PLANE TESSELATION WITH MUSICAL-SCALE TILES AND BIDIMENSIONAL AUTOMATIC COMPOSITION ABSTRACT We present a method for arranging the notes of certain musical scales (pentatonic, heptatonic, Blues Minor and Blues Major) in the plane by using the idea of plane tessellation with specially designed musical-scale tiles. The motivation of such a representation is described, as well as the mathematical analysis of the possibility of its realization. As a main application of the idea we introduce a bi-dimensional designed automatic composition algorithm, at which we also explore the target-note improvisation paradigm, by using Markov Chains conditioned in certain events. 1. INTRODUCTION Musical instruments can be separated in two classes regarding their input interfaces: one- or bi-dimensional. In the first class we can put the piano, the flute, the the xylophone, the trombone, etc. In the second, most of the string instruments (with more than one string), button accordions, and others. There are still instruments with both interfaces, like the piano accordion, whose melodic interface is one-dimensional and the bass interface is bi-dimensional. So, in the first class, notes are associated with points displayed on a line, and in the second, on a plane. Typically there are no redundancy of notes in one-dimensional instruments: a note with, say, fundamental frequency f, can only be triggered at one specific position. On bidimensional instruments the contrary is the default: most values of f would have two or more bi-dimensional points associated with. Bi-dimensional instruments are normally tuned in fourths (the guitar, for example) or in fifths (like the mandolin). In both cases, when playing the chromatic scale using more than one row of the matrix of notes, we can see patterns (like tiles) which repeat themself along the instrument interface. In this work we analyze such tiles for instruments tuned in fourths. We will show how the idea of plane tessellation can be applied for musical scales other than the 12-notes chromatic scale. More specifically: the Blues Minor and Major (6-notes) and the generic pentatonic (5-notes) and heptatonic (7-notes) scales. A simple proof of the possibility of such tiling is presented. As a main application of the idea we introduce a bidimensional designed automatic composition algorithm, at which we also explore the target-note improvisation paradigm, by using Markov Chains conditioned in certain events. There are, actually, three Markov Chains, one for each of the melody, meter and harmony components. The remaining of this paper is organized as follows. In section 2 we briefly mention related works that can be found in the literature. In section 3 the method of plane tessellation using musical-scale tiles is described and analyzed. The bi-dimensional automatic composition algorithm based on those tilings is detailed in Section 4. Section 5 contains some results (including a score), and conclusions are addressed in Section 6. 2. RELATED WORK The automatic composition algorithm we will describe shortly has three main aspects: it is bi-dimensional designed, it has a Markovian Process engine, and such process obeys certain restrictions. All these subjects have already been exploited in computer assisted composition. Computational models using Markov Chains, for example, are used since at least 1959, according to [7], and ideas using them keep emerging (see [6], for instance). The idea of constraint-composition has been used in [2]. To enable real-time composition, the solution of the related combinatorial problem is searched for a limited amount of time, after what the current approximation is used. We will apply a similar idea in our method. Regarding bi-dimensional composition, [4] (Section 4) mention a work of Xenakis, where Brownian motion of gas particles (in 2D) is combined with Bernoulli s Law of Large Numbers to work as engine for automatic composition. To build the interface for bi-dimensional improvisation we will explore the idea of tiling the plane with musicalscale tiles. Tilings also have been applied to computer assisted composition [3, 5, 1], but we have not found works using that theory for constructing bi-dimensional interfaces for automatic composition. Moreover, to our knowledge the use of Markov Chains with restrictions have not been explored yet on bi-dimensional
interfaces (with notes displayed according to the tiling method we will describe) for the purpose of automatic composition. 3. MUSICAL-SCALE TILES Roughly speaking, our algorithm for composition consists of walking randomly on a matrix of points and playing the musical notes associated with them. We now describe how to build such matrix. Let s start by looking at Figure 1, where notes with the same number have the same corresponding fundamental frequency. The representation of Figure 1-right appears naturally in instruments tuned in fourths, like the guitar, for example. This means the note immediately above the note 0 (i.e., note 5 in Figure 1-right) is it s perfect fourth; that the note immediately above the number 5 (i.e., note number 10) is the perfect fourth of note number 5; and so on. Figure 3. Bi-dimensional representation of heptatonic (left) and pentatonic scales (right). Gray-filled circles are the scale roots. The corresponding tile is shown in Figure 4-left. It s worth remembering that the Blues Minor scale notes are: scale root, minor third, perfect fourth, augmented fourth, perfect fifth and minor seventh. Figure 1. Arrangement of notes from the chromatic scale at the piano interface (left) and in instruments tuned in fourths (right). Figure 2 shows three examples of heptatonic scales (Major, Natural Minor and Harmonic Minor) in the representation of Figure 1-right. Figure 2. Gray filled points represent the Major, Natural Minor and Harmonic Minor scales, according to the representation of Figure 1-right. The idea is to take off those circles that are not filled, since they represent notes out of the scale, which have not (and normally must not) be played. This way we arrive at the representation showed in Figure 3-left, where this time the gray-filled note represent the scale root. The order is preserved, that is, from the tonic note (scale root), left to right and bottom to top. Analogously, we propose for the pentatonic scales the representation shown at Figure 3-right. In view of tiling the plane with scale-tiles like those shown in Figure 3 it is necessary to state precisely some geometrical measures. Here, we will use as example the Blues Minor scale, the process for the other scales being similar. Figure 4. On the left, Blues Minor scale tile. Dark gray filled note: tonic note. Light gray filled note: blue note (augmented fourth). On the right, tiling of the plane with the Blues Minor scale tile. The blue note has special highlight in this representation. Given a tile, the next step is tiling the plane as shown in Figure 4-right. Figure 5 shows the octave relation in the tessellation. Again, it is similar the one that appears naturally in instruments tuned in fourths. After a tessellation, what remains is to subtract the area that actually will be used in the algorithm. For simplicity, such region will normally have a rectangular shape. In the case of the Blues Minor scale tessellation, an example is shown at figure 6. We have studied the shape of tiles for the Blues Major and Minor Scale, as well as general heptatonic and pentatonic scales. We just described how to tile the plane using Blues Minor scale tiles. For the other mentioned scales, the procedure is analogous, tiles being the ones showed in figure 7. Notice that all tiles have a common L-like shape, as shown in figure 8-left. The corresponding tessellation must satisfy the condition that corner x of some tile coincide with corner y of the adjacent tile (Figure 8-right). The tiling is completed by coupling side by side the bands shown in Figure 8-right (see Figure 9), what is possible due to the coincident metrical relations of adjacent boundaries (shown in figure 8-right).
Figure 5. Octave relation in the Blues Scale tiling. Tiles in the same band are such that the fundamental frequencies associated with points are the same if the points have the same relative position in the corresponding tile. Notes of tiles at the region B are one octave above the ones in the tiles of region A, and so on. Figure 9. Geometrical proof of the possibility of tessellating the plane with L-like tiles such that corners x and y (see Figure 8) meet. 4. BIDIMENSIONAL AUTOMATIC COMPOSITION Figure 6. A rectangular region of the Blues Minor scale tessellation. Figure 7. Tiles for pentatonic, heptatonic and Blues Major scales. Figure 8. All presented musical scale tiles have a common L-like shape (left), and the corresponding tessellation is such that corners x and y meet (right). In this section we present the algorithm to generate a music sample and the related probabilistic tools. In a few words, a random sample of music was chosen as a Markov Chain conditioned to specific events. The finite Markov Chain state-space was defined as E = H R M, where H is the space of possible chords, whose sequence determines the music harmony. In the specific implementation, H = {I 7,IV 7,V 7 } (where I, IV and V are the root, sub-dominant and dominant chords, respectively), but it could be much more general, as well the next particular choices. R is the space of rhythm patterns for melody, including the silence figure. We have used five different states, corresponding to silence (rest), whole, half, third and quarter notes. Lastly, M is the space of possible notes, namely, the scale. Here is where bi-dimensional composition appears, since M is a rectangular subset of a tessellation as described in the previous section. More precisely, in our Blues-like style experiments we have used the Minor pentatonic scale tessellation, as shown in Figure 10. The Markov Chain (X n,y n,z n ), for n = 1,...,N is indexed with the beat. The blues has, generally, a quaternary score, and the music twelve bars (that s the well known 12- bar Blues). So we fix N = 48. The Markov Chain of harmony, (X n ), and (Y n ), the Markov Chain of rhythm, and independent. On the other hand, (Z n ), which gives the choice of notes, is strictly dependent of X n and Y n. The dependency on harmony, X n, is natural from the fact the melody must follows harmony rules. On the other hand, the dependency on Y n, the sequence of time figures, comes from the fact the number of notes is determined by it, and can be even none, in the case of the silence figure. The conditioning on specific events mimics the behavior
of playing two half-notes or four quarter notes is high, etc. Figure 11 illustrates this situation. The case of the sequence (Z n ) is analogous, with the states being the row and the column of the points in the bi-dimensional representation of the scale that we have introduced. Actually, there are two independent Markov Chains controlling the sequence of notes, one for the row index and the other for the column index, the transition probabilities of them being shaped as shown in figure 11. Figure 10. Rectangular subset of the (minor) pentatonic scale tessellation. Numbers represent MIDI-note codes. of a musician, that when improvising, pursues a target note in meaningful chords. This is what we call the target-note improvisation paradigm. In a naive case, when in the V 7 chord preceding the final chord I 7, it is natural to finalize with the note of the music tonality, which will combine with the next chord, I 7. In our implementation, besides the previous conditioning for the melody we have conditioned the first note of the 12-bar series as being the root note (modulo the octave). Regarding chords we sample one for each bar. The first is fixed as being the root chord (I 7 ), and the last the subdominant (V 7 ). For the others, we have conditioned a chord IV 7 for the fifth bar and V 7 for the ninth. Let be A the set of sequences which satisfy the rules chosen. The method to simulate the conditioning of the Markov Chain on A is the very well known Rejection Method, which consists simply in sampling the Markov Chain, and if the sample belongs to the set A, keeping that one. If not, we resample until we get an allowed sample. In computational terms, the number of samples until an allowed sample be obtained can be very large. For this reason, we limited the number of trials. If no allowed sample is found, the last one is chosen. Of course doing this we do not simulate exactly the conditioned Markov Chain defined above. However, this way the algorithm imitates musician s errors, when he doesn t reach the target note, what can eventually happen. Summarizing: each time a new 12-bar series will begin we sample three Markov Chains as described above until the mentioned conditions are satisfied or the maximum specified number of trials is reached, what come first. We have used the uniform distribution as initial distribution of both (X n ), (Y n ) and (Z n ) sequences. The transition probabilities for (X n ) was set as uniform, i.e., being at state I, the next state could be IV or V with equal probability, and so on. For (Y n ) we have chosen M-like functions centered in the current sample. This means that if at the current beat the chosen figure is three thirds, in the next beat the probability of playing the same figure is small, the probability Figure 11. Shape of the transition probabilities for music figures and notes. The probability of remaining in the same state is small of that of going to the nearest states. Then, the farther the state, the lower the probability of it to be the next state. 5. RESULTS Figure 12 shows the score corresponding to a 12-bar sample output of our method. The algorithm has many parameters and we have found good results with the ones cited in the previous section. Regarding Markov Chain restrictions, the more the number of target notes, the more the number of trials the algorithm has to do satisfy the restrictions. We have seen that for two target notes an upper bound of one thousand trials is never reached i.e., the algorithm always find a satisfying solution before the thousandth trial. However, in some tests we have conducted, for more than 4 or 5 solutions that upper bound is easily passed. We could in this case raise up the upper bound to, say, 10,000. But in this case when the number of trials is high (near the upper bound) the time consumed is such that the algorithm can not work in real time (for tempos around 120 beats per minute). 6. CONCLUSIONS In this work we have presented a method for automatic composition which has three main aspects: it is bi-dimensional based, it uses a Markov Process engine and the composition tries to follow the target-note improvisation paradigm. The bi-dimensional nature of the algorithm comes from the fact that the melodic line is sampled according to a random walk in a matrix of points, whose associated notes are
Figure 12. A 12-bar sample from the automatic composition algorithm we have implemented. modeled according to the idea of tiling the plane with music scale tiles. We have implemented the method for the 12-bar blues style, using the pentatonic minor scale for the melodic line. Technically speaking, such an algorithm is not difficult to implement, given the simplicity of the Markov Model and of the bi-dimensional representation of musical scales we have used. Regarding musical quality, the method produce nice jazzlike improvisations. Of great importance here is the fact that only the notes of some determined scale are played. We believe bi-dimensional inspired automatic composition is more adequate to simulate certain kinds of improvisations. The guitarist, for example, thinks about the scale at which he is improvising, but it also uses more the notes which are near the current note. This means that the geometry and the dimension of the instrument is of great importance, and it s worth noting when building automatic composition systems. As future work it would be interesting to use Machine Learning techniques to estimate the transition probabilities of notes for different styles, specially those when the instrument used for improvisation is bi-dimensional. For more examples of music samples built using the method described here the reader can refer the project website (to appear in the final version of this text). There he can also download the software we have used to obtain the hosted results. 7. REFERENCES [1] E. Amiot, M. Andreatta, and C. Agon, Tiling the (musical) line with polynomials: Some theorethical and implementation aspects, in International Computer Music Conference, 2005. [2] T. Anders and E. Miranda, Constraint-based composition in real time, in International Computer Music Conference, 2008. [3] M. Andreatta, C. Agon, and A. E., Tiling problems in music composition: Theory and implementation, in International Computer Music Conference, 2002. [4] P. Doornbusch, A brief survey of mapping in algorithmic composition, in International Computer Music Conference, 2002. [5] F. Jedrzejewski, Permutation groups and chord tessellations, in International Computer Music Conference, 2005. [6] F. Pachet, The continuator: Musical interaction with style, Journal of New Music Research, vol. 32, pp. 333 341, Sep 2003. [7] G. Papadopoulos and G. Wiggins, Ai methods for algorithmic composition: A survey, a critical view and future prospects, in AISB Symposium on Musical Creativity, 1999, pp. 110 117.