Fibonacci Mod 12 System – Performance Instructions
Casey Mongoven
I. General Information
This composition is for 12 solo singers, the make-up of the voice-parts is
unimportant. Each performer sings the same pitch for the whole
composition at different times during the piece. The composition is in two
movements and the duration can be chosen by the performers under
requirements to be explained later. The work is based on a collection of
ten Fibonacci sequences under the modulus 12 with some very
interesting properties. In the course of the work, these mathematical
properties are converted into musical ones.
The Fibonacci sequence begins {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, …}. Each number is the sum of the previous two. Using a modulus,
we divide each number in the sequence by twelve and take the
remainder. So the sequence above becomes {0, 1, 1, 2, 3, 5, 8, 1, 9, 10,
7, 5, 0, …}. Other sequences are possible in which each number is the
sum of the two previous. The Lucas sequence, for example, begins {2, 1,
3, 4, 7, 11, 18, 29, 47, 76, …}. Using the modulus 12, ten such
sequences are possible:
These sequences are periodic; notice at the end of F1 {5, 9, 2, 11, 1, …},
that 11 + 1 = 0 using mod 12, making the next element in the sequence
0. This lands us back at the beginning of F1 {5, 9, 2, 11, 1, 0, 1, 1, 2, 3,
5, 8, 1, 9, 10, 7, …}. This is an important feature of this composition.
It is a fascinating fact that every ordered dyad occurs in the collection
shown above. This means that if one were to name any two pitches, 3
and 7 for example, you could find these pitches adjacent to each other at
some point in the collection. Look, for example, in the collection shown
above for the appearance of 5 and 6 next to each other, then choose any
two numbers yourself and look for them. The musical impact of
exhausting all ordered dyads (in harmony and melody) is profound: it
gives the composition great unity and variety. For more information on
the properties of the collection, see my essay Fibonacci Pitch Sets
(2000).
For this piece, the number 0 in the collection above is represented by the
note E (329.6 Hz). This makes 1 equal the note F, 2 means F#, 3 means
G and so on.
This composition gives the conductor or performers a lot of freedom to
make choices about the nature of the performance. Before the
instructions for performing individual movements are given, it is best to
first clarify some more things about the general nature of the composition
and performance.
II. Syllables And Language
Syllables are attached to each note. The syllable for each note should
not change during the course of the piece. The conductor or performers
may choose the syllables, but there are guidelines. Each syllable has to
be a number in a different language that has only 1 syllable; 12 different
languages must be used, and the numbers 0 through 11 must be
represented. The note E should be sung on a syllable meaning “zero” in
some language, the note F should be sung on “one,” the note F# should
be “two,” G should be “three” and so on. It is preferable that the language
is still in common use. This is, however, not a criterion of this work. Old
languages can be used, as long as it is possible to find some reasonably
authentic pronunciation.
III. Arrangement Of The Pitches
In the course of this work, each performer sings his pitch about 36 times.
The conductor is free to choose the octave in which each pitch occurs;
he may choose any sort of arrangement of all 12 pitches he desires, and
assign each singer a pitch. This means that he should find a chord with
all 12 pitches in it occurring once and only once. Any method for
constructing this chord is acceptable. The voicing of this chord plays a
significant role in characterizing the sound of the piece. The conductor
orchestrates this chord by assigning the different tones in the chord to
the performers of his choosing. The chord can be sung out loud as an
exercise for the performers, or for the conductor to test out different
possibilities. Different syllables can be experimented with as well. Once
chosen, the pitches and syllables should stay the same for each singer
throughout the piece.
IV. Positions Of Performers
There are several options for the positions of the performers in this
piece. The positions should stay the same for the whole composition.
Each singer must be either 1, 5, 7, or 11 semitones away from people on
their left and right. So the order of performers has to be one of the
following in any case: (0,1,2,3,4,5,6,7,8,9,10,11),
(11,10,9,8,7,6,5,4,3,2,1,0), (0,5,10,3,8,1,6,11,4,9,2,7), or
(7,2,9,4,11,6,1,8,3,10,5,0). In the diagrams below, the
(0,1,2,3,4,5,6,7,8,9,10,11) set-up is used for practicality. The can be
replaced with any of the above. The following formations are acceptable:
LINE
Performers are placed anywhere in the audience or around it. Singers
are evenly spaced and facing the same direction. Placement in front of or
behind the audience, or on the left or right is acceptable as well. For the
line the 0 must be on the left or right side.
CIRCLE
Performers are placed around the audience facing the center of the
circle. The singers can also be in the audience. It is not preferable to
have them in front of the audience in a circle. Placement of 0 is
unimportant.
TRIANGLE
The performers are situated in a triangle facing the center of the triangle.
The triangle should be equilateral. As with the circle the performers can
be in the audience or around it. The 0 should be at one corner.
SQUARE
The performers are in a square in or around the audience, facing the
center of the square. The 0 should be at one corner.
In all cases the distance between the singers should be the same, as
shown in the diagrams above.
V. Movement One: Melodic
In the first movement, all ten sequences are heard one after another, in
an order chosen by the conductor. As explained before, each singer
sings one tone, playing a role similar to that of a bell player, who just
plays a single note in a larger piece. In between each sequence there
should be a pause of at least 3 seconds, but not more than 10 seconds.
The pauses between sequences should not differ from each other. The
conductor may choose any point in the sequences as a starting point, but
it is important that the first note of each sequence sung is repeated at the
end of each sequence. If the singers start in the middle of the sequence,
when they come to the end of the line, the next member of the sequence
that is heard should be the first on that line, and they should continue
singing until they arrive at the first pitch sung again. F1 could be
performed as {2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0,
1, 1, 2}, for example – F1 will always have 25 pitches, because the first
pitch is repeated at the end. The last note should not be held; each note
should be exactly as long as the others. The tempo should stay steady
throughout; any tempo is acceptable. The dynamic level should be
steady throughout. Any sort of articulation is acceptable as long as each
note is articulated the same within the movement. The singers should try
to match each other in loudness and articulation. There are a total of 154
tones in this movement.
VI. Movement Two: Harmonic
In the first movement, the complete Fibonacci pitch class sets mod 12
were expressed melodically; in this movement, each sequence is layered
on top of itself, creating a sequence of intervals (two-note chords). This
means that when F1 is performed for example, there is another “layer” of
F1 performed simultaneously. Due to the nature of the Fibonacci
sequence, the resulting sequence of intervals is always a Fibonacci
sequence. At any given moment, two people are singing, with the
exception of the occurrence of the interval 0. The two sequences being
heard at once are always offset by 1, meaning for example, that if {0, 1,
1, 2, 3, 5, 8, 1, 9, 10, 7, …} is used, the sequence {1, 1, 2, 3, 5, 8, 1, 9,
10, 7, 5, …} is heard simultaneously. The performers do not need to
piece together the sequences, the structure just described is already in
the notation. The performers’ task is to figure out where to sing their
notes.
When the interval 0 occurs, a singer is in “unison” with himself. This will
happen twelve times in the collection, once for each singer (unless a
sequence is started on a unison, in which case the singer sings the
unison twice). When a unison occurs the singer should compensate in
volume slightly, to match the steady volume of the movement – his
volume should match that of two singers. The articulation, however,
should not change. Every sequence should be heard, and just as before
the first chord of each sequence sung should be repeated at the end.
Notice that the sequence F0 is essentially the same in both movements,
differing only in volume and articulation (if the conductor chooses).
In this movement, the conductor can choose a new order for the
sequences and new starting points if he desires. The pitches and
syllables stay the same as the first movement. Each sequence should be
heard once and only once – this means that there are 154 X 2 = 308
notes in this movement (counting each unison as 2 notes). The tempo
and dynamic level can differ from the first movement but should stay
steady throughout. Just as in the first movement, an articulation should
be chosen and used throughout – it need not be the same as in the first
movement. As in the first movement, a length of pause should be
selected to come between each sequence and it should stay the same
length throughout the movement.
VII. Number Of Pitches And Intervals Sung In Each Sequence
For clarity, here are the number of terms of each sequence used – the
number of pitches in the first movement and the number of chords in the
second:
F0: 2
F1, F1’, L1, L1’, and F2: 25 each
F3, F3’: 7 each
F4: 9
F6: 4
This makes for a total of 154 tones in the first movement, and 154
intervals in the second movement.
VIII. More Information
Visit www.caseymongoven.com/catalogue/b168.html for more
information and files for this composition.
February 16, 2005