Investigating morphing algorithms for generative music

Ren Wooller, Andrew R. Brown Queensland University of Technology, Australia

r.wooller, a.brown @qut.edu.au

This paper presents our research into morphing algorithms for generating musical transitions, including

digitally inspired composing techniques as well as models of existing creative processes. We show that it is

possible to generate effective morphs between contrasting musical passages and describe some ways in

which iterative developments seem to have improved the musicality of the generated material. The paper

begins by outlining the potential of musical morphing processes and introducing important considerations.

Relevant existing research and music theory is then summarised. Following this, our developments of

morphing algorithms are detailed and some evaluation criteria are proposed. The article concludes by

discussing the apparent results achieved by these morphing algorithms and what they indicate about

directions for further research.

Keywords: music, morph, metamorphosis, generative, algorithmic, adaptive, game, multimedia, non-linear

1. Introduction 1.1. Motivations The music of most non-linear digital genres such as computer games, multimedia art and websites is based

upon recombination of pre-produced sections. While these can be designed to adequately reflect the

aesthetics of a particular world-state, there is the more complicated problem of how to shift from one to

another in a musically satisfying way. Manually composing the transitions can be unworkable due to

combinatorial explosions - consider a game with fifty scenes and from each one the user can move to any

other, thus requiring a total of almost two and a half thousand transitions!

Currently the most successful automated transition is the cross-fade, which is metaphorically well suited to

the action of moving away from one place and into another and may also be the perfect technique for

diegetic or source-music. However, cross-fading does little to enhance the internal emotional transitions of

the player/user required by adaptive music, and is especially unsatisfying for musical passages with

conflicting elements. We explore the alternative possibility of formalizing various methods employed by

composers in order to generate more meaningful bridge sections and also propose that effective new

morphing composition techniques can be inspired through artistic exploration of automated processes.

1.2. Introducing musical morphing and related research Morphing is a pattern-merging process paying particular attention to smoothness and coherent combinations

of form. The topic of this paper is compositional morphing, focusing on musical constructs such as pitch and

rhythm, rather than sonic or graphical elements. Unless stated otherwise, morphing will refer to the

generation of musical transitions as mentioned in 1.1. Thus, morphing algorithms take start and end patterns,

hereafter named and , and produce a transition, . In functional terms we could have ),( morph= . The central problems of morphing emerge from the features that differentiate it from simple transitioning:

how is smoothness and coherence defined or measured? What ways exist to combine the form of and ?

In generative music systems, little research has gone towards these questions. Nevertheless, new

developments in the area can draw from many sources of potential inspiration. Some of the above questions

have been explored through related perspectives of music theory, music cognition and composition itself.

More general morphing processes could also be adapted from other fields such as graphics and sound;

however there is not sufficient space here to investigate all these. If the algorithm designer is a generative

music composer, personal intuitions could also be applied, in which case the algorithms also have the

potential to express a new musicality that may be satisfying to some.

2. Previous systems for musical morphing 2.1. Compositional morphing There have been a small number of explorations into musical morphing algorithms to date. Berger [1]

published a brief abstract on the topic a decade ago but seems not to have conducted any further research.

Edlund [2] recently developed a commercial prototype, with sound demos to show that it can be effective.

The technique apparently is to interpolate parameters such as timbre, volume, articulation, melodic material

for single note melodies and bass, rhythms, meters and tempo, however, it is unavailable for examination.

The only published software for compositional morphing was MMorph by Daniel Oppenheim [3], which has

ceased development and is unavailable. MMorph allowed four separate MIDI files to be morphed together in

amounts controlled by a user through the location of a point on a two dimensional plane. Users could select

between two different styles of morphing: interpolation or weighted selection, which are explained in more

detail below. The morph could be applied independently for rhythm, pitch, velocity, duration, and MIDI

channel. MMorph did not utilise analytic or generative processes, focusing instead on transformations. The

breadth of contextual information used by algorithms is limited to the immediate musical data. For detailed

discussion on our use of these terms when discussing algorithmic music systems, see [4].

2.2. Parameter morphing It has been more common for morphing in computer music research to focus on timbre, which can include

interpolating filter coefficients, FM modulation ratios and depths, or other synthesis parameters. Most of this

synthesis work is beyond the scope of this paper, however, parametric interpolation in general is an approach

to musical morphing and transformation that can inform this research. It was first discussed by Babbitt [5]

and reviewed recently by Momeni and Wessel [6] and Bencina [7] when presenting their musical interfaces.

The Metasurface [17] enables the user to morph between any number of parameter states positioned on a

two-dimensional plane using natural neighbour interpolation. Although the parameters would generally be

classified as sonic rather than musical, many of these are border-line. For example an LFO mapped to

amplitude will effectively control the rate of occurrence of a sound event in a musically rhythmic way.

Momeni and Wessels approach utilises a Gaussian kernel at each state, enabling greater control of the

influence that the state has upon the interpolated result. Of particular relevance is the Beat Space software [6]

where the parameters effectively control probabilities of beat generation within a certain eighth-note slot.

2.3. Morphing in music theory Oppenheim [3] provided a neat summary of morphing in music composition that bears repeating:

Classical composers experimented with the new sonata form that incorporated two contrasting themes.

Faced with a new compositional challenge of unifying a work containing contrasting elements, they began to

emphasize the bridge and developmental sections. One could argue that Beethoven was the first composer to

consciously attempt musical morphing often, this bridge section gradually introduced the musical elements

of the target theme as characteristics from the previous were placed in the background.

Despite this observation that morphing is an established compositional strategy there is no definite area of

musicological literature devoted to it, moreover, problems are addressed tangentially in various topics.

Bridge sections are rarely discussed exclusively and when mentioned, the overriding emphasis is on

harmonic techniques [8: 19, 21, 35, 38, 9]. Thus, key modulation is the most well-developed morphing

technique in music theory and occupies an accordingly significant portion of this section.

2.3.1. Modulation as Morphing Modulation is often framed as a kind of morph. For example Davidson and Welsh [10] set a task to find an

appropriate key modulation from a starting key to the tri-tone and back again. In terms of morphing, the

resulting modulation could be seen as , with the starting key as and the tri-tone as . Schoenberg discusses modulation extensively and implicitly refers to the morphing problems mentioned above [11: 165].

We will discuss direct modulations, pivots and the use of dominants in modulation.

Direct modulation cuts directly into the new key without any obvious transition section. It is common to

have a cadence or some form of end-phrase just before the new key occurs, to signal that a change is coming.

Drabkin defines a pivot as: a chord (or a note) having different harmonic (or melodic) functions in two

different keys, this property being used to effect a smooth transition from one key to the other [12]. The

distance between two keys is defined as the number of pivot notes possible. After the key change, and thus

harmonic function of the pivot, it is common to insert a cadence to reinforce the new tonality.

Another technique is to change the chord function so that it becomes the dominant of . For any chord in the major key, we could substitute it with the dominant chord-type, Mm7, and then shift to the new tonic

which would be Maj7 and a fifth below. The change in function warns that a new key is immanent and hints

what it is going to be. In jazz theory this is extended, so that instead of changing the function of the chord a

second, which leads to the new dominant and finally the new tonic (ii-V-I). Often, the second is changed to a

secondary-dominant, meaning that it has the function of a dominant that leads into another dominant chord.

Although this overview is brief, some interesting approaches have emerged that partially answer some of the

questions posed by morphing. The general natures of these are summarized here with more specific

examples:

Smoothness:

o Creation of sub-morphs through a series of states that are more similar to each other than is to . The ii-V-I can utilise pivots of ii-V and V-I that are not in ii-I.

o Applying quantifiable measure of distance between and . Key-distance is the number of pitches that are common.

Coherence: o Signals within that imply forthcoming change. The ii in a ii-V-I. Also when the end-

phrase is applied before a direct modulation.

o Reinforcing the change to by accentuating differences with . For example cadences in that involve I, IV and V will collectively highlight all of the pitches within the new key.

Combination: o Common elements from and , now referred to as , being used to create . In key

modulation, are pitches. o The same having different functions within and . For example, the I of might be the

IV of . Some of the more general ideas could be applied to elements of music other than pitch.

2.3.2. Rhythmic morphing Rhythm relates to the organisation of music in time and the identity of a musical section is tied to this

organisation. In discussions about rhythm and rhythmic variation in music, three aspects of rhythm are

dominant;

rhythm as duration o The duration of notes and their onset spacing provide cues to the pace and density.

rhythm as element-succession o The rate of change of elements dictates its stability or rate of variation. For example, the

frequency of chord progressions in a song can be referred to as its harmonic rhythm.

rhythm as grouping o The grouping of notes as rhythms allows phrase identification and underpins metric

structure. Maintenance of rhythmic grouping contributes to perceptions of musical character.

In addition to discussions about the direct influence of rhythm and meter in music Wallace Berry emphasises

the significance of accent as a stylistic cue which arrises from a relationship between rhythmic placement (in

the metrical structure), note duration, and dynamic [13]. These elements can be used as identifiers of

musical structure in the morphing algorithms such that the generated transition maintains identifiable

cohesion with the existing musical segments. Notions of similarity with regard to statistical overlap between

the rhythmic elements of musical sections can inform the generation of morphs, as with pitch similarity.

3. Preliminary developments Our current explorations of musical morphing algorithms have led us to explore three different strategies to

morphing, each undertaken with a view to overcoming observed limitations with the previous. These

techniques were interpolation, weighted selection and probabilistic generation. The first two are mentioned

briefly by Oppenheim [3], while the latter is a new development which, in particular, is directed at the

morphing problem of combination, mentioned in section 1.2.

3.1. Interpolation Interpolation, generally, is the technique of estimating appropriate transitional values between known points.

For example, drawing a straight line between them is one approach, called linear interpolation, and applying

a curved line model is another, called non-linear, polynomial or spline interpolation. If we view the musical

data as a collection of numeric parameters, it is possible to apply interpolation techniques between them.

Given this apparent simplicity, a major consideration becomes how to represent or map the musical data into

a space where interpolation along each dimension enacts a smooth and coherent progression. That is, the

music at any sub-section should be more similar to the music in adjacent sub-sections than any other part

of . Put this way, the problem is one of defining similarity measures, as discussed below and in [14].

Another problem when mapping music to continuous dimensions is the discrete or symbolic nature of note

events. Imagine that has three notes and has ten. Oppenheim [3] alluded to an effective technique for interpolating rhythm; however descriptions of his process proved difficult to find. Our initial approach was

to recombine rhythmic and metric structure first then interpolate the tonal pitch centre at each cycle of the

recombination.

Figure 1 Two one-bar patterns (bar 1 and bar 4), morphed over two bars using rhythmic recombination and pitch interpolatio

If the music is quantised, a simple technique to overcome the problem of note-events is to define a sin

parameter for each of the possible note start-times, interpolating the desired parameters for each one. Th

effectively a form of layering.

3.2. Layering While interpolation transforms parameters to obtain intervening values, layering superimposes element

and into . Any point n in the transition, n , must be somehow have been derived from n or n . example, cross-fading and . The advantage of layering is that the form of both patterns is unchanmaking them easily coherent within . The disadvantage is that , as a pattern itself, contains minformation than and on their own and some of it may be conflicting. Classic examples are fadbetween incompatible keys, or divergent rhythms.

A way to partially avoid this problem is through weighted selection, which probabilistically reveals a

section of either or , depending on the weighting, that represents their prominence in w . Throughthe transition, the value of will gradually increase, starting with and w 0=w = , endwith and1=w = . Over this period, when 25.0=w , there is 0.25 probability of being unmasked, the inverse for . This way, neither will occur simultaneously, thus reducing the conflicts between musn gle

is is

s of

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ing

sub

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ing

and

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elements that might occur when they both occupy the same n . There is a trade-off however, as the individual structures of and within are fragmented and the form of is random.

Figure 2 The layering technique of weighted selection (left) as compared with cross-fade (right), applied to two patterns, a and b

To overcome this, we analyse and for similarities and increase the chances that sub-sections used in follow coherently from each other. For example, if 11 = nn and 1n is dissimilar to 1n , the chances are that nn = will follow more smoothly than nn = and thus it would make sense to increase the probability of this selection. However, this seems only to slightly increase the effectiveness of weighted selection on

certain patterns that fit well together already. A logical extension is to concatenate subsections that appear to

follow each other well regardless of onset, in which case n is not necessarily constituted from n or n and so it is no longer considered layering, rather a form of probabilistic generation.

3.3. Probabilistic generation Although probabilistic models have been employed to generate melodies through note-by-note prediction

since the beginning of computer music history [15] they have not been applied to morphing. The standard

process is note-by-note prediction, where the entire sequence is generated through a series of decisions about

which note should follow the current one. The current or seed note influences a probability distribution in

some way, from which a next note is drawn at random. The newly selected note is fed into the algorithm

recursively. Mozer [16] formally tested the compositional output of Markov chains and an Artificial Neural

Network (ANN), aiming to explore the feasibility of note-by-note prediction (the ANN would create a

probability distribution at each step). Although rigorous surveys found that global coherence is lacking from

the compositions, there is a possibility that, for short transitions, this attribute is less important and adequate

music can be generated, as seems to be the case when Markov chains are applied to the problem of

continuation [17].

3.3.1. Markov refresher A Markov model or Markov chain describes the probabilities for generating a string of symbols that can be

used to represent any sequential process. The probabilities can be automatically derived through statistical

analysis of sequences; the more frequently one element follows another, the greater probability is assigned.

The probability values are stored in a table we will refer to as a Markov matrix.

Let be a matrix (table of numbers), where each cell ( i is the row, is the column) contains the

probability of transitioning from the state represented by column to the state represented by the row

(some texts are opposite). The states could be mapped to anything, for example, every pitch class would

mean an 11x11 matrix with the notes C, C# etcetera through to B. Thus we could generate a sequence of

pitches by starting with a seed, for example C (1). The probability distribution for next pitch would be the

numbers in the column . If there would be an 80% chance of the next pitch being a C also and

A jiA , j

j

i

1,...A 8.01,1 =A

2.01,2 =A would indicate a 20% chance of the next pitch being a C#. Because these sum to one, there would be no chance for other pitches to be selected. Markov chains are also represented graphically. In this case:

Fig. 1: Diagram of a segment of a simple Markov chain for pitch

3.3.2. Markov morphing Our technique of Markov morphing involves two matrices, andA B , which are created to represent the notes and transition probabilities of and respectively, using statistical analysis of the sequences. Note-by-note prediction as described above is employed to select the note at each time-step and the onset or start

time of the selected note is adjusted to match the position in the bar that is being generated n . The matrix

that generates the note at each particular time-step is either A and B , as chosen via weighted selection. The process of prediction can be summarized:

1. )),,,(( 1= nn wBAselectpredict One problem with this is obvious: what happens when a seed note does not exist in the currently selected

matrix? We are dealing with sections of music, not large databases, and so this result, null prediction, can be

a frequent occurrence.

The way we deal with it is to fall back on weighted selection of layers, that is, simply insert

n or n depending on which matrix was selected. This has the benefit of refreshing , and thus the next seed, with data that is structurally sound. 1 2 3 4 5 6

n

An rid

3.3.3. Similarity of notes ed consists of pitch, onset, duration and dynamic (loudness) attributes. The

hen pitch is applied directly, C4 bears more similarity to C4# and B3 (difference of a semitone) than to C5,

(twelve semitones) even though, in most instrumental musical traditions, the opposite is true. To overcome

Figure 3 Example Markov-morph over four bars (2 to 5), with a %50 null-prediction rate, falling back on weighted selectio

other way is to select the most similar note and use it as a seed instead or, extending this, build a hyb

probability distribution based on adding together the probability distributions for similar notes and weighting

them on their level of similarity. Both of these techniques are incorporated into our algorithm at present and

if the similarity is below a threshold, the fallback technique is employed. But how is similarity determined?

The note representation employ

similarity measures for each of these are combined under user-controlled weightings. When using the

Markov morph in practice, unless there is a wide range of dynamic and duration values within both the

source and target patterns, these parameters detract from the accuracy of similarity. While similarity for both

duration and dynamic can be taken directly from differences in value, the same cannot be said for pitch and

onset.

W

0.8 0.2

C C#

this, one can transform pitch into a multi-dimensional space, for example a helix [18], and apply a Euclidian

distance measure, as employed by Mozer [16]. There are many other psychological representations of music

that have been conceived and a good reference is [19]. The current algorithm takes the difference in absolute

pitch and combines this with the difference in pitch-class (one dimensional pitch torus with twelve semitone

values). The result is put through a function that exaggerates the combined similarities and differences. The

sensitivity of this function can be adjusted. 1 2 3 4 5 6

Figure 4 Markov morph as before, but with exaggerated similarity of pitch and hence a lower null prediction rate of %16.

A similar technique is used for start time. Absolute difference in onset is combined with difference within

the cycle of two bars (so the last beat in the second bar is only one step away from the first beat in the first

the first to propose a criterion for musical morphing algorithms Morphing can be said to

ate the listener can still perceive the relation to both the source and target

ation of transitions, where the weighting is a

nction of time. In this case, there are no perceivable morph states within the continuous transition, only

of the morph.

r this section describes some

fi at was noted by the authors when conducting informal tests as part of the

bar), one bar, half a bar, three beats, two beats and one beat. Each of these can be weighted differently within

the combination and a squashing function as with pitch is also applied.

4. Discussion 4.1. Evaluation criteria Oppenheim was

work well when at any given st

music [3: 479]. This criterion works well for Oppenheims original context, that is, where the weighting

between the source and target is time-independent so that the listener has the opportunity to examine a

complete cycle of the music at any particular weight value.

However, this measure does not apply so well to the situ

fu

moments of time, and it would be impossible for all moments in time to relate to both source and target.

Instead, the criteria we employ looks at two traits in particular:

Smoothness: how well each section within the morph is related to the adjacent sections. Coherence: the logic of the musical syntax or structural form4.2. Musical results Thorough testing of the musical results is beyond the scope of this paper howeve

initial ndings, based on wh

development cycle. When testing the various morphing techniques against these criteria we transitioned

between musical patterns that were similar in musical style, and used the same channels and instruments.

None of the test patterns were shorter than a bar (measure) or longer than four bars and most patterns were

two bars long. Three classes of musical morphing were tested; random selection, weighted selection, and

weighted Markov selection. Overall, the perceived results seemed to support the previous views that

different algorithmic processes have noticeable musical effects and also that note-by-note prediction tends to

lack coherence. Some more detailed insights were also gained that will inform future developments.

The length of the morphing section is significant, in that the longer a morph period is, the smoother it tends

iven reasonably similar patterns, pure weighted selection will yield a fairly smooth morph however, it lacks

g involves the algorithmic generation of a musical transition between set patterns.

Morphing Music at a high-level of structure. 1995, CCRMA. site: http://www-

to be. Morphs of 32 or 64 beats long are noticeably smoother than those of 8 or 16 beats. The musical

coherence of the transition was strongly affected by the point within the cycle at which it began, indicating

the importance of phrasing. It was the most effective when the cycle of the source pattern finished just

before the transition was commenced, rather than beginning mid-way through.

G

coherence as a musical segment. Weighted selection is particularly suitable for rhythmic material but less

effective for melodic material. Applying the same patterns to our Markov morpher, and fine tuning the

parameters to suit the patterns, ensures that there is more likelihood of coherence in melodic material,

however, rhythmic structure may not be significantly improved. Rhythmic parts, such as drums, sound more

natural when the onset time was given priority weighting in the Markov selection; this makes the Markov

process more similar to weighted selection. Conversely, the melodic contour of pitched parts, such as bass

and pads, morphed well when pitch was given priority in the Markov selection probabilities, and responded

well to modest increases in the Markov depth despite the increase in null conditions that were resolved by

similarity substitution. It was found that in the case of very contrasting patterns, which mostly gave low

similarity ratings, it was better to induce a null-prediction and fall back on weighted selection. This was

achieved by adjusting the similarity sensitivity (see above) so that differences were exaggerated and values

that were close to zero became zero. Increasing the order or depth of the Markov chain would also increase

the rate of null predictions, which varied from 0% in morphs of similar patterns and with a depth of one, to

70% in contrasting patterns and depths of three or more. Full morph examples can now be downloaded [20].

5. Conclusion Musical morphin

Morphing is useful for non-linear digital media systems where the timing and, perhaps, duration of state

transitions is difficult to predict. We present morphing as an approach that operates at the compositional

level of note structure, rather than the more commonly used strategy of volume or timbre cross fading

between musical patterns. We have presented an overview of some of the issues involved in designing

musical morphing algorithms and analysed potential morphing strategies including linear interpolation,

layering and probabilistic generation. In order to evaluate these, or other, musical morphing techniques we

have proposed a three dimensional contextual criteria consisting of smoothness, coherence and aesthetics, of

which the latter two compare the generated transition material against the start and end patterns. Our initial

results indicate that effective musical morphs are possible and that the development of new morphing

techniques responds well to informed iterative elaboration. Therefore, we look forward to further testing of

our existing techniques and further research to design musical morphing techniques that are more convincing

aesthetically, are effective across a wider range of musical styles, and that contribute further to our

understanding musical structures.

References [1] Berger, J.,

ccrma.stanford.edu/~brg/research/morph/morph.html accessed on: 6th of September.

[2] Edlund, J., Music Morphing. 2005, InterAmus Music Systems. site: http://www.interamus.com accessed on: 6th of September.

[3] Oppenheim, D.V. Demonstrating MMorph: A System for Morphing Music in Real-time. in ICMC95 International Computer Music Conference. 1995. Banff Canada: ICMA.

[4] Wooller, R., et al. A framework for comparing algorithmic music systems (in press). in Symposium on Generative Arts Practice (GAP). 2005. University of Western Sydney.

[5] Babbitt, M., The use of computers in musicological research. Perspectives of New Music, 1965. 3(2): p. 74-83.

[6] Momeni, A. and D. Wessel. Characterizing and Controlling Musical Material Intuitively with Geometric Models. in Proceedings of the 2003 Conference on New Interfaces for Musical Expression NIME2003. 2003. Montreal, Canada.

[7] Bencina, R. The Metasurface - Applying Natural Neighbour Interpolation to Two-to-Many Mapping. in New Interfaces for Musical Expression NIME05. 2005. Vancouver, BC, Canada.

[8] Carpenter, P., Grundgestalt as tonal function. Music Theory Spectrum, 1983. 5: p. 15-38.

[9] Kamien, R. and N. Wagner, Bridge Themes within a Chromaticized Voice Exchange in Mozart Expositions. Music Theory Spectrum, 1997. 19(1): p. 1-12.

[10] Davidson, L. and P. Welsh, From collections to structure: the developmental path of tonal thinking, in Generative Processes in Music, J.A. Sloboda, Editor. 1988, Oxford University Press: Oxford. p. 260-285.

[11] Schoenberg, A., Theory of Harmony. 3rd ed. 1978, Berkeley: University of California Press.

[12] Drabkin, W., Pivot. 2005, Grove Music Online. site: http://www.grovemusic.com accessed on: 6th of August.

[13] Berry, W., Structural Functions in Music. 1976, NJ: Prentice Hall.

[14] Rolland, P.-Y. and J.-G. Ganascia, Musical pattern extraction and similarity assesment, in Readings in Music and Artificial Intelligence, E.R. Miranda, Editor. 2000, Hardwood Academic Press: Amsterdam. p. 115-144.

[15] Hiller, L. and L. Isaacson, Musical Composition with a High-Speed Digital Computer, in Machine Models of Music. 1958.

[16] Mozer, M., Neural network music composition by prediction: Exploring the benefits of psychoacoustic constraints and multiscale processing. Connection Science, 1994.

[17] Pachet, F. The Continuator: Musical Interaction with Style. in ICMC. 2002. Gteborg, Sweden.

[18] Shepard, R.N., Geometical approximations to the structure of musical pitch. Psychological Review, 1982. 89: p. 305-333.

[19] Deutsch, D., ed. The psychology of music. 2nd ed. 1999, Academic Press: New York.

[20] Wooller, R., Morph Music Demos. 2005, Queensland University of Technology. site: http://www.adaptivemusic.net/ and http://student.ci.qut.edu.au/~n2501350/morphDemos/ accessed on: 4th of November.

Investigating morphing algorithms for generative musicIntroductionMotivationsIntroducing musical morphing and related researchPrevious systems for musical morphingCompositional morphingParameter morphingMorphing in music theoryModulation as MorphingRhythmic morphingPreliminary developmentsInterpolationLayeringProbabilistic generationMarkov refresherMarkov morphingSimilarity of notesDiscussionEvaluation criteriaMusical resultsConclusionReferences

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