Evolving Four Part Harmony using a Multiple Worlds Model

Marco Scirea

and Joseph Alexander Brown

Center for Computer Games Research, IT University of Copenhagen, Copenhagen, Denmark

Artiﬁcial Intelligence in Games Development Lab, Innopolis University, Innopolis, Republic of Tatarstan, Russia

Keywords:

Genetic Algorithms, Procedural Content Generation, Music.

Abstract:

This application of the Multiple Worlds Model examines a collaborative ﬁtness model for generating four part

harmonies. In this model we have multiple populations and the ﬁtness of the individuals is based on the ability

of a member from each population to work with the members of other populations. We present the result of two

experiments: the generation of compositions, given a static voice line, both in a constrained and unconstrained

harmonic framework. The remaining three voices are evolved using this collaborative ﬁtness function, which

looks for a number of classical composition rules for such music. The evolved music is found to meet with

the requirements placed on it by musical theory. Using the data obtained while running our experiments we

observe and discuss interesting qualities of the solution space.

1 INTRODUCTION

A great variation of techniques have been used for

the generation of music material. A large number

of algorithms have been researched and explored in

this particular application. This might be because

music is, to a certain degree, quantiﬁable and ab-

stract. There exists artiﬁcial intelligence systems that

can produce very high quality music. (Cope, 1991)

is focused on codifying music styles is very inter-

esting and many more approaches are described by

(Miranda, 2013). These algorithms can range from

creating entire pieces of music, to solving speciﬁc

problems (for example how to do a transition from a

phrase A to a phrase B), to being a compositional aid

and many more applications. Evolutionary algorithms

have been a popular technique to use in this kind of

application as this approach, based on the concept of

random variation and selection, can efﬁciently search

a large solution space. Notable works using this ap-

proach are (Dahlstedt, 2007) on evolving complete

piano pieces, (Hoover et al., 2011) using interactive

evolution of accompaniments, and (Miranda, 2003)

in the evolution of music in artiﬁcial life societies.

The Multiple Worlds Model (MWM) (Brown and

Ashlock, 2010) is an evolutionary approach which

uses a multiple population approach with a collabora-

tive ﬁtness function between the populations. It uses

multiple evolving populations acting upon the same

target optimization, not unlike island models (Whitley

et al., 1997). However, island models have the goal of

providing a single solution to a problem and pass ge-

netic traits to each other by an explicit migration step.

The chromosomes in an island model are part of the

same species as they breed amongst themselves. The

multiple worlds model separates these populations on

the genetic level, they are separate species under the

biological species deﬁnition — they do not interbreed,

due to biological infertility, or behavioral differences.

The goal of the MWM is not to provide a single good

solution, but a set of interacting solutions.

Previous examples of MWM have been demon-

strated to split and join sets of iterated prisoners

dilemma playing agents and produce new agents

(Brown, 2012; Brown, 2013; Brown, 2014). (Ashlock

and McEachern, 2011) used a MWM in order to sim-

ulate the biological networks produced by species of

bacterium using game theory. Further, it has been

used to model the splits in a radio station market

based on the preferences of listeners in the demo-

graphic area (Brown, 2014). Each of the populations

represents a radio playlist and ﬁtness is deﬁned by

a set of listeners who change stations based on an

enjoyment value. These studies of the MWM have

been competitive, each of the species is ﬁghting over

food resources. This study has a collaborative aspect,

with each of the species, a single musical voice, hav-

ing to construct a which avoids conﬂict with other

voices. Similar process of divergence are seen in

roots, leaves, and seed structures of plants with a his-

220

Scirea, M. and Brown, J..

Evolving Four Part Harmony using a Multiple Worlds Model.

In Proceedings of the 7th International Joint Conference on Computational Intelligence (IJCCI 2015) - Volume 1: ECTA, pages 220-227

ISBN: 978-989-758-157-1

tory of coexistence (Tilman and Snell-Rood, 2014).

This process is known as Adaptive radiation. The

properties of this species were ﬁrst fully examined

by (Lack, 1947) and were not used by Darwin’s ex-

aminations due to a number of misclassiﬁcations in

the species caused by poor record keeping (Sulloway,

1982). Finches were found to have large changes to

their phenotypic traits in even short periods of time

due to changes in food sources (Grant and Grant,

1979; Grant and Grant, 1982; Grant and Grant, 1983).

Furthermore, behavioural modiﬁcation can also

lead to specialization in food sources through a pro-

cess of niche partitioning. Hanson in Feathers gives

an anecdotal account of studying the behaviours of

North American bird actions in a forest: “Nuthatches

foraged mostly on the trunks, Chickadees dominated

the main branches, and Kinglets spend their time ﬂit-

ting about in the side branches” (Hanson, 2011). The

MWM aims to use such principles of inter-population

competition with intra-population evolution to guide

a process of partitioning into models.

The niche effect is intrinsic to MWM. It does

not require an explicit calculation of phenotype dis-

tance, or a crowding measure, the novel method of

ﬁtness evaluation is implicitly making crowding un-

desirable. It has also been shown to increase the

diversity of ﬁnal solutions of the creation of multi-

ple models (Brown, 2014). This diversity has been

seen in studies of mixture vs. monoculture plants in

(Zuppinger-Dingley et al., 2014), which examined the

results of eight years of experimental growth in Jena,

Germany. It showed there was an increased interspe-

ciﬁc difference to those plants grown in mixture types

compared (P < 0.05) and intraspeciﬁc distance within

mixture types on traits was increased (P = 0.101).

They attribute a difference in relative speciﬁc leaf area

(P = 0.073) and height (P = 0.074) to specialization

into a niche. While these ﬁndings where marginally

signiﬁcant correlations, the authors claim that these

traits are representative of relevant niche dimensions,

and that further study is warranted looking at the pro-

cesses of change. (Tilman and Snell-Rood, 2014) ex-

amine this study, and the previous mentioned ﬁnch

studies, to question if such studies can experimentally

demonstrate a divergence of species.

This study examines the application of the adap-

tive radiation by the MWM on the creation of a four

part harmony in which one of the voices is known.

The remainder of this paper is organized as follows:

Section 2 examines the problem of creating a four part

harmony as an evolutionary process, with Section 3

explaining the compositional rules used by the gener-

ator. The Multiple World Model of evolutionary algo-

rithm is examined in Section 4, with special attention

Figure 1: Ranges for the four voices used in four-part har-

mony.

to the collaborative model of ﬁtness. Section 5 de-

scribes the experimental setting of the system used to

demonstrate the method. The results of these experi-

ments are examined in Section 6. Section 7 gives con-

clusions sets out further directions for this process.

2 FOUR PART HARMONY

Four-part harmony deﬁnes the category of music writ-

ten for four voices (which could be singers or musi-

cal instruments) where the four parts produce a note

for each chord in the piece. The four voices are gen-

erally called soprano (or treble), alto (contralto or

countertenor), tenor, and bass (Boldrey, 1994). These

voices all have some classically deﬁned ranges (see

ﬁgure 1), the origin of which comes from the limits

of human singers: most singers have a vocal range

of two octaves (McKinney, 1982), which means that

most bass singers won’t be able to reach the higher

notes of a tenor or of a soprano. Typically the higher

voice will perform a melody while the lower three

will harmonize it, in our study any melody arising in

the higher voice is going to be purely a by-product

of evolution, as we do not interfere with its creation.

Nonetheless, it’s worth noting that this doesn’t mean

that interesting melodies cannot emerge.

This type of music (also called chorale) has slowly

evolved from Gregorian chant, which was unison

choir), becoming predominant in the Renaissance era,

where its role as sacred music in Western Europe

made it the main type of formally notated music. It

has been explored by innumerable artists and later

(Baroque and Classical music) accompanied by var-

ious types of instrumentation. Examples are Bach’s

Mass in B minor, Mozart’s Mass in C minor, Haydn’s

The Creation and The Seasons. While all these works

are thought to be for use in sacred ceremonies, the

use of choirs (and chorale) soon after expanded to

the concert stage, ﬁrst examples of this tendency are

Berlioz’s Te Deum and Requiem, and Brahms’s Ein

deutsches Requiem.

We chose to explore the usage of evolutionary al-

gorithms in creating this style of compositions as the

composition style is commonly used as a teaching ex-

Evolving Four Part Harmony using a Multiple Worlds Model

221

ercise for composers to learn to manage and create

harmonies (Sessions, 1951). This exercise is partic-

ularly effective as the student must control both the

horizontal and vertical dimensions of the texture. This

strictly limited style of composition also provide us

with a good benchmark to test the Multiple Worlds

Model in the ﬁeld of music generation.

We though it was interesting to consider each

voice as a single individual, by using this assumption

the MWM approach seemed a very interesting way to

evolve cooperatively such compositions. These indi-

viduals are part of different populations that evolve at

the same time, but without exchange of genetic mate-

rial between them. We expected the populations (ini-

tialized randomly) to then achieve speciation, evolv-

ing in a way that the individuals from one population

would evolve to “work well” with the voices belong-

ing to the other populations.

3 SELECTED RULES FOR FOUR

PART HARMONY

In this section we describe the rules of four-part har-

mony composition we decided to apply in this study,

these are going to form the ﬁtness function for the

evaluation of the evolved individuals (see Section

4.3). We should note how the rules that we chose are

just a subset of four-part harmony rules; many of the

rules presented in various composition manuals are to

be considered “guidelines” for “good” composition,

so they can, in particular cases (stylistic, harmonic,

or melodic reasons), be ignored (Schoenberg, 1978).

The rules we chose consist of the strongest rules, the

ones that should always, apart from extreme cases, be

followed (Sessions, 1951).

Rule 1: for each beat each of the four voices plays

one of the notes of a triad, with only one of the

voices doubling a note one of the other voices are

playing. This rule prevents usage of alterations,

practically limiting the expression of chords to the

diatonic triad appropriate for the degree consid-

ered (eliminating for example possible diminished

chords). This rule would be considered “strong”

mostly in a pedagogical context, but for this ﬁrst

study we believed it would be enough to consider

the smaller problem space deﬁned by this rule.

Rule 2: avoid voice mixing. By voice mixing we

mean the situation where a higher voice plays a

note lower in pitch than what a lower voice is

playing. For example in the case where the so-

prano plays a C

while, at the same time, the

alto plays a D

. This situation is possible, as the

ranges of the voices overlap, but in this type of

composition it is not recommended.

Rule 3: no parallel octaves. Parallel octaves happen

when two voices, which have between each

other an interval of an octave, move by parallel

motion to two new notes that still create an

interval of an octave. This type of motion must

be avoided, as it destroys the independence of

the voices, by creating the sense of not two

voices, but of one voice doubled at the octave.

Forbidden

Rule 4: no parallel perfect ﬁfths. Parallel ﬁfths

work exactly like the previously described

parallel octaves, obviously with intervals of a

ﬁfth. In the same way they convey the sense

of a single voice doubled and should then be

always avoided. A parallel ﬁfth movement

can be accepted if it goes from a perfect 5th

to a diminished 5th if the notes of the dimin-

ished 5th resolve. However, as we already

explained, rule 1 prevents the creation of dimin-

ished intervals, so we can ignore this exception.

Forbidden

4 MULTIPLE WORLDS MODEL

4.1 Collaborative Fitness Model

In this study there is a change to the ﬁtness evaluation

from previous models, while previous models were

looking at a competitive this application is collabora-

tive. It implements the collaborative model of ﬁtness

proposed in (Brown, 2014), that each member of a

world is scored the same as the world score, rather

than the individual being given points only if it cap-

tures the modeled point better than the other models.

This makes the ﬁtness not just dependant upon a sin-

gle voice, but on how it interacts with other voices in

the model.

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

222

for some number of generations do

randomize the worlds

for all worlds do

for all datapoints do

(competitive) award the point to the model with the best ﬁtness

(collaborative) award the same ﬁtness to each model based on their com-

bined ability to solve the problem

end for

for all populations do

Select breeding pairs based on ﬁtness

Apply Crossover/Mutation

end for

Figure 2: Demonstration (collaborative models) and Pseu-

docode (competitive and collaborative models) of the Mul-

tiple Worlds system.

4.2 Genome Representation

The genome representing an individual (or voice line)

is composed of an array of n integers, where n is the

amount of notes of the given bass line or a user de-

ﬁned number if the bass line is left free to evolve

with the other voices. These integers represent the

amount of semi-tones from the lower note of the

voice’s range, for example this means that the so-

prano’s genes can go from 0 (C

) to 19 (A

We can express each note belonging to the range,

allowing us to evolve melodies ﬁtting any key, even

if the rules we selected (see Section 3) prevent alter-

ations inside a speciﬁc key.

4.3 Fitness Evaluation

The rules discussed in Section 3 are implemented as

ﬁtness penalties. The evolution aims to minimize the

number of violations of this rule set, each with an as-

sociated penalty score. A ﬁtness score of zero is an

arrangement with no rules violations. Therefore, it is

a ‘valid’ composition.

f itness =

∑

i=0

(Double(s

+ 10 ∗ Mixing(s

) + 30 ∗ Stuck(s

))+

n−1

∑

i=0

(10 ∗ Fi f th(s

) + Octave(s

))

(1)

As you can see from equation 1 we have all the four

rules we discussed before: we check for each set of

notes to form a triad with one voice doubling one

of the others (Double), we check for voice mixing

(Mixing), for parallel ﬁfths (Fi f th) and parallel oc-

taves (Octave). We also had to add an extra part in

the ﬁtness function: Stuck. This function checks if a

voice is in the situation where, between the voice

above and the one below, there is less than 7 semi-

tones. If this is so, we can have the situation where

the two other voices are occupying pitches which are

“good” (they satisfy the doubling rule) and the middle

voice is stuck in a loop trying to reach one of the two

pitches that are already occupied effectively bringing

evolution to a dead end(note that most of the problem

comes from the voices having limited, different and

partly overlapping ranges). With the introduction of

this part of the ﬁtness function we can avoid these sit-

uations, and as it is a very unwanted situation it has a

weight modiﬁer of 30.

The Double function creates a proximity matrix

of the notes that are being evaluated in respect to the

possible pitches that can compose the triad, then it

calculates and returns the minimum distance (in semi-

tones) from an acceptable solution. For each couple

of adjacent voices Mixing checks if they have over-

lapped, so for example if the alto is playing a note

above the soprano or below the tenor. Octave and

Fi f th check for each possible couple of voices if they

for a parallel octave or ﬁfth between the notes at the i

index and the i + 1. Mixing, Stuck, Octave and Fi f th

are boolean functions, yet they represent violations of

very strong rules, which is why they have a fairly high

weight modiﬁer.

Evolving Four Part Harmony using a Multiple Worlds Model

223

Parents

Children

Mutations of the Children

Figure 3: Example of breeding (crossover and mutation)

between the representations of notes. Parent one is the ﬁrst

note sequence of size ten in light blue. Parent two is the

second note sequence of size ten in the darker red. A one-

point crossover then occurs between the two parents at the

third loci creating child one and two. The mutations of the

children then happen in child one at the third position and

the second child at position four, labeled in dark gray.

4.4 Selection

First, an elitist strategy is applied, copying to the new

generation the best individual of the current genera-

tion, the rest of the population is ﬁlled with offspring

from individuals selected by a simple tournament se-

lection algorithm (Miller and Goldberg, 1995). Fi-

nally, a mutation chance for each individual of the

new generation (1%) is applied. This is a low mu-

tation rate of the best individuals that this should help

to avoid inbreeding and promote exploration of the

problem space.

4.5 Variation Operators of Crossover

and Mutation

In this section we discuss how we have implemented

the production of a new generation; remember that we

have three (or four, depending on the setup) popula-

tions, the following method is applied to all of these.

We apply a simple one-point crossover: given two

parents we select randomly a point in their genomes

and create two new individuals containing the data

from one parent up to the crossover index and the data

from the other parent afterwards (see Figure 3).

When we mutate an individual, we give each gene

a chance of

NumberO f Genes

to mutate, effectively ob-

taining in general only one gene mutation per indi-

vidual mutation, but allowing for more (or no) muta-

tion. As we discussed in Section 4.2, each gene rep-

resent a note; when a gene mutates we take a random

point from a Gaussian distribution bound in the inter-

val [−5,5] and transform the note by adding the value

obtained in semi-tones. For example, if our gene rep-

resents a C

and we obtain a -2 from our random se-

lection, we subtract two semitones from C

obtaining

an A]

/B[

5 EXPERIMENTAL SETTINGS

We have conducted two trial run consisting of 100

evolutions of a four-part harmony composition given

an already written bass line. This choice was mo-

tivated ﬁrstly to try to simulate typical pedagogical

exercises in composition (where generally one of the

voices is already given to the students, which then

have to write the remaining three) and secondly be-

cause it seemed a more complicated problem for our

evolutionary approach. The difference between the

two trial was the introduction of a constraint: forcing

the note played by the bass to be always the root of

the chord. The introduction of the constraint allows

us to explore if we can generate variations within the

same harmonic framework. Indeed, we have observed

(but will not discuss in this paper) that the evolution-

ary process has much more success if given freedom

to also evolve the bass line. This stands to reason, as

giving a static bass line restricts the solution space,

effectively giving less freedom to the evolution.

We have obtained 85 solutions from the evolution

of compositions without the constraint, while out of

the 100 runs with the constraint we have only ob-

tained 8 solutions, those with a ﬁtness score of 0, de-

noting no rules violations. You can access the midi

of solutions/non-solutions for the two experiments at

msci.itu.dk/fourpart. Often the evolution gets stuck in

local optima from which it cannot move out with our

current approach.

6 RESULTS AND DISCUSSION

The mean ﬁtness of the populations proceeds to reach

a plateau after 120 generations, this seems to indi-

cate that evolution progresses correctly to a optimum.

Sadly most of the times in the constrained experiment,

instead of ﬁnding a solution we get stuck in a local op-

timum. We should note that the evolution is also not

very expensive in time usage criteria: it takes about

one to two seconds to reach the plateau. This seems

to indicate how the solution space for our constrained

problem is composed by many suboptimal points.

We have calculated distance matrixes between

the solutions/non-solutions and between non-

solutions/non-solutions. These give us information

on how the solution space is composed. The distance

metric we used is fairly naive, when confronting two

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

224

different evolved compositions we check how distant

(in amount of semitones) each note of each voice is

compared to it’s counterpart in the other composition.

We are aware that there are better distance measures

(like the Tonal Pitch Step Distance (De Haas et al.,

2008)), but most focus on harmonic distance. In our

constrained case this would make little sense between

solutions, as we deﬁned a harmonic constraint on the

given bass’s notes always being the root of the chord.

In this paper we will focus on the data gathered

through the constrained experiment, as it gives us

some more interesting information on the solution

space of variations with the same harmonic structure.

6.1 Constrained Experiment Results

We have conducted an unpaired t test on the distances

between the solutions and between the non-solutions.

The mean minimum distance between the eight so-

lutions ( ¯x = 34.5, σ = 26.2) and the minimum dis-

tance from one of erred pieces the to one of the solu-

tions ( ¯x = 55.3, σ = 17.7) was found to be statistically

signiﬁcantly lower (p = 0.0304) using a two-tailed t-

test between means. Thus, there is in general more

similarity between the closest correct pieces to each

other than the errors being close to the known solu-

tions. This implies that either the known solutions are

valleys in the search landscape which more than one

non-conﬂicted song exists in with a number of local

optima to bypass or that there are other unknown good

songs which are also moving the population. A pos-

sible explanation for this phenomenon might be given

by the harmonic constraints we have introduced. We

expect that by lifting such a constraint we will have

well distributed solutions, that might still cluster de-

pending on the harmonic qualities of the composi-

tions.

In the eight solutions, there are two pairs of ar-

rangements with close features, less than twenty units

of the distance measure. Yet, the same arrangement

was never found twice, demonstrating the diversity

of solution which is present due to the genetic ap-

proach. This makes for eight unique songs meeting

with the baseline, see Fig. 4 for all the solutions, we

highlight two very similar ones and two very different

ones. For comparison we also present a human-made

solution from the Genevan Psalter (Fig. 5). Note that

the notes that appear in the solution with a diesis (])

correspond in this key not to a A] but to a B[, which

appears normally in Fig. 5, because the B[ is included

in the key notation. They all share some common

chords and characteristics. This is due, as already dis-

cussed, to the harmonic limitations we have enforced

on the compositions. Yet, we obtain some different

Figure 4: The eight solution obtained by the constrained

experiment. The distances between solution 7 and solution

6 is 76 (semitones), while the distance between the solution

7 and solution 8 is only 6.

notes dispositions, that show how we can obtain many

variations even considering this limitations. We can

especially see this between the second and third solu-

Evolving Four Part Harmony using a Multiple Worlds Model

225

Figure 5: An example of human composed four part har-

mony using the same bass line and harmonic sequence,

from the Genevan Psalter: Old 124th.

Figure 6: Representation of the distance matrix calculated

for all runs in the unconstrained experiment calculated us-

ing Multidimensional Scaling. In green you can see the so-

lutions we found.

Figure 7: Representation of the distance matrix calculated

for all runs in the constrained experiment calculated using

Multidimensional Scaling. In green you can see the solu-

tions we found.

tion which, while being very similar still have man-

aged to present some small variation. The variation

in question appears at the 7th crotchet (which, in this

case, corresponds with the 7th beat), where we have

(from the bass to the soprano) {D, F, A, D} for the

second score and {D, D, A, F} for the third one. Even

if the voices are still doubling the root of the chord,

the three free voices sing different notes.

6.2 Unconstrained Experiment Results

Out of 100 runs we obtained 85 solutions when we

allow the algorithm to choose which voice plays the

root of the chord. This means that, as we still are giv-

ing a static bass line, the algorithm can choose one

of three possible chords to build per each bass note.

This is because the bass note can then be interpreted

as the root of the chord, as its third or its ﬁfth; so if

the bass note is F, the possible chords would be F,

Dminor and B[. It should be noted that these com-

positions might present weird chord sequences, as the

choice of which of the three possible chords to choose

is stochastic and the algorithm has no information or

rules about chords. We have not conducted an study

to investigate these aesthetic questions yet, but we be-

lieve most compositions sound pleasant (you can ﬁnd

the solutions at msci.itu.dk/fourpart).

We found no statistically signiﬁcant relationship

between distances of solutions and non-solutions as

in the Constrained experiment. This seems to suggest

that these solutions occupy more evenly the solution

space and the same seems true for the non-solutions

(see Figure 6). We believe these result make sense:

we are exploring a wider search space and our algo-

rithm then is able to ﬁnd more solutions without get-

ting stuck in local optima. This also helps us interpret

the results from the Constrained experiment: that ex-

periment had some constraints that limited the accept-

able solutions to a speciﬁc part of the solution space,

which would explain why they seems to be grouped

in a valley. That might be an indicator that our ﬁt-

ness function has issues navigating through the solu-

tion space when the individual is far away from any

possible solutions.

7 CONCLUSION

In this paper we present a novel approach to gen-

erating four-parts harmonies by using the Multi-

ple Worlds Model evolutionary approach. We have

shown how we can generate multiple compositions

correctly, which present variations within and outside

a harmonic constraint. This study also allowed us

to notice the solution space, for constrained music,

appears to be composed by many suboptimal points.

Their is a solution valley where some of the most sim-

ilar found cluster.

At this time we cannot say that we can generate

better harmonies than other approaches, in fact our

ruleset is restricted.While rule based expert systems,

e.g. CHORAL (Ebcio

glu, 1988), or the use of plan-

ning (Yi and Goldsmith, 2007) might create just as

good compositions or even better compositions. The

variety of solutions offered by our approach is very

interesting. A user using this system to harmonize a

voice line would, instead of having to be happy with

the one solution the system returns, be presented with

a variety of variations could be selected.

One of the major questions raised by this work

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

226

is the issue of human competitive performance of

the result. This can be ﬁrst demonstrated by show-

ing that the system given a baseline will produce the

same composition as a human composer. The best

method of testing the claim to human competitive per-

formance would be for this system along with a group

of student musicians to be given the same set of base-

lines in a larger study. The generated harmonies being

played in a random order for an audience in a Turing

test or Imitation Game (Turing, 1950), attempting to

decide between human and computer generations.

In conclusion we believe that this work, while

needing much improvement, has given us a very in-

teresting glance in how the solution space of these

composition is made and has shown that we can use

collaborative evolution to achieve speciation between

the voices.

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