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Mathematical Scaling Operations for
Composing Music from Numbers
Presented by Jonathan Middleton
Assistant Professor of Composition and Theory
Eastern Washington University
Pacific Northwest Chapter, College Music Society
2006 Annual Meeting - Douglas College - Vancouver, British Columbia
Mathematical content from:
“Composing Music from Numbers”
By Jonathan Middleton and Diane Dowd
Compositions:” Redwoods Symphony”
and “Spirals” by Jonathan Middleton
Web-based application:
Topic Goals
Introduce Basic Scaling Processes
Numbers Source Set
» Scaled to »
Instrument Range Set
A. Division Operation
B. Modulo Operation
Listen to the effects of each scaling process
A. Redwoods Symphony
B. Dreaming Among Thermal Pools and Concentric Spirals
Pitch Range in Numbers
The scaling process for mapping a
source set of numbers to an
instrument’s range relies on a numeric
representation of the range of each
Here, we see the keyboard range 1-88.
The numbers for piano keys can be
used for scaling pitch ranges for other
instruments, since most instrument
ranges are encompassed in the piano
range. The alternative would be to use
the standard MIDI range of 128 pitches
(middle C is 60).
Scaling Operations
Two basic methods for mapping a set of numbers to new numeric range are 1) the
Division Operation, and 2) the Modulo Operation. We can see at a glance that the
division operation follows the contours of the Dow averages.
The Division Operation
The division operation is based on a
proportionate representation of the source set
of integers.
The division operation works by a process of
expansion or reduction from a source range of
numbers to a destination pitch (or rhythmic)
That is, the source number distance e.g. 0-9
can expand to a broad instrument range (0-88
for the piano), or reduce to a small motivic
range 0-3. If we take for example the number
set [64, 81, 100, 121, 200] and apply a division
operation to the pitch range 28-52, the numbers
from the set correspond to pitches 28, 31,34,
38, 52 [DEMO upon request]
[offline apple tab to import algorithm]
The Division Operation is anchored by a Scaling Ratio
Division Calculation
• the expression to calculate the pitch p for a given
source number s is the following:
Trans to Mod
Like the division operation,
the modulo scaling
process requires that a
fixed range of pitch
numbers be designated.
The number of pitches in
the range defines the
cycle number to be used,
e.g., a pitch range of 3
pitches would define the
cycle number 3.
Note: Modulo represents a
cyclical form of counting.
Mod “3” refers to the
number of numbers in a
span of 3, i.e. 0-2, or 1-3,
or 99-101. In the span
28-52 there are 25
numbers (not 24).
Modulo Operation
Modular arithmetic offers a cyclical approach to translating source numbers into pitches.
The entire operation is based on a system of counting (cycling) from the lowest to the
highest number in the cycle number range. For example, cycle number 3 creates the cyclic
pattern: 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, …. If a sequence of numbers is translated to a
cycle of 3 numbers, all of the numbers will be congruent () to 1’s, 2’s or 3’s.
Modulo in Detail
In the modulo operation, we can find a pitch for any source number by locating the pitch number
that is congruent to the source number. This process of matching pitches with source numbers by
congruence must begin by establishing a cyclical pattern of counting within the pitch range.
As an example, again consider the set of source numbers [64, 81, 100, 121, 200]. If we wish to
scale these numbers to the pitch range 28 to 52 (called mod 25), we proceed through the following
64: p = 14
81: p = 6
100: p = 0
121: p = 21
200: p = 0
+ 28
+ 28
+ 28
+ 28
+ 28
= 42
= 34
= 28
= 49
= 28
The pitches, within the range 28 to 52, that are associated with the source numbers
[64, 81, 100, 121, 200] are the pitches (or modular numbers) [42, 34, 28, 49, 28].
[DEMO upon request]
[offline apple tab to import algorithm]
Modulo Calculation
Redwoods Symphony
Movement 1
Main theme in xylophone
from tctcaagcac ataaaaaggc cattcgaaga
gctgctgtca attcatttgg ttacattgct
cgtgctcttg gtcctcaaga tgtacttcaa
gtcttgctca ccaatttgcg agttcaagaa
The DNA comes from LOCUS AY562165 of
the National Center for Biotechnology
Information (NCBI), a databank for
research in genetics.*
Each DNA letter A-T-C-G was associated
with a number A=0, T=1, C=2, and G=3.
*AUTHORS Bruno,D. and Brinegar,C.“Microsatellite markers in coast redwood (Sequoia sempervirens)”
JOURNAL Mol. Ecol. Notes 4 (3), 482-484 (2004)
Redwoods Symphony
Movement 1, page 2
The Redwood DNA was used to create a
theme for xylophone by mapping the DNA
“numbers” 0-3 to the numeric range 41-59
which falls within the xylophone’s numeric pitch
range 40 (C4) to 76 (C7). There are three
ranges to choose from depending on the
model of the instrument.
A scaling process called the division operation
was used to convert, or map, numbers 0-3
(DNA input) to numbers 41-59 (xylophone
output). The pitch output was modified so that
53 becomes zero and 47 becomes 46. [DEMO]
CMS2006/ConPresentation/Redwood DNA.rtf
[Offline apple tab to DNA algorithm]
The DNA source numbers were also used for
scaling range of durations : 1-3 with a modulo
Commissioned by John Marshall
and Lynne Feller Marshall
uses a set of integers drawn from spiral
coordinates. The integers were found at a
Web site called the Online Encyclopedia of
Integer Sequences (see URL:
num=A033988). The ID Number is A033988
and the description is as follows : “write
0,1,2,... in clockwise spiral, writing each digit
in separate square; sequence gives numbers
on positive y axis.” The authors provide the
following example:
From Spiral Numbers to Cello
The 104 numbers were scaled by a division
operation within the range of the cello to
create walking bass line. The cello range
was 16 (low C2) to 59 (high G5).
The first 28 numbers go from zero to zero:
The pitch output is:
Explore how other Composers Create Music from Numbers
D. Cummerow’s Website
T. Dukich’s Website
T. Johnson, Self-Similar Melodies (Paris: Editions 75, 1996).
I. Peterson article MathTrek
Introductory Information on Algorithmic Composition
K. H. Burns, “The History and Development of Algorithms in Music Composition, 1957-1993,” Ph.D.
Dissertation (Ball State University, 1994).
C. Roads, The Computer Music Tutorial (Cambridge: MIT Press, 1996).
Connections between Music and Mathematics
G. Assayag, H. G. Feichtinger and J. F. Rodrigues, eds., Mathematics and Music: A Diderot Mathematical
Forum (New York: Springer-Verlag, 2002).
J. Flauvel, R. Flood and R. Wilson, eds., Music and Mathematics: From Pythagoras to Fractals (Oxford:
Oxford University Press, 2003).
For information on
“Composing Music from Numbers”
or commercial recordings of
Redwoods Symphony and Spirals
contact me at
[email protected]
Redwoods Symphony will be available in June through ERM Media
“Masterworks of the New Era”
Kiev Philharmonic