eight state Markov chain transition matrices. The coordinates of the ‘information space’ are entropy rate (hµ), redundancy (ρµ), and predictive information rate (bµ). The points along the redundancy axis correspond to periodic Markov chains, where as those along the entropy axis produce uncorrelated sequences with no temporal structure. As can be seen, processes with high PIR are to be found at intermediate levels of entropy and redundancy. PIR is low both for regular processes, such as constant periodic sequences, and low for random processes, where each symbol is chosen independently of the others.
Abdallah and Plumbley note how this balance between predictability and unpredictability of the PIR is reminiscent of the inverted ‘U’ shape of the Wundt curve(Abdallah & Plumbley, 2009). Already entioned in section 2.6.4 of the background chapter, the Wundt curve suggests that stimuli are most pleasing at intermediate levels of novelty or disorder, where there is a balance between ‘order’ and ‘chaos’.
An inverted ‘U’ shape is visible in the upper envelope of the plot in fig. 3.2a. The distribution of the transition matrices in this space form a thin curved sheet, and is hollow inside. This comparison with the Wundt curve a suggestion that the PIR might be related to the Berlyne’s ‘hedonic value’(Berlyne, 1970).
It was observed how the natural distribution of Markov processes plotted along entropy rate and redundancy formed a triangular shape as shown in fig. 3.2b. It is this triangular shape that then formed the interface that became the Melody Triangle, this is explained in greater detail in 3.3.
3.3 Constructing The Melody Triangle
The Melody Triangle is musical interface that is designed around the natural distribution of Markov chain transition matrices in the information space of entropy rate (hµ) and redundancy (ρµ), the right-angled triangle, as illustrated in Fig. 3.2b.
The right-angled triangle is rotated and stretched to form an equilateral triangle with the ‘redundancy’/‘entropy rate’ vertex at the top, the ‘redundancy’ axis down the right-hand side, and the ‘entropy rate’ axis down the left, as shown in fig. 3.3. This is the Melody Triangle and forms the interface by which the system is controlled.
The corners correspond to three different extremes of predictability and unpredictability, which could be loosely characterised as ‘periodicity’, ‘noise’ and ‘repetition’. Melodies from
3.3. Constructing The Melody Triangle 100 Less R ed un da ncy Repetition Periodicity Noise Low er En tropy Rat e Hig her Entro py Rat e Mo re R ed un da ncy High Predictive Information Rate
Low Predictive Information Rate
Figure 3.3: The Melody Triangle with its relationship to entropy rate (maximum at the bottom left), redundancy (maximum at the bottom right) and predictive information rate (maximum in middle bottom). The ‘repetition’ corner maps to sequences of a single repeated note. The ‘peri- odicity’ corner maps to long loops, and ‘noise’ corner maps to completely random sequences.
the ‘noise’ corner (high hµ, low ρµ and low bµ) have no discernible pattern; those along the ‘periodicity’ to ‘repetition’ edge are all cyclic patterns that get shorter as one approaches the ‘repetition’ corner, until each is just one repeating note. Those along the opposite edge consist of independent random notes from non-uniform distributions. Areas between the left and right edges will tend to have higher predictive information rate (bµ), and it is hypothesised that, these will be perceived as more ‘interesting’ or ‘melodic’. These melodies have some level of un- predictability, but are not completely random. Or, conversely, are predictable, but not entirely so.
3.3.1 The Melody Triangle as a Conceptual Space of Predictability
In arranging Markov chains according to information measures in this way, the Melody Triangle provides a mapping between the parameters of the Markov processes, and how predictable it will be perceived to be. Conversely if one were to plot the Markov chains according to their parameter values, as defined by the values in their transition matrices, some chains that are close to each other in parametric space would not have similar perceptual properties, while some far
3.3. Constructing The Melody Triangle 101
apart in parameter-space would have very similar properties. There is nothing about the euclidean distance between those parameter values that would correspond to similarity.
G¨ardenfors’ theory of conceptual spaces(G¨ardenfors, 2004b), outlined in section 2.4.2 of the background, is a framework for the representation of concepts, and supports spatial reasoning in terms of similarity; objects that are phenomenally similar are close to each other in the conceptual space defined by a number of quality dimensions.
The Melody Triangle, with its axes of entropy rate, and redundancy displays many of the characteristics of a conceptual space. Markov chains spatially close to each other in this space have similar phenomenal characteristics, and certain geometries of the triangle could be said to correspond to ‘natural concepts’. The right edge, for instance, represented the natural concept of the ‘perfect loop’, the top corner represents the natural concept of ‘repetition’, the left edge corresponds to natural concept of ‘perfect randomness’. The left-right direction could be said to correspond to the ‘quality dimension’ of predictability, and the up-down axis corresponds to the quality-dimension of ‘many/fewness’, as the further down one goes, the more symbols/notes are involved in the pattern on average.
There are however some significant caveats. The information measures that define the Melody
Triangle assume a constant rate of symbols, and thus the output sequences proceed at a constant,
uniform rate. Although the placing of events in time and rhythm has a strong effect on expec- tations, surprise and satisfaction in music, the system does not address this temporal dimension. Additionally the system does not address the culturally defined expectations of melodic structure that result from our exposure to tonal music; all symbols are considered equal, regardless of what note in a scale they are mapped to.
Nevertheless it could be said that the Melody Triangle represents a ‘conceptual space of predictability’ for first-order Markov processes, and as such, it affords reasoning about the pre- dictability of these processes in spatial and geometric terms.
3.3.2 Usage and Sonification
The different incarnations of the Melody Triangle – the interactive installation, the desktop appli- cation and the mobile app – all work in fundamentally the same fashion. Thousands of Markov chains are generated, and then mapped according to their entropy rate and redundancy values6.
6Although the triangle is 2d, the third dimension, the PIR (b
µ), is present implicitly, as the transition matrices along the centre line will tend to higher PIR.
3.3. Constructing The Melody Triangle 102 P: position in triangle (x,y) information measures (redundancy rate, entropy rate) transition matrix P: tempo P: notes / beat P: scale choice (pentatonic, diatonic, harmonic) symbol to note mapping symbol note note to audio sample mapping P: instrument selection P: register audio P: volume audio sample
Figure 3.4: Input parameters and how they lead to the generation of audio. Parameters are indicated by P. The position in the triangle map to information measures. These in turn cause a Markov transition matrix with these information properties to be selected. A sequence of symbols is then generated based on the matrix, at a rate based on the tempo and notes per beat parameters. These are mapped to notes based on the scale parameter. The notes are mapped to audio samples based on instrument selection and register parameters.
A user selects a point within the triangle, and this position maps into the information space of entropy rate and redundancy. How this selection is made depends on the incarnation: for the in- stallation this is done by placing one’s body in the space, for the desktop application and mobile app this is done by dragging a round token into the triangle through a graphical interface.
The nearest Markov chain in this information space is used to generate a sequence of symbols which are then sonified either as pitched notes or percussive sounds. This process is outlined in figure 3.4.
In each of the three incarnations of the Melody Triangle, the musical sounds are encoded and generated in the same way. The Melody Triangle has a core tempo (which can be adjusted by the user). When a Markov chain has been selected by specifying its position in the triangle, at each time-step, a symbol is generated. The sequence of symbols correspond to the chain being ‘played out’. Each generated symbol is fed-back as input to the transition matrix, which defines the probability of the subsequent symbol, which in turn is the input to the matrix for the subsequent symbol, and so on. The subjective predictability of the generated sequence is thus determined by the position in the triangle. The symbols are generated at a rate that is an integer multiple of the core tempo; a ‘notes per beat’ value associated to each token, and controllable by the user.
The audio is generated by then mapping each of these symbols to an audio sample. In the installation and the desktop application, this is implemented by sending MIDI messages that trigger audio in an external audio application (Apple’s Logic). Whereas in the mobile app, the