Figures

Figure 1-1 An illustration of “relatedness ratings” for each pitch-class in the context of the musical key A major.

Figure 1-2 A three-dimensional spiral can simultaneously capture the linear arrangement of pitch and the circular arrangement of pitch-class on a piano keyboard.

Figure 1-3 Using circles of minor seconds to explain the tritone paradox.

Figure 2-1 An example artificial neural network that, when presented a stimulus chord, responds with another chord.

Figure 2-2 The logistic activation function used by an integration device to convert net input into activity.

Figure 2-3 The Gaussian activation function used by a value unit to convert net input into activity.

Figure 3-1 An example major scale, and an example harmonic minor scale, represented using multiple staffs.

Figure 3-2 The circle of minor seconds can be used to represent the pitch-classes found in the A major and the A harmonic minor scales.

Figure 3-3 Architecture of a perceptron trained to identify the tonic notes of input patterns of pitch-classes.

Figure 3-4 The connection weights between the 12 input units and any output unit in the scale tonic perceptron.

Figure 3-5 The connection weights between the 12 input units and an output unit in the scale tonic perceptron, showing only those connections that have a signal sent through them when the output unit’s major scale pattern is presented to it.

Figure 3-6 The connection weights between the 12 input units and an output unit, showing only those connections that have a signal sent through them when the output unit’s harmonic minor scale pattern is presented to it.

Figure 3-7 The active connections to the output unit for pitch-class A when the network is presented the G major scale.

Figure 4-1 A multilayer perceptron, with two hidden units, that detects whether a presented scale is major or minor.

Figure 4-2 The hidden unit space for the scale mode network.

Figure 4-3 The connection weights between the 12 input units and each hidden unit in the scale mode network.

Figure 4-4 Connection weights between input units and each hidden unit.

Figure 4-5 Spokes in a circle of minor seconds used to represent the pitch-classes that define major and minor keys.

Figure 4-6 A two-dimensional and a three-dimensional multidimensional scaling solution that arranges scales associated with different keys in a spatial map.

Figure 4-7 The structure of Hook’s (2006) Tonnetz for triads.

Figure 5-1 A multilayer perceptron that uses four hidden units to detect both the mode and the tonic of presented major or harmonic minor scales.

Figure 5-2 A hypothetical two-dimensional hidden unit space for key-finding.

Figure 5-3 A three-dimensional projection of the four-dimensional hidden unit space for the key-finding multilayer perceptron.

Figure 5-4 A different three-dimensional projection of the four-dimensional hidden unit space for the key-finding multilayer perceptron.

Figure 5-5 The results of wiretapping each hidden unit, using the 24 input patterns as stimuli.

Figure 5-6 A perceptron that can be used to map key profiles onto musical key.

Figure 6-1 The top staff provides examples of four different types of triads built on the root note of A: A major (A), A minor (Am), A diminished (Adim) and A augmented (Aaug).

Figure 6-2 A multilayer perceptron with local pitch encoding that learns to identify four types of triads, ignoring a triad’s key and inversion.

Figure 6-3 The connection weights from the 28 input units for pitch to Hidden Unit 1 in the network trained to classify triad types for different keys and inversions.

Figure 6-4 The connection weights from the 28 input units for pitch to Hidden Unit 2 in the network trained to classify triad types for different keys and inversions.

Figure 6-5 The connection weights from the 28 input units for pitch to Hidden Unit 3 in the network trained to classify triad types for different keys and inversions.

Figure 6-6 The connection weights from the 28 input units for pitch to Hidden Unit 4 in the network trained to classify triad types for different keys and inversions.

Figure 6-7 The geography of the piano.

Figure 6-8 Using the number of piano keys as a measure of the distance between pitches.

Figure 6-9 The circle of minor seconds.

Figure 6-10 The circle of major sevenths.

Figure 6-11 The circle of perfect fourths.

Figure 6-12 The circle of perfect fifths.

Figure 6-13 The two circles of major seconds.

Figure 6-14 The two circles of minor sevenths.

Figure 6-15 The three circles of minor thirds.

Figure 6-16 The three circles of major sixths.

Figure 6-17 The four circles of major thirds.

Figure 6-18 The four circles of minor sixths.

Figure 6-19 The six circles of tritones.

Figure 6-20 The 12 circles of octaves, or circles of unison.

Figure 6-21 Added note tetrachords in the key of C major.

Figure 6-22 A multilayer perceptron that classifies tetrachords into four different types.

Figure 6-23 The hidden unit space for a multilayer perceptron trained to identify the four types of tetrachords.

Figure 6-24 The connection weights from the 12 input units to Hidden Unit 1.

Figure 6-25 The connection weights of Figure 6-24 re-plotted so that weights from pitch-classes a minor third apart are stacked on top of one another.

Figure 6-26 The connection weights of Figure 6-24 re-plotted so that weights from pitch-classes a minor third apart are stacked on top of one another.

Figure 6-27 The weights of the connections from the input units to Hidden Unit 3.

Figure 6-28 Three different combinations of four Hidden Unit 3 weights that produce net inputs close enough to µ to generate high activity.

Figure 6-29 The connection weights from the 12 input units to Hidden Unit 2.

Figure 6-30 The connection weights from the 12 input units to Hidden Unit 2, with balanced weights stacked on top of each other.

Figure 6-31 The input patterns in their position in the hidden unit space are illustrated on the left.

Figure 6-32 A two-dimensional hidden unit space for the input patterns created by removing the Hidden Unit 2 coordinate from Figure 6-31.

Figure 7-1 Musical notation for 12 different types of tetrachords, each using C as the root note.

Figure 7-2 Pitch-class diagrams of the 12 tetrachords from the score in Figure 7-1.

Figure 7-3 The architecture of the multilayer perceptron trained to identify 12 different types of tetrachords.

Figure 7-4 The jittered density plot for Hidden Unit 1 in the extended tetrachord network.

Figure 7-5 The connection weights and the jittered density plot for Hidden Unit 1.

Figure 7-6 Three patterns of tritone sampling for tetrachords. A and B are patterns that turn Hidden Unit 1 on; C is a pattern that generates weak activity in Hidden Unit 1.

Figure 7-7 The connection weights and the jittered density plot for Hidden Unit 2.

Figure 7-8 The connection weights and the jittered density plot for Hidden Unit 4.

Figure 7-9 The connection weights and the jittered density plot for Hidden Unit 7.

Figure 7-10 The connection weights and the jittered density plot for Hidden Unit 6.

Figure 7-11 The connection weights and the jittered density plot for Hidden Unit 5.

Figure 7-12 The connection weights and the jittered density plot for Hidden Unit 3.

Figure 8-1 The mapping between input units used for pitch-class encoding and a piano keyboard.

Figure 8-2 The keyboard layout of four different minor seventh tetrachords.

Figure 8-3 The mapping between input units used for pitch encoding and a piano keyboard.

Figure 8-4 The ii-V-I progression for each possible key.

Figure 8-5 Voice leading for two versions of the ii-V-I progression.

Figure 8-6 Lead sheet encoding of tetrachords.

Figure 8-7 A perceptron trained on the ii-V-I progression task.

Figure 8-8 The connection weights from the 12 input units to the output unit representing the pitch-class A.

Figure 8-9 The two-dimensional MDS solution from the analysis of the similarities between output unit weights.

Figure 8-10 The first three dimensions of a five-dimensional MDS solution for the analysis of output unit weight similarities.

Figure 8-11 A map of the Coltrane changes’ chord roots created by combining the circle of perfect fifths (the inner ring of pitch-classes) with the circles of major thirds.

Figure 8-12 The portion of the Figure 8-11 map that provides the Coltrane changes for the key of C major.

Figure 8-13 Using three circles of major thirds to define the Coltrane changes for the key of C major.

Figure 8-14 The circles of major thirds for generating the Coltrane changes in any key.

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