Fibonacci Suite
for retuned piano, seven hands

by David Canright

The world premiere of this piece was 2001 April 7 at the MICROFEST 2001 Conference and Festival of Music in Alternate Tunings , performed by Bruce Brode, Rick Tagawa, Steve Lockwood, and Brian Vessa (in seating order: high to low), conducted by the composer (see photo).


  • Program Notes
  • Photos
  • Scores
  • Tuning Notes

  • Program Notes

    These notes were excerpted from the full MICROFEST 2001 Concert 4 Program

    -- P R O G R A M --

    Fibonacci Suite for retuned piano, seven hands [*world premiere]David Canright
    Piano (Primo)Bruce Brode
    Piano (Secondo)Rick Tagawa
    Piano (Terzo)Steve Lockwood
    Piano (Quarto)Brian Vessa
    1. Canon (quasicrystal)
    2. Fantasy (birdcalls)
    3. Fugue (redder)

    Fibonacci Suite. Leonardo Fibonacci was a twelfth-century Italian mathematician, one of whose discoveries was a sequence of numbers that shows up in patterns of nature (such as plant forms), probability, and other surprising places. The Fibonacci sequence begins: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... where, after the beginning, each number is the sum of the previous two (e.g., 13 = 5 + 8). The ratios of successive Fibonacci numbers approach the "Golden Mean" (1.6180339887...), considered by many throughout history as the most esthetic proportion (it appears in the design of the Parthenon). The three movements of "Fibonacci Suite" each utilize rhythms and forms based on Fibonacci numbers.

    The form of the "Canon (quasicrystal)" was inspired by Indonesian gamelan music, which is built of layers of rhythms, where melodic motion in one layer is twice as fast as the layer below (and half as fast as the layer above). [see Fibonacci Gamelan Rhythms] The Canon has seven layers, one in each octave of the piano. Each layer plays the same melody of long and short notes, with durations based on successive Fibonacci numbers, with lower levels using larger Fibonacci pairs. So the top voice plays in ones and twos, the next in twos and threes, etc., in such a way that lower voices always line up with upper voices. The basic 21-note melody repeats, in whole or in (13-note) part, forward or backward, as required by the alignment pattern. The dynamics vary to bring out the various layers and their relations. A recently discovered form of matter, called quasicrystal, has atoms arranged with spacings that are not strictly repeating but instead alternate long and short, with the same pattern as in the Canon.

    The "Fantasy (bird calls)" consists of phrases, each with a descending (or ascending) line and a repeating note, inspired by a haunting bird call. Each phrase part has notes of a constant Fibonacci duration (some Fibonacci number of them), so this movement features the resulting cross-rhythms (e.g., three quarter notes versus two dotted quarter notes). Each total phrase duration is also a Fibonacci number, and the whole movement is the length of the slowest phrase.

    The "Fugue (redder)" treats the Fibonacci numbers sequentially rather than simultaneously as in the previous two movements. (This same approach was used in an earlier work, "Rosier Sands," for Partch instruments.) In each section, each voice consists of simple phrases (often single notes) of successive Fibonacci durations, hence it gets slow quickly. Each later voice enters such that at some point two phrases of the later voice fit into one phrase of the earlier voice (based on the Fibonacci property that each number is the sum of the previous two). This same idea also appears in reverse, where voices speed up and drop out. The sections, punctuated by percussive sounds, are themselves of successive Fibonacci lengths, until the middle, when the pattern reverses. Hence the overall form of the movement is a palindrome.

    The tuning uses just intonation, based on the harmonic series of overtones of an ideal string. (The overtones of piano strings go progressively sharp of the harmonic series, due to string stiffness. [see On Piano Retuning] ) Standard Western harmony utilizes (or approximates) intervals up to the sixth harmonic. This tuning extends to the seventh harmonic, and so includes some nonstandard ("bluesy") intervals. (The frequency ratios are 1/1, 28/27, 9/8, 7/6, 5/4, 4/3, 45/32, 3/2, 14/9, 5/3, 7/4, 15/8.)

    David Canright is a mathematics professor and self-taught musician. He first became interested in just intonation through a copy of Lou Harrison's Music Primer, and later through Genesis of a Music by Harry Partch. He has refretted two guitars to just intonation, and has composed for guitar, the Partch ensemble, and jazz-rock band. His interests include the mystery of musical affect, hunting wild mushrooms, and rock climbing.

    Bruce Brode is an accomplished performer on both piano and French horn. He has played in a wide variety of ensembles, stretching stylistically from historical European concert music and music for wind ensembles to jazz, blues, and rock. He studied music and composition at UCLA and has composed several works. His musical interests include all types of music, including American folk music, music from other cultures around the world, etc.

    Rick Tagawa studied composition with Elliot Carter and Luciano Berio at the Juilliard School and later did graduate studies in ethnomusicology at UCLA. It was there that he developed an interest in Ugandan traditional music with its sophisticated use of a 5-tone quasi-equal temperament. (In this music the seconds can vary from 190 to 279 cents over a 3 1/2-octave xylophone.) He has published a composition for percussion and has numerous other works to his credit, including a large-scale work for orchestra using Kiganda musical techniques and 72-tone equal temperament. His interests include the intonational nuances in American popular music, Indian music, and other musics.
    [see also Rick's 72-Tone Equal Temperament Site]

    Steve Lockwood is a classically trained jazz musician. He finished his conservatory training at the University of Missouri at Kansas City Conservatory as a piano performance major in 1971. He formed his own group in Minneapolis, where, in 1974, he met and performed with theater director, vocalist, and composer Meredith Monk. He moved to New York in 1976 to perform with her and her group and his own. With her group, he played major theater venues in New York, Europe, and Japan, and appears on three of Meredith's recordings for ECM records: Dolmen Music, Turtle Dreams, and Atlas: An Opera in Three Acts. With his own, he recorded and toured the East Coast and Europe. He was solo accompanist for Meredith during the sacred music festival at the Getty Museum in October 1999. He has been a fixture in the Los Angeles new music/jazz scene for ten years. He will release a new CD with his ensemble this year, and is happy to perform Samuel Barber's Excursions for piano with the San Pedro City Ballet.

    Brian Vessa is a drummer, keyboard player, and composer, largely self-taught, with experience in jazz, rock, blues, and other styles. He is a professional audio engineer with decades of experience, and is currently producing a CD of original works with his band. His musical interests are broad and include the harmonies of just intonation. His other interests include home brewing; his brews with Mr. Brode have won numerous medals.


    Deepest thanks to Rick Tagawa for his many long days of painstaking work in preparing and perfecting the scores. His mastery of notation and of Finale aided tremendously in making the piece playable.

    The links below provide the full scores for the three movements (in Adobe Portable Document Format [PDF]). Anyone wishing to perform this work should contact the composer for scores of the parts suitable for performance.

  • Canon (quasicrystal) [207 KB]
  • Fantasy (birdcalls) [233 KB]
  • Fugue (redder) [128 KB]

  • Tuning Notes

    Thanks to David Vanderlip of Pomona College for his outstanding retuning of the Steinway B grand piano to the scale shown below.

    Tuning Table for Fibonacci Suite

    [also see the harmonic lattice with letter names, or the Harmonic Lattice Diagram and the Harmonic Melodic Diagram from my EPS for JI page]
     Note #   Freq. Ratio   Harm. val     MIDI        Cents     P.Bend   Rel. P.Bend
    |  1    |   1.00000  |     1/1     |  C       |      0   |   8192   |    0      |
    |  2    |   1.03703  |    28/27    |  C#/Db   |   -37.0  |   6675   |   -1517   |
    |  3    |   1.12500  |     9/8     |  D       |     3.9  |   8352   |    160    |
    |  4    |   1.16666  |     7/6     |  D#/Eb   |   -33.1  |   6835   |   -1357   |
    |  5    |   1.25000  |     5/4     |  E       |   -13.7  |   7632   |   -560    |
    |  6    |   1.33333  |     4/3     |  F       |    -2.0  |   8112   |   -80     |
    |  7    |   1.40625  |    45/32    |  F#/Gb   |    -9.8  |   7792   |   -400    |
    |  8    |   1.50000  |     3/2     |  G       |     2.0  |   8272   |    80     |
    |  9    |   1.55555  |    14/9     |  G#/Ab   |   -35.1  |   6755   |   -1437   |
    |  10   |   1.66666  |     5/3     |  A       |   -15.6  |   7551   |   -641    |
    |  11   |   1.75000  |     7/4     |  A#/Bb   |   -31.2  |   6915   |   -1277   |
    |  12   |   1.87500  |    15/8     |  B       |   -11.7  |   7712   |   -480    |

    last updated 2001 April 13

    David Canright --