Symphony No. 4, "Souvenir des Ming"

In form, the symphony is a passacaglia with eight variations and 105-part fugue on Ming (1368–1644) or Ming-style melodic fragments, such as may be found in the temple hymns transcribed in Joseph S. C. Lam's State Sacrifices and Music in Ming China (1998). The fragments link up as the complete Theme in G minor, eight phrases of four notes each, variously orchestrated. Then follow the eight variations of the passacaglia:

• Variation I. Against the 60-part imitative figures in the strings, the Theme is heard twice, first forwards then backwards.
• Variation II. The imitative texture thins down for flutter-tongued flutes, oboes, and 'cellos, under a shimmer of tremolando string chords, to which clarinets and bassoons add rapid, light arpeggios.
• Variation III. Two-bar phrases for lower brass and lower strings alternate with single-bar phrases for high pizzicati and winds.
• Variation IV. The Theme, elaborated into fleet 12/32 triplets scherzo-fashion, combines with its own inversion, staccato in the strings, legato in the woodwinds.
• Variation V. Without letup from the ongoing string and woodwind triplets, brass chords restate the Theme in E^b, accompanied by a repeatedly broadening rhythm in the percussion. The brass eventually join the ever-thickening triplets in retrograde and retrograde inversion, while the brass chords transfer, in a striking change of texture, to trilling violins in retrograde.
• Variation VI. A 12-tone variation launched by a cadenza for the 2 harps. The row is constructed out of distinctive patterns of the Theme, its 4 forms played in 4 permutations of 4-part counterpoint in oboes, English horns, clarinets, and bass clarinets, and also used to generate durations and dynamics. The harmonic accompaniment on the 2 harps, also serial in derivation, is especially elaborate, involving arpeggios, broken octaves, and full chords in Harp 1, and a complete G minor statement of the Theme in stark octaves in Harp 2, the original intervals somewhat distorted.
• Variation VII. Four times the Theme is given in E minor by the timpani in stately, sarabande tempo. Over it, various counterpoints are heard singly, and then in increasingly dense combinations and scoring. After the quiet, central 'Trio' for flutes, clarinets, glockenspiel, harps, and violins over a timpani roll on low E, the entire sarabande is repeated al rovescio (backwards).
• Variation VIII. A leisurely modulating canon in eight parts begins in C minor on solo double bassoon, and continues through D minor, E^b minor, F^# minor, B^b minor, F minor, E minor, and B^b minor.

The eight subjects of the fugue derive from the eight individual phrases of the Theme, and are expounded—in a kind of miniature Art of Fugue—in separate fugatos and fugues before all being combined in ten permutations in invertible counterpoint:

• Fugato I on Subject I (5 parts)—oboes, English horn, bassoon
• Fugato II on Subject II (6 parts)—brass
• Fugato III on Subject III (4 parts)—piccolos and flutes
• Fugato IV on Subject IV (4 parts)—horns and tuba
• Fughetta I on Subject V (4 parts)—clarinets
• Fughetta II on Subject VI (4 parts)—bassoons
• Fugue I on Subject VII (4 parts)—harps and celesta
• Fugue II on Subject VIII, and then also I-VII (54 parts)—tutti, starting with solo violins and glockenspiel
• Fugue III on Subjects I-VIII (105 parts)—tutti, with aleatory stretto after the 10th permutation, over which the Theme is superimposed by two trumpets and trombone playing from an auditorium balcony, while within the orchestra proper each player is directed to play earlier fugal material in any order at will.

The composition is dominated by fractal number series, preponderantly the Lucas (1, 3, 4, 7, 11...1364), but occasionally also the Fibonacci (1, 2, 3, 5, 8...), sequences in which each term is the sum of the previous two. Variants are created by simple multiplication, e.g., Lucas x 3, which produces the new series 3, 9, 12, 21... All these are used to generate the precise duration of each variation (so that each lasts the added lengths of the previous two), most of the vertical and horizontal patterns (since the numbers may be made to correspond to pitches, ascending or descending, played successively or as chords), and also many of the rhythmic motifs.

Another fractal value, the Golden Section, demarcates the end of the passacaglia and the beginning of the fugue. (The Golden Section is the point at which the ratio of the remaining area to the larger one preceding, is the same as the ratio of the larger area to the whole.)

The Lucas series determines that, in general, each fugato or fugue is in length the sum of the preceding two. In the last two fugues, the keys and octave registers of the principal subject entries follow the Lucas series, while the accretion of parts in Fugue II (from one to fifty-four) follows the Fibonacci aggregates.