In this paper, we introduce our system HarmonyMapper, a system for generating diverse musical sequences. By combining the MAP Elites algorithm with insights from music theory, specifically Neo-Riemannian Theory, HarmonyMapper is able to generate sets of chord sequences with a high-level diversity in terms of the emotions they are expected to elicit. Our intention is that this system will form the basis for a mixed-initiative musical composition tool focused on generating diverse music for digital games. While HarmonyMapper is limited in the musical complexity it can produce, our pilot experiments showed that it is capable of generating large numbers of varied sequences with no input data and only limited set up and computational resources required.
AIIDE Workshop on Experimental Artificial Intelligence in Games, October 08, 2023, University of Utah, Utah, USA * Corresponding author. † These authors contributed equally. $ s.cardinale@qmul.ac.uk (S. Cardinale); o.withington@qmul.ac.uk (O. Withington) saracardinalemusic.com/ (S. Cardinale); owithington.co.uk (O. Withington) 0000-0002-seven00seven-5193 (O. Withington)
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and compound NROS employed by composers and the specific events or emotions they aimed to portray [ 12]. By 2.1. Music Generation Systems identifying the NROs and CNROs used in a progression we can to an extent infer the likely emotional response The generation of chord progressions has been investi- of a potential listener. gated for the automatic generation of musical harmony NRT has been used as the basis of generative music to accompany various forms of media. Bernardes et. al. for games before in the work of Cardinale and Colton [6] implemented the D’accord harmony generation sys- [13]. They introduced procedural NRT and suggested tem which worked over a perceptually motivated tonal that it could be used in conjunction with Lehman’s obserinterval space. Monteith et al. [7] generated music to in- vations to write event-based music for media, including duce targeted emotions, using statistical techniques such games. This led to the development of their GENRT as HMMs, and applied this in [8] to produce afective system which used NRT to procedurally generate musimusic to accompany the audio of fairy tales being read. cal sequences which matched with a given specification
Numerous generative systems have been used to cre- such as matching a specified tempo, or by incorporating ate soundtracks for videogames. For instance, the game changes in the music’s mood using Lehman’s descriptors GhostWriter (1998) [9] uses a generative approach to at specific time stamps. We expand on these ideas by create the music. The system implements a rule-based generating sets of diverse sequences rather than individalgorithm that maps the in-game tension to the music. ual sequences, though the current version of our system To create a chord progression, the system implements produces simpler music than GENRT which is capable of Schoenberg’s Theory of Harmony [10]. While the com- generating sequences with a melody, harmony, bassline, positional aspect is conducted by a generative system, and percussion rather than just a chord sequence. the detection of in-game tension is not an automated process: a director identifies and maps the level of suspense in the music to the in-game events. 2.3. MAP-Elites and Music Generation
More recently, Lopez Duarte [11] introduced an algorithmic music generator that makes use of a second or third-order Markov model to manage instrumental range and quantity, RNN for melody and bass, and an Augmented Transition Network (ATN) to provide a tonal grid. Duarte states that a system which combines generative and algorithmic (rule-based systems, deterministic approaches) could provide extensive musical variety in the structure and musicality while supporting the style and ofering continuity [11].
The algorithm at the centre of this work’s music generation system is the ‘Multi-dimensional Archive of Phenotypic Elites’ algorithm, more commonly referred to as MAP-Elites. MAP-Elites is the most prominent algorithm within a family of relatively modern algorithms referred to as ‘Quality-Diversity’ (QD) algorithms. These algorithms work similarly to traditional genetic search methods, which aim to find high-performing solutions to problems by combining and mutating the best solutions found so far, in a way heavily influenced by real-world evolution. Where QD algorithms distinguish themselves 2.2. Neo-Riemannian Theory is by generating populations of solutions rather than fit individuals, and by aiming to maximise the phenotypic Neo-Riemannian Theory (NRT) is a branch of transforma- diversity of this population. tional theory that provides analytical methods to study MAP-Elites achieve quality and diversity by discretischromatic chord progressions that deviate from tradi- ing the search space for solutions into an evenly spaced tional tonality. Conventional music theory, which relies grid, with each cell in the grid defined by a range of calon keys and modes to analyze harmonic developments, culable phenotypic traits (Phenotype/phenotypic in this often struggles to efectively analyze these types of pro- context meaning characteristics of the actual generated gressions. NRT introduces Neo-Riemannian Operators artefact, in our case a musical sequence. This is opposed (NROs) to examine chord changes. These operators de- to genotype/genotypic which refers to the encoded form scribe the transitions between chords in a sequence, al- of the artefact). Classically this grid is defined by two lowing for the analysis of chord progressions that go dimensions, though there are implementations that work beyond traditional tonal frameworks. These NROs can in higher dimensions [14]. Each cell in the grid is only be applied individually to a chord to produce a new chord, allowed to contain a fixed number of solutions, and the or several can be applied at once to produce a Compound membership of the cell is determined by a fitness funcNRO (CNRO) which is more distant from the initial chord. tion. If a new solution is generated and its phenotypic
NRT does not just allow for the description of non- traits mean it should be added to a cell that is already traditional progressions; it also allows them to be anal- at capacity, then the fitness of all solutions is calculated ysed. Through the analysis of diverse film scores using and the least fit solution is removed. Each loop of the NRT, Lehman observed a link between the chosen NROs algorithm previously generated solutions are selected from the archive, new solutions are generated from them and then they are added to the archive. By repeating this process until a stopping condition is met such as a certain number of loops, the intention is to arrive at a final grid populated with a set of solutions that are both high performing and diverse in terms of designer-selected traits.
MAP Elites is a relatively new algorithm, having been introduced by Mouret and Clune in 2015 [15]. However it and its variants have already seen significant success in domains as varied as generating robot designs [16], generating digital artworks [17], controlling swarms of micro-robots [18], and many others. It has also seen significant popularity in the generation of content for digital games, especially game levels [19, 20, 21].
A domain that has seen little use of QD algorithms is the domain of music generation (that the authors are aware of). However, they have seen success in similarly creative tasks such as digital drawing. McCormack et al.[22] employ quality-diversity search techniques to investigate a creative generative system based on an agentbased line drawing model. The application of qualitydiversity search allowed for the discovery of multiple high aesthetic value phenotypes within the system.
Berker and Colton have recently argued that QD algorithms for music generation could be a good fit for the ifeld [ 23]. We agree that it is a potentially good fit for multiple reasons. There is a substantial amount of music theory that can be leveraged to produce interesting phenotypic metrics with which to evaluate generated music.
Music can also be encoded genotypically in a form that makes it easy to manipulate using conventional genetic operators, especially when using NRT as we do in this paper. chord in the form of an array of midi note numbers. A midi note number is a widely used encoding form within music in which each note is assigned a number from 0 to 127, with middle C found at number 60. For example, a C Major chord in the 4th octave composed of the notes C4, E4, and G4 can be encoded as [60, 64, 67]. Each NRO in traditional NRT can be described as a set of additions and subtractions to the notes of a trichord in midi note number form (see Table 1). This makes applying an NRO to a trichord a mathematically trivial operation. However, after applying an NRO to the midi note number, its form may have to be reordered to place the trichord in its root position. This is also complicated by the fact that each NRO letter operates on note numbers diferently depending on whether it is applied to a Major or a Minor triad.
3. Methodology As discussed in Section 2.3, a central decision when implementing MAP-Elites is on the choice of phenotypic 3.1. Musical Encoding metrics to use to determine where generated artifacts are placed within the grid archive. One of the contributions To generate our musical sequence we need a genotypic of this work is the introduction of metrics that can be form for them which can be easily manipulated by ge- calculated for a musical sequence. These metrics are innetic operators, while also being convertible into midi formed by musical theory and are intended to capture audio with limited efort and compute. This is where NRT specific characteristics of musical pieces which are meanworks well. NRT describes a series of Neo-Riemannian ingful to listeners, and therefore characteristics that a Operators (NROs), conventionally labeled with a letter, game designer or composer might like to have diversity which can operate on a trichord to produce a new tri- in to fit diverse game settings. chord. These NROs can be combined together and applied in sequence to a starting chord to produce a Compound 3.2.1. Fitness Function: Key Prediction NRO (CNRO). The musical encoding we use in this version of our system is to store tracks as an ordered array of CNROs along with a starting chord. From this we can generate the series of chords produced by applying these CNROs in order, forming a musical sequence that can be converted to midi at any point.
To apply these CNROs to a chord, we need to store the
we experimented with several alternatives before arriving at the option that gave the most appealing results: the strength of fit between a chord sequence and its predicted key. To analyse the predicted key of a piece we use the Krumhansl-Schmuckler Key-Finding Algorithm [24].
By comparing the distributions of pitches in a musical Algorithm 1 Illustration of the Process for Generating a List of Chords from a Start Chord and a List of Compound NRO Objects Require: _, _ ℎ_ = [] ℎ_ ← ℎ_ + _ℎ for CNRO IN CNRO_List do for NRO IN CNRO do _ℎ ← ℎ(ℎ) _ ← (_ℎ) if _ then
_ℎ ← ( , _ℎ) else
_ℎ ← ( , _ℎ) end if _ℎ ← ( _ℎ) ℎ_ ← ℎ_ + _ℎ end for end for return ℎ_ sequence with those of the fixed key profiles which they as you increase the size of the CNROs used, the resulting developed, one can predict the key of any piece of music. chord tends to be further away from the original key. In
In this work, we use Music21 [25], a popular mu- a series of experiments carried out by Krumhansl [26], it
sicology Python Library, and their implementation of was shown that tones less closely related to the tonality
Krumhansl-Schmuckler’s Algorithm. This not only gives are less stable than tones closely related to the tonality.
us a possible key of a piece, which is used later for pre- As tonality is inferred through harmonic context and
dicting the mode of our generated music, but it also gives the interplay of tension and release among surrounding
us the correlation coeficient of the confidence level of chords in relation to a specific tone or chord base [
4.0.1. Average CNRO Shift
Average NRT Shift. This metric is based on the observations of musicologist Frank Lehman [12] that there is a relationship between how many NROs are used between two chords in a sequence and how unsettling a piece is. This could be attributed to the fact that when using only one NRO, the resulting chord is more likely to be in the same key or a closely-related key, whereas
As discussed in Section 1, music theory describes seven
musical modes. These are highly useful for our purposes
as prior research has found that they can be ordered
from brightest-sounding, Lydian, to darkest-sounding,
Locrian, as shown in Figure 1 [
There are several steps involved in HarmonyMapper’s
process for predicting the mode of a generated piece
of music. First, we take the predicted key of the piece
using the same Krumhansl-Schmuckler’s Algorithm [24]
which we use in calculating fitness. For this key, we
then generate the scale of that key for each of the seven
modes, giving us seven scales, each containing 7 notes, to
able to pick a mode and NRO shift that most represent the
in-game situation or the situation they want the player
to perceive. For example, in the song At Doom’s Gate
[
child solutions from pairs of parent tracks, namely a mutation operator and a crossover operator both designed to work on a sequence of CNROs. This parent pair are selected at random from the MAP-Elites archive archive population as in conventional MAP-Elites [15].
The mutation operator works by replacing a CNRO in a sequence with a new generated CNRO. This CNRO is generated by first stochastically selecting a CNRO length with a range between 1 and the max length for a CNRO which is specified for the MAP-Elites run as a whole. We then generate a sequence of NROs by selecting NROs at random until we have enough to produce a CNRO of the specified length. This new CNRO then replaces the selected CNRO in the original sequence. This process has a chance of being applied to every CNRO in a sequence, so a mutation rate of X% means that every CNRO in a selected sequence has an X% chance of being replaced with a newly generated one.
The crossover operator used is a one-point crossover.
First, we generate a random crossover point for a selected
pair of parents which can be anywhere in the range 1
to the fixed track length minus 1. Two child CNRO
sequences are then generated by taking the portion of
parent 1s CNRO sequence up until the crossover point and
concatenating it with the portion of parent 2s following
the crossover point, and vice versa for child 2.
check. We then loop through every chord in the piece of
music and count each chord whose notes are all contained
within each mode’s scale. At the end of this process, we
then select the mode for which the most chords fit within. 4.2. Duplicate Chord Fitness Adjustment
In the case that there the chords fit within two or more An issue with our method for generating CNROs
stochasscales equally, we then check the frequency rate of the tically is that there is a chance that when a series of
appearance of a character note of each scale within the NROs are applied to a chord it can produce a loop that
chords. This note can be considered to be the signature produces the same chord as the initial chord. This can
note of a scale, and the one that most represents that be especially problematic in conjunction with the NRT
scale (see Table 2) [
We again argue that this mode prediction would be might be to a listener based on the magnitude of the potentially very useful to a game designer or composer, shifts between sequential chords. Due to this looping as similarly to NRO shift, being able to target a variety of issue, however, it is possible to have long NRO sequences moods would be very useful for many games and game which would cause higher scores on the NRT shift metric, genres. While some games may not change their intended while producing duplicate chords which do not produce mood much at all during play, especially traditional ar- the desired discomfort. cade titles, these are in the minority in contemporary video gaming. The composer or game designer would be
To address this issue we opted to check each newly
generated track for the existence of duplicate chords in
sequence. Where a duplicate chord is found we force the performance of the system when it comes to the number
iftness of the track to be 0. This means the track can still of elites stored, but also the impact on system usability.
be stored in the MAP Elites grid, but it will always be One can imagine circumstances in which having a large
discarded when a suficient number of tracks without range of similar compositions with similar traits would
duplicates are generated within the same cell location. be useful to present a user with a suficient choice, but
While this approach is commonly used to handle infea- also circumstances in which a choice of any kind could be
sible individuals in genetic search and it undoubtedly redundant. However, for these initial experiments, this
achieves the goal of avoiding them being present in the decision was made purely based on system performance.
ifnal generated track set, it has also long been argued For the dimensions of the map, the predicted mode
against as an approach due to the loss of genetic informa- dimension was split into 7 increments, one for each
postion involved in efectively discarding these individuals sible mode, whereas the NRT Shift dimension was split
which may have made for useful parents [
culate two features: Average Grid Fitness and Coverage.
To calculate the Average Grid Fitness we first sum up the fitness scores of the fittest tracks found in each grid cell, with empty cells receiving a fitness of 0. This number is then divided by the total number of cells. To calculate the Coverage we count the number of cells which have any solution at all, and divide this count by the total number of cells. Coverage is intended to measure how efectively we are exploring the overall space, whereas Average Grid Fitness looks at performance more generally and requires both exploration but also the finding of high-performing tracks. Coverage is commonly calculated in MAP Elites implementations including in the paper which introduced the algorithm [15], and Average Grid Fitness is similar to the Reliability score which appears in that paper and others, except that we do not have an alternative algorithm to compare to.
To conduct our initial experiments to explore this system and its capabilities we first needed to decide on how the system would be parameterised. Where possible we based these decisions on quantitative data from pilot experiments to indicate parameterisations that appeared to lead to the best system performance, but where this was not possible a more arbitrary decision was made. 5.0.1. Musical Parameterisation
selves, each track starts with a tri-chord in the 4th octave.
This tri-chord is randomly selected for each track when generating the starting population, and then inherited from one of the two parents for tracks generated during a run. Each track is composed of 20 CNROs giving a total of 21 chords including the initial chord. The maximum CNRO length was capped at 5. Decisions on track and max CNRO length were made arbitrarily, and it is here 5.1.1. Run Length where we expect that designers and composers using fu- In terms of experiment length, we let the process run ture iterations of this system would have the most control. for 5,000 loops. This cutof point was selected after iniHowever, it is more than possible there is a performance tial experimentation suggested that we were unlikely impact induced by these choices, something we aim to to see Average Grid Fitness gains after this point. Each examine in future work. loop involved the selection and generation of two tracks to support the use of the crossover operator. Therefore 5.0.2. MAP Elites and Genetic Operator 10,000 tracks were generated in total. To seed the ini
Parameters tial map a starting population of size 500 was generated The most important performance-afecting parameters for each run. This starting population was generated are the mutation and crossover rates, as well as the num- through purely stochastic generation and concatenation ber of elites that are stored per cell in the MAP Elites grid. of CNROs, as well as the selection of a random 4th-octave Initial experiments using a grid search suggested that a start chord for each starting track. mutation rate of 0.2, a crossover rate of 0.5, and storing a maximum of 3 elites per cell led to the best overall perfor- 5.1.2. Computational Resources Used mance. We note that in future iterations of this system All experiments were run on a Dell laptop with an i5that are designed to meet the needs of game designers 10310U CPU with 16.0 GB of RAM. With the above conand composers, we may have to consider not just the of chords with a high level of tonal consistency.
The system has the advantage of being fairly
computationally light and it was able to generate and evaluate
over 10 thousand chord sequences on a modestly
powered laptop in only 92 minutes. However, the process
would be too slow to be used in a generative system
which required quick iteration and generation such as a
mixed-initiative generative system with a human in the
loop (see [
As is evident in Figure 3 there were combinations of metric values which the system struggled to find tracks for. At the extreme low and high ends of the possible NRT Shift values the system visibly struggles to further populate the MAP Elites grid, especially in the 4.6 to 5 range in which it found 0 tracks. While one would expect that the outlying locations of the possibility space would naturally be harder to reach, we expect that this could also be the result of the high mutation rate used and the nature of our mutation operator. With the mutation rate used of 20% for each of a track’s 20 CNROs, and with the 80% chance that a generated CNRO will have less than 5 NROs there is significant pressure pulling newly generated tracks back to the centre of the grid.
By inspecting the rules of our system, we can see that it cannot produce chord progressions that are fully in the Locrian mode. This is because it is impossible to produce Figure 3: Figure showing the final MAP Elites archive in the its signature half-diminished interval tonic using only form of a heat map. The color of each cell is determined by conventional NRO shifts. The best tracks found in this the fittest track found for the cell part of the archive are comprised of chord sequences that do not fit comfortably into any key when analysed with Krumhansl-Schmuckler’s Algorithm but happen to ifguration on this hardware, a full generative run took 1 contain chords that conform best to the Locrian scale hour and 32 minutes. for the best-fitting key. However, this does not mean
The system including all of its code, data for the results, they are actually in the Locrian mode. This highlights
as well as the generated chord progressions from this pa- the need for a fitness cutof before any generated chord
per’s experiment in both textual and .midi form is avail- sequences are actually selected for use, as the lower the
able at github.com/KrellFace/harmony-mapper/. fitness of a track, it both conforms less to a key, but also
less likely to be in the correct mode. The inability of
6. Results and Discussion the system to produce sequences in the Locrian mode is
a limitation we aim to address on as discussed later in
Here we discuss the overall performance of the system Section 7. This limitation could be improved by using the
including its strengths and weaknesses of it which are extended version of NRT by Colton and Cardinale [
In terms of systems performance, we are cautiously diminished. optimistic. It is able to produce sets of tracks with signif- The music produced by the system consists of chords icant amounts of diversity present in terms of our two played on a piano. Each chord has an equal duration metrics. The relatively high final average fitness score of and the instrumentation does not change throughout the 0.786 indicates that the system is able to produce series chord progression. This is a simplified version of the system as proof of concept. In Section 7 we discuss the future addition of melody components and improving the phrases that often represent a concept or character in complexity of the music generated by HarmonyMapper. soundtracks. Firstly we aim to explore the generation
In terms of the metrics themselves, future listening of diverse harmonies that all include the same leitmotif, studies are required to explore whether or not we are suc- mirroring how they are used in traditional composition cessfully producing emotionally diverse chord sequences. to evoke diferent moods while still referencing the same However, our initial impressions are that the NRT shift character or concept. More excitingly perhaps, we aim metric did not perform as expected. We were aware that to investigate whether we can use HarmonyMapper to there were limitations to it as a heuristic due to the way produce tracks that are blends of two leitmotifs by develNROs operate. To take an example from one of the re- oping a heuristic for evaluating how much a composition views of this paper, R and SSLL are equivalent in terms of confirms to either or both of two specified leitmotifs. This the chord change they produce but they difer massively could allow us to generate musical sequences that both in length. We were aware of this limitation but hoped evoke diferent moods but also evoke two contrasting that larger shifts would still tend to indicate tracks with characters or concepts to a lesser or greater degree, simichord transitions which would be perceived as more jar- lar to how composers write music to unite two or more ring. To the authors’ ears, however, this was not always leitmotifs. the case. As a result, improving on this distance metric is going to be a focus of our future work.
HarmonyMapper is the musical complexity it is able to generate. Specifically, we aim to extend the system with the generation of melodies, bass lines and percussion to pair with the diverse harmonies that are currently generated. One of the potential end goals for this system is for non-musicians to be able to use HarmonyMapper musical sequences in their games without the need for any alteration. In terms of how these additions are generated, we could take inspiration from Cardinale and Colton’s GENRT and generate them stochastically based on a set of specifications [ 13]. Naturally, these musical additions are also another facet of the music in which we can try and promote interesting diversity. For instance, we plan to produce deliberate diversity during the melody generation process, by using similar heuristics to the ones introduced in this paper to make the melody more or less voice-led.
We also plan to explore using diferent evaluation
metrics for our system. One substantial change we plan
to make is to use a diferent distance metric, such as
Lerdhal and Krumhansl’s model for tonal tension [
Another potential extension to HarmonyMapper which we are keen to explore is the introduction of the concept of leitmotifs. Leitmotifs are repeated musical
system for generating diverse chord sequences using MAP Elites, and have argued that a Quality-Diversity approach for music generation with music theory-inspired metrics is a valuable basis for game music generation systems. While the current output is limited to simple chord progressions, their intentional diversity in terms of the mood they convey to a listener makes them valuable as a musical foundation for generating more complex compositions. We look forward to expanding on this approach to develop a controllable system for generating diverse and appropriate music for games.
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