Abstract
This paper focuses on a
successful attempt to eliminate the fitness bottleneck in
GenJam, an
interactive genetic algorithm that performs jazz improvisation, by eliminating
the need for fitness. This was
accomplished by seeding an initial population of melodic ideas with phrases
selected from a database of published jazz licks, and employing an intelligent
crossover operator to breed child licks that tend to preserve the musicality of
their parents. After a brief
overview of the changes made to GenJam’s architecture, the paper
describes the mapping of licks to measure and phrase individuals in GenJam
Normal Form. The intelligent
crossover operator on phrases is then described, along with a discussion of
measure mutations performed during a solo to insure that repeated measures are
altered in interesting ways. The
paper concludes with a discussion of the implications of removing fitness from
a genetic algorithm and whether the result still qualifies as a genetic
algorithm.
1 BACKGROUND
Fitness in most
genetic-algorithm-based music and art systems derives from aesthetic
judgements. Since aesthetic
judgements are not often amenable to algorithmic implementation, these systems
usually end up as interactive genetic algorithms (IGAs), which depend on the
feedback of a human mentor [Bentley 1999]. The mentor must experience all the individuals in a
population of competing works or pieces and somehow indicate each one’s
artistic or musical merit. In
other words, the objective function in these domains is typically a subjective function.
IGAs suffer from a
bottleneck caused by the need for the mentor to experience and provide feedback
for each individual. This fitness
bottleneck
is particularly narrow in temporal domains like music, where individual
population members must be experienced one-at-a-time and in real time [Biles
1994]. This can set practical
limits on population size and number of generations, which in turn often leads
to the design of unusual genetic operators, all in an attempt to evolve
acceptable individuals before the mentor gives up.
1.1 GenJam REVISITED
This is certainly the
case with GenJam [Biles 2001], an IGA that performs jazz improvisations in two
modes. In the first mode, GenJam
learns to improvise full-chorus solos by evolving hierarchically interrelated
populations of measures and phrases under the interactive guidance of a human
mentor. Individuals in the measure
population map to sequences of eighth-note-length events that make up a measure
of melodic content. Three types of
events can be represented. Rest events turn off the
note currently being played, if there is one. Hold events hold the current note, if there is one. New-note events turn off the
current note and begin a new note, using the event number as an index into an
array of pitches derived from the current chord in the tune. In this way, GenJam never plays a
theoretically wrong note.
Individuals in the phrase population map to sequences of four measures
from the measure population.
Melodic material that has been encoded as measure and phrase individuals
is referred to as being in GenJam Normal Form (GJNF).
To create a GenJam
soloist, the mentor starts with a random measure and phrase population, listens
to GenJam improvise on some tune to the accompaniment of a synthesized rhythm
section, and supplies feedback by indicating “good” or
“bad” whenever so moved during the improvisation. New measure and phrase individuals are bred
in both populations using tournament selection, single-point crossover,
musically meaningful mutation, and replacement with a 50% generation gap. The mentor usually evolves several
soloists in various musical styles prior to performing, and a typical GenJam
gig with the author’s Virtual Quintet will feature a dozen or so
pre-trained soloists.
In its second
improvisational mode, GenJam trades fours with a human in real time during a
performance by using its genetic representations and mutation operators to
evolve the human’s last four measures into what it then plays in the next
four measures. GenJam listens to
the human performer play a four-measure phrase, maps what it heard to a phrase
and four measure chromosomes, mutates those chromosomes, and immediately plays
the result in the next four measures as its response. The mutation operators, then, act to develop the human’s
phrase in musically stimulating ways.
The details of GenJam’s architecture, chromosome structures and
genetic operators have been documented elsewhere [Biles 2000 & 2001].
1.2 IMPROVING GenJam SOLOS
While GenJam improvises
quite successfully in both improvisational modes, its performance when trading
fours is superior to its performance taking full-chorus solos. This is partly due to the spontaneous
interactivity inherent in trading fours--a musical conversation is usually more
stimulating than a musical monologue.
However, another difference between GenJam’s full-chorus solos and
the material it plays when trading fours stems from the origin of that
material. Individuals that make up
the measure and phrase populations of an evolved soloist began their evolution
as random strings. After evolving
for several generations, these musical ideas reach a level that is competent
and often pleasing but seldom compelling.
However, the phrase and four measure chromosomes that are mutated when
trading fours are a fairly close approximation to a four-measure phrase played
by a presumably accomplished human jazz musician.
The “presumably
accomplished” part is important because it makes it possible for GenJam
to evolve responses without the use of a fitness function in the 15
milliseconds or so allotted to it when trading fours. If the four-measure phrase played by the human is good and
the mutation operators are safe, then the mutated phrase will be good, and
there is no need to assign fitness.
If the human performer is bad, however, then all bets are off, and
GenJam’s responses will likely sound bad as well. Garbage in, garbage out, as the saying
goes.
GenJam’s success
trading fours prompted the author to make another attempt to ease
GenJam’s fitness bottleneck for this paper. Previous attempts have included trying to develop
neural-network-based fitness functions [Biles et al 1996], which did not work,
and seeding the initial measure population with individuals generated by a
fractal generator, which produced individuals that statistically resembled
measure individuals from a mature, trained soloist [Biles 1998]. The fractal generator certainly helped
by making the initial population more musical, which made training easier. However, when soloists were evolved in
the absence of feedback, they ended up markedly less pleasing than
mentor-trained soloists, so fitness was still deemed necessary [Biles 1999].
This paper describes a
successful attempt to create a fitness-free or autonomous GenJam. By seeding the initial population with
licks derived from human performance, breeding new licks with an intelligent
crossover operator, and mutating repeated measures during performance, autonomous
GenJam (AutoGenJam) is able to play full-chorus solos that are generally
superior to those played by soloists trained by a mentor. We begin with a description of AutoGenJam’s
revised architecture, then discuss the mapping of licks to GJNF, our
intelligent crossover operator, and the role of mutation in keeping a soloist
fresh. We conclude with a
discussion of whether AutoGenJam is still a genetic algorithm, once fitness is
no longer necessary.
2 AUTONOMOUS
GenJam
Figure 1 shows AutoGenJam’s
system architecture. As described
elsewhere [Biles 2001], the Rhythm and Head Sequence files are MIDI files that
provide rhythm accompaniment and arranged harmony parts for the tune being
played, and the Chord Progression and Choruses files describe the structure of
the tune. The MIDI Params file
provides parameters to configure the tone generator GenJam and the rhythm
section will use. The human
Performer trades fours with GenJam, as described above. Missing from previous versions of
GenJam are the human Mentor and the two Population files, one for measures and
the other for phrases. The Mentor
is unnecessary because there is no need for fitness, and the stored population
files have been replaced by a Licks Database, which supplies four-measure licks
from which AutoGenJam creates a measure population of 64 individuals and a
phrase population of 48 individuals.
These populations use the same chromosome structures as previous
versions of GenJam, but they are generated fresh for each tune and are not
stored.
Figure 1: Autonomous
GenJam System Architecture
2.1 POPULATIONS FROM LICKS
The creation of the
measure and phrase populations from the Licks Database is summarized in Figure
2. The database itself is made up
of four-measure licks encoded from a recently published vocabulary of 1001 Jazz
Licks
[Shneidman 2000], hereafter referred to as 1001 Licks. Multiple licks databases can be created
to reflect different jazz styles.
Shneidman includes a generous portion of “essential” licks,
along with swing, bebop, hard bop, post bop, non-harmonic, and fusion/funk
stylistic licks. All the licks are
four measures long (how convenient!) and are presented in the context of
specific chord progressions, including licks for the first four measures of 20
standard tunes.
|
Randomly select 16
licks from a licks database |
|
Read those 16 licks to get 64 measure
individuals |
|
Initialize first 16 phrase individuals to
represent licks |
|
Create 32 child phrases via crossover on
original licks |
Figure 2: Algorithm for Creating
a Soloist
Licks that are selected
for inclusion in a licks database are mapped to GJNF by hand. Since GJNF represents measures and
phrases independent of any chord progression, the items in a licks database are
actually more general than the licks in 1001 Licks and can be played against
any chord sequence.
Table 1: Chord/Scale Mappings Using C as Chord
Root with Restored Avoid Notes in Italics
|
Chord |
Scale |
|
CMaj7,
C6, C |
C
D E F
G A B |
|
C7,
C9, C13 |
C
D E F
G A Bb |
|
Cm7,
Cm9, Cm11 |
C
D Eb F G Bb |
|
Cm7b5 |
C
Db
Eb F Gb Ab Bb |
|
Cdim |
C
D Eb F Gb Ab A B |
|
C#5 |
C
D E F# G# A B |
|
C7#5 |
C
D E F# G# A# |
|
C7#11 |
C
D E F# G A Bb |
|
C7alt |
C
Db D# E Gb G# Bb |
|
C7#9 |
C
Eb E G A Bb |
|
C7b9 |
C
Db E F G Ab Bb |
|
CmMaj7 |
C
D Eb F G A B |
|
Cm6 |
C
D Eb F G A |
|
Cm7b9 |
C
Db Eb F G A Bb |
|
CMaj7#11 |
C
D E F# G A B |
|
C7sus |
C
D E F G A Bb |
|
CMaj7sus |
C
D E F G A B |
The chord-scale mappings
used to convert new-note events to actual pitches were altered somewhat for AutoGenJam
by restoring some “avoid notes” to accommodate the more standard
seven-note scales implied by the licks in 1001 Licks. The modified chord scale mappings are illustrated in Table
1, with italic entries representing notes restored for AutoGenJam.
The two criteria for
including a lick in a licks database represent a modest fitness function. Licks are included if they have at
least one new-note event in each measure and if the rhythm can be mapped to
eighth-note multiples without losing the essence of the rhythmic structure of
the lick.
The first criterion
eliminates long silences and assures that notes won’t be held through too
many chord changes. Specifically,
a few licks in 1001 Licks end with a long note that begins at the end of the
third measure and ties into the last measure. In many four-measure phrases, the chord in the third measure
creates dissonance, which is then resolved by the chord in the fourth measure. When GenJam performs notes that are
tied across chord boundaries, the scale used for selecting the pitch for that
note comes from the measure in which the note began, which in this case often
results in a dissonant note being held through a consonant measure. By insuring that each measure has at
least one new-note event, any dissonance has a better chance of resolving. 121 of the 1001 Licks were eliminated
by this criterion.
The rhythmic criterion
for inclusion in the licks database eliminates licks whose rhythmic structure
cannot be mapped to the eighth-note multiples that GenJam can represent in its
chromosome structure. Most of the
licks in 1001 Licks contain only eighth-note multiples and map easily. Some licks contain sixteenth-note or
eighth-note-triplet ornaments that can be simplified to eighth notes without
losing the essence of the melodic contour. However, some phrases are built on sixteenth or triplet
structures that cannot be simplified without losing the essence of the phrase,
so those licks were eliminated.
This criterion eliminated virtually all the fusion/funk licks, nearly
half of the hard-bop licks and occasional licks in other styles. The fusion/funk licks were no great
loss to the author because he prefers not to play in that style anyway…
Figure 3 shows two licks
from 1001 Licks to illustrate their mapping to GJNF. These licks will also be used to illustrate the intelligent
crossover operator described in the next section. The licks selected were numbers 563 and 569, both of which
came from Swing Progression # 1.
The numbers beneath each lick are the GenJam event numbers encoded from
the notes in the licks. 0
represents a rest event; 15 represents a hold event; and 1-14 represent new
note events, where 1 maps roughly to D above middle C, 7 maps roughly to C an
octave above middle C, and 14 maps roughly to C an octave above that. The specific pitches for each scale
degree were given in Table 1. The
first lick is structurally interesting in that it is made up of a two-measure
motif that is repeated with minor variations. This lick is also diatonic, meaning that all the notes are
in scales suggested by the chords in the accompanying progression.
Figure 3: Sample Licks with
Event Mappings
The second lick in
Figure 3 is chromatic, which means that there are notes that do not come from
the scales suggested by the chords.
Specifically, there are chromatic runs in the first half of the second
measure and the second half of the third measure. Since GenJam maps repeated new-note events to chromatic
passing tones when a measure is performed, chromatic runs can be represented in
GJNF by repeated new-note events, hence the repeated 10’s and 9’s
in the second measure and the repeated 10’s and 11’s in the third
measure. However, the C# in the
third measure, which is a chromatic approach tone not found in the scale
suggested by a Dm7 chord, cannot be represented in GJNF, so it was coded as a diatonic
approach tone. In this case the C#
was encoded as a 7, which will be performed as a C, a note that is in the scale
suggested by a Dm7 chord.
|
0 |
0 |
1 |
2 |
3 |
|
1 |
4 |
5 |
6 |
7 |
Phrase Individuals
|
0 |
0 |
6 |
7 |
9 |
11 |
9 |
7 |
6 |
|
1 |
9 |
15 |
15 |
10 |
0 |
0 |
0 |
0 |
|
2 |
0 |
5 |
7 |
9 |
11 |
9 |
7 |
5 |
|
3 |
9 |
15 |
10 |
8 |
0 |
0 |
0 |
0 |
|
4 |
0 |
4 |
5 |
6 |
7 |
8 |
9 |
11 |
|
5 |
9 |
10 |
10 |
9 |
9 |
7 |
5 |
7 |
|
6 |
9 |
7 |
8 |
9 |
10 |
10 |
11 |
11 |
|
7 |
12 |
10 |
8 |
5 |
7 |
5 |
6 |
11 |
Measure Individuals
Figure 4: Final GJNF for
the Licks in Figure 3
While mapping a lick to
GJNF, a transposition often needs to be performed to normalize the lick to a
suitable range for GenJam. The
fourteen possible new-note events map to just under two full octaves, which is
less than the nearly three-octave range used for the licks in 1001 Licks. In our example, the second lick is
already in a suitable range, but the first lick is a bit low. To correct that, the first lick was
transposed up a third by adding 2 to each new-note event while leaving the hold
and rest events unchanged. The
resulting phrase and measure individuals for these two licks are shown in
Figure 4, which assumes that these two licks are the first two processed in
creating the measure and phrase populations, hence the indices beginning at 0.
2.2 INTELLIGENT CROSSOVER
One third of the 48
phrases that represent a soloist are GJNF approximations of licks from a licks
database, like our two examples.
These 16 original licks are made up of 64 measure individuals, which
comprise the entire measure population for the soloist. The remaining two thirds of the phrases
are children of pairs of the original 16 phrase individuals, where pairs of
children are created via an intelligent crossover operator. Crossover points in phrase individuals
can occur only at interior measure-pointer boundaries, which means there are
only three potential crossover points in a phrase individual, as illustrated in
Figure 5.
1 2 3
|
4 |
5 |
6 |
7 |
Figure 5: Three Potential
Crossover Points in Phrase 1
The crossover operator
performs either single or double-point crossover at one or two of the potential
crossover points. The six possible
crossovers, then, are three single point crossovers at points 1, 2 and 3, and
three double-point crossovers at points 1 and 2, 1 and 3, and 2 and 3. The choice of which of these possibilities
will be performed is determined by the algorithm in Figure 6.
|
At each potential
crossover point |
|
Compute horizontal intervals in both
parents |
|
Compute horizontal intervals in both
children |
|
Compute interval differences both ways |
|
Return smaller difference as
“fitness” for that point |
|
Select crossover point with smallest
difference |
|
If a second crossover point has same
difference |
|
Select it too |
|
Perform crossover(s) at selected point(s) |
Figure 6: Algorithm for Intelligent
Crossover
The algorithm determines
which crossover point(s) produce children that most closely preserve the
horizontal intervals at the crossover point(s) in the two parents. A horizontal interval is the interval between
two adjacent notes and is defined here to mean the difference between the first
new-note event of the measure following the crossover point and the last
new-note event of the measure preceding the crossover point. Since there are two parents and two
children, there are two possible comparisons between parents and children. Both comparisons are computed, and the
one that yields the smaller total difference is selected as the difference for
that crossover point.
The differences for the
three potential crossover points are then used to select the actual crossover
point(s). If one difference is
less than the other two, it is selected as the only crossover point, and a single-point
crossover is performed at that point.
If two crossover points tie for the smallest difference, they are both
selected, and a double-point crossover is performed. If all three differences are identical, a double-point
crossover is performed at points 1 and 3.
|
2 |
4 |
1 |
2 |
3 |
|
3 |
0 |
5 |
6 |
7 |
Figure 7: Child Phrases Showing
Crossover Point
In our example, which
can be worked out by the reader with not enough to do, the differences are 0
for crossover point 1, 6 for point 2, and 6 for point 3, so a single point crossover
was performed at crossover point 1.
The resulting phrase individuals are shown in Figure 7, and the final
child licks, as played over the original chords are shown in Figure 8. Notice that the chromatic runs
mentioned earlier have been altered slightly in the second phrase due to
GenJam’s somewhat idiosyncratic
handling of repeated new-note events as chromatic passing tones.
Figure 8: Child Phrases Played
Over Original Chords
2.3 MUTATIONS TO INSURE FRESHNESS
Since a given measure
occurs in three different phrases in a soloist, it may be repeated during the
course of a solo on a specific tune.
Furthermore, since building blocks of two or three measures tend to
survive in child phrases, sequences of two or three measures could be repeated
in a solo. This is more likely to
occur in tunes with longer forms or tunes where GenJam is playing double time,
but it often occurs even in standard 32-bar forms, which only consume eight of
the 48 available phrases per chorus.
Even though the chords, and thus the scales used to determine actual
pitches, will likely be different the second or third time a measure individual
is played, the listener may detect the repetition and conclude that the soloist
has run out of new ideas and is simply reusing old ones.
To combat this, AutoGenJam
mutates measures that have already been played as part of an earlier phrase in
the tune. The mutations used are
the subset of GenJam’s musically meaningful measure mutations that are
used when trading fours with a human [Biles 1998]. The specific operators used are a random transposition up or
down one to three steps, a range-corrected inversion that essentially plays the
measure upside down, a retrograde that reverses the events in the measure, and
a retrograde inversion that performs both the inversion and reverse.
These mutations are some
of the same compositional devices used by human composers and arrangers to
develop melodic material. As such,
they tend to develop measures in musically pleasing ways, which is important in
an environment with no fitness available to identify and weed out poor
mutations. These mutation
operators are considered safe, then, because they reliably develop good
measures into good measures, thereby preserving the musicality of the original
measures.
Actually, these
mutations are the last in a series of steps that insures a fresh soloist for
every tune. The 16 licks chosen
for a soloist are selected randomly from a database of two to several times
that number, so a different subset is chosen for each tune. Once a set of 16 licks is chosen to be
1/3 of the phrases, the other 2/3 of the phrases are created by randomly
breeding pairs of the original 16 phrases. This would result in different children phrases even if the
same 16 original licks were chosen over again. Finally, measures that are repeated in a solo are mutated the
second and third time they are used so that they will be less recognizable but
still acceptable when repeated.
2.4 AutoGenJam VS. GenJam
Anecdotal evaluations by
listeners indicate that the solos played by AutoGenJam are generally superior
to those played by GenJam using well-trained soloists. AutoGenJam solos were characterized as more
interesting, more coherent, more human, less random, and less mechanical. The author also finds AutoGenJam’s
solos to be more engaging, and definitely less repetitive. The author plays several gigs per month
and practices with GenJam frequently, so he knows his trained soloists all too well. This induces in the author a sense of
boredom with GenJam solos that, fortunately, is not usually shared by audience
members who haven’t heard GenJam before. However, in most gigs a certain song style will be played
more frequently than the others, which results in the soloist trained on that
style being heard more often. This
can lead to a sense of déjà vu, even in audience members who have
not heard GenJam before. While the
author does train new GenJam soloists occasionally to supplement or replace soloists
that have gotten too familiar, doing so takes time, which the author typically doesn’t
have when a gig is imminent.
AutoGenJam provides a
vastly expanded pool of superior melodic ideas without having to evolve them
from scratch. Since a new soloist
is generated for each tune from this much larger pool, the solos are fresher
from one tune to the next, even if the same licks database is used over and
over. In addition, the licks used
to generate each soloist are more coherent and musical than the evolved licks
of a mature trained GenJam soloist, so the solos are not only fresher but
better.
One nagging issue, at
least to the author, is the use of non-original licks to seed AutoGenJam’s
populations. While the standard
pedagogical practice for apprentice jazz musicians is to listen to, transcribe,
and memorize classic jazz solos, the author has always eschewed this tradition
in an effort to avoid sounding like someone else. Since GenJam starts with random strings, any licks that it
evolves are its own, or at least the mentor’s own, since the mentor must
have liked the survivors enough to not breed them out. AutoGenJam is more reminiscent of some jazz
musicians the author has encountered at jam sessions who seem to have memorized
every solo by their personal favorite and play solos that are essentially
strings of quotes from their hero.
These musicians often get a good response from the audience, usually
because they are quoting great music, but their abilities as an improviser are
suspect. AutoGenJam at least
develops its memorized licks, both harmonically, with its chord-scale mappings,
and melodically, with its intelligent crossover and mutation operators.
An alternate way of
seeding the initial population is currently being developed by the author to
address this originality issue.
When the human performer takes a full-chorus solo, AutoGenJam will listen
to the human’s solo and build measures and phrases in the same way it
does when trading fours. It then can
mate these with each other or with licks from 1001 Licks and generate
populations that integrate ideas from the human’s solo into the mix. AutoGenJam would still be memorizing
the licks of others, but at least it wouldn’t be doing it strictly by the
book.
The biggest problem
being worked on now is, yes, a fitness function to determine if a human’s
four in a solo is worth keeping and breeding. This is not an issue when trading fours because the
occasional bad four is gone as soon as it is played, but a bad lick breeding in
a fitness-free environment could ruin a soloist. Hopefully, future papers will document the success of this
approach.
3 CONCLUSIONS
One question remains: Is
AutoGenJam still a genetic algorithm?
Most authorities list fitness evaluation as a requisite step in a
genetic algorithm. Bentley [1999],
for example, lists “initialisation, mapping, evaluation, selection,
fertility, reproduction and termination” as stages used by a simple
genetic algorithm. GenJam clearly
has all these stages, but AutoGenJam has no evaluation stage, and the selection
and fertility stages are uniform and random rather than preferential.
Goldberg [1999] lists encoding
to chromosomes, determining “fitness to purpose,” using a “population
of encoded solutions,” employing “selection, recombination and
mutation,” and preferring “better solutions to worse ones.” Again, GenJam clearly does all this,
but AutoGenJam does not determine fitness, other than the elimination of licks that
do not meet the two criteria for inclusion in a licks database. AutoGenJam also does not prefer better
solutions to worse ones because all solutions are deemed good, once they make
it into a licks database.
If genetic algorithms
are viewed from the perspective of a generate-and-test strategy, however,
AutoGenJam begins to look more like a GA. Genetic algorithms use initialization, recombination, and mutation
to generate
new solutions that are then tested by fitness. In classic GA’s, these operations are random or at
least “dumb,” and any knowledge of the problem domain resides primarily,
if not entirely, in the fitness function.
This is what makes GA’s attractive for problems that are not well understood,
because all that is usually necessary for the GA machinery to work is the
ability to tell whether one individual is better than another. However, many GA developers seed their
initial populations with promising individuals, and knowledge-based crossovers
and mutations are certainly nothing new.
These techniques effectively decentralize domain knowledge and
distribute it across initialization, recombination and/or mutation. AutoGenJam simply takes this to the
logical extreme by removing all domain knowledge from fitness and placing it in
the initialization, crossover, and mutation operators. Fitness, then, serves no useful purpose
and can simply be eliminated.
The lack of a fitness
function and the resulting lack of selection pressure in AutoGenJam also serves
a useful purpose in that it eliminates the convergence problem that has plagued
GenJam [Biles 1998]. Since the
search in both systems is for a good soloist, not for a good solo, and
certainly not for a super phrase or measure, convergence on a small number of
highly fit phrases and measures is disastrous. Without selection pressure, convergence is no longer a
problem.
Our conclusion, then is
that AutoGenJam is still a genetic algorithm, albeit an unusual one. It uses the evolutionary paradigm to explore
a melodic space of jazz licks in such a way that it never takes a truly wrong
turn. If only it were so easy for
aspiring human jazz musicians…
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