most often occurs in conjunction with the completion of sig-nifi cant structures such as phrases, periods, and larger formal units. In many instances, chromatic completion occurs progressively and terminates in places similar to where one would fi nd the basic cadences in traditional tonal music.

Chromatic completion by modes from a single modal family can occur in two ways. If they share a common tonic, the second of the two modes must contain the missing chromatic pitches. This allows for several combi-nations, two of which are shown here.

C Ionian C–D–E–F–G–A–B

This sort of modal combination is unique in that it allows a kind of chromatic completion that retains a strong modal sense because of the shared tonic. A second way of combining two single-family modes is by using transpositions of the same or diff erent modes. This usually results in a texture where the bimodality is easy to perceive because of the diff erent transpositional levels of individual modes. Since the number of combinations is so vast, the following example must suffi ce to demonstrate the general principle.

C Ionian C–D–E–F–G–A–B–C

C–C#–D–D#–E–F–F#–G–G#–A–A#–B C# Ionian C#–D#–E#–F#–G#–A#–B#

Chromatic completion may also be achieved by using modes from dif-ferent modal families. Again, the bimodality of the resulting texture is even more discernible because there are two diff erent modes at diff erent

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transpositional levels. The number of possible combinations is also so vast that one of the most favored modal combinations of this kind, that is, a pentatonic/diatonic one has been chosen to demonstrate this kind of combination.

C# Pentatonic C#–D#–F#–G#–A#

C–C#–D–D#–E–F–F#–G–G#–A–A#–B C Ionian C–D–E–F–G–A–B–C

In the Forest, one of Rebikov’s “Trois Idylles,” Op. 50, represents a devel-opmental stage in the technique of bimodal completion of the chromatic cycle. While the bimodal combination of the “black key” pentatonic [C#–

D#–F#–G#–A#] accompaniment and the “white key” diatonic [C–D–E–

F–G–A–B] melody together produce a static musical texture, each part is unfolded in a series of triads that though not functional, still convey a strong tonal sense. The invariable accompaniment for the fi rst half of this unmea-sured piece is an arpeggiated F#–A#–C# triad. Against this accompaniment, the initial melodic statement, unfolded as a pair of parallel arpeggiated tri-ads [G–B–D and F–A–C], generates a diatonic hexachord [F–G–A–B–C–D].

Together, the melody and accompaniment generate a gapped chromatic seg-ment [F–F#–G–( )–A–A#–B–C–( )–D] that is clearly bimodal.⁸

The following long melodic ascent completes the diatonic white note col-lection [E–F–G–A–B–C–D] establishing what seems to be an E tonal cen-ter. The completion of the diatonic pitch collection brings the entire texture one note closer to the completion of the chromatic segment, now missing only two notes [E–F–F#–G–( )–A–A#–B–C–C#–D–( )]. A counterbalanc-ing melodic descent establishes a new C tonal center that is confi rmed in the central section. The C tonality is decipherable only by the assertion of the

“tonic” at end of each melodic statement. The missing D# and G# are sup-plied by the accompaniment of the new middle section which points to the completion of the pentatonic black key collection laid out in scalar order [C#–D#–F#–G#–A#]. The diatonic melody is chromatically interlocked with the accompaniment to produce a series of adjacent semitone pairs that move parallel to one another, unfolding a cyclic segment in scalar order [C–C#–D–D#–E–F–F#–G–G#].

The miniature ends with the opening accompaniment while the mel-ody unfolds the complete diatonic collection in a series of parallel triads [E–G–B, D–F–A, and C–E–G] that elucidate the C tonality. Despite the tonal underpinnings, the gradual bimodal completion of the chromatic cycle forms the compositional basis of the piece and is implemented in such a way that it helps to delineate the formal structure of the work.

The identical bimodal combination has been favored by many composers throughout the 20th century. While Rebikov’s cyclic realization originates from a modal perspective, that of later 20th century composers stems from a purely cyclic perspective. In Ligeti’s fi rst piano etude Désordre (1983),

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the black note pentatonic, white note diatonic combination redefi nes the completion of the chromatic cycle in a context of confl icting tonal identities (C and F#) and neither pentatonic nor diatonic modes used in the combina-tion retains a consistent tonal identity.

The formal structure of Désordre is based on a series of phases articu-lated by a gradual metric misalignment and subsequent realignment of the bar lines of the upper and lower parts. In the fi rst few measures (mm. 1–4), the upper and lower parts are aligned note for note. At the fi fth measure, the lower part adds an extra eight note while the upper part starts a new measure creating a misalignment so that the two parts are out of phase.

The process is continued so that the individual parts move further and further apart only to be realigned when the process has run its full cycle (m. 33 of the upper part). The texture is arranged in such a way that the piano modal collections, whether complete or incomplete, are punctuated by forte octaves that mark the misalignment of the two parts (Figure 6.3).

The upper part consists exclusively of the white note diatonic pitch col-lection [C–D–E–F–G–A–B] while the lower consists exclusively of the black note pentatonic collection [C#–D#–F#–G#–A#]. Combinatorial comple-tion of the interval-1/11 cycle occurs within the fi rst four measures and is highlighted by the complete scalar unfolding of both modes. However, the punctuating octaves outline a series of regularly alternating tritones that disrupt any sense of tonality.

Upper octave: B C B D C A

Interval: 8 6 8 6 4 6

Lower octave: D# F# D# G# G# D#

When the modes are fi nally laid out (m. 4), the intervals between the notes of the two parts describe a symmetrical convergence from the tritone to the semitone that reveals the symmetrical identity of the chromatic scale rather than the tonal one.

Upper diatonic collection: A B C D E F G

Interval: 6 5 4 4 3 2 1

Lower pentatonic collection: D# F# G# A# C# D# F#

Further evidence that the two pitch collections are seen as chromatic subc-ollections rather than two individual modes is supplied by the next phrase that ends in precisely the same way as the fi rst. The intervallic sequence described by the now misaligned octaves outlines all but one of the tones contained in the complete chromatic cycle. The only remaining tri-tone [B–F] cannot be generated by the bimodal combination because it is the only tritone in which neither pitch class is contained in the pentatonic collection. The two collections are therefore linked exclusively by the inter-val-6/6 cycles.

Upper octave: A A D C E D G Interval: 6 3 6 6 6 4 6 Lower octave: D# F# G# F# A# A# C#

When the two parts reach a synchronized realignment (m. 33 of the upper part), the octaves once more outline a tritone [F#–C] confi rming that the completion of the chromatic scale is intended as a true transformation from bimodal combination to interval cycle.

CYCLIC TRANSFORMATION OF BIMODAL COMBINATIONS