CHAPTER 8: EXTENDED TETRACHORD CLASSIFICATION Chapter 7 introduced the notion of strange circles: using various circles of musical intervals as equivalence classes to which input pitch-classes are assigned. It illustrated this concept by ana- lyzing the internal structure of a network that was trained to identify four different types of tetra- chords. Chapter 8 provides a more complex example of this concept. It describes additional formulae that can be used to define twelve different types of tetrachords for each of the twelve major musical keys. It then reports the training of a multilayer perceptron that learned to classify an input tetrachord into these different tetrachord types. This is a more complex network, requir- ing seven hidden units to converge on a solution to this classification problem. However, this more complicated network can still have its internal structure interpreted. One reason for this is because it, like the Chapter 7 network, organizes input pitch-classes using strange circles. We provide an interpretation of this network, introducing an additional interpretative technique (exam- ining bands in jittered density plots). We then illustrate how the structure of this extended tetra- chord network provides an elegant example of coarse coding. 8.1 Extended Tetrachords .................................................. 2 8.2 Classifying Extended Tetrachords .............................. 5 8.3 Interpreting the Extended Tetrachord Network .......... 7 8.4 Bands and Coarse Coding ......................................... 22 8.5 References ................................................................... 25 © Michael R. W. Dawson 2014 Chapter 8 Classifying Extended Tetrachords 2 8.1 Extended Tetrachords Figure 7-27. Musical notation for twelve different tetrachord types, each using C as the root note. 8.1.1 Extended Chords Therefore the formula for the C major triad is 1-3-5. Adding the seventh note of the scale, Chapter 7 described training a multilayer B, produces the C major seventh tetrachord, perceptron to classify four different types of which follows the formula 1-3-5-7. This was tetrachords (major 7, minor 7, dominant 7, one of the added note tetrachords that we minor 7 flat 5), and presented a detailed studied earlier in this chapter. analysis of the internal structure of this net- work. The point of that example was to use More chords can be created by manipu- a fairly simple musical problem to illustrate lating formulae like those provided in the how an artificial neural network can organize previous paragraph. For instance, one input pitch-classes in terms of their mem- could flatten the third and the fifth note in the bership in various ‘strange circles’ (i.e. inter- formula 1-3-5-7. This produces the formula val-based equivalence classes). In this 1-♭3-5-♭7; if C is the root this is the set of chapter we turn to a more complicated mu- sical problem, involving a larger set of differ- notes [C, E♭, G, B♭], which defines the C ent types of tetrachords. As this problem is minor seventh tetrachord. Note that the flat- more complex, the multilayer perceptron that tened third and seventh notes do not belong solves it requires more hidden units. How- to the C major scale. ever, these hidden units also organize inputs into a variety of strange circles which assists In jazz one often finds extended chords, the interpretation of the network’s internal which use formulae that add notes that fall structure. beyond the octave range of a major scale. For example, if one adds the D that is an The four tetrachords that we explored in octave higher than the second note in the C Chapter 7 were all examples of added note major scale to the C major triad, then one tetrachords. That is, each tetrachord could produces the Cadd9 tetrachord (C, E, G, D). be described as being constructed from a The formula for this chord is 1-3-5-9. triad based upon different notes that be- longed to a scale, with an fourth note added Figure 8-1 provides the musical notation, on top of this triad (see Figure 7-15). The and the musical chord symbol, for twelve fourth note also belonged to the scale. different types of tetrachords. Each of these example tetrachords uses C as the root note Another general approach to building tet- of the chord. Four of these tetrachord types rachords produces a greater variety of chord were used to train the multilayer perceptron types. One begins with a triad formula. For that was described earlier in this chapter. instance, if one takes the first, third, and fifth The other eight are new; the formula for notes of the C major scale (C, E, G, see each is provided in Table 8-1. Figure 7-15) the result is the C major triad. © Michael R. W. Dawson 2014 Chapter 8 Classifying Extended Tetrachords 3 Tetrachord Type Formula Example Forte Number Diminished Seventh 1-♭3-♭5-♭♭7 C7 4-28(3) Minor Seventh 1-♭3-5-♭7 Cm7 4-26(12) Minor Sixth 1-♭3-5-6 Cm6 4-27 Minor, Major Seventh 1-♭3-5-7 Cm(maj7) 4-19 Minor Added Ninth 1-♭3-5-9 Cm(add9) 4-14 Seventh, Flat Fifth 1-3-♭5-♭7 C7♭5 4-25(6) Augmented Seventh 1-3-♯5-♭7 C+7 4-(24(12) Seventh 1-3-5-♭7 C7 4-27 Sixth 1-3-5-6 C6 4-26(12) Major Seventh 1-3-5-7 Cmaj7 4-20(12) Added Ninth 1-3-5-9 Cadd9 4-22 Seventh, Suspended Fourth 1-4-5-♭7 C7sus4 4-23(12) Table 8-1. The names and formulas for twelve different types of tetrachords. An example of each is provided in Figure 8-1. The formulae that are provided in Table 8-1 are designed to work in the context of When we train a multilayer perceptron to any major scale. The numbers in each for- classify the twelve different types of tetra- mula refer to a note’s position in a particular chords in Table 8-1, we will again be using scale. That is, 1 is the first note in a particu- pitch-class representation. Because of this, lar scale, 3 is the third note in a particular notes in extended chords like the added scale, and so on. This means that there are ninth chord will be moved back into the 12 different versions of each of the chord range of a single octave. As well, when we types listed in Table 8-1: one for each of the interpret the internal structure of the net- twelve possible major scales. work, we will be exploiting the properties of some of the circles of intervals that were When these formulae are used to create introduced in earlier sections of the current tetrachords in different keys, some interest- chapter. ing relationships between chords arise. Consider the 6 chord, whose formula is 1-3- For these reasons it is useful to repre- 5-6. In the context of the C major scale this sent the various tetrachords in another visu- produces the C6 chord whose notes are [C, al format. In particular, we can illustrate a E, G, A]. Now consider applying the formula tetrachord in a circle of pitch-classes (in par- for the minor seventh tetrachord (1-♭3-5-♭7) ticular, a circle of minor seconds) by drawing in four spokes that represent which four in the context of the A major scale. This notes are present in a particular chord. produces the Am7 chord whose notes are Drawing such a diagram will illustrate a par- [A, C, E, G]. Note that these notes are iden- ticular chord in the context of a specific ma- tical to those of C6; musically speaking Am7 jor key. However, this diagram represents is identical to an inversion of C6. Similarly, the structure of a tetrachord type for any the dominant seventh chord is the inversion key: if one rigidly rotates the spokes do a of a minor sixth tetrachord in a different key. different position in the circle, then it will provide the notes for the same type of tetra- In other words, the same set of four chord, but relative to some other musical pitch-classes can have two different chord key. names. If we train a network to identify tet- rachord types, then it must be trained to Pitch-class diagrams of the first six tetra- generate both of these chord names to one chords provided in Figure 8-1 or in Table 8-1 set of four input pitch-classes. Table 7-8 are provided in Figure 8-2. Figure 8-3 pro- also provides the Forte numbers of each of vides similar diagrams for the other six tet- these chord types. Note that tetrachords rachord types. that are related by inversion – or tetrachords that can represent different names for the same set of input pitch-classes – have the same Forte number. © Michael R. W. Dawson 2014 Chapter 8 Classifying Extended Tetrachords 4 Figure 8-2. Pitch-class diagrams of the first Figure 8-3. Pitch-class diagrams of the sec- six tetrachords from the musical score in Fig- ond six tetrachords from the musical score in ure 8-1. Figure 8-1. All of the chords presented in Figures 8-2 and 8-3 are created in the context of the C major scale. The structure of the spokes in the diagrams provides an interesting per- spective on the similarities and differences between various tetrachord types. For in- stance it is immediately apparent that both the diminished tetrachord and the seventh flattened fifth tetrachord include two pairs of notes that belong to the same circle of tri- tones, because both diagrams include two long spokes that bisect the circle.