changeset 62:2cd533f149b7

Finished sec 4 intro and 4A.

Binary file draft.pdf has changed

--- a/draft.tex	Fri Mar 16 22:50:48 2012 +0000
+++ b/draft.tex	Fri Mar 16 23:18:35 2012 +0000
@@ -823,8 +823,8 @@
 %predictability vs unpredictability
 %s model, this criteria thus becoming a means of interfacing with the generative processes.  
 
-The tools of information dynamics provide a way to constrain and select musical
-materials at the level of patterns of expectation, implication, uncertainty, and predictability.
+%The tools of information dynamics provide a way to constrain and select musical
+%materials at the level of patterns of expectation, implication, uncertainty, and predictability.
 In particular, the behaviour of the predictive information rate (PIR) defined in 
 \secrf{process-info} make it interesting from a compositional point of view. The definition 
 of the PIR is such that it is low both for extremely regular processes, such as constant
@@ -850,8 +850,8 @@
 three of the information measures discussed in \secrf{process-info} for several thousand 
 first-order Markov chain transition matrices generated by a random sampling method. 
 The coordinates of the `information space' are entropy rate ($h_\mu$), redundancy ($\rho_\mu$), and
-predictive information rate ($b_\mu$). The points along the 'redundancy' axis correspond
-to periodic Markov chains. Those along the `entropy' produce uncorrelated sequences
+predictive information rate ($b_\mu$). The points along the `redundancy' axis correspond
+to periodic Markov chains. Those along the `entropy' axis produce uncorrelated sequences
 with no temporal structure. Processes with high PIR are to be found at intermediate
 levels of entropy and redundancy.
 These observations led us to construct the `Melody Triangle' as a graphical interface
@@ -883,15 +883,15 @@
 
 The Melody Triangle is an exploratory interface for the discovery of melodic
 content, where the input---positions within a triangle---directly map to information
-theoretic measures of the output.  The measures---entropy rate, redundancy and
-predictive information rate---form a criteria with which to filter the output
-of the stochastic processes used to generate sequences of notes.  These measures
+theoretic properties of the output. 
+%The measures---entropy rate, redundancy and
+%predictive information rate---form a criteria with which to filter the output
+%of the stochastic processes used to generate sequences of notes.  
+These measures
 address notions of expectation and surprise in music, and as such the Melody
 Triangle is a means of interfacing with a generative process in terms of the
 predictability of its output.
 
-The triangle is `populated' with first order Markov chain transition
-matrices as illustrated in \figrf{mtriscat}.
 
  \begin{fig}{mtriscat}
 	\colfig{mtriscat}
@@ -905,27 +905,34 @@
 	not visible in this plot, it is largely hollow in the middle.}
 \end{fig}
 
+The triangle is populated with first order Markov chain transition
+matrices as illustrated in \figrf{mtriscat}.
 The distribution of transition matrices plotted in this space forms an arch shape
-that is fairly thin.  It thus becomes a reasonable approximation to pretend that
-it is just a sheet in two dimensions; and so we stretch out this curved arc into
-a flat triangle.  It is this triangular sheet that is our `Melody Triangle' and
-forms the interface by which the system is controlled.  Using this interface
-thus involves a mapping to information space; a user selects a position within
-the triangle, and a corresponding transition matrix is returned.  
-\Figrf{TheTriangle} shows how the triangle maps to different measures of redundancy,
-entropy rate and predictive information rate.
+that is fairly thin. Thus, it is a reasonable simplification to project out the 
+third dimension (the PIR) and present an interface that is just two dimensional. 
+The right-angled triangle is rotated and stretched to form an equilateral triangle with
+the $h_\mu=0, \rho_\mu=0$ vertex at the top, the `redundancy' axis down the right-hand
+side, and the `entropy rate' axis down the left, as shown in \figrf{TheTriangle}.
+This is our `Melody Triangle' and
+forms the interface by which the system is controlled. 
+%Using this interface thus involves a mapping to information space; 
+The user selects a position within the triangle, the point is mapped into the 
+information space, and a corresponding transition matrix is returned. The third dimension,
+though not visible, is implicitly there, as transition matrices retrieved from
+along the centre line of the triangle will tend to have higher PIR.
 	
-
 Each corner corresponds to three different extremes of predictability and
 unpredictability, which could be loosely characterised as `periodicity', `noise'
-and `repetition'.  Melodies from the `noise' corner have no discernible pattern;
-they have high entropy rate, low predictive information rate and low redundancy.
-These melodies are essentially totally random.  A melody along the `periodicity'
+and `repetition'.  Melodies from the `noise' corner (high $h_\mu$, low $\rho_\mu$
+and $b_\mu$) have no discernible pattern;
+Melodies along the `periodicity'
 to `repetition' edge are all deterministic loops that get shorter as we approach
-the `repetition' corner, until it becomes just one repeating note.  It is the
-areas in between the extremes that provide the more `interesting' melodies.
-These melodies have some level of unpredictability, but are not completely random.
- Or, conversely, are predictable, but not entirely so.
+the `repetition' corner, until each is just one repeating note. The
+areas in between will tend to have higher PIR, and we hypothesise that, under
+the appropriate conditions, these will be perceived as more `interesting' or 
+`melodic.'
+%These melodies have some level of unpredictability, but are not completely random.
+% Or, conversely, are predictable, but not entirely so.
 
 \begin{fig}{TheTriangle}
 	\colfig[0.9]{TheTriangle.pdf}
@@ -940,21 +947,24 @@
 interactive installation where a Kinect camera tracks individuals in a space and
 maps their positions in physical space to the triangle.  In the latter each visitor
 that enters the installation generates a melody and can collaborate with their
-co-visitors to generate musical textures---a playful yet informative way to
-explore expectation and surprise in music.  Additionally visitors can change the 
-tempo, register, instrumentation and periodicity of their melody with body gestures.
+co-visitors to generate musical textures. This makes the interaction physically engaging
+and (as our experience with visitors both young and old has demonstrated) more playful.
+%Additionally visitors can change the 
+%tempo, register, instrumentation and periodicity of their melody with body gestures.
 
 As a screen based interface the Melody Triangle can serve as a composition tool.
-A triangle is drawn on the screen, screen space thus mapped to the statistical
-space of the Melody Triangle.  A number of tokens, each representing a
+%%A triangle is drawn on the screen, screen space thus mapped to the statistical
+%space of the Melody Triangle.  
+A number of tokens, each representing a
 melody, can be dragged in and around the triangle.  For each token, a sequence of symbols with
 statistical properties that correspond to the token's position is generated.  These
-symbols are then mapped to notes of a scale%
-\footnote{However they could just as well be mapped to any other property, such
-as intervals, chords, dynamics and timbres.  It is even possible to map the
-symbols to non-sonic outputs, such as colours.  The possibilities afforded by
-the Melody Triangle in these other domains remains to be investigated.}.
-Additionally keyboard commands give control over other musical parameters.  
+symbols are then mapped to notes of a scale or percussive sounds.
+However they could easily be mapped to other musical processes, possibly over
+different time scales, such as chords, dynamics and timbres. It would also be possible
+to map the symbols to visual or kinetic outputs. 
+%The possibilities afforded by the Melody Triangle in these other domains remains to be investigated.}.
+Additionally keyboard commands give control over other musical parameters such
+as pitch register and note duration.  
 
 The Melody Triangle can generate intricate musical textures when multiple tokens
 are in the triangle.  Unlike other computer aided composition tools or programming