Linear Recurrence Relations in Music

1
Linear Recurrence Relations in Music

• By Chris Hall

2
The Aim of My Project

• The goal was to take a composition by Beethoven
  and generate a linear recurrence relation that
  best represents the tonality of the music.
• Sonata No. 1, Op. 12 in D Major

3
Linear Recurrence Relations

• xn a1xn-1 a2xn-2 akxn-k

4
Premises

• only the melodic line was used no chords. This
  allows the counting of only a single note at a
  time.
• The range of notes used was restricted to the
  audible range as determined by MIDI Z128.
• The value of a single note was determined by MIDI
  format with middle C being 60. All notes were
  sequentially valued based on their pitch relative
  to the pitch of the notes immediately above and
  below (i.e. if E is 20, D is 19 and F is 21).

5
What was investigated

• The degree of the recurrence relation
• The range for coefficients
• The best algorithm for finding the relation
• The best-fit solution

6
Establishing the quality of results

• The standard deviation of the original set of
  notes was considered the maximum allowable error
  for the notes being generated by the relation.
• The calculated errors considered for comparison
  were the square errors of the generated notes
  from the original notes.
• The average square error of all the notes
  generated was then compared to the standard
  deviation of the original note set. Only those
  errors below the standard deviation were
  considered acceptable.
• The highest quality results were considered to be
  acceptable results with the farthest distance
  from the standard deviation.

7
The Method

• The first avenue taken was to consider the same
  number of equations as unknowns, where the
  unknowns were the coefficients of the recurrence
  relation.
• Like all following methods, MATLAB was used to
  generate notes, errors and comparisons.
• The MATLAB program was written to allow the user
  to enter in their desired degree of relation
  (which corresponds to the number of unknowns).
  For the sake of comparison, more than one degree
  could be entered at a time.
• The program then produces a series of average
  square errors on a graph that correspond to the
  number of equations used, the theory behind which
  will be discussed a bit later.

8
The First Program
9
(No Transcript)
10
Using more equations

• In order to vary the number of equations used,
  the least squares method was used.
• Because the rank of each matrix was full, a
  unique solution could be obtained.
• Moreover, since the function considered was the
  sum of the square errors, the Hessian Matrix of
  the function was always a multiple of the product
  of the matrix with its transpose.

11
A simple 4x2 example
12
Other definitive information

• I know that the point zero is a local minimum
  because the Hessian Matrix of my original
  function is positive definite.

13
The Discreet Case

• The next path for investigation was to analyze
  coefficients with discreet rather than continuous
  ranges.
• It should be noted that these ranges included but
  were not limited to finite fields.
• Since the notes themselves were elements of
  finite range (remember, Z128 ) the ranges for the
  coefficient vectors investigated were covered by
  the range of the notes

14
The Discreet Case contd

• Another MATLAB program was written that allowed
  the user to input a desired range of coefficients
  and the desired number of equations.
• For reasons to be discussed shortly, only degrees
  two and three were used.
• In the degree two case, the square errors were
  calculated in arrays for all possible
  combinations of the range of coefficients and
  then stored in a square matrix that was
  Range(C)xRange(C) for comparison. The output is
  the coefficient vector along with the least
  square error. A graph is also generated that
  compares the original note set along the total
  number of notes use with the generated notes.
• The difference in the degree three case was in
  the storage of the square errors. In this case,
  three dimensional storage was required.

15
The Discreet Program
16
Output for Range 3 with 10 Equations
17
Output for Range 3 with 10 Equations
18
Testing With an Actual Recurrence Relation

• An actual recurrence relation was written to test
  the program.
• When the chosen range covered the range of the
  original coefficients, the program generated the
  sequence exactly. Otherwise, it produced a best
  approximation.

19
Test with 1,4,6 as the Coefficient Vector and a
Range of 5 with 7 Equations
20
Test with 1,4,6 as the Coefficient Vector and a
Range of 7 with 7 Equations
21
Test with 1,4,6 as the Coefficient Vector and a
Range of 7 with 7 Equations
22
Future Work

• For more discreet analysis, writing a MATLAB
  program that allows for the continual expansion
  of the degree should be done.
• Applying the same methods to other, perhaps less
  complex, pieces of music (portions of Pachelbels
  Cannon in D Major, for instance).
• For musical analysis, writing a MATLAB program
  that would export the generated notes, replacing
  the originals in the original MIDI file. We might
  not be producing Beethoven, but it might still be
  musically interesting.

23
References

• Steven J. Leon, Linear Algebra With Applications.
  7 ed. pp.51, 382-383,234-244 (2006).
• C.W. Groetsch and J. Thomas King, Matrix Methods
  Applications. pp.115-118, 283 (1988).
• Carla D. Martin, PhD, James Madison University.
• Ken Schutte, Massachusetts Institute of
  Technology
• Mei Chen, PhD, The Military College of South
  Carolina