An Ant Colony Optimizer for Melody Creation with Baroque Harmony

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1 n nt Colony Optimizer for Melody Creation with Baroque Harmony Michael Geis, Member, IEEE and Martin Middendorf, Member, IEEE 2 Bioinformatics Group, Department of Computer cience and Interdisciplinary Center for Bioinformatics, University of Leipzig, Haertelstr. 6-8, D-0407 Leipzig, Germany. 2 Parallel Computing and Complex ystems Group, Department of Computer cience, University of Leipzig, Johannisgasse 26, D-0409 Leipzig, Germany. bstract We propose an algorithm that is based on the nt Colony Optimization (CO) metaheuristic for producing harmonized melodies. The algorithm works in two stages. In the first stage it creates a melody. This melody is then harmonized according to the rules of Baroque harmony in the second stage. This is the first CO algorithm to create music that uses domain knowledge and the first employed for harmonization of a melody. Index Terms nt Colony Optimization; Genetic algorithms; Composition; Harmonization; I. INTRODUCTION The computer assisted creation of music fascinates researchers and composers alike since several years. While composing is essentially algorithmic and therefore at least in part programmable, great composers did break the rules of composition, thus pushing the existing boundaries. It is not obvious that computers can be programmed to achieve this. The techniques employed for the computer based creation and exploration of music are many, ranging from grammars [], neural networks [2], and Evolutionary Computation (EC) [3] to multi-agent systems [4] and artificial chemistries [5], to name a few. Miranda [3] subsumes the multitude of applications in three categories: engineering, creative and musicological approaches. The first category addresses specific engineering problems in Music Technology, such as creating synthesizers [6] or generating grains in granular synthesis [7]. Creative approaches produce music by employing composition rules [8], [9], providing training data [0] or interacting directly with a user [4]. The musicological approach employs computer simulations to investigate the origins of music. Within the creative category, a vast number of projects have employed Evolutionary Computation (EC) and related metaheuristics. Well known examples include Cellular utomata (C) [], Genetic lgorithms (G) (see [2] for an overview), and swarm algorithms [3], and [4]. o far one CO has been proposed [5]. Evolutionary and related methods are expected to be successful in composition, when a lack of intentionality is desired as well as when musical structure is sought as an emergent property of a system. The CO is a metaheuristic for combinatorial optimization problems. It is inspired by the behavior of ants that rely on pheromone trails to find the shortest route from their nest to a food source. The key concept behind the CO is the cooperation of a set of relatively simple artificial ants through effective indirect communication mediated by the environment (stigmergy). The CO method has been formulated as a metaheuristic by Dorigo and coworkers [6] and has been successfully applied to more than 00 problems including the Travelling alesman Problem (TP) [7], the Quadratic ssignment Problem (QP) [8] and the Vehicle Routing Problem (VRP) [9]. This paper pursues the approach of Gueret et al. [5] that uses CO for creating music further. We modify the CO graph structure as well as the timing of the pheromone update and introduce a selection process for promising solutions. The graph structure is designed to optimize both melody creation as well as harmonization of a given melody. While different voices are independent in Gueret et al., we codirect them via the rules of Baroque harmony. The resulting ant music is the outcome of an optimization process that is guided by compositional rules. We analyze this optimization process and discuss the influence of varying parameter sets on the optimization behavior as applied to melodies and harmonizations. ection II presents the musical background necessary to understand the algorithm, which is described in ection III. The rules for melody creation and harmonization along with their implementation are explained in ection IV. The experimental set up and the parameter choices are described in ection V. Results are presented in ections VI and??.. Notes II. MUICL BCKGROUND Music is notated as a collection of notes and rests. Notes can be grouped into the 2 pitch classes C, C#, D, D#, E, F, F#, G, G#,, # and B. The # (sharp) sign raises the note that it preceeds in staff notation by a semitone or half step. Consecutive pitch classes are separated by a semitone. The ratio between the fundamentals of consecutive notes is.05945, where the fundamental is the strongest frequency present in the note as well as the lowest frequency of its so called overtone series. s = 2, the frequency of the fundamental between notes doubles every 2 half steps. This tonal distance (in musical language referred to as interval) is

2 Fig.. Interval names (on top) and sizes (below), names are abbreviated, where stands for first, 2 for second, etc., sizes are given in half steps Fig. 3. Correlation between pitches and MIDI values. Contrary to convention the bass key is placed above the violin key to follow the progression of MIDI values Fig. 2. The set of triads in C major plus the dominant chord V7 called an octave. Notes separated by a multiple of one octave, i.e. 2 semitones, sound similar and are hence given the same name, i.e. pitch class. B. cales scale is defined by its key and its mode. The key is a pitch class and denotes the pitch class of scale step one. cale steps start at one and are counted upwards. Note that the first scale step is also referred to as the tonic, while steps four and five are referred to as the subdominant and the dominant. The most common modes in Western music are major and minor. Major scales are defined by the following succession of intervals (in half steps): This changes into for harmonic minor scales. The succession is for ascending and for descending melodic minor scales. The sum of half steps in a scale is 2, as it repeats itself in every octave. For the sake of simplicity, this study will confine itself to the C major scale. C. Intervals The tonal distance between two notes is referred to as an interval and is counted in half steps. Intervals are named by the number of scale steps between them including the original note. Thus, the interval comprised of the same note being repeated is called a first, a scale step is called a second, two scale steps are a third, etc.. Figure shows all the intervals referred to in this paper along with their sizes in half steps. s scale steps can be full steps or half steps or a combination of both, intervals are more specifically named if the exact number of half steps need to be known. This is done with the attributes of a diminished, minor, perfect, major, or an augmented interval. Each of these are one half-step apart, i.e. a minor third is one half step smaller than a major third, which is a half step smaller than an augmented third and so on. Note however, that only firsts, fourths, fifths and octaves can be perfect. minor second, third, sixth or seventh that is increased by a half step becomes major, not perfect. D. Chords In this paper, we confine ourselves mostly to chords of three notes, called triads. triad in root form is made up of the root note, after which it is named, followed by the third and fifth above it. Thus, the C major triad in root form consists of the notes C, E, and G. The F major triad of F, and C. Major triads differ from minor triads by the size of their thirds. major triad has a major third at the bottom followed by a minor third. minor triad reverses this order. In the first inversion of both types of triad, the lowest note is the third of the root chord, followed by the fifth and then the root. The second inversion starts with the fifth of the root chord, followed by the root and the third. Note that in determining a chord, the relative pitch of notes does not matter, only the pitch classes. Figure 2 shows all chords relevant to C major. In addition to listing triads, it also introduces V7. V7 is called the dominant chord and consists of the G major triad (G, B, D) plus the seventh, which is an F. E. MIDI The Musical Instrument Digital Interface (MIDI) specification was created in the early 80 s to standardize communication between electronic music instruments. It breaks a musical performance down into events that can be sent in real time or be stored in files for later playback. The MIDI protocol represents note pitches as integer values ranging from 0-27, with 60 being assigned to middle C, that is the C between the violin and bass clef. When considering a succession of notes, we will occasionally refer to them by their MIDI pitch values. This simplifies the encoding of notes and the calculations performed by the functions that evaluate compliance with compositional rules. Figure 3 shows the MIDI values that are most commonly used in this paper and the pitches they stand for. The output of our algorithm is given in MIDI format, because MIDI files can be played back and are the recognized standard for recording performances. ee [20] for further information on MIDI.. The CO Metaheuristic III. LGORITHM CO as a metaheuristic can be used for solving problems from many different problem domains. In general, in CO artificial ants move along a graph, whose edges constitute solution components to the problem. solution is given by a path through the graph. The solution construction of the ants is guided towards good solutions through the reinforcement of pheromone trails highlighting good solutions that have been found before. This mechanism is twofold: ants finding good solutions are allowed to update the pheromone trail along the

3 T 64 T 64 T B 48 B 48 B T T 64 T 65 Fig. 4. Melody graph with the edges connecting to middle C B 48 B 48 B Fig. 5. Voice ranges of the soprano, alto, tenor and bass voices path they just took. Furthermore, pheromone evaporates, thus decreasing the influence of former good solutions that did not receive recent pheromone updates. To adapt the CO to a problem domain, a graph structure reflecting the structure of solutions, an objective function and constraints for the construction of a path are required. B. n CO for Composition and Harmonization The algorithm we propose has two modes. One creates a melody, the other harmonizes a melody. Both modes employ the same algorithm, but their graphs, objective functions and constraints differ. For the creation of a melody, the vertices of the graph are the set of all pitches allowed in the melody. Transitions of ants between vertices are possible in both directions, so every vertex is connected to every other vertex by two directed edges, one in each direction. In addition, each vertex is connected to itself by a single directed edge to allow pitch repetitions. For the harmonization of a melody, a vertex of the graph is an ordered tuple of four notes. Each note belongs to one of four voices, in the order of bass, tenor, alto and soprano, which play these notes simultaneously. We are placing the melody in the soprano voice. To limit the range of pitches and therefore the number of vertices in the graph, we further impose range constraints on the four voices. Naming middle C as C4, the bass is between E2 and C4, tenor between D3 and G4, alto between G3 and D5, and soprano between D4 and G5 as given in Figure 5. To reduce evaluation of note tuples that are known to lead to bad harmonizations a priori, we exclude two types of vertices. T B 65 T 65 T B 47 B 50 Fig. 6. Harmony graph of transitions from the C major triad (48,64,72,79)=(C,E,C,G) to nodes with (D) in the melody voice. The selected node (45, 65,, 8) = (, F, D, ) is the only chord in the set: the second inversion of the D minor chord. Vertices that do not constitute a chord. 2. Vertices, whose pitch values are not ascending from bass to soprano. For paths through this graph to be a harmonization of the melody, the succession of pitches in the soprano voice along every ant s path must constitute the melody. We will refer to this as the harmonization constraint. This suffices to determine the graph structure. To satisfy the harmonization constraint, the last pitch in the tuple must be a pitch that actually occurs in the melody. The other three enitres of the tuple must lie in the voice ranges specified for the three remaining voices and not fall into the categories listed in. and 2. above. ll vertices are connected. To satisfy the harmonization constraint, ants are only allowed to make transitions to vertices whose soprano pitch succeeds the soprano pitch of the ant s current vertex in the melody. This, together with demanding that the soprano voice at the starting vertex in a path is the first pitch of the melody, is sufficient to satisfy that constraint. Figure 6 illustrates a small subset of transitions from the chord (48, 64, 72, 79) in the center which is the C major triad with doubled root (C, E, C, G). In the example, the melody is in the alto, which progresses to a (D), hence all neighbors share the in the alto voice. The top row varies

4 only the soprano voice, but keeps tenor and bass unchanged. The middle row adopts the 8 () in the soprano that is also shared by the D minor chord. It varies the tenor while the bass stays the same. The bottom row chord, finally, share all but the bass note with the D minor chord. Given a graph and constraints, we describe the actual algorithm. Each edge is initialized with a constant pheromone value τ 0. The starting point of each ant s path is the tonic when building a melody. For the harmonization, there is no specific starting point. It is determined by the transition rules below but substituting.0 for the pheromone values. This is done because there is no pheromone value for the transition to the first chord, as there is no previous chord from which the transition is initiated. Then each ant constructs a melody or harmonization. This is done until the note sequence of each ant has the duration desired for the melody or harmonization. In each step of the construction process, every ant adds one edge to its path. The transition probability p ij between two vertices V i and V j is determined by equation : p ij = k N C i τ ij η β ij τ ik η β ik, <= i, j <= n () The relative transition probability in the denominator is given by the product of the amount of pheromone τ ij on the edge from V i to V j with the desirability of taking this step. The desirability is the fitness value given to the partial melody or harmonization obtained when V j is added as the next note or chord. It is the output of the objective function for the considered sequence of notes. The exponent β > 0 determines the importance of the desirability with respect to the pheromone values. Ni C denotes the set of neighbors of V i that satisfy the harmonization constraint. Thus, relative transition probabilities are normalized by dividing by the cumulative relative probabilities of all neighbors of V i satisfying that constraint. When each ant has constructed a solution, these solutions are evaluated by the objective functions described in ection IV. The ant that found the best solution is allowed to update the pheromone values along its path. But before that the pheromone values are evaporated. This is done by multiplying the pheromone for each node transition by a constant factor ( ρ), 0 < ρ < as shown in Equation 2. τ ij = τ ij ( ρ), <= i, j <= n (2) Then the ant that found the best solution in the iteration deposits a constant value τ 0 of pheromone on each edge along its path. If the fitness of this solution is better than all solutions found in previous iterations, it is stored in memory as the best solution found so far. Then the next iteration of ants starts. fter a prescribed number of iterations, the algorithm terminates and returns the best solution found during any of its iterations. Figure 7 illustrates the construction of a melody. tarting at the tonic of the key of the piece, the ant successively visits vertices, building the first measure of Joy to the world. Fig. 7. Partial path of an ant walking the first measure of Joy to the World. Melody Rules IV. COMPOITION RULE Melodies are scored according to 5 rules: moothness Rule, Contour Rule, Resolution Rule, Ratio Rule, and End-On-Tonic Rule. The qualitative aspect of the moothness, Contour and Resolution Rule are described in [2], the Ratio Rule was proposed in [9]. The End-On-Tonic Rule, added by us, simply enforces that the melody ends on the tonic of the melody s key. The total score of the melody is obtained by averaging the scores for the 5 melody rules. In ection IV-B details about scoring rules in a multi tiered approach are described. We apply this to melody rules in a simpler way: unless the moothness Rule scores at least.8, all other melody rules score 0.0 by default. This is a reflection of the necessity to have a smooth note progression before optimizing other aspects of the melody starts to make sense. ) moothness Rule: melody is expected to exhibit a regular flow. Therefore, most note transitions should be by step. fraction of 80% stepwise motion with 20% of intervals larger then seconds is considered ideal [2]. Intervals larger than a fifth resolve inward by step. First the ratio of stepwise note transitions within the total number of note transitions is calculated. This reatio is converted into a score, s, via the function in Figure 8, which is the result of a linear interpolation between the points (0, 0), (0.4, 0) (, ) (, ). If there are no intervals larger than fifths to resolve, this is the overall smoothness score. Otherwise, the fraction of correctly resolved intervals larger than fifths out of the total of such intervals, r, is calculated. The overall score for the smoothness function is then.5 (s + r). 2) Contour Rule: melody is required to follow either an ascending/descending ramp or an arch/inverted arch contour. The arch starts low, gradually rises to the highest note of the melody close to its midpoint and then descends again. The inverted arch simply inverts this shape, starting at the highest point, going down and reascending. The ascending ramp begins at the lowest note and gradually ascends towards the highest note at the end of the melody. The pitch drops between ascensions, letting each ascension start and finish

5 Fig. 8. Fig coring function for moothness tep Transitions in % coring function of smoothness rule melody starting with an arch, taken from a Nocturne slightly above the start and end of the previous ascension. The descending ramp inverts this pattern, starting at the highest and finishing at the lowest note. The ascents and descents for arch and inverted arch are allowed to repeat notes of the same pitch. Ramps, however, must be strictly ascending or descending. Examples of an arch and ramp contour are given in Figures 9 and 0 respectively. melody s contour score is the fraction of its notes that are part of a contour. djacent contours are concatenated. Thus, a twelve note melody consisting of a six note arch followed by a six note descending ramp receives a score of 0.5 for each contour, totalling.0. 3) Resolution Rule: cale step four and seven are called tendency tones as step seven demands a resolution to the tonic, step four resolves to step three. s an alternative, step seven can resolve down by step if it is part of a downward scale passage from the tonic to the fifth, as shown in Figure. The score for the resolution rule is the ratio of correctly resolved tendency tones out of the total number of occurring tendency tones requiring resolution. If there are no tendency tones that require resolution, the score is.5. This is to reward melodies exhibiting proper tendency tone resolution compared to those not possessing tendency tones. 4) Ratio Rule: set of ideal ratios for notes of the categories tonal center, color notes and avoid notes was proposed in [9]. The tonal center comprises the notes that constitute chords in a key, i.e. C, E and G in C major. Color notes are all other notes in the scale, i.e. D, E, and B. void notes are all notes outside of the scale. In C major these are essentially the notes with accidentals. The proposed ideal ratios are 60 %, 35 % and 5 % for tonal center, color notes and avoid notes respectively. s we are here not using avoid notes, we drop avoid notes but maintain the ratio between tonal center notes and ratio notes. This yields ideal percentages of 63 % and 37 %. imilar to the moothness rule we linearly interpolate between the points (0, 0), (0.63, ) (, 0) to obtain the function converting a tonal center ratio to a score. Note that replacing the maximum Fig. 0. The first line of Harlequin (Op. 42, No. ) by John Thompson largely follows an ascending ramp contour Fig.. Possible resolutions of scale degree seven, that is B in C major point by (0.37, ) and using the color note ratio yields the same scoring function. The curves can be seen in Figure 2. 5) End-On-Tonic Rule: This rule awards points for melodies that end on the tonic or are likely to end on the tonic as they are extended. With n being the desired melody size, scoring is as follows:. Paths of size n ending on the tonic score Paths of size n ending within a major third from the tonic score Paths of size n 2 ending within a perfect fourth from the tonic score ll other paths score 0.0. B. Harmonization Rules TBLE I IGNMENT OF HRMONIZTION RULE TO TIER Rule Chord Rule Voice Distance Rule Voice Leading Rule Progression Rule moothness Rule Resolution Rule TBLE II Tier I I II II III III PERCENTGE REQUIRED FOR TIER FOR THE NEXT TIER TO BE EVLUTED Tier % required I 80 II 80 III N/ The reason we chose four part harmonization is that its rules have been clearly laid out and can be readily applied. The classical reference for Baroque harmony is [22], which is based on observing the works of eminent Baroque composers like Bach and abstracting the rules they have been employing. The rules we are using are listed in I. Following [8], we evaluate rules in a multi tiered fashion. This is because if a harmonization attempt fails to address certain basic requirements, it is of low value, irrespective of how well it addresses more advanced requirements. It is essential for a harmonization to score high on a given

6 Fig. 2. coring function for Ratio Rule Color Note Ratio in % coring function for ratios coring function for Ratio Rule Tonal Center Ratio in % TBLE III CORING FUNCTION FOR CHORD Feature Root doubled Fifth doubled Root position First inversion Points awarded Fig. 3. The four types of movement in voice leading: contrary, oblique, similar and parallel motion tier before scores for rules in the next tier even need to be evaluated. Thus, chords and voice distance need to be solid first. Then, it is crucial to establish good voice leading and progressions before addressing smoothness or resolutions. Table II gives the percentages required at a given tier before rules for the next tier are considered. Unless 85% of the maximal combined score for the tier I rules are obtained, the tier II rules score 0.0 by default. imilarly, unless the combined tier I scores and the combined tier II scores, both reach.85, the tier III functions also score 0.0 by default. Given that the number of good harmonizations is very small compared to number of possible harmonizations, this saves computation time, as only reasonably good harmonizations require all rules to be evaluated. This is indeed helpful, as the number of each node s neighbors is much greater than the number of candidates leading to good harmonizations. ll of these neighbors need to be evaluated before an ant decides which edge to move along. The total harmonic score of a chord sequence is obtained by averaging the scores for all the harmonic rules. For transition probabilities between vertices, however, we only compute the score for the chord, respectively the score for the transition to the chord that is about to be added to the harmonization. That means, the averaging over all chords in a sequence is omitted in this case. ) Chord Rule: For the four parts to sound harmonious, they must form a chord. That means they must play notes from the exact three pitch classes of the chord. The admissible chords are determined by the key of the piece and are listed in ection II. s there are four parts but only three pitch classes to a triad, two parts play notes from the same pitch class. This is called doubling. For reasons of stability, doubling the root is preferred, doubling the fifth is acceptable, and doubling the third is avoided. imilarly, chords in root position are awarded full points, chords in first inversion are acceptable and chords in second inversion are avoided. Thus, points are awarded cumulatively according to Table III. If however, the chord is either not properly spelled, or the third is doubled or is in second inversion, the score is 0.0. The total score for a note sequence is obtained by averaging scores over vertices. 2) Voice Distance Rule: The following should be observed for the distances between voices:. oprano and alto should at most an octave apart. 2. lto and tenor should at most an octave apart. 3. Tenor and bass should be at most 2 steps apart. 4. oprano and bass should be at most three octaves apart. For each chord,.0 points are awarded if all of these are observed, 0.0 otherwise. The total score is obtained by averaging over all chords in the note sequence. 3) Voice Leading Rule: Voice Leading is the art of coordinating the movement between voices. Following [8], we consider the movement between soprano and bass on the one hand and the movement between alto and tenor on the other. There are four types of movement: contrary, oblique, similar and parallel motion, as illustrated by Figure 3. In contrary movement, the two voices move in opposite directions by any distance. In oblique movement, one voice repeats its previous note and the other moves up or down. Two voices move similarly if they move in the same direction but by different intervals. They move in parallel if both the direction and the intervals match. Contrary movement is awarded.5, oblique movement 0.25 for each of the two pairs soprano/bass and alto/tenor. No points are awarded for similar or parallel movement. s usual, the final score is obtained by averaging over the sequence of chords. 4) Progression Rule: For a harmonization to sound interesting, the succession of chords should not be repetitive. We require that successive chords have different roots. Each chord transition that satisfies that condition receives.0, 0.0 otherwise. The average of these values makes up the total for the Progression Rule. 5) moothness Rule: None of soprano, alto and tenor should move by more than a perfect fourth. The bass should not move by more than a perfect fifth with the exception of jumping an entire octave. chord scores.25 for each voice that progresses smoothly. s usual, the overall score is obtained by averaging. 6) Resolution Rule: Both the V7 and vii 0 chords contain the notes B and F. The distance between these notes is 3 steps, called a tritone, and creates a tension that demands to be resolved. This is done by replacing B and F with C and E, turning the tritone into a third. Both the C major and the minor chords contain C and E and are hence valid resolutions of the tritone in the V7 and vii 0 chords. The score

7 Fig note melody with a total score of 2 and subscores smoothness (0.96), contour (0.2), resolution (0.2), ratio (5), and endon-tonic (0.2) is the quotient of correctly resolved tritones by the number of occurring tritones. V. EXPERIMENT In our experiments, we constructed a 2 tone melody between pitches 50 (D3) and 69 (4). The starting pitch was fixed to 60 (C4) and only notes of the C major scale were added to the melody. For harmonizations, each voice had to observe the pitch boundaries specified by Figure 5. We chose the first 4 notes of Joy to the World transposed to C major as melody, which we placed in the soprano voice. This piece is a very well known example of four part harmony. Unless otherwise specified, tests were conducted using a standard value set as given in [5]. These values are β=.0, ρ=0.0 and τ 0 =0.. We further decided to terminate the algorithm after 5000 solutions are produced. With these constraints, populations of 00 ants achieved the best results for melodies while 25 ants were best for harmonizations. We adopted these as the respective default values for the population size. To compare the performance of the standard parameter set with an exhaustive search, we switched off the pheromone component. In that mode, ants only take desirability into account when choosing transitions. We conducted 500 test runs per parameter set for melodies and 200 for harmonizations. For the analysis of the optimization behavior, solutions were recorded after every iteration Pheromones no Pheromones Iterations Fig. 5. Optimization process of melodies for the standard value set with and without pheromone information. The exhibited increase, however, is significantly less than when no pheromone is used as illustrated in Figure 8. Furthermore, its overall scores are higher than when no pheromone is used. This indicates, that combining pheromone information with melody rules leads to better results than using melody rules alone. The scores also increase with the population size. This increase is steep for populations smaller than 25 but levels out for larger populations. Figure 4 shows one of the highest scoring solutions found by the algorithm. It scores 2, with suboptimal scores for smoothness and note ratios only. The contour is an inverted arch followed by a descending ramp. The occuring tendency tone is properly resolved and the melody ends on the tonic as desired. It also satisfies qualitative demands. Progression is predominantly stepwise, the contour is flowing and the forward momentum that builds towards the highest note is released during the descent to the tonic. VI. REULT Figure 5 compares the optimization process for the standard parameter set when pheromone is and is not used, respectively. It shows the best solutions obtained at given iterations averaged over 500 runs. The graph illustrates that relying on pheromone information induces a marked improvement in the solution quality. On average, the score obtained without pheromones after 50 iterations is surpassed in iteration six when pheromones are used. To study the influence of different parameters on the optmization behavior, we made several tests where one parameter was varied at a time. The remaining parameter values were taken from the standard parameter set. The results can be observed in Figure 6. It is notable that for most parameters, the values in the standard parameter set obtain scores that are the best or are close to the best among the values that were tested. Pheromone deposits should be in [.05,.]. maller values lead to steep decreases while larger values cause a gradual decrease in performance. The fraction of evaporating pheromone should be below Larger evaporation of pheromone leads to loss of information from previously constructed solutions. s increase with increasing β. This underscores the importance of relying on composition rules in addition to pheromone Population Evaporation Desirability Pheromone Fig. 6. of best melody found after generation of 5000 solutions when different parameter values are varied and all other parameter values are as in the standard value set; bars indicate the standard error The optimization processes with and without pheromone information for chords are shown in Figure 7. s with

8 0.9 Pheromones no Pheromones Iterations Fig. 7. Optimization process of harmonizations for the standard value set with and without pheromone Desirability Fig. 8. of best melody found after generation of 5000 solutions when β is varied and no pheromone is taken into account. The remaining parameter values are as in the standard value set; bars indicate the standard error melodies, the optimization process benefits significantly from the use of pheromones though the effect is not as marked. On average, the final value of the optimization without pheromones is surpassed at Iteration x when pheromones are used. The results for varying single parameters in the standard value set are displayed in Figure 9. INERT HERE PR- GRPH ON THEE VLUE ND OPTIML RNGE. imilar to the case of melodies, Figure 20 illustrates that the influence of the β parameter is less marked when pheromone is used compared to when no pheromone is used. This shows, that the communication of the ants through pheromone indeed improves solution quality. It supports the hypothesis that the CO heuristic is suited for the problem domain of melody construction and harmonization. Figure 2 shows an example of a harmonization of the first 4 notes of Joy to the World. It scores 0.933, with perfect subscores of 6 except in voice leading (0.25), progression (0.54) and smoothness (0.54). Even though the voice leading score is lowest, the qualitative aspect of voice leading is achieved rather well. Out of 26 transitions, contrary, oblique and similar motion occur 3 times, 9 times and 4 times respectively. Considering that only similar motion is to be avoided, about 85 % of all transitions show desired behavior. This compares favourable to ratios between 50 % and 75 % in [23] and [8] Population Evaporation Desirability Pheromone Fig. 9. of best harmonization found after generation of 5000 solutions when different parameter values are varied and all other parameter values are as in the standard value set; bars indicate the standard error Desirability Fig. 20. of best harmonization found after generation of 5000 solutions when β is varied and no pheromone is taken into account. The remaining parameter values are as in the standard value set; bars indicate the standard error VII. CONCLUION In this paper we have proposed an nt Colony Optimization (CO) algorithm that creates melodies and harmonizes them. Melodies and harmonizations are constructed by relying on pheromone information about previous solutions as well as compositional rules. The obtained results exhibit not only high fitness values, but also display the qualitative features that the objective function tries to capture. This is a marked improvement on the only previous CO that creates music, in which voices are not evolved codependently and no compositional rules guide the development of a melody. Future work may involve extending the musical criteria to include more refined aspects of baroque harmony or even entirely different musical styles. pplying a similar graph and rule structure as the one presented in this paper to durations of notes constitutes another natural continuation of this study. REFERENCE [] D. Cope, Computers and Musical tyle. Oxford, UK: Oxford University Press, 99. [2] C.-C. Chen and R. Miikkulainen, Creating melodies with evolving recurrent neural networks, 200.

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