The score for Leaf Piece (above) is designed to illustrate the concept for the composition. Each circle represents a key center related to Eb major/Ab lydian and is sized to reflect the amount of time the computer and improviser spend exploring that tonal center. The improviser returns back to Eb (represented by the first brown circle) after visiting each new tonal center.
The tonal centers are, in order of appearance, Eb, Bb, c, Db, Eb, f, g, and the final encircling figure represents Ab lydian. The color scheme reflects the symbolism of a progression from a stable major tonal center (Eb, the brown soil) through shades of green (plant life) to a “transcendent” lydian tonal center (Ab, the yellow sun).
In this image the meaning of the title is apparent. The final (lower right) image from the score is superimposed with a line drawing that follows the improviser’s “path” through the series of tonal centers. The internal veins show the movement from Eb to successive tonal areas and back, the leaf shaped outline represents the bitonal harmonic concept.
The form of Leaf Piece is based on three principal goals:
1. use the rational frequency relationships found in the diatonic set to dictate both the harmony and time scale of a piece.
2. allow the improvisation to have some temporal flexibility while adhering to the above ratios (just as plant merisetms allow flexibility in growth rates and form while maintaining the essential phenotype of the plant).
3. create a piece wherein computer interaction, improvisation, and visual effects are integrated.
The basic pure intervals, represented by the ratios 1:2, 2:3, and 4:5, are the basis for the form of Leaf Piece. Various tuning systems aim to maintain these simple ratio intervals.
One of these is the Pythagorean system wherein pitches are compounded using a 3:2 ratio. Using note letters this system quickly generates C, G, D, A, E, and B. To complete the diatonic set F is derived as 2:3 of C. Some say this points to Lydian as the principal diatonic scale, as it has the lowest, thus most “fundamental,” pitch in the Pythagorean derivation as its root. Thus in concept there is a tension between a stable major key and its lydian mode; which is more powerful, more “tonic?” Leaf Piece explores this tension by thoroughly exploring a major key center, Eb, but finally resolving to an Ab lydian sound. The overall harmonic scheme is in both Eb major and Ab lydian. While these represent the same pitch set, the goal is to create a bitonal effect.
Another system concerned with pure intervals and the diatonic set uses the major triad and its ideal ratios 4:5:6. I will call this tuning with pure thirds. Pythagorean tuning yields no 4:5 pure major thirds. Tuning with pure thirds generates the diatonic set as follows: F, A, C, E, G, B, D. The tonic, subdominant, and dominant triads are all tuned purely. This system of generating a diatonic system is very elegant and, as it is based on the triad, has an affinity for the western tonal system. This leads directly to the issue of the relative minor and an interesting conflict. Using tuning with pure thirds “pure” minor triads with a 6:5:4 frequency relationship can be found between the relative minor tonic a-c-e and dominant e-g-b, but the pitch “d” in the subdominant d-f-a must be tuned slightly differently for the principal minor triads to be pure. This yields two pitches named “D” which have a ratio of 81:80, the syntonic comma. While pure triads are surely pleasing and a suitable basis for a triadic tuning system, the comma reveals a disconnect between western tonal music and a natural or scientific “purity” of sound. This disconnect leads to tempered tuning, which approximates the pure intervals. Leaf Piece uses concepts found through tuning with pure thirds to create the timescale of the composition, and correlates the phenomenon of the comma to its own temporal flexibility.
In tonal improvisation, especially jazz improvisation, related key centers are frequently used to construct a harmonic scheme for the player to negotiate. The time scale of the piece, however, tends to be based on some variation of song form, often employing 4 and 8 measure phrases and a predictable rhythmic formula. As an improviser I am interested in the possibilities presented by a different system for negotiating harmonic “changes” that eschews strict timekeeping. Leaf Piece uses triadic harmony and a set progression, but, rather than employing a strict harmonic rhythm, the computer plays triads for a period of time based on the triad root’s relationship to the tonic key(s) (remember this is meant to be bitonal), then listens for a specific pitch to be sounded in the improvisation, a cue for the next tonality. These listening points (henceforward “nodes”) are similar to the meristematic nodes found in plants, which “listen” for appropriate climactic and other conditions before initiating growth of a new branch.
The progression moves by steps and fifths and always returns to Eb, except at the very end as, follows:
Eb>[F(7)]>Bb>[Bb(7)]>Eb>[G(7)]>c>[Bb(7)]>eb>[Ab(7)]>Db>[Do>]eb>[Bb(7)]>Eb>[C(7)]>f>[Bb(7)]>Eb>[D(7)]>g>Ab
Bracketed chords are the nodes and have no set duration. The computer is set to listen to the improvisation for the third or seventh degree of these chords before sounding the next triad. The eb minor sound and brief arrival at Db aim to establish a sense that Ab is a possible tonic through use of borrowed chords from Ab major and ab minor. The use of eb minor also hints at the broader world of chromatic harmony, inviting improvisational allusion to Gb major etc.
The piece’s time scale uses ratios derived from the frequency ratios found in tuning with pure thirds and moves as follows:
Eb 40 sec>[node]>Bb 27 sec (2/3 the Eb duration)>[node]>Eb 20 sec (1/2 initial Eb duration)>[node]>c 24 sec (4/5 of Ab)>[node]>eb 20 sec (1/2 initial Eb duration)>[node]>Db 23 sec (1/4 of 3/2 of Ab>[node]>eb 20 sec (1/2 initial Eb duration)>[node]>Bb 27 sec (2/3 the Eb duration)>[node]>Eb 20 sec (1/2 initial Eb duration)>f 18 sec (2/3 Bb (which is 2/3 Eb))>[node]>Eb 20 sec (1/2 initial Eb duration)>[node]>g minor 16 sec (1/2 of 4/5 of Eb)>Ab lydian 30 sec (1/2 of 3/2 of Eb)
[node] has no set duration. While sometimes approximate, the time ratios above can be directly related to the pitch ratios found in tuning with pure thirds. Note that the ratios are based on difference in vibrational length of a string (thus G is 4/5 of Eb) rather than frequencies in Hz (where G is 1.25 Eb), and any time duration can be halved because with vibration this ratio (1:2) yields an octave which we hear as the same pitch class.
