Figures 3-5, 7: Fibonacci Gamelan Rhythms

Figure 3: traditional gamelan structure

Figure 3. Graphical representation of how the note attacks (represented by vertical lines) in the different layers of gamelan music are related over time (in the horizontal direction). Where they line up vertically, the layers reinforce each other.

layer 
(note length)   time->

1 (1)  | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
2 (2)  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
3 (4)  |       |       |       |       |       |       |       |       |
4 (8)  |               |               |               |               |

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Figure 4: Fibonacci gamelan choices

Figure 4. Graphical representation of the relations between note attacks (vertical lines) for Fibonacci gamelan layers made up of short and long notes (labeled S and L). To get reinforcement and the same pattern on each level, the pattern must start L S L, but then there are choices.

layer 
(note lengths)   time->

1 (1,2)  | L |S| L |  ?  |    ?    |       ?       |            ?            |
2 (2,3)  |  L  | S |  L  |    ?    |       ?       |            ?            | 
3 (3,5)  |    L    |  S  |    L    |       ?       |            ?            | 
4 (5,8)  |       L       |    S    |       L       |            ?            | 
5 (8,13) |            L            |       S       |            L            |

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Figure 5: Fibonacci gamelan schemes

Figure 5. The three simplest schemes for resolving the choices give different, but similar, patterns.

(a) long beat first:

layer 
(note lengths)   time->

1 (1,2)  | L |S| L | L |S| L |S| L | L |S| L | L |S| L |S| L | L |S| L |S| L |
2 (2,3)  |  L  | S |  L  |  L  | S |  L  | S |  L  |  L  | S |  L  |  L  | S | 
3 (3,5)  |    L    |  S  |    L    |    L    |  S  |    L    |  S  |    L    | 
4 (5,8)  |      L        |    S    |       L       |       L       |    S    | 
5 (8,13) |            L            |       S       |            L            |

(b) short beat first:

layer 
(note lengths)   time->

1 (1,2)  | L |S| L |S| L | L |S| L |S| L | L |S| L | L |S| L |S| L | L |S| L |
2 (2,3)  |  L  | S |  L  | S |  L  |  L  | S |  L  | S |  L  |  L  | S |  L  | 
3 (3,5)  |    L    |  S  |    L    |  S  |    L    |    L    |  S  |    L    | 
4 (5,8)  |      L        |    S    |       L       |    S    |       L       | 
5 (8,13) |            L            |       S       |            L            |

(c) long beat near most deeply reinforced beat:

layer 
(note lengths)   time->

1 (1,2)  | L |S| L |S| L | L |S| L | L |S| L |S| L | L |S| L | L |S| L |S| L |
2 (2,3)  |  L  | S |  L  | S |  L  |  L  | S |  L  |  L  | S |  L  | S |  L  | 
3 (3,5)  |    L    |  S  |    L    |  S  |    L    |    L    |  S  |    L    | 
4 (5,8)  |      L        |    S    |       L       |    S    |       L       | 
5 (8,13) |            L            |       S       |            L            |

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Figure 7: harmonic structure of "Canon..."

Figure 7. The "harmonic tree" of the 21-note melody of "Canon for Seven Hands." The diagonal and vertical connecting lines mean these notes are struck together on different layers. Each note of the melody appears once.

                         1/1
                          |
                         3/2
                          |
                         5/4
                   ____/     \___
               7/4                9/8
             /     \               |
         7/6        2/1          15/8
        /   \        |          /    \
    14/9    5/3     9/8      5/3      5/4
    /  \     |     /  \      /  \      |
28/27  4/3  4/3  3/2  7/4  2/1  3/2  45/32

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David Canright -- DCanright@NPS.Navy.mil