an experiment with algorithmic composition · 2019. 12. 1. · orbits of conservative systems do...

An Experiment with Algorithmic Composition

Jingwei Liu

Abstract

Many algorithmic composition methods are either developed by imitating the styles and struc-

tures of existing corpus, or only able to produce raw materials for composers or other systems to

utilize. In order to overcome these shortcomings, a conservative dynamical system – double pendu-

lum was modeled to generate compositions in its own right due to some modifications applied by

the algorithm itself. The original data sampled from a double pendulum system was transformed

into staff notes through a natural 2-D to 1-D mapping. This note sequence was then rhythmized

by pattern recognition methods and modified according to some basic rules in music theory. The

melody generated was accompanied by a chord progression in bass clef, which was generated auto-

matically by the rules of common chord progressions and voice leading of common tone approach.

The compositions produced by this method could achieve a high level of variety because firstly, the

chaotic system is sensitive to initial conditions, which means a subtle change of parameters could

change the melody dramatically; secondly, the model is highly adjustable – any key signatures and

time signatures can be specified. Due to the modifications, even sampled from the same system,

two compositions with different key signatures or time signatures can be distinct from each other

significantly.

Keywords: computer music, algorithmic composition, mathematical music, double pendulum,

computational creativity

Extenuating Circumstances Statement

I want to give some explanations about the circumstances of me writing this paper. One year

ago, I got enrolled in Basic Concepts of Music Theory at the UW-Madison. This was my first

music-theory course and also the cause for my career shift. Shortly after I got enrolled, I started

my algorithmic-composition experiment. At that time, I had little theoretical knowledge, but I

was excited, because I knew it was the thing I had been looking for and I desired to see what I

could do. Based on our textbook Theory Gizmos:Fundamental Tools to Understand, Analyze, and

Build Music (Henke, 2011), I started building my compositional system. As you can see, the book

only covers some basic concepts like rhythm, scales and chords. I didn’t even learn consonance

and dissonance at that time. I tried my best to use the knowledge I’ve learned, such as “starting

and ending a period on tonic”, “antecedent and consequent”, “common chord progressions” to set

up the framework of my algorithms. Some of the results look ridiculous now, but at that time it

worked for me. Actually, I sort of circumvented music theory on purpose, since my intention was to

explore computer creativity and see to what extent the computer could do. Besides, I was told that

there were no strict rules for music in the twentieth century. Therefore, I chose to minimize the

restrictions of music theory, allowing the system to reflect the model (chaotic system) properties as

much as possible. At that time, this project was a success for me. I adopted lots of methods from

various fields of study and they fitted right in place. My idea was implemented as imagined.

However, in the next semester, as soon as I got enrolled in another course, Musica Practica

1, I knew my project messed up. Those computer-generated compositions countered every rule I

learned in this course. I was too naive at that time. However, I was also glad that music is so

complicated and well-formed, so that I can exploit my other skills to study it, hopefully producing

more rigorous and meaningful results. When I was taking this course, unfortunately, my work on

algorithmic composition has been closed. Instead of going back and changing everything, I chose

to let it go. This paper has its own value, and what matters most, it is my first paper. I decided

to keep it as it was regardless of the imperfectness. Hope you can find something interesting here.

Please enjoy!

Content

Introduction 1

1 Previous Work Summary 2

2 Double Pendulum System 2

3 Rhythm Patterns Distribution 4

3.1 Time Series Data Mining Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1.1 Local Pattern Recognition: Collision Matrix . . . . . . . . . . . . . . . . . . 5

3.1.2 Global Pattern Recognition: Perceptually Important Points (PIP) . . . . . . 8

3.2 Form Duration Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Step 1: Identify Adjacent Similar Patterns . . . . . . . . . . . . . . . . . . . 10

3.2.2 Step 2: Identify Locally Distinct Motifs . . . . . . . . . . . . . . . . . . . . . 11

3.2.3 Step 3: Identify Globally Similar PIP Modes . . . . . . . . . . . . . . . . . . 12

3.2.4 Step 4: Randomize Rhythm for Remaining Parts . . . . . . . . . . . . . . . 14

3.2.5 Data Adjustment Based on Music Theory . . . . . . . . . . . . . . . . . . . 15

4 Building Chord Progression 16

4.1 Triads and Seventh Chords: Cyclic Group . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Common Chord Progression: Longest Weighted Tree . . . . . . . . . . . . . . . . . 18

4.3 Voice Leading of Common Tone Approach: Common Tone Distance . . . . . . . . . 21

5 Model Evaluation 22

5.1 Control Variable Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Variety: Sensitive to Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.3 Generality: Highly-Adjustable System . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.4 Other Pros and Cons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References 31

CONTENT An Experiment with Algorithmic Composition

Introduction

Inspired by (Beyls, 1991) and (Dubnov & Surges, 2014), in which a distinctive idea about

chaos and creativity was put forward, I started my journey in finding chaotic dynamical models for

musical composition. Many traditional channels in algorithmic composition are either developed

by imitating the styles and structures of existing corpus, or only able to produce raw materials for

composers or other systems to make use of. However, two observations led to the consideration of

totally different methods. First, expert systems are always biased by the knowledge of program-

mers. The biggest difference between art and science is, there is no standard or absolute solution

to a problem in the art field. Every work in art is a product of individual – it all depends on the

creator’s style, experience, preference, societal situation, etc. Even if there is a general formula,

artists would never give up on trying to break it and developing new rules. Second, in program

music, composers get their inspirations from everything in life – nature, poems, romantic experience

and even dreams. This follows the recurring concept along the history of mankind called ”Musica

Universalis” (Castilho, 2015). Everything can be translated into notes on staff papers by talented

composers. As science and technology have developed rapidly since the 20th century, new possi-

bilities are also open to different ways of composing music. Among all of the systems and models,

the complex dynamical system became an alternative due to its simple forms of expression and

surprisingly complex overall behavior.

The article is structured as follows: in the first section, I would briefly summarize the previous

work related to chaotic systems when composing and analyze the pros and cons of these models.

The second section is dedicated to the model we selected – double pendulum system. The system

would be described from the perspective of composition and a map from the 2-D data sampled from

the system to 1-D notes on the keyboard is defined. In the third section, two methods would be

introduced to recognize similar patterns in the note sequence. According to these patterns, four

steps would be devised to distribute durations to the note sequence by allocating the same durations

to similar patterns. The fourth section is meant to construct a chord progression accompanying

the melody generated by the last steps. Based on the principles of common chord progressions, a

chord sets tree would be built. By determining the longest weighted path from the root to a leaf of

this tree, we could find the desired chord progression for this piece. To place these chords, guided

by the voice leading of common tone approach, a new distance is defined to determine the chord

positions. In the last section, the model would be evaluated from multiple perspectives. Its shining

points would be pointed out and the further improvements would be discussed as well.

1

2 DOUBLE PENDULUM SYSTEM An Experiment with Algorithmic Composition

1 Previous Work Summary

Most of the work of musical composition involving chaotic systems makes use of dissipative

chaotic systems (Gogins, 1991) (Pressing, 1988) (Herman, 1993). As we know, chaotic systems can

be classified into two basic types – dissipative and conservative. The dissipative system dissipates

its energy to its surroundings as the system evolves, while the energy of a conservative system is

conserved as a constant no matter how the system evolves with time. Since dissipative systems

lose energy over time, their phase space - that is, the area or volume in which the possible states

of the system take place - shrinks with time. It is this shrinkage that leads to the formation of an

attractor after an initial transient phase. Among all the attractors, what draws the attention of

music researchers most is the chaotic strange attractor, which is sensitive to initial conditions. The

main reason why this particular type of attractor is popular is that they present self-similarity and

fractal structures. ”In general, self-similar musical patterns have multiple levels of structure, with

pleasing regularities but also dotted with sudden changes” (Fernández & Vico, 2013). However,

The methods rooted in dissipative systems are widely regarded as not suitable to produce melodies

or compositions in their own right, but as a source of inspiration or raw material (Bidlack, 1992).

This is mainly caused by the globally stable characteristic of strange attractors. Once passing the

transient phase, the sequence will stay around the attractor (or attractors). This phenomenon is

doomed to cluster the sampled notes, which would make them indistinguishable and unmusical.

However, conservative systems, on the other hand, maintain a constant phase space. Orbits

of conservative systems do not go through a transient phase, nor are they drawn to an attractor.

These properties give us an opportunity to generate compositions directly from the chaotic models.

2 Double Pendulum System

The conservative chaotic system used in this article is the double pendulum system. The orbits

of this system wander erratically almost throughout the phase space, which enables the system to

produce melodies in its own right as the notes won’t cluster. What’s more, this model is sort of

periodic in a rough sense but also has variations in details. These repetitions and variations are

exactly what we need in compositions. This section is dedicated to a brief introduction to the double

pendulum system and a detailed description of the sampling and mapping methods to illustrate

how to turn this dynamic system into a sequence of notes.

Here we apply the fourth Runge-Kutta method to simulate the motion of double pendulum.

When sampling from the trajectory of the lower pendulum, in order not to make the sampling points

too dense or too sparse, I chose to calculate the time for the upper mass to go from the leftmost

to rightmost(or vice versa) in its own motion for the first time. Then sample 6 points evenly in

2

2 DOUBLE PENDULUM SYSTEM An Experiment with Algorithmic Composition

respect to time in the lower pendulum path – which determines the time interval we use to sample

the points afterwards. Here are the first few points I sampled using this principle (Figure 1). As

we can see, these points would neither be too dense to distinguish when mapped to notes nor too

sparse to depict the original shape of its motion.

Figure 1: First Few Points from Sampling

Next we consider the mapping. The points we sampled are on the x-y plane, namely, they are

2-dimensional. However, the notes from keyboard are 1-dimensional. In order to define a reasonable

natural map from 2-D to 1-D, we consider as follows.

As the lower mass moving from left to right, we model this movement as it goes through

the keyboard, with pitches increasing. Under such consideration, we divide the interval [−(L1 +L2), (L1 + L2)] on x-axis evenly, to 12 parts for example, then map part of the keyboard which is

composed of 12 notes to the interval. In order to compose tonal music, we choose 12 notes only from

certain key signature sequentially, rather than 12 notes half step away from each other in chromatic

scale.

As for the y-axis, the intuitive sense reading the graph is – the pitch should go higher when

the value of y gets larger. However, the increase of pitches has been realized by x-axis. Here to

make use of values from y-axis and make the map natural, we consider a way to translate the note

sequence mapped to x-axis to different parts on the keyboard as y variates. To be specific, I divided

y-axis on interval [−(L1 +L2), (L1 +L2)] into several intervals, as the values of y of these samplingpoints change, for example, from an lower interval to its upper one, the corresponding part of the

keyboard mapped to x-axis will translate parallel to the right by 2 or 3 notes. In this way, we get

the mapping note sequence as Figure 2.

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3 RHYTHM PATTERNS DISTRIBUTION An Experiment with Algorithmic Composition

Figure 2: Note Sequence from Sampling

3 Rhythm Patterns Distribution

Since the note sequence has been obtained, the next step is to modify it based on basic rules

of music theory, using tools borrowed from math and computer science, and to achieve our goal of

composing a complete piece fully automated and sounds musical.

The first task is to assign durations to every note. Rhythm is the vitality of music, we cannot

call a piece melody without rhythm. In order to make rhythm patterns fit the time signature (2/4

meter in this example), instead of using random duration allocation for each note (which turned

out to be a failure as it’s too difficult to put the composition produced into measures, letting alone

apply basic concept of music theory to modify it), I adopted a so called predefined rhythm pattern

set as shown in Figure 3.

In the predefined rhythm set, we give the rhythm patterns for 1,2,3,4 notes separately. For

patterns discovered consisting of more than four notes, it’s easy to generate their durations from

the combinations of these basic rhythm patterns. For example, the rhythm of a pattern including

5 notes can be generated by ”2 + 3” or ”3 + 2” pattern directly.

In this section, two time series data mining methods are introduced – collision matrix method

and perceptually important points method – to recognize similar patterns locally and globally

respectively. Then applying the characteristics we discovered from these two methods, four steps

are devised to form the duration array. These steps respectively are: identifying similar patterns,

discovering locally distinct motifs, identifying globally similar PIP modes and randomizing rhythm

4


Figure 3: Predefined Rhythm Patterns

for remaining parts, which are ranked carefully due to certian principles. In the meantime, we

also applied some basic rules in music theory to modify the note sequence to make it classical and

melodious. The Accomplishment of the rhythm distribution process represents a piece of melody

has been successfully composed out of our model.

3.1 Time Series Data Mining Methods

It seems incredible to link time series data mining to the rhythm distribution at first glance

as they come from totally different systems. However, let’s plot a graph to joint these notes from

Figure 2 first. The graph is shown in Figure 4, as you may notice, there is a subtle difference

between the original note sequence and the notes represented here. We can tell that there are three

vertical lines in Figure 4 which give the tonic tones for this piece (E major) and the first note

in this graph starts on the tonic, which is not the case in Figure 2. Actually here we did some

preprocessing for the original data by truncating its first few notes to make it start from tonic while

keeping everything remaining the same. This conduct also can be seen in Algorithm 7 when we

discuss the data adjustments based on music theory.

Figure 4 somehow looks like the data sequence from stock market right? Inspired by this

discovery, and following the basic intuition in finding repetition and motifs in music, we adopted

two methods from time series data mining to distribute durations.

3.1.1 Local Pattern Recognition: Collision Matrix

Definition 3.1. (Time Series) A time series is a series of data points T =< t1, t2, . . . , tm > indexed

(or listed) in time order, where m is the length of the time series.

Suppose sp is a subsequence of the time series T with length w, i.e. |sp| = w. DenoteS = {s1, s2, . . . , sq} to be the set of all subsequences of T with length from 1 to m (Tang & Liao,2008). For any 1 ≤ i < j ≤ q, si, sj are the subsequences of T with equal length w, namely

5


Figure 4: Math Representation of Note Sequence

si =< si, si+1, . . . , si+w−1 >

sj =< si, sj+1, . . . , sj+w−1 >

Since each subsequence is consecutive and of equal length, we can name it by its first element

without confusion. Now we observe that as patterns to be compared, these two subsequences

shouldn’t overlap with each other, namely j − i ≥ w.

Definition 3.2. (Collision Matrix ) The collision matrix of T is a q×q matrix, and for each elementof the matrix denoted as eij ∈ M , eij = collisionHit(si, sj) where collisionHit is the similaritydegree of two subsequences, for example, si and sj. In other words, the collision matrix records the

similarity degree between any two subsequences si, sj ∈ S, 1 ≤ i, j ≤ q.

However, in regard to comparison, we only define collisionHit between two sequences with

same length in this problem. Besides, as we mentioned before, without loss of generality, we assume

1 ≤ i < j ≤ q and in order to avoid overlapping, we set j − i ≥ w, where w is the length ofsubsequences si and sj. Consequently, all terms in the collision matrix not satisfying these criteria

would be dismissed (represented by NaN for example).

In this problem, the similarity degree is measured by the Euclidean distance – the difference

between sequential notes in mathematical representation. This intuition comes from a music concept

called sequence (Henke, 2011).

6


Definition 3.3. (Motif (music)) A short rhythmic idea repeated in different ways throughout a

composition is called a motif.

Definition 3.4. (Sequence (music)) When a motif is repeated using the same pattern of intervals

but starting on a different pitch, it is called sequence.

Since the pattern doesn’t depend on which pitch it starts with, only the shape of each pattern

matters regardless of its position. The shape of each pattern can be depicted by the difference

of successive points. If the difference sequence is exactly the same between two patterns, their

similarity degree is accurate. However, we don’t need two patterns exactly the same to say they

have similar shapes. Here we define 3 similarity degrees: Accurate, medium-similar and loosely

similar. The difference sequence B = (b1, b2, . . . , bm−1) of the note sequence P = (p1, p2, . . . , pm) is

defined as: for ∀i, 1 ≤ i ≤ m− 1,bi = pi+1 − pi

The procedure to find collision matrices is described in Algorithm 1. Instead of forming a big

sparse matrix as described in Definition 3.2, we’d love to disassemble it into several small and dense

matrices according to the lengths and similarity degrees of similar patterns. The result is shown in

Table 1.

Algorithm 1 Collision Matrix

Input: Difference sequence B = (b1, b2, . . . , bm−1)

Output: Collision matrices of pattern length 2, 3, 4 . . . for each similarity degree accordingly

1: Give 3 similarity degrees: accurate, medium-similar and loosely similar

2: for each similarity degree do

3: Traverse B again and again to find all similar subsequences of length 2, 3, 4 . . . until there is

no similar subsequence in given length.

4: end for

Table 1: Collision Matrices Enumeration

Similarity Degree Collision Matrices (Pattern Length Subscript)

Accurate A2, A3, . . . , Am

Medium B2, B3, . . . , Bn

Loose C2, C3, . . . , Cl

Here all these matrices have 2 rows – the upper and lower terms of the same column represent

similar patterns (the length of the patterns are denoted by the subscript of the matrix), and the

number of columns are determined by the number of similar patterns of the given length.

7


3.1.2 Global Pattern Recognition: Perceptually Important Points (PIP)

The identification of Perceptually Important Point (PIP) is first introduced by (Chung, Fu, Luk,

& Ng, 2001) and used for pattern matching of technical (analysis) patterns in financial applications

(Fu, Chung, Luk, & Ng, 2008). The concept of data point importance is defined by the influence

of a data point on the shape of the time series. A data point that has a greater influence to the

overall shape of the time series is considered as more important. The data point with importance

calculation is named as perceptually important point (PIP) and the process to identify the PIPs

is as follows: given time series P , the first two PIPs are given by the first and last points in P .

The next PIP will be the point in P with the largest distance to the first two PIPs. The fourth

PIP will then be the point in P with the largest distance to its two adjacent PIPs, either between

the first and second PIPs or between the second and the last PIPs. The process of locating the

PIPs continues until we find enough number of PIPs we demand. The PIP identification process is

described as pseudo code in Algorithm 2.

Algorithm 2 PIP Identification

Input: Time series P = (p1, p2, . . . , pm)

Output: PIP list L = (l1, l2, . . . , ln), n ≤ mSet l1 ← p1, l2 ← pmrepeat

Traverse P to identify pi with maximum distance to its adjacent points in PIP list.

Append pi to L

until Length(L) = n

To calculate the distance of the points in between two adjacent PIPs to these two points,

there are three types of distance called Perpendicular Distance (PD), Vertical Distance (VD) and

Euclidean Distance (ED) as shown intuitively in the figures below. In this algorithm we use VD

to be our measurement, let pi(xi, yi) denotes the test point, pil(xil, yil), pir(xir, yir) represent its

left-adjacent point and right-adjacent point in the current PIP list, then the measure is written as

dist(pi; pil, pir) = |yil +yir − yilxir − xil

(xi − xil)− yi| (1)

Where dist(pi; pil, pir) denotes the distance of pi to pil and pir. Through this procedure, we

identified PIPs nearly every 8 points per PIP in our note sequence Figure 2, the result is shown in

Figure 8 where PIPs are represented by ”*”.

8


Figure 5: Perpendicular Dis-

tance

Figure 6: Vertical Distance Figure 7: Euclidean Distance

Figure 8: PIP Identification

9


3.2 Form Duration Array

We initialize our duration Array d to be an array of the same length as the note sequence with

default values 0. Then we take the following 4 steps to fill in d gradually according to the pattern

characteristics we discovered in section 4.1.

3.2.1 Step 1: Identify Adjacent Similar Patterns

Among all the similar patterns found by collision matrices, we pay particular attention to

those adjacent ones. When we listen to music, it’s always the consecutive repetitions that draw our

attention mostly. Here we use this intrinsic property to distribute our rhythm. For two adjacent

similar patterns with default duration sequences, we randomize one duration sequence based on the

predefined rhythm set (Figure 3) and assign its value to the other one. See Algorithm 3.

Algorithm 3 Identify Adjacent Similar Patterns

1: for all collision matrices of 3 similarity degrees do

2: Find adjacent similar patterns in every matrix, e.g. s1 and s2

3: Make sure their corresponding duration subsequences d1 and d2 are in default setting.

4: Randomize d1 according to the predefined rhythm set.

5: Set d2 ← d16: end for

7: return The newly updated duration array d.

In Algorithm 3, step 3 seems too strict. For example, if d1 is determined previously while d2

is in default value, why not assign d1 to d2? The process wasn’t set to operate this way mainly for

two reasons:

• To insure the duration of d2 is generated by the predefined rhythm set so that it can be fitinto measures. Actually d1 can be a combination of the latter part of one measure and the

former part of its next measure, which may not belong to the predefined rhythm set. What’s

worse, giving the value of such a d1 to d2 would probably cause the rhythm mess up around

d2 and its following compositions.

• It doesn’t matter if we discovered some adjacent patterns but failed to assign same durationsto them. We can apply other characteristics using methods through later steps.

We also need to point out that in this duration array formation process, once the duration of a

note is assigned, it’s locked. In other words, the operations won’t change the existing values. Only

the default values would be modified through each step until every element in d is changed from

default value.

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3.2.2 Step 2: Identify Locally Distinct Motifs

As defined in Definition 3.3, we want to locate the short rhythmic patterns recurring in a com-

position. Looking into the collision matrices we got earlier, it’s instinctive to see that patterns with

higher similarity degrees and longer lengths are more distinguishable than shorter, low-similarity

sequences in a melody. Guided by this discovery, we rearrange the matrices in Table 1 as

Am, Bn, Cl, Am−1, Bn−1, Cl−1, Am−2, Bm−2, . . . (2)

We enumerate these matrices from the longest length of every degree from high to low re-

spectively and decrease the lengths sequentially. We exclude the case of pattern length 2 here as

we don’t consider the similar patterns of length 2 to be distinct. Consequently, when one of the

subscripts in (2) decreases to 3, without loss of generality, we assume it’s A3. Then the sequence

would be

Am, Bn, Cl, . . . , A3, Bn−(m−3), Cl−(m−3), Bn−(m−3)−1, Cl−(m−3)−1, . . . (3)

As the rearrangement rotates within A,B and C in decreasing subscripts, when the subscript

reaches 3, this type of matrices disappear and the rotation continues within the remaining types

until the subscripts of all 3 types of matrices decrease to 3. The method of distributing durations by

using this rearranged matrices sequence is like what we did in step 1. We randomize the duration

for one pattern and assign its value to its similar pattern given the condition that they are both at

default settings. The detailed process is described in Algorithm 4.

Algorithm 4 Identify Locally Distinct Motifs

Input: All collision matrices found by Algorithm 1 as listed in Table 1

1: Rearrange these matrices to be Am, Bn, Cl, Am−1, Bn−1, Cl−1, Am−2, Bm−2, . . .

2: for every matrix in the rearranged order do

3: for every pair of similar patterns listed in the matrix do

4: Make sure their corresponding duration subsequences d1 and d2 are in default setting.

5: Randomize d1 according to the predefined rhythm set.

6: Set d2 ← d17: end for

8: end for


The reason for designing Step 4 in Algorithm 4 is the same as we mentioned in Algorithm 3.

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3.2.3 Step 3: Identify Globally Similar PIP Modes

Observing the PIPs found in Figure 8, we notice that the shapes determined by PIPs are

somehow repeated. In other words, we can find pairs of PIPs of similar shapes around them. In this

way, we are able to distribute durations due to the global trend led by PIPs of the note sequence.

In implementation, we first apply sign function to the difference sequence B = (b1, b2, . . . , bm−1).

When defining shapes globally, we only care about the trend – it’s increasing, decreasing or invariant.

We ignore the information about how much it changes. It’s much relaxed than the similarity degrees

we gave above. In this way, we define similar PIPs only by the similarity of its sign shapes around

them. Then we are ready to assign durations from a looser requirement and global view to the

remaining default positions.

Through the operations in Step 1 and Step 2, the duration array is filled little by little. Maybe

it’s not rigorous to phrase this way. Due to the pattern recognition process and the predefined

rhythm set, the duration distribution is actually part by part but not note by note, which also

causes the remaining notes waiting to be assigned durations gathered together – they also appear

as patterns, several notes sequentially, not scattered. To assign durations to the remaining parts

according to similar PIPs, we design two steps.

• Find Patterns of Exact Length Fitting the Vacant Space

First of all, we need to locate these remaining default subsequences. From the perspective of

these vacant subsequences, for every one, we traverse the whole sequence to find the patterns

fitting the predefined rhythm set and of the same length as this vacant part. If the pattern

found and the given part are from the same segment of similar PIP shapes, we assign the

pattern’s duration to the default part. If no pattern satisfying the above requirements exists,

it’s ok to move on to the next vacant one and leave this part waiting for the following steps

to determine its values.

In order to find the patterns fitting the predefined rhythm set, namely, the patterns need to

be able to fit into measures themselves, we introduce a concept called ”Isolated Pattern”.

Definition 3.5. (Isolated Pattern) A subsequece of duration array with default values right

ahead and after it is defined as an isolated pattern. An isolated pattern is not adjacent to

any assigned values.

The method to find patterns of exact length fitting the blank space is described in Algorithm

5. We repeat this process several times before moving to the next stage.

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Algorithm 5 Patterns of Exact Length

1: Find the location and length of blank space e1, e2, . . . , en in duration array d.

2: for i = 1 : n do

3: repeat

4: Continue traversing d from current location (initial location: d(1)) until we find an isolated

pattern g of same length as ei.

5: if g doesn’t exist then

6: Break;

7: else

8: Find the PIPs pi, pj closest to g and ei respectively

9: if (pi and pj are similar PIPs) & (g and ei are at the same position w.r.t. pi and pj) &

(g has at least one nonzero element) then

10: Set ei ← g11: Break;

12: else

13: Continue;

14: end if

15: end if

16: until d is traversed.

17: end for


The method to find patterns of exact length fitting the blank space is described in Algorithm

5. We repeat this process several times then move to the next stage.

• Find Patterns of Shorter Length Fitting the Blank Space

There is an obvious omission in Algorithm 5. If there is a huge vacant sequence, for example,

a sequence of length 13 by chance, what should we do? It’s difficult to find an isolated pattern

of length 13 so this space would probably be left empty after the last step. In order to address

this problem, we choose to insert the longest isolated pattern shorter than 13, which comes

from the same segments of similar PIPs as the vacant part, into the most suitable part of

the vacant space to complete the rhythm distribution process. The procedure is described in

Algorithm 6. We repeat this process several times as well.

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Algorithm 6 Patterns of Shorter Length

1: Find the location of blank space e1, e2, . . . , en in duration array d.

2: for i = 1 : n do

3: Set l = length(ei)

4: for q = l − 1 : −1 : 2 do5: repeat

6: Continue traversing d from current location (initial location: d(1)) until we find an

isolated pattern g of length q.

7: if g doesn’t exist then

8: Break;

9: else

10: Find the PIPs pi, pj closest to g and ei respectively

11: if (pi and pj are similar PIPs) & (g and ei are at the same position w.r.t. pi and pj)

& (g has at least one nonzero element) then

12: Insert g into ei. The position of g in ei is adjusted by its relative position to the

PIP.

13: Break the loop and continue from the next iteration of Step 2.

14: else

15: Continue;

16: end if

17: end if

18: until d is traversed.

19: end for

20: end for


3.2.4 Step 4: Randomize Rhythm for Remaining Parts

After all of the operations through Step 1-3, now our duration array d is almost full. For the

remaining sporadic default parts R = {r1, r2, . . . , rk}, we simply randomize their durations due tothe predefined rhythm set. Here we still split the operations into two steps:

• If two elements in R are of same length and they belong to the same segments of similar PIPs,we randomize the duration of one element by the predefined rhythm set (Figure 3) then give

its value to the other element.

• After performing the last step, if there still are elements left in R not assigned, we simply ran-domize their durations according to the predefined rhythm set. Then the rhythm distribution

14


procedure is finally completed.

Remark. We have mentioned that through the duration distribution process, once one value

is assigned, it’s fixed, namely whatever operations we take afterwards, this value is not viable to be

changed. This property indicates that the order of operations is vital. We need to rank all these

operations reasonably to make sure the distribution process works methodically. In general, we

follow these principles: ranking from most important to least important, from local to global, from

scarce to plentiful, from strict to loose. We analyze the reasons why we design Steps 1-4 in the given

order in Table 2.

Table 2: Analyze the Rank of Each Step

Step 1: Identify adjacent similar patterns Adjacent similar patterns are rare and signifi-

cant; This is the characteristic we want to high-

light most

Step 2: Identify locally distinct motifs These motifs are local and important; This is

a very conspicuous local characteristic in note

sequence

Step 3: Identify globally similar PIP modes These modes are more general and of looser re-

quirements; It depicts the global feature of note

sequence

Step 4: Randomize duration for remaining parts This is the final complementary part; It applies

the fewest characteristics in note sequence

3.2.5 Data Adjustment Based on Music Theory

According to some basic concepts of music theory and composition (mostly by the general rules

of classical music), we made some adjustments to the note sequence P and duration array d, to get

the whole piece start from the tonic and end on the tonic at every period. We also accelerated the

third measure and retarded the forth measure of every phrase to build the contrast of tension and

peace as well as propelling the melody to move more smoothly. The details of conducting these

adjustments are listed in Algorithm 7.

15

4 BUILDING CHORD PROGRESSION An Experiment with Algorithmic Composition

Algorithm 7 Adjustment Based on Music Theory

1: Truncate first few points of data sequence P = (p1, p2, . . . , pm) s.t. P starts from tonic. Denote

the new sequence as P̃ .

2: Apply duration array forming procedure to P̃ to get duration array d. Then we accelerate the

third measure of every phrase to build the tension and retard the forth measure of each phrase

to cause a sense of rest or ending.

3: Add one tonic note to the end of each period. This note takes up the whole ending measure of

the period and its position is determined by minimizing the distance from its previous note.

4: We get a new note sequence P ∗ and the adjusted duration array d∗.

Figure 9 shows what we got from the original note sequence (depicted in Figure 2) after the

complete rhythm patterns distribution process .

Figure 9: Melody in Treble Clef

4 Building Chord Progression

The melody part has been completed as shown in Figure 9, the next step we’d love to add

chord progressions in bass clef to accompany the treble clef harmonically. The process of forming

the desired chord progression is accomplished through three steps. We first map the music scale to

Z7 group and use 7 elements of Z7 to represent all notes in the melody and all triads and seventhchords. Next, we use combination to find out all chord sets harmonizing with the melody in each

measure, then determine the best chord progression by picking up the longest weighted path in

16


the chord sets tree according to the common chord progression. Finally, in order to position these

chords properly, we applied a metric derived from the principle of voice leading of common tone

approach, and we adjusted these chords to be placed within one octave from the melody line, which

completed the chord progression building process.

4.1 Triads and Seventh Chords: Cyclic Group

It can be perceived directly that the structure of octave in tonal music is isomorphic to Z7cyclic group. If we use numerals 1 ∼ 7 to replace letters A-G which represent pitches in one octave,we can write all triads and seventh chords as following:

Mapping: tonic → 1; supertonic → 2; mediant → 3; subdominant → 4; dominant → 5; submedi-ant → 6; leading tone → 7

Triads: {1, 3, 5}; {2, 4, 6}; {3, 5, 7}; {4, 6, 1}; {5, 7, 2}; {6, 1, 3}; {7, 2, 4}

Seventh chords: {1, 3, 5, 7}; {2, 4, 6, 1}; {3, 5, 7, 2}; {4, 6, 1, 3}; {5, 7, 2, 4}; {6, 1, 3, 5}; {7, 2, 4, 6}

Here we use ”set” but not ”array” because the chords can be in any order due to inversions.

As to how to position these chords, we will discuss it later in Section 6.3.

Now we use this method to mark all the notes in our adjusted note sequence P ∗. They will

be represented by 1 ∼ 7 seven elements of Z7. Besides, according to duration array d∗, we woulddivide these notes into measures and put all notes in the same measure into one set (here we don’t

merge same elements in the set). As we accelerated the third measure in each phrase as described

in Algorithm 7 Step 2, we’d love to add two chords in the third measure, so here we divide the

notes in this measure into two sets of same durations. In this example, As shown in Figure 9, the

set of numeral sets for the first period is

S = {{1, 7, 2}, {5, 5}, {3, 2}, {4, 5, 5}, {2, 6}, {5, 2, 1}, {6, 2, 4, 1}, {4, 1, 3, 2}, {7, 5}, {1}} (4)

Then we use combination to find the triads or seventh chords that match each set in S. For

si ∈ S, we program to find the largest subsets of si that equal or belong to some triads sets orseventh chords sets. For example, the first set in S is {1, 7, 2}, whose largest subset is itself, so wetraverse the sets of triads to see if there is any match. However, there is no {1, 7, 2} in triads set. Itturns out that there is no set in seventh chords that takes the set {1, 7, 2} as a subset as well. Thenwe decrease the size of subsets to 2. There are 3 subsets {1, 7}, {1, 2}, {7, 2}. We match them tothe sets of triads. There is no match for {1, 7} and {1, 2}, but for {7, 2}

{7, 2} ∈ {5, 7, 2} & {7, 2} ∈ {7, 2, 4}

17


As a result, the chord set for the first measure {1, 7, 2} is {{5, 7, 2}, {7, 2, 4}}. However, if weonly have {1, 2} in this measure, there is no matching triads. Now we would look out for sets inseventh chords, then we found match {2, 4, 6, 1}, which is the only set that includes elements 1 and2. Notice that we can always find a match for any 2-element sets in seventh chords, so the chord

sets corresponding to every numeral measure set in S would never be empty. Besides, according

to our adjustment before (Algorithm 7), the ending measure of each period is formed just by one

tonic tone. We set its corresponding chord to be {1, 3, 5} always. The corresponding chord sets for(4) is listed in Table 3.

Table 3: Corresponding Chord Sets

Measure Chord Sets

{1, 7, 2} {5, 7, 2}, {7, 2, 4}{5, 5} {1, 3, 5}, {3, 5, 7}, {5, 7, 2}{3, 2} {3, 5, 7, 2}{4, 5, 5} {5, 7, 2, 4}{2, 6} {2, 4, 6}{5, 2, 1} {1, 3, 5}, {5, 7, 2}{6, 2, 4, 1} {2, 4, 6, 1}{4, 1, 3, 2} {2, 4, 6, 1}, {4, 6, 1, 3}{7, 5} {3, 5, 7}, {5, 7, 2}{1} {1, 3, 5}

4.2 Common Chord Progression: Longest Weighted Tree

In music theory, there is a chart called Common Chord Progression as shown in Figure 10.

The term ”common” means most chord progressions in classical music follow the rules listed in this

chart like I → IV → V → I. Besides, I chord can move to any chords as the dashed line in Figure10 shows.

The Roman numerals here correspond to the numeral representation we developed in the last

section directly. The numeral of the root for each chord (the first element of each chord set) indicates

its Roman numeral representation. For example, the chord {1, 3, 5} corresponds to Roman numeralI. We notice that the Roman numerals are case-sensitive. The reason why they present the way

shown in Figure 10 is that this example is in E major, so chords I, IV, V are major, ii, iii, vi are

minor and viio is diminished.

In order to determine the best chord progression for the piece, we devised an algorithm using

tree structure and we turned this optimization problem into finding the longest weighted path from

18


Figure 10: Common Chord Progression

root to leaf in the chord sets tree. To form the chord sets tree following the rules of common chord

progression, we first set the root to a triple (I, 1, 0), which represents the Roman numeral, repeated

times and weight separately. It’s possible for a chord to repeat itself especially when there are more

than 5 notes in one measure set. As we can tell, if one chord shows up more frequently than others,

it’s supposed to fit the melody better. Besides, there are chord progressions wider-used than others.

For example, the chord progression I → IV → V → I is more popular than the progression I → iii→ vi → I. As a result, we assign different weight to different chords according to its popularity incomposition. Thus, the repeated times and weight assigned to each chord are used as indicators of

the chord’s suitability for each measure.

Back to the construction of the chord sets tree. We set the root to (I, 1, 0), which indicates

that the chord for the first measure is forced to be I, which agrees with the melody part starting

at the tonic (Algorithm 7 Step 1). Then we move to the second measure. Since chord I can go

to any chords next, we concatenate all chords in the chord sets for this measure to the root as

its child notes, with repeated times and weight computed. Next, we move to the third measure.

Here we have to examine all chords concatenated in the last step, taking them as parent nodes,

then matching them with chords in the current measure according to the common chord progression

chart (Figure 10). If the progression is displayed in the chart, we concatenate it to its corresponding

parent node; if not, we leave the parent node alone and move to the next chord. Obviously, if there

is no matching chord in the current measure to one of the parent nodes, this node would be left as

a leaf thus this branch would be abandoned. However, if there is no matching chord for all of the

parent nodes, the tree would be cut off, which means the chords for the following measures cannot

be appended to the tree. To avoid this from happening, under such situation, we would concatenate

all chords in the current chord sets to every parent node, then continue the process above until the

last measure. Applying this procedure to our example in Table 3, we get the chord sets tree for the

first period of our melody as illustrated in Figure 11.

Now we want to determine the longest weighted path from root to leaf in the chord sets tree.

Utilizing the second and third terms of the triple, namely the repeated times and weight assigned

19


Figure 11: Chord Progression for the First Period

20


to each chord, we can easily compute the total weight of each complete path covering all measures

from root to leaf. Comparing all the weights computed, we find out the ”best” chord progression

with the biggest weight for this piece.

4.3 Voice Leading of Common Tone Approach: Common Tone Distance

Since we already determined the chord progression along the bass clef, now the only thing left

is to position these chords appropriately. Although we have triads and seventh chords, considering

the practical way of playing instruments (piano for example), we would only position 3 notes for

each chord in bass clef. For the seventh chords, as the first note in treble clef of each measure would

be played simultaneously with its accompanying chord, if this note belongs to this seventh chord,

we can simply take it as a component of the chord and position the rest 3 notes in the bass clef; if

not, we eliminate the fifth of this chord and display the rest 3 notes as a chord, which is generally

considered as a way to preserve the features of seventh chords mostly.

Here I want to follow the voice leading of common tone approach in composition and devise

an algorithm to achieve this rule to the greatest extent. In order to translate this approach into

mathematical language, we give the definition of common tone distance first.

Definition 4.1. (Common Tone Distance) The distance between a note and its previous chord is

defined as the minimum distance of the note to every note of this chord.

To be more specific, we denote our note as X and its previous chord composed of 3 notes as

(A,B,C). Then the distance between this note and the 3 chord notes can be represented as a vector

(a, b, c). To aggregate the distance vector as a scalar, the most common way would be

Arithmic Mean :|a|+ |b|+ |c|

3

or

Quadratic Mean :

√a2 + b2 + c2

3

However, the common tone approach means that we want to keep one tone invariant throughout

several measures. To achieve this goal, we introduce the distance as

Common Tone Distance : min{|a|, |b|, |c|}

In this way, once there is any possibility for the note to get to the same tone as one of the

chord’s notes, the distance would automatically be 0, which achieves the optimum since we would

minimize the distances to determine the best position for this note. This strategy gives a way to

realize the voice leading of common tone approach.

21

5 MODEL EVALUATION An Experiment with Algorithmic Composition

The common tone method gives the relative position among chords. Nevertheless, we still need

to figure out chord positions in relation to the melody line. We neither want these chords too close

to the melody line to separate from each other, nor see them too far apart to be an integral whole.

Based on the chord positions given by common tone approach, we modify them to fit our demands

in the way of putting the highest pitch of each chord within one octave of the lowest pitch of its

corresponding measure of the melody. In this way, the chords company part is finally completed.

Our final result is shown in Figure 12.

5 Model Evaluation

There are some distinctive advantages of this model – the generality and variety it achieved

are quite impressive. With respect to these properties, An thorough analysis with demonstrating

examples are provided in this section. As well, there are also other pros and cons of this method.

We will cover them in the last part.

5.1 Control Variable Method

For all examples given below, we followed the control variable method to the greatest extent.

The control group is given as Figure 13. This example is another composition generated by the

pre-described process with different parameters for the double pendulum system. These control

experiments provided the best demonstration for the generality and variety of our method.

5.2 Variety: Sensitive to Initial Conditions

The notes are sampled from chaotic system, one of whose most distinct features is sensitive to

initial conditions. This indicates that with a subtle change of initial data, we are promised to get a

totally different sequence of notes, which means the system can generate infinite distinctive pieces

of melodies.

For example, we only made a change to the initial positions of the double pendulum and

kept every other parameter unchanged, going through the same procedure, a new composition was

produced as shown in Figure 14.

5.3 Generality: Highly-Adjustable System

The model is highly adjustable. The users are allowed to choose any key signature (major and

minor), any time signature and change the predefined rhythm patterns. The users can also change

22


Figure 12: Final Piece

23


Figure 13: Control Group

24


Figure 14: Contrast: Change of Initial Positions

25


the weight of chord progressions according to personal preference. All these functions have added

more flexibility to the possible melodies produced by the raw data sequence.

• Key Signature

Here we give an example of a composition generated by this model in D minor (harmonic

minor), in contrast with the original example which is in E major. Other conditions would

stay the same (Figure 15).

Figure 15: Contrast: Composition in D Minor

• Generalized Rhythm Set

We can change the predefined rhythm patterns as well. As you may notice, the rhythm

set we used in the original example (Figure 3) is quite limited. There are only a few of

the rhythm combinations composers often use in practice listed in this figure and they are

pretty conservative. Here we want to generalize the predefined rhythm set by adding more

26


syncopation into it to make our composition versatile. The generalized set is shown in Figure

16.

Figure 16: Generalized Rhythm Set

Here we eliminated the 4-note patterns since we want them to be more multifarious, thus

these patterns as well as patterns with more notes are going to be produced by combinations

of the fundamental 1,2,3-note patterns. For example, patterns consisting of 4 notes can be

generated by ”1+3” or ”2+2” modes.

A composition generated under the generalized rhythm patterns is given in Figure 17. As we

can see, there is a new pattern called triplet appearing in the melody and more syncopation,

which have added the composition’s diversity.

• Time Signature

Our original example is in 2/4 time signature. Here we give an example in 3/4 (Figure 18)

produced through the same process described in this article.

5.4 Other Pros and Cons

Another advantage of the model is automation. The composition process is totally automated,

which differentiate this method from many other algorithmic composition methods which require

expert knowledge and human creativity. The system itself is capable of composing a whole piece

without human intervention. In this way, the composition produced won’t be biased by personal

style and music taste, which means this method actually has delegated all creativity to the computer.

Besides, the principles from music theory used to adjust the sequence are indeed general and

widely accepted. In other words, these adjustments only give a frame to the piece but not a certain

specific style. These modifications are necessary to make the piece integral and musical but it won’t

affect the major role played by the original sampled data.

27


Figure 17: Contrast: Composition of Syncopation

28


Figure 18: Contrast: 3/4 Time Signature

29


Apart from all the superiorities above, however, some of the rules applied in the algorithm

are indeed rigid and determined to some extent, such as ending every period on the tonic. These

principles are widely used but not permanent. If the computer wants to compose master pieces,

it has to be taught to break the rules and to achieve the real level of creativity, which can be

considered as one of the possible improvements to the system in the future.

30

References An Experiment with Algorithmic Composition

References

Beyls, P. (1991). Chaos and creativity: The dynamic systems approach to musical composition.

Leonardo Music Journal , 31–36.

Bidlack, R. (1992). Chaotic systems as simple (but complex) compositional algorithms. Computer

Music Journal , 16 (3), 33–47.

Castilho, P. L. (2015). Chaotic systems as compositional algorithms mus-15.

Chung, F.-L., Fu, T. C., Luk, R., & Ng, V. (2001). Flexible time series pattern matching based on

perceptually important points.

Dubnov, S., & Surges, G. (2014). Delegating creativity: Use of musical algorithms in machine

listening and composition. In Digital da vinci (pp. 127–158). Springer.

Fernández, J. D., & Vico, F. (2013). Ai methods in algorithmic composition: A comprehensive

survey. Journal of Artificial Intelligence Research, 48 , 513–582.

Fu, T.-c., Chung, F.-l., Luk, R., & Ng, C.-m. (2008). Representing financial time series based on

data point importance. Engineering Applications of Artificial Intelligence, 21 (2), 277–300.

Gogins, M. (1991). Iterated functions systems music. Computer Music Journal , 15 (1), 40–48.

Henke, J. (2011). Theory gizmos:fundamental tools to understand, analyze, and build music. Text-

book Consortia.

Herman, M. (1993). Deterministic chaos, iterative models, dynamical systems and their application

in algorithmic composition. In Proceedings of the international computer music conference

(pp. 194–194).

Pressing, J. (1988). Nonlinear maps as generators of musical design. Computer Music Journal ,

12 (2), 35–46.

Tang, H., & Liao, S. S. (2008). Discovering original motifs with different lengths from time series.

Knowledge-Based Systems , 21 (7), 666–671.

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