Mathematical Scaling Operations for Composing Music from Numbers

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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 http//musicalgorithms.ewu.edu/
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I. Introduce Basic Scaling Processes
Numbers Source Set Scaled to
Instrument Range Set A. Division
Operation B. Modulo Operation
Topic Goals

• Listen to the effects of each scaling process
• A. Redwoods Symphony
• B. Dreaming Among Thermal Pools and Concentric
  Spirals

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Pitch Range in Numbers

• The scaling process for mapping a source set of
  numbers to an instruments range relies on a
  numeric representation of the range of each
  instrument.
• 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).

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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.

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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)
  range.
• 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
• http//musicalgorithms.ewu.edu/
• offline apple tab to import algorithm

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The Division Operation is anchored by a Scaling
Ratio
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Division Calculation

• the expression to calculate the pitch p for a
  given source number s is the following

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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).

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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 1s, 2s or 3s.

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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
  steps
• 64 p 14 28 42
• 81 p 6 28 34
• 100 p 0 28 28
• 121 p 21 28 49
• 200 p 0 28 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
• http//musicalgorithms.ewu.edu/
• offline apple tab to import algorithm

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Modulo Calculation
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Redwoods SymphonyMovement 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 A0, T1, C2, and G3.

AUTHORS Bruno,D. and Brinegar,C.Microsatellit
e markers in coast redwood (Sequoia
sempervirens) JOURNAL Mol. Ecol. Notes 4 (3),
482-484 (2004)
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Redwoods SymphonyMovement 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
  xylophones 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
• http//musicalgorithms.ewu.edu/algorithms/DNAseq.h
  tml
• Offline apple tab to DNA algorithm
• The DNA source numbers were also used for
  scaling range of durations 1-3 with a modulo
  operation.

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Xylophone theme from Redwood DNA
?
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Spirals work-in-progressCommissioned 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 http//www.research.att.
  com/projects/OEIS?AnumA033988). 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
• 131416...
• 245652...
• 130717...
• 121862...
• 101918...

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From Spiral Numbers to Cello

• Sequence 0,5,1,4,3,7,8,0,4,7,7,1,2,6,2,1,8,7,4,2
  ,6,1,8,9,2,7,6,0,6,5,1,2,0,4,1,5,8,5,1,8,8,8,2,1,2
  ,3,2,4,9,0,2,8,9,9,3,3,2,0,3,7,9,3,4,2,8,8,4,7,1,5
  ,5,3,7,4,5,9,7,5,6,5,9,8,7,1,5,3,7,8,4,0,8,5,6,9,9
  ,3,1,0,9,8,1,1,6,9,9
• 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
  0,5,1,4,3,7,8,0,4,7,7,1,2,6,2,1,8,7,4,2,6,1,8,9,2,
  7,6,0
• The pitch output is 16,39,20,35,30,49,54,16,35,49
  ,49,20,25,44,25,20,54,49,35,25,44,20,54,59,25,49,4
  4,16

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Explore how other Composers Create Music from
NumbersD. Cummerows Website http//www.geocitie
s.com/Vienna/9349/index.html T. Dukichs
Website http//tomdukich.comT. Johnson,
Self-Similar Melodies (Paris Editions 75,
1996).I. Peterson article MathTrek
http//www.sciencenews.org/articles/20050917/matht
rek.aspIntroductory Information on Algorithmic
CompositionK. 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
MathematicsG. 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).
Bibliography
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For information on Composing Music from
Numbersor commercial recordings ofRedwoods
Symphony and Spiralscontact me
atjmiddleton_at_ewu.eduRedwoods Symphony will be
available in June through ERM MediaMasterworks
of the New EraKiev Philharmonic