Chapter 7: Tetrachords and Strange Circles

CHAPTER 7: TETRACHORDS AND STRANGE CIRCLES

The interpretation of one multilayer perceptron detailed in Chapter 6 revealed two hidden units that exhibited tritone equivalence: pitch-classes separated by the musical interval of a tritone were assigned the same connection weights. Chapter 7 takes the notion of tritone equivalence and extends it to all possible musical intervals. That is, it considers grouping pitch-classes together in circles where each adjacent note in the circle is separated by a particular musical interval. Then it proposes that one could treat each circle as picking out an equivalence class in which each pitch-class that belongs to the same circle is identical. We call these ‘strange circles’. After introducing the different strange circles that are possible, we describe training a network to identify four different types of four-note chords called tetrachords. We then interpret the internal structure of this network, showing how its hidden units use strange circles to organize input pitch- classes.

7.1 Circles of Intervals and Strange Circles ...... 2 7.2 Added Note Tetrachords ...... 11 7.3 Classifying Tetrachords ...... 14 7.4 Interpreting the Tetrachord Network ...... 16 7.5 Summary and Implications ...... 27 7.6 References ...... 28

7.1 Circles of Intervals and Strange Circles Chapter 6 described training multilayer perceptrons to classify triad types. The pitch-class encoding of this problem revealed two hidden units that exhibited tritone equivalence: pitch-classes separated by the musical interval of a tritone were assigned the same connection weights. When a “piano keyboard’ encoding was used to present different inversions of the triads, pitch- class equivalences involving different musical intervals were revealed.

The current chapter explores interval Figure 7-1. The geography of the piano. The equivalence in more detail, because it is a top illustration provides the pitch-class property that is frequently discovered when names for adjacent keys in one region of the artificial neural networks are trained on tasks keyboard. For the black keys, enharmonically involving musical harmony (Yaremchuk & equivalent names are provided in parenthe- Dawson, 2005, 2008). We begin by using ses. The bottom illustration shows that dif- music theory to generate the possible inter- ferent pitches that belong to the same pitch- val equivalences that might be discovered class occur at regular intervals along the inside a network. keyboard.

7.1.1 Piano Geography The layout of piano keys is quite regular. This is evident in Figure 7-1 from the ar- One prominent piano technique book rangement of black keys, which alternate in provides exercises that are intended to in- groups of twos and threes across the figure. crease the player’s familiarity with the geog- The pattern of twelve differently named pi- raphy of the keyboard (Merrick, 1958). We ano keys at the top of Figure 7-1 repeats can use this geography to generate geomet- itself again and again along the keyboard. ric representations of pitch-class relationships. Later in this chapter these represen- While every piano key plays a differently tations appear in the internal structure of pitched note, that note belongs to one of the artificial neural networks trained to classify twelve pitch-classes of Western music that particular harmonic entities, called tetra- we have already encountered. Therefore chords. several different piano keys play different pitches that all belong to the same pitch- The foundation of our geometric repre- class, and they occur at regular intervals sentations of pitch-class relationships is a along the piano keyboard. This is illustrated physical artifact, the piano keyboard. A at the bottom of Figure 7-1, which highlights modern piano has 88 different keys, 52 the locations of four different instances of white and 36 black (Isacoff, 2011). The lay- the pitch-class C. Modern pianos are tuned out of a subset of these keys (only 51 are using a system called equal temperament illustrated) is provided at the top of Figure 7- (Isacoff, 2001). This means that adjacent 1. Each piano key, when struck, produces a notes (e.g. adjacent piano keys) differ in unique pitch. For instance, the lowest (left- pitch by a semitone. This means that near- most) shaded note at the top of Figure 7-1 est neighbors on the keyboard that belong to corresponds to the pitch ‘middle C’, which is the same pitch-class are separated by a sometimes designated as C4, is the pitch span of twelve adjacent piano keys (see produced by a sine wave whose frequency Figure 7-1). This distance is equivalent to is 261.6 Hz. twelve semitones, or a musical interval of a perfect octave.

7.1.2 Distance and Intervals

With respect to our piano geography, tance along the piano as being arranged in a what is the distance between two notes? circle. The circle that results when a dis- For instance, what is the distance between tance of four piano keys is used starts with the highlighted notes C and E at the top of C, moves next to E, moves next to G#, and Figure 7-2? We will measure this distance then returns to C. It is the fact that after a in terms of the number of piano keys that few moves we return to the pitch-class that separate the two notes. we started from (C in this example) that mo- tivates the idea to arrange this set of pitch- Examine the top illustration in Figure 7-2. classes in a circle. We literally come full cir- If one starts at the highlighted C and moves cle back to the pitch-class from which we up in pitch (i.e. to the right along the key- started. board), then the first key encountered is C#, the second is D, the third is D#, and the If we use the same distance between fourth is E. Therefore the distance between notes, but start at a different piano key, we C and E is four piano keys. Alternatively, we can define a different circle of pitch-classes. can say that the distance between C and E For instance, if we start at C# and move up in this figure is four semitones, which is a four keys at a time, our circle will only in- musical interval of a major third. clude the pitch-classes C#, F, and A. With a between-note distance of four piano keys, there are four different circles of three pitch- classes can be defined.

Interestingly, we can encounter the same subset of pitch-classes by starting at C and counting up a different distance. The bottom part of Figure 7-2 shows that G# is eight piano keys higher than C, and that E is eight piano keys higher than G#. With this different distance we encounter the same pitch- classes highlighted in the middle of Figure 7- 2, but encounter them in a different order.

The fact that the same subset of pitch- classes are encountered when one moves different distances along the keyboard indicates that we can consider these pitch- classes as being separated by two different musical intervals. For instance, C and E can .Figure 7-2. Using the number of piano keys be considered to be a major third apart (four as a measure of the distance between pitches. See text for details piano keys) or a minor sixth apart (eight piano keys). This is why in Chapter 6 we de- We can identify sets of pitches that are scribed hidden units as representing multiple spaced the same distance apart by continu- musical intervals between pitch-classes. ing to count up the same number of keys to For instance, Section 6.3.3 described Hid- the next note, as illustrated in the middle den Unit 1 of the triad classification network part of Figure 7-2. So, the distance up from as being sensitive to intervals of a major C to E is four piano keys; if we move the third or of a minor sixth. same distance up from E we reach G#. If we move up four piano keys from G#, we Clearly there is a musical interpretation reach another C. In other words, if we start for each distance between pitches meas- at C, and always move four piano keys up, ured in terms of number of piano keys. Re- we will only encounter three different pitch- specting the notion of pitch-class, there are classes: C, E, and G#. thirteen different distances between piano keys that are available to us: zero keys, one It is convenient to think of a set of pitch- key, two keys, and so on up to twelve keys. classes picked out by moving a fixed dis- Each distance can also be expressed in

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 4 terms of semitones; each of these semitone Helmholtz’ theory increases in roughness distances represents a musical interval. The produce decreases in tonal consonance. names of the different intervals in Western music, and their associated distance be- The roughness values presented in Ta- tween pitches, are presented in the first two ble 7-1 show that, in terms of tonal conso- columns of Table 7-1. nance, not all of the musical intervals are the same. The most consonant intervals are Distance Be- Interval Roughness perfect unison and the perfect octave. The tween Pitches Name of Interval next most consonant intervals are the per- In Semitones 0 Perfect Unison 0 fect fifth and the perfect fourth, followed by 1 Minor Second 76 the major third. Not surprisingly these more 2 Major Second 25 consonant intervals are the ones most 3 Minor Third 24 commonly used in Western harmony. 4 Major Third 18 5 Perfect Fourth 3 7.1.3 Single Circles of Intervals 6 Tritone 18 7 Perfect Fifth 1 8 Minor Sixth 22 In our discussion of note distance, we 9 Major Sixth 22 noted that there were only thirteen possible 10 Minor Seventh 24 distances between piano notes when pitch- 11 Major Seventh 48 classes were considered. A subset of these 12 Perfect Octave 0 distances will pick out all twelve different Table 7-1. Distances between pitches with their pitch-classes, and arrange them in a single corresponding named musical interval and its roughness. See text for details. set of note names. This section describes these single circles of pitch-classes. For many years researchers have stud- ied the perceptual properties of the different To begin, let us consider starting on a pi- intervals listed in Table 7-1 (Bidelman & ano keyboard at a note named C, and mov- Krishnan, 2009; Guernsey, 1928; Helmholtz ing up from this note (i.e. to the right along & Ellis, 1863/1954; Krumhansl, 1990; the keyboard) a distance of one piano key. Malmberg, 1918; McDermott & Hauser, The first note that we will encounter is C#. 2004; McLachlan, Marco, Light, & Wilson, Moving the same distance up from it we will 2013; Plantinga & Trehub, 2014; Plomp & next encounter D. Moving along the key- Levelt, 1965; Seashore, 1938/1967). Each board in this fashion it will turn out that we of them is associated with a degree of tonal will encounter every possible pitch-class consonance. “Tonal consonance refers to before we finally reach another note named the attribute of particular pairs of tones that, C. The set of pitch-classes that we encoun- when sounded simultaneously in isolation, ter, and the order in which they are encoun- produce a harmonious or pleasing effect” tered, can be represented in a single circle (Krumhansl, 1990, p. 51). Tonal conso- that is provided in Figure 7-3. (To make this nance is important in harmony, because the figure easier to compare to other versions, musical pleasantness of two pitches sound- instead of drawing the circle, we arrange the ed simultaneously is related to their conso- pitch-classes in a circle, and then draw a nance. radius to the center of the circle to place each pitch-class on a “spoke”.) A variety of studies, using different meth- odologies, have found the same patterns of Because the distance between adjacent the judgments of tonal consonance of differ- notes in this circle is one piano key (one ent intervals. The final column of Table 7-1 semitone), the interval between adjacent notes is a minor second. Therefore we can provides one measure of consonance, nds roughness (Helmholtz & Ellis, 1863/1954). name Figure 7-2 the circle of minor 2 . Helmholtz defined roughness in terms of the Note that moving in a clockwise direction beats produced by the interference between around this circle is equivalent to moving up the sinusoidal waves that define different (i.e. to the right) along a piano keyboard; pitches. The greater is the number of beats, moving in a counterclockwise direction is the greater is the roughness. According to equivalent to moving down along a piano keyboard.

Figure 7-3 is a mirror image of Figure 7-4, the inversion of an interval of a minor 2nd produces an interval of a major 7th.

We can use the piano keyboard to make the same point about inversion. Imagine playing two piano notes simultaneously, a C and the C# immediately above it. This can be described as playing a minor 2nd interval. However, one can play the inversion of this interval by taking its lowest note (the C) and making it the highest note in the interval by moving it an octave higher. Now when these two notes are played, one is playing a major 7th interval because the distance be- Figure 7-3. The circle of minor 2nds. tween C# and the closest C above it is 11

semitones. Earlier we noted that by choosing a different distance to move along the piano We have seen that distances of one or keyboard we will encounter the same set of eleven piano keys pick out all twelve pitch- pitch-classes, but will do so in a different classes, which can be arranged in a single order. This is the case if we start at a note circle to represent the order in which pitch- named C and move up the keyboard a dis- classes are encountered. Another distance tance of eleven keys (or eleven semitones), that picks out all twelve pitch-classes is one which corresponds to a musical interval of a of five piano keys (or semitones). Moving major 7th. The order of notes that are en- up a piano keyboard at this distance pro- countered is illustrated in Figure 7-4 as a duces the circle of perfect 4ths which is illus- circle of major 7ths. trated in Figure 7-5. Note that this circle

arranges pitch-classes in a very different order than we saw in the preceding two circles of Figures 7-3 and 7-4.

Figure 7-4. The circle of major 7ths

An inspection of the circle of major 7ths indicates that it picks out the same pitch- classes as does the circle minor 2nds, but Figure 7-5. The circle of perfect 4ths. does so in a different order. Indeed, the two circles are complementary: the circle of ma- ths jor 7 is a mirror reflection of the circle mi- nds As was the case with the circle of minor nor 2 . In our discussion of the formal the- 2nds, the circle of perfect 4ths has a comple- ory of pitch-classes (Forte, 1973) in Chapter mentary circle that is its reflection. It is the 3 (e.g. Figures 3-8 and 3-9) we noted that circle of perfect 5ths that is produced when when pitch-classes are represented in a cir- one starts at C and moves up a piano key- cle, the mirror reflection of the circles inverts board a distance of seven piano keys or the pitch-classes. In other words, because

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 6 seven semitones. The circle of perfect 5ths dimensional manifolds, in the sense that you is illustrated in Figure 7-6. Note that if one can trace the entire shape with a finger were to move counterclockwise around the without having to lift the finger up before the circle of perfect 4ths, one would encounter trace is completed. These manifolds are pitch-classes in the same order as moving embedded in a higher-dimensional space. clockwise around the circle of perfect 5ths. For instance, each of these manifolds is de- This is because when a musical interval of a picted in a two-dimensional space, the plane perfect 4th is inverted, the result is a musical of the page in which each figure is drawn. In interval of a perfect 5th. short, each of the circles that we have dis- cussed is a one-dimensional manifold, circular in shape, embedded in a two- dimensional space.

The shape of a manifold is important, because it constrains how one moves from location to location along the manifold’s surface. That is, to move from one location to another, one must never leave the manifold’s surface. This property of manifolds has been used in theories of visual perception and visual imagery (Farrell & Shepard, 1981; Shepard, 1984) to model the appearance of three-dimensional objects as they move or as they are mentally rotated.

ths Figure 7-7. The circle of perfect 5 . In describing the single circles of intervals as manifolds, we are saying that their Of the four circles presented in this sec- shape and layout places certain constraints tion, the one most frequently seen in music ths on how one can move from one note (i.e. is the circle of perfect 5 . Students are from one point on the manifold’s surface) to taught how to use this circle to determine another. For instance, on the circle of per- the number of sharps or flats to use in the fect 5ths, to move from C to D one must nec- key signature for a particular musical key. essarily pass through an intermediate loca- Later we will see how this circle can be used tion, G. This is because G occupies an in- as a map to guide a musician who wishes to termediate location between C and D on the play the chords of a particular jazz progres- manifold’s surface. sion, the ii-V-I. The circle of minor 2nds is also commonly encountered; for instance it Interpreting each of the circles as a mani- is frequently found in geometric discussions fold has further implications concerning the of musical regularities (Tymoczko, 2011). notion of distances between notes (Tymoczko, 2011). We derived each mani- All of the circles presented in this section fold by moving along a piano keyboard at were defined musically: they were created set distances. However, after creating a by moving upwards along a piano keyboard manifold, we could measure the distance in steps of a set distance. However, these between notes along the manifold’s surface. circles – as well as those presented in fol- For instance, in the circle of perfect 5ths, G is lowing sections – can be viewed as repre- one unit of distance away from C because it sentational or mathematical objects called is next to C on the manifold; similarly D is manifolds. two units away from C on the same manifold. A manifold is a surface upon which objects are represented as points. Manifolds From this perspective, the distance be- have specific shapes, and are represented tween notes depends upon a particular con- in a space of set dimensions. For instance, text: the specific manifold being considered. all of the manifolds described in this section In the circle of perfect 5ths, G is one distance are circular shapes. They are one- unit away from C. In the circle of minor 2nds,

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 7 the shortest distance between C and G is of the twelve available pitch-classes. If we five distance units. The idea that the dis- repeat this process, but start on C# instead tance between notes can be measured of C, we will create a second circle of major along a manifold, and that the size of this 2nds that complements the first because it distance depends on the particular manifold captures the remaining six pitch-classes. being considered, is strongly related to Dimi- These two circles of major 2nds are illustrated tri Tymoczko’s idea of measuring distance in Figure 7-7. between notes in the context of different musical scales (Tymoczko, 2011).

Importantly, if one uses a musical manifold or a musical scale as the context in which to measure the distance between pitch-classes, then one is tacitly assuming that different points on the manifold represent different pitch-classes. Of particular interest to us in this chapter is the fact that in nds some instances artificial neural networks Figure 7-7. The two circles of major 2 . captures pitch-classes that can be repre- nd sented in a manifold, but does not treat If one takes a major 2 interval (e.g. by playing C and D simultaneously) and inverts these pitch-classes as being different. For th these networks the manifold is an equiva- it the result is a minor 7 . This is because lence class, because all the pitch-classes after inverting the major second interval, the that belong to it are the same. However, in distance between the low D and the higher order for equivalence classes of this sort to C would be 10 semitones. From this we be useful there must be more than one, so should expect that we should be able to produce two circles of minor 7ths that com- that some pitch-classes belong to one nds equivalence class, but not others. In the plement the circles of major 2 illustrated in next section we consider circles of intervals Figure 7-7. Indeed, if we begin at C and that pick out different and complementary move upwards along the piano keyboard 10 subsets of pitch-classes. keys at a time we produce a circle of minor 7ths that is a reflection of the first circle of nds 7.1.5 Pairs of Circles major 2 in Figure 7-7. If we instead start at C#, we produce a manifold that comple- ths In the previous section, we detailed four ments the first circle of minor 7 , and re- flects the second circle of major 2nds in Fig- different musical manifolds that are single ths circles of intervals. They are single in the ure 7-7. These two circles of minor 7 are sense that a one manifold captures all provided in Figure 7-8. twelve pitch-classes on its surface.

Now we will describe some new manifolds, each of which only captures half of the available pitch-classes. As a result, two different versions of the same process – moving along the piano keys – are required to build two complementary manifolds which, when combined, capture all twelve pitch- ths Figure 7-8. The two circles of minor 7 . classes. 7.1.6 Trios of Circles Consider starting at a C note on a piano, and moving upwards a distance of two keys In this section we define sets of three to the next note, which will be D. Following complementary manifolds; each of which this procedure, we will next encounter E, F#, captures four pitch-classes; all three com- G#, and A# before encountering another C. bined contain all twelve pitch-classes. We can build a manifold of these notes – a circle of major 2nds – but it will only hold six

Imagine starting on the piano at some C note, and moving upwards along the keyboard a distance of three keys. The note encountered is D#. Moving the same distance upwards, one encounters F#, then A, and then encounters another C. Thus this defines a circle that captures four of the twelve pitch-classes. This manifold is a circle of minor 3rds because if two adjacent notes are three semitones apart, they are separated by an interval of a minor third.

To define other, complementary, mani- ths folds we must move the same distance Figure 7-10. The three circles of major 6 . along the keyboard but from different starting points. If we start at C# instead of C, we 7.1.7 Quartets of Circles define a second circle of minor 3rds; if we start at D instead of C, we define a third cir- Imagine starting on the piano at some C cle of minor 3rds. The three possible circles note, and moving upwards along the key- of minor 3rds are illustrated in Figure 7-9. board a distance of four keys. The note encountered is E. Moving the same distance upwards, one encounters G#, and then encounters another C. Thus this defines a circle that captures only three of the twelve pitch-classes. This manifold is a circle of major 3rds.

Three other circles of major 3rds are possible, and are required to capture the remaining pitch-classes. They are created by moving the same distance along the piano keyboard, but from different starting points: C#, D, and D# respectively. The four circles rds of major 3 are illustrated in Figure 7-11. Figure 7-9. The three circles of minor 3rds.

If one takes a minor third and inverts it, one produces an interval of a major sixth whose notes are nine semitones apart. Not surprisingly, if we start at each of the three notes used above (C, C#, D) and move along the piano nine keys at a time, we produce three different complementary circles of major 6ths, shown in Figure 7-10. Each of these circles is a reflection of one of the cir- rds cles of minor 3 illustrated in Figure 7-9. rds Figure 7-11. The four circles of major 3 .

If one takes a major third and inverts it, one produces an interval of a minor sixth whose notes are separated by eight semitones. If one starts at each of the four starting points used to create the manifolds of Figure 7-11 (C, C#, D, D#) and moves along the piano eight keys at a time, then one produces the four complementary circles of mi-

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 9 nor 6ths, each of which is illustrated in Figure 7-12. Each of these circles is a reflection of one of the circles illustrated in Figure 7-11.

Figure 7-14. The twelve circles of octaves, or circles of unison.

If one inverts an octave interval by rais-

ths ing the lower note an octave, the result is Figure 7-12. The four circles of minor 6 . two identical notes – the distance between

them is zero semitones. This musical inter- 7.1.8 Sextets of Circles val, perfect unison, produces exactly the

same set of twelve manifolds that are pro- If one starts at some C note on the piano vided in Figure 7-14. keyboard and moves up six piano keys, one encounters F#. Moving up another six piano 7.1.10 Strange Circles keys reaches another C. This defines a simple manifold that contains only two We introduced single circles of intervals points. To capture the remaining pitch- nds in Section 7.1.3 (e.g. the circle of minor 2 , classes requires starting from five additional ths and the circle of perfect 5 ). Then we in- notes on the keyboards. This produces six troduced a number of circles of intervals in different circles of tritones which are illus- which more than one circle existed for the trated in Figure 7-13. nds same interval: the two circles of major 2 , ths the two circles of minor 7 , the three circles of minor 3rds, the three circles of major 6ths, the four circles of major 3rds, the four circles of minor 6ths, the six circles of tritones, the twelve circles of perfect octaves and the twelve circles of unison.

Figure 7-13. The six circles of tritones. As was the case for the single circles of

intervals, each of these multiple circles can The inversion of a tritone is itself a tri- be considered to be a manifold. For in- tone, because this interval is defined by six nds stance, on one of the circles of major 2 semitones which is exactly half an octave. the distance between C and D is one unit. Thus there are no other circles of intervals that reflect those illustrated in Figure 7-13. However, each of these manifolds can

also be interpreted in a different way: as an 7.1.9 Dodecal Circles equivalence class. For instance, for some

networks, the pitch-classes C, D, E, F#, G#, Choose some note C on a piano, and and E# are all equivalent because they all move twelve keys – a perfect octave – up- belong to the same circle of major 2nds. wards. You reach another C (see Figure 7- There equivalence is established because 1). This produces a special case manifold, a their connections to a hidden unit are all as- “circle” that represents a single pitch-class signed the same weight. As far as this hid- as a single point. Obviously eleven other den unit is concerned, these pitch-classes such manifolds are required to capture the cannot be distinguished from one another remaining pitch-classes. The twelve “cir- because they are all assigned the same cles” of perfect octaves are illustrated in name (connection weight). We will see sev- Figure 7-14.

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 10 eral examples of this later in the current the sense that it treats pitch-classes that we chapter, and also in Chapter 8. view to be different as being the same. Again, this notion of equivalence is just as We often call equivalence classes based valid as the notion of octave equivalence upon circles of intervals strange circles. that motivated our discussion of pitch-class These circles are strange in two ways. First, representations in Chapter 3. However, it is while entities like the two circles of major rarely encountered in the literature. One seconds or the three circles of minor thirds exception is their use as an input encoding are proper components of music theory, (Franklin, 2004); this topic will be considered they are rarely encountered. The circle of in more detail in a later chapter. fifths, and the circle of minor seconds, is frequently encountered by music students. Let us now turn to describing networks Most of the other circles of intervals that we whose interpretation reveals that they treat a have described above are encountered by number of different pitch-classes as being students much less frequently, and are the same, because they are assigned the strange in that respect. same connection weight. Whenever this occurs, we find that the equivalence class When used to define an equivalence that they belong to is one of the strange cir- class, a circle of intervals is also strange in cles that have been detailed above.

7.2 Added Note Tetrachords

Figure 7-15. Added note tetrachords in the key of C major. The top line provides the C major scale. The middle line provides the triads built upon each of the notes of this scale. The bottom line provides the tetrachords for this scale created by adding one note to each triad. See text for details.

7.2.1 Tetrachords ping over notes. For instance, the C major triad is C-E-G, which skips over D and F. The major and minor scales that serve as Similarly, the D minor triad is D-F-A (skip- the foundation for much of Western music ping over E and G), and so on. This pro- are rooted in musical formalisms invented by cess produces three different major triads the ancient Greeks. The foundation of (C, F, G), three different minor triads (Dm, Greek music was not the scale, but was in- Em, Am) and one diminished triad (B). stead the tetrachord. The Greek tetrachord was a set of four different notes, the lowest The last line of the score in Figure 7-15 always separated from the highest by an converts each of these triads into a tetra- interval of a perfect fourth. Two additional chord by adding yet another note from the C notes were placed between these two, carv- major scale. Again, the added note is two ing the tetrachord’s perfect fourth into three scale notes higher than the highest note in smaller intervals. There were three main the triad. For instance the Cmaj7 tetrachord types of tetrachords, depending upon the is C-E-G-B (skipping over the A to add the choice of the inner two notes (Barbera, B). Similarly, the Dmin7 tetrachord is D-F-A- 1977; Chalmers, 1992). Our modern major C, and so on. Each of these tetrachords is and harmonic minor scales can be de- an example of a seventh chord. There are scribed as being constructed from two adja- different types of these tetrachords created cent Greek tetrachords. from this process: two major seventh chords (Cmaj7, Fmaj7), three minor seventh chords The modern definition of tetrachord in- (Dm7, Em7, Am7), one dominant seventh cludes a much wider variety of chords than chord (G7), and one minor seventh flat fifth does the Greek definition. Modern tetrachord (Bm7♭5). chords are any chord that includes four different pitches. Figure 7-15 illustrates the construction of a subset of modern tetra- Each of these different types of seventh chords. The top line of this score provides chord has a different sonority, and therefore the notes of the C major scale. In the mid- a different role in a musical composition. dle line each of these notes serves as the Furthermore, the same approach to chord root of a triad. The notes that are added are construction can be applied to any major always two scale notes higher than the low- scale, producing a set of 7 different chords er note, so triads are constructed by skip- for each key.

Importantly if we create these 7 different chords of the type illustrated in Figure 7-15, chords for each of the 12 major keys, then and must assign each to one of four different we will not be creating 84 different chords. tetrachord classes. Prior to describing this This is because when a pitch-class repre- network, let us consider the musical proper- sentation is used the same chord will appear ties of these chords. in different musical keys. For example in the set of 84 chords each min7 chord will ap- In Chapter 6 we noted that each different pear three different times, and each maj7 type of triad was defined by a particular pat- chord will appear twice. As a result our total tern of musical intervals between adjacent set of 84 chords will include 48 unique tetra- notes. The same is true for the different chords and 36 duplicates of some of these tetrachords. For instance, consider the chords. Cmaj7 tetrachord in root position, whose notes (in order) are C, E, G, and B. There is 7.2.2 Tetrachord Properties an interval of a major 3rd from C to E, of a minor 3rd from E to G, and of a major 3rd In order to illustrate networks that solve from G to B. This pattern of intervals distin- musical problems by assigning notes to guishes this type of tetrachord from the oth- equivalence classes based upon circles of er three types as can be seen from the third intervals, we will consider a multilayer per- column of Table 7-2 below. ceptron that is presented modern tetra-

Forte Prime IC Type Example Intervals Between Adjacent Notes Number Form Vector Major 7 [C, E, G, B] Major 3rd - Minor 3rd - Major 3rd 4-20(12) 0,1,5,8 101220

Minor 7 [D, F, A, C] Minor 3rd - Major 3rd - Minor 3rd 4-26(12) 0,3,5,8, 012120 Dominant 7 [G, B, D, F] Major 3rd - Minor 3rd - Minor 3rd 4-27 0,2,5,8, 012111

Minor7♭5 [B, D, F, A] Minor 3rd - Minor 3rd - Major 3rd 4-27 0,2,5,8, 012111

Table 7-2. Musical properties of each type of tetrachord in Figure 7-15. See text for details

Of course, if one considers the distances to differentiate tetrachord types, as detailed between nonadjacent notes in a tetrachord, below. then there are more intervals than those presented in the third column of Table 7-2. Recall that in an ic vector the first num- In order to obtain a deeper understanding of ber indicates how many minor 2nd/major 7th the structure of these tetrachords we used intervals occur in a musical object, the sec- the procedures described in Chapter 3 to ond indicates the frequency of ma- determine each tetrachord’s Forte number, jor2nd/minor 7th intervals, the third indicates prime form, and interval-class vector (ic vec- the frequency of minor 3rd/major 6th intervals, tor). The results of this analysis are provid- the fourth indicates the frequency of major ed in the final three columns of Table 7-2. 3rd/minor 6th intervals, the fifth indicates the frequency of perfect 4th/perfect 5th intervals, Two particularly interesting findings and the sixth indicates the frequency of tri- emerge from this set-theoretic analysis. tones. First, both the dominant seventh and the minor seventh flat fifth tetrachords have the The ic vectors in the final column of Ta- same prime form, and the same ic vector. ble 7-2 reveal, for instance, that a major This means that a network may only be able seventh tetrachord is the only one that has a to differentiate these two tetrachords by minor 2nd or major 7th interval, and the only considering the specific order in which the one that does not have a major 2nd or a mi- component musical intervals occur. Sec- nor 7th interval in its structure. Neither the ond, the ic vectors for each tetrachord type major nor the minor seventh tetrachords provide some indication of the musical regu- contain a tritone, but the other two types of larities that a network may be able to exploit chords do. The minor seventh tetrachord shares individual ic vector values with each

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 13 of the other types of tetrachords; this means distinguished from them by detecting the that it can only be distinguished from them absence of a tritone interval. by considering several interval types at the same time. For instance, it can be distin- Now let us turn to describing the training guished from the major seventh by the pres- of a multilayer perceptron to detect these ence of a major 2nd or minor 7th interval, but four different types of tetrachords, regard- this property is shared by the other two tet- less of the musical key in which they occur. rachord types. A minor seventh can only be

7.3 Classifying Tetrachords

Figure 7-16. A multilayer perceptron that classifies tetrachords into four different types. See text for details.

7.3.1 Task quired three hidden value units to converge to a solution to this problem. It used 12 in- Our goal was to train an artificial neural put units to represent input tetrachords in network, when presented with four notes the same pitch-class representation that was that defined a tetrachord constructed from used for the training of the networks in sev- the notes of a major scale (Figure 7-15), to eral previous chapters. Figure 7-16 illus- identify the type of tetrachord (major sev- trates the presentation of the C major seventh, minor seventh, dominant seventh, or enth tetrachord (grey input units), resulting minor seventh flat fifth) ignoring the key of in the ‘Major Seventh’ output unit activating. the tetrachord (i.e. independent of what major scale had been used to construct, or in- 7.3.3 Training Set dependent of what note of that scale was the tetrachord’s root). The training set consisted of 84 stimuli: the seven different tetrachords that could be At the end of training, the multilayer per- created for a major scale (see Figure 7-15); ceptron used for this task turned one output these tetrachords were constructed for each unit ‘on’ to identify tetrachord type, and of the twelve different major scales. As was turned the remaining three output units ‘off’, noted in Section 7.2.1 within this set of 84 when presented a tetrachord. Thus this stimuli there were 36 duplicate patterns network had four different output units, each (each maj7 tetrachord appeared twice, and one dedicated to representing a particular each min7 tetrachord appeared three times). tetrachord type. For the current network this simply means that these two different types of tetrachords 7.3.2 Network Architecture received more training than did the other two types. This difference is not relevant to the Figure 7-16 presents the architecture that point that the network is being used to illus- was used to accomplish this tetrachord clas- trate: the presence of strange circles in the sification task. It used four output value connection weights of its hidden units. units to represent tetrachord type, and re-

Each tetrachord was encoded as input analyzed in the next section required a fair pattern in which four input units were acti- amount of patience during training! vated with a value of 1, and the remaining eight input units were all activated with a Importantly, all of the networks trained on value of 0. Each input pattern was paired this problem developed patterns of connec- with an output pattern that required one out- tivity that reflected equivalence classes de- put unit to activate with a value of 1, and the fined by circles of intervals. We simply fo- other three output units to activate with a cus on the smallest of these networks be- value of 0. The output unit that was trained cause with only three hidden units it is easi- to activate was the one that represented the er to consider some of its properties, such input pattern’s correct tetrachord type. as its hidden unit space.

7.3.4 Training

The multilayer perceptron was trained with the generalized delta rule developed for networks of value units (Dawson & Schopflocher, 1992) using the Rumelhart software program (Dawson, 2005). During a single epoch of training each pattern was presented to the network once; the order of pattern presentation was randomized before each epoch.

All connection weights in the network were set to random values between -0.1 and 0.1 before training began. In the network to be described in detail below, each µ was initialized to 0, but was then modified by training. A learning rate of 0.01 was employed. Training proceeded until the network generated a ‘hit’ for every output unit for each of the 84 patterns in the training set. Once again a ‘hit’ was defined as activity of 0.9 or higher when the desired response was 1 or as activity of 0.1 or lower when the desired response was 0.

We explored a number of different network architectures with this training set. When four or five hidden value units were employed, the problem was very easy and was often solved in a few hundred epochs. However, in order to get a three hidden unit network to converge, each µ had to be modified during training. On some occasions a three-hidden unit network would converge very quickly. For instance, the network to be described in more detail in the next section converged after 11,566 epochs of training. However, on many occasions a three-hidden unit network would settle to a local minimum and fail to converge to a solution even after more than 20,000 epochs of training. In other words, the network to be

7.4 Interpreting the Tetrachord Network

Figure 7-17. The hidden unit space for a multilayer perceptron trained to identify the four types of tetrachords.

7.4.1 Hidden Unit Space right edge of the Figure 7-17 cube. All of the dominant seventh tetrachords fall in two re- How does this multilayer perceptron gions in the lower left hand corner of the identify the four different types of tetra- cube. All of the minor seventh flat fifth tetrachords? Let us begin by considering the chords fall either in a single tight area locat- hidden unit space for this network, which is ed in the upper back region of the cube, or illustrated in Figure 7-17. It can be seen that in a similar location in the lower left hand this space is very sparse, because the dif- corner at the front of the cube. ferent instances of tetrachord types are placed very near one another in the space. Importantly, all of these locations of tet- Indeed, in many cases different tetrachords rachord types in the hidden unit space can occupy the same coordinates in this space, easily be separated from the other tetra- which is why it appears to have so few sym- chords by two parallel planes that are posi- bols illustrated, even when there are 84 dif- tioned by each output value unit. Now let us ferent input patterns positioned in this turn to considering the tetrachord properties space. that are detected by each hidden unit that provide this representation of input patterns In this space, all of the major seventh tet- to each hidden unit. rachords are positioned in two general locations at the cube’s upper left. All of the mi- The fact that all of the different types of nor seventh tetrachords fall along the front tetrachords are placed near one another,

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 17 typically in two different areas of the hidden unit space, suggests that each type of tetra- In order to understand these different chord produces a small number of different patterns of activity, and why particular tetra- patterns of activity in the hidden units. We chords produce specific activities in specific confirmed this by taking each type of tetra- hidden units, let us turn to describing the chord and examining the hidden unit activi- pattern of connectivity between the twelve ties produced by each. Three of the tetra- pitch-class input units and the three hidden chord types produced two distinct patterns units. Once we have a sense of the regu- of hidden unit activity, while the fourth (the larities in these connection weights, we can minor sevenths) produced three distinct pat- use this knowledge to explain the regulari- terns of hidden unit activity. ties of Figure 7-17 and Table 7-2.

Table 7-3 presents the activity in each 7.4.2 Hidden Unit 1 hidden unit, averaged over all of the hidden units that fall together in a particular type. To begin, let us examine the pattern of Each row of hidden unit activities represents connections between the twelve input units the general coordinates of tetrachord loca- and Hidden Unit 1. These are plotted in tions in Figure 7-17, and confirms our visual Figure 7-18. This figure provides a strong inspection of that figure. indication that this hidden unit has classified input pitch-classes in terms of the circles of Table 7-3 indicates that each subtype of tritones described in Section 7.1.8. That is, a tetrachord shares some similarities in Figure 7-18 exhibits tritone equivalence: terms of some hidden unit activities, but is pitch-classes that are a tritone apart are as- differentiated from each other in terms of the signed essentially the same connection other activities that they produce. For in- weight. stance, consider the three different subtypes of the minor seventh tetrachords. Each of these subtypes is similar in producing very high activity in Hidden Unit 1, and in producing very low activity in Hidden Unit 2. The three are differentiated in terms of the activity that each produces in Hidden Unit 3: one produces very high activity in this unit, another produces near zero activity, and the third produces weak activity.

Type Pat # H1 H2 H3

1 16 0.00 0.00 0.63 Major 7 2 8 0.00 0.48 0.95

1 12 0.96 0.04 0.95

Minor 7 2 12 1.00 0.00 0.09 Figure 7-18. The connection weights from the 3 12 1.00 0.01 0.27 twelve input units to Hidden Unit 1.

1 8 0.00 0.00 0.29 Dominant 7 Tritone equivalence in this hidden unit is 2 4 0.32 0.00 0.02 important because at the end of training its µ 1 8 0.00 0.01 0.05 had been assigned a value of 0.00. We Minor7♭5 might expect that the presence of a pair of 2 4 0.32 0.95 0.94 pitch-classes that are a tritone apart would Table 7-3. The different patterns of hidden unit activity produced by different subsets of each type of tetrachord. produce an extreme net input, turning Hid- The Pat column simply names the subset, and the # col- den Unit 1 off. The ic vectors in Table 7-2 umn indicates how many tetrachords belong to that sub- might lead us to predict that Hidden Unit 1 set. The H1, H2, and H3 columns provide the activity produced in each hidden unit, averaging over the activities would therefore turn on to either major sev- produced by those patterns that belong to each subset of enth or minor seventh tetrachords (which do tetrachords. not include a tritone. However, this predic-

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 18 tion is not supported by the data in Table 7- Our earlier discussion of circles of minor 3. Major seventh tetrachords never activate 3rds indicated that, unlike two pitch-classes Hidden Unit 1. There must be something separated by a tritone, one pitch-class is a more sophisticated within the Hidden Unit 1 minor third away from two other pitch- connection weights. classes. For example, an examination of the first circle of minor 3rds in Figure 7-9 indi- In networks described in earlier chapters cates that C is not only a minor third away we observed a phenomenon that we called from A, but it is also a minor third away from tritone balance. In tritone balance, pitch- D#. If Hidden Unit 1 is truly balancing minor classes a tritone apart had connection thirds, then we expect to find that one pitch- weights that were equal in magnitude, but class is balanced with two others, and not opposite in sign. As a result, if both pitch- just one. classes were present they would cancel each other out. An examination of Figure 7- This appears to be the case for Hidden 18 reveals that these connection weights are Unit 1. Figure 7-20 presents yet another balanced, but not in terms of tritones. In- depiction of its connection weights from Fig- stead, this hidden unit balances minor thirds. ure 7-18. However, in this second figure Pitch-classes that are a minor third apart each connection weight is stacked against have connection weights that are equal in the other pitch-class that it is a minor third magnitude, but opposite in sign. away from (i.e. the pitch-class that it was not stacked against in Figure 7-19). Once This relationship is made explicit in Fig- again, this figure is very symmetrical, alt- ure 7-19, which plots exactly the same con- hough the balancing is not quite as perfect nection weights from Figure 7-18, but stacks as that shown in Figure 7-19. Together, the weights from pitch-classes separated by Figures 7-19 and 7-20 reveal that Hidden a minor third on top of one another. The Unit 1 balances a pitch-class with either of symmetry of this bar graph provides the evi- the pitch-classes that are a minor third away dence that these connection weights bal- from it. ance minor thirds.

Figure 7-19. The connection weights of Fig- Figure 7-20. The connection weights of Fig- ure 7-18 re-plotted so that weights from pitch- ure 7-18 re-plotted so that weights from pitch- classes a minor third apart are stacked on top classes a minor third apart are stacked on top of one another. of one another. Note the difference in stack- ing between this figure and Figure 7-19. See text for details.

sented in this format, each contains two Our investigation of Hidden Unit 1 con- pitch-classes that are a minor second apart nection weights has revealed that they as- (A and A# in A#maj7, D and D# in D#maj7). sign pitch-classes a tritone apart to the This property is true of every major seventh same equivalence class, and they balance tetrachord in the training set. minor thirds. Major seventh tetrachords do not include a tritone, and do include pitches The presence of adjacent pitch-classes that are a minor third apart (Table 7-2). in any major seventh input pattern causes Why, then, do these tetrachords fail to acti- Hidden Unit 1 to turn off. This is because vate Hidden Unit 1? any four connection weights that include adjacent pitch-classes cannot be balanced The answer to this question comes from to produce a net input near 0 to activate this the pitch-class representation that is used unit. Instead, major seventh tetrachords that for the multilayer perceptron. This represen- belong to Pattern 1 from Table 7-3 will in- tation rearranges the structure of the various clude two balanced weights (e.g. D and F for tetrachords, and it is this rearranged struc- A#maj7), one near zero weight (e.g. A# for ture that the hidden units must process. A#maj7), and one extreme weight that is out of balance with the other three (e.g. A for Table 7-4 presents the pitch-class repre- A#maj7). Major seventh tetrachords that sentation of two major seventh tetrachords belong to Pattern 2 from Table 7-3 combine in its first two rows; one is an example of four weights that are even more unbalanced, Pattern 1 from Table 7-3 and the other is an causing a more extreme net input (e.g. the example of Pattern 2. The key property to D#maj7 chord in Table 7.4. observe in both is that after being repre-

Chord A A# B C C# D D# E F F# G G# Net H1

A#maj7 1 1 0 0 0 1 0 0 1 0 0 0 1.44 0.00

D#maj7 0 1 0 0 0 1 1 0 0 0 1 0 3.28 0.00 F#min7 1 0 0 0 1 0 0 1 0 1 0 0 0.12 0.95

Dmin7 1 0 0 1 0 1 0 0 1 0 0 0 0.00 1.00

Gmin7 0 1 0 0 0 1 0 0 1 0 1 0 0.00 1.00 Table 7-4. Example pitch-class representations of two major seventh and three minor seventh tetrachords, along with the net input they provide to Hidden Unit 1 and its resulting activity. See text for details.

Minor seventh tetrachords do not include small net inputs, and high Hidden Unit 1 ac- adjacent pitch-classes in their pitch-class tivities, as presented in Table 7-4. Every representation for the network, and as a re- other minor seventh tetrachord in the train- sult always include four connection weights ing set also exhibits this property. In other that when combined produce near zero net words, for minor seventh tetrachords Hidden input to turn Hidden Unit 1 on. Three exam- Unit 1 behaves as expected from our inter- ple pitch-class representations of minor sev- pretation of connection weights! enth tetrachords (one for each pattern in Table 7-3) are also presented in Table 7-4. Let us now briefly turn to explaining the activity produced in Hidden Unit 1 by the two Each of the three minor seventh tetra- other types of tetrachords, the dominant chords presented in Table 7-4 contains two seventh and the minor seventh flat fifth. pairs of pitch-classes that are a minor third Table 7-3 reveals that the majority of both apart and which have balanced weights; all types of these chords produce zero activity of these balanced pairs are illustrated in in Hidden Unit 1. This is consistent with our Figures 7-19 and 7-20. For F#min7 they are analysis of this unit’s weights. We noted [A, F#] and [C#, E]. For Dmin7 they are [A, that these weights exhibit tritone equiva- C] and [D, F]. For Gmin7 they are [A#, G] lence. Therefore pitch-classes a tritone and [D, F]. The balancing of each of these apart cannot cancel each other’s signal out, pairs of connection weights produces very because both pitch-classes are associated

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 20 with the same connection weight. Indeed, in [C, F#] in G#7, [D, G#] in Dmin7♭5, and [B, the pitch-class representation of each of the F] in Bmin7♭5. However, other intervals dominant seventh and the minor seventh flat that are present moderate the effect of the fifth chords that turns this unit off one finds unbalanced tritone. For example B7 in- two pitch-classes a tritone apart. Their cludes the pitch-classes [A, F#] which pro- combined weights produce an extreme net vide a balanced minor third (Figure 7-20). input that is very far from µ. The same is true for the pitch-classes [C, D#] in C#7, [F, G#] for Dmin7♭5, and [D, B] What is surprising about Table 7-3 is that in Bmin7♭5. The remaining two connection a minority of both of these types of tetra- weights together produce a less extreme net chords produce weak activity in Hidden Unit input (see Table 7-5) that produces moder- 1. How is this possible if these stimuli in- ate Hidden Unit 1 activity. In other words, clude a tritone? for this subset of tetrachords, Hidden Unit 1

compromises its activity because it detects Table 7-5 presents the pitch-class repre- one interval that should turn it off (a tritone), sentation of four example tetrachords that but another that should turn it on (a minor produce this surprising behavior in Hidden third). Unit 1. All four of these chords include a pair of tones a tritone apart: [A, D#] in B7,

Chord A A# B C C# D D# E F F# G G# Net H1

B7 1 0 1 0 0 0 1 0 0 1 0 0 -0.61 0.32 G#7 0 0 0 1 0 0 1 0 0 1 0 1 0.60 0.32

Dmin7♭5 0 0 0 1 0 1 0 0 1 0 0 1 0.60 0.32 Bmin7♭5 1 0 1 0 0 1 0 0 1 0 0 0 -0.61 0.32 Table 7-5. Example pitch-class representations of two dominant seventh and two minor seventh flat fifth tetrachords, along with the net input they provide to Hidden Unit 1 and its resulting activity. See text for details.

7.4.3 Hidden Unit 3

Let us next consider Hidden Unit 3, whose connection weights are illustrated in Figure 7-21. As was the case with Hidden Unit 1, Hidden Unit 2 assigns input pitch- classes to equivalence classes related to circles of intervals. For Hidden Unit 2 these equivalence classes involve the three circles of minor 3rds.

All four pitch-classes that belong to the first of these circles in Figure 7-9 are assigned the same negative weight (-0.43) in Figure 7-21. All that belong to the second circle of Figure 7-9 are assigned the same weak positive weight (0.07) in Figure 7-21.

Finally, all of the pitch-classes that belong to Figure 7-21. The weights of the connections the third circle in Figure 7-9 are assigned the from the input units to Hidden Unit 3. same stronger positive weight (0.33) in Fig- ure 7-21. Unlike Hidden Unit 1, Hidden Unit 3 does not exhibit any obvious balancing between pairs of weights a particular musical interval apart. However, the weights that it assigns to the three different equivalence classes reveal some very interesting properties. If

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 21 one considers combinations of four different four weight values are summed, the result- weights, then one discovers specific pat- ing net input is approximately -0.04, which terns that cancel net input signals and cause generates activity of approximately 0.95 in high activity in Hidden Unit 3. Hidden Unit 3. This pattern appears in four different major seventh chords: Emaj7: [B, First, it is important to recognize that the D#, E, G#], C#maj7: [C, C#, F, G#], Gmaj7: value of µ for Hidden Unit 3 is -0.08. This [B, D, F#, G] and A#maj7: [A, A#, D, F]. No means that for an input pattern to generate a other tetrachords, including the other major maximum response in this hidden unit, the seventh chords, exhibit this pattern. net input generated by this pattern will be slightly negative. Interestingly, the other major seventh chords produce moderate activity in this hid- An examination of different combinations den unit (0.63 as shown in Table 7-3). They of four weights from Figure 7-21 reveals that exhibit a slightly less optimal combination of there are three different patterns which ac- four weights than the three shown in Figure complish this. The three patterns are illus- 7-22. This involves one weak positive trated in Figure 7-22, which attempts to ar- weight, one negative weight and two strong- range the four different bars representing er positive weights or one weak positive connection weights in such a way that the weight, one stronger positive weight, and balance between negative and positive two negative weights. Any of these combi- weights is apparent. nations produces a net input that ranges between -0.47 and 0.30 depending upon which particular weights are included.

The second combination of four Hidden Unit 3 weights that produces high activity involves two pitch-classes that have strong negative weights, and two pitch-classes that have strong positive weights. This pattern is illustrated with the group of four bars in the middle of Figure 7-22. This pattern only appears in four different minor seventh tetrachords: F#min7: [A, C#, E, F#], Cmin7: [A#, C, D#, G], Amin7: [A, C, E, G] and D#min7: [A#, C#, D#, F#].

The third combination of four Hidden Unit 3 weights that produces high activity involves one pitch-class that has a strong negative weight, and three pitch-classes that have weak positive weights. This pattern is illustrated with the stack of four bars at the Figure 7-22. Three different combinations of right of Figure 7-22. This pattern only ap- four Hidden Unit 3 weights that produce net pears in four different minor seventh flat fifth inputs close enough to µ to generate high activity. See text for details. tetrachords: G#min7♭5: [B, D, F#, G#], D min7♭5: [C, D, F, G#], Fmin7♭5: [B, D#, F, The first combination occurs when a tetrachord contains only one member from the G#] and Bmin7♭5: [A, B, D, F]. equivalence class assigned a negative weight, only one member from the equiva- One type of tetrachord that does not pro- lence class assigned a strong positive duce high activity in Hidden Unit 3 is the weight, and two members from the equiva- dominant seventh. The highest activity is lence class assigned a weak positive weight. produced by an input pattern like A#7: [A#, This pattern is represented as a stack of four D, F, G#]. Note that this pattern includes bars on the left of Figure 7-22. When these three weak positive weights (D, F, G#), but

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 22 the fourth weight is a stronger positive weight (A#). Because most of these weights are weak, this type of input pattern produces a relatively small net input (0.54). However, this net input is extreme enough to reduce Hidden Unit 3 activity to about 0.29. This pattern is true of eight of the twelve different dominant seventh chords. The other four dominant seventh chords include three of the extreme negative weights balanced by only a single weak positive, producing a net input of -1.23 and essentially turning Hidden Unit 3 off.

Finally, while some minor seventh and some minor seventh flat fifth tetrachords cause Hidden Unit 3 to activate, most do Figure 7-23. The connection weights from the not. All of these tetrachords include a pair of twelve input units to Hidden Unit 2. pitch-classes that are minor third apart. As a result of Hidden Unit 3 exploiting minor third equivalence, these pitch-classes do not Why might we say that the pattern of cancel their signals out. Instead they pro- weights in Figure 7-23 is less musically gen- duce a more extreme net input for Hidden eral than those that we have seen earlier in Unit 3, reducing its activity. Importantly, the this chapter? One reason is that the weights combined effect of a pair of such pitch- in Figure 7-23 do not exhibit any systematic classes is not uniform: [A#, C#] will be more assigning of pitch-classes to equivalence extreme than [B, D] because the former pair classes. For instance, Figure 7-23 begins has more extreme connection weights than by suggesting tritone equivalence because does the latter pair (see Figure 7-21). This the weight for A is nearly identical to the explains why some tetrachords that include weight for D#. However, the weights for the at least one minor third can still produce mild next tritone (A#, E) are not equivalent, nor activity in Hidden Unit 3 (e.g. 0.27 produced do they balance. No other systematic by some minor seventh input patterns). equivalences based upon circles of intervals are apparent in this figure either. 7.4.4 Hidden Unit 2 We saw earlier that both Hidden Units 1 Let us finally consider Hidden Unit 2, and 3 organized pitch-classes using interval- whose connection weights are illustrated in based equivalence classes, but also bal- Figure 7-23. Although these connection anced other combinations of pitch-classes weights have a very regular appearance, related by different intervals. Hidden Unit 2 they are not as musically general as were balances several different pairs of pitch- the weights for both Hidden Units 1 and 3. classes as well. This is illustrated in Figure This is because Hidden Unit 2 fulfills a very 7-24, which presents the same weights that specialized task for the tetrachord classifica- are in Figure 7-23, but stacks balanced tion network. weights on top of each other to highlight their symmetry.

F] is a tritone apart. In short, the connection weights for Hidden Unit 2 seem to balance specific pairs of pitch-classes, and do not balance specific types of musical intervals.

Why does Hidden Unit 2 exhibit properties that seem less musically general than those exhibited by the other two hidden units? An answer to this question comes from considering the role of Hidden Unit 2 in arranging input patterns in the hidden unit space.

To begin, let us consider the hidden unit space in the context of output unit functions. Figure 7-25 attempts to make this context explicit. On its left is a copy of the hidden

unit space that was presented earlier in Fig- Figure 7-24. The connection weights from the ure 7-17. On its right is the same space, but twelve input units to Hidden Unit 2, with bal- with four different planes that have been anced weights stacked on top of each other. added. These four planes illustrate the fact that in this three dimensional hidden unit Once again, though, the balancing in space all of the input patterns that belong to Figure 7-24 is not musically systematic. For a particular tetrachord type are aligned instance, the balanced pair [A, C#] is a ma- along a two-dimensional plane that passes jor third apart, as is the balanced pair [C, E]. through the space. However, the balanced pair [A#, F#] is a minor sixth apart, while the balanced pair [B,

Figure 7-25. The input patterns in their position in the hidden unit space are illustrated on the left. Four different planes that intersect each of the four different input pattern types have been added to the same hidden unit space on the right.

The planes drawn on the left part of Fig- close together, because a value unit is sen- ure 7-25 are important in terms of output unit sitive to a very narrow range of net inputs. function for this particular network. Recall The single planes illustrated in Figure 7-25 that each output unit is a value unit. This are important, because they will fall between type of unit carves a three-dimensional hid- the two parallel planes that an output unit den unit space into decision regions by plac- carves through this hidden unit space. This ing two parallel planes that cut through this permits the output unit to correctly turn on to space. Any input patterns that fall between these patterns, and to correctly turn off to these two planes are patterns that that turn any other patterns that do not fall between the output unit on. These planes are very the two cuts.

patterns using Hidden unit 1 and 3 activities What is Hidden Unit 2’s role in arranging as coordinates. In other words, it is the hid- patterns in this space? To answer this den unit space that would exist if Hidden question, we can redraw the three- Unit 2 was not present in the multilayer per- dimensional hidden unit space in Figure 7- ceptron. This new, Hidden Unit 2 free 25 as a two-dimensional hidden unit space. space, is illustrated in Figure 7-26. This two-dimensional space arranges input

Figure 7-26. A two-dimensional hidden unit space for the input patterns created by removing the Hidden Unit 2 coordinate from Figure 7-25. The space plotted on the left shows that this space permits the correct identification of minor seventh, dominant seventh, and minor seventh flat fifth tetrachords. The same space plotted on the right shows that this space does not permit the correct classification of major seventh tetrachords. See text for details.

When an output value unit is confronted permit the major seventh tetrachords to be with a two-dimensional hidden unit space, it correctly identified. This is illustrated on the does not carve it with parallel planes. In- right side of Figure 7-26. This graph is the stead, it carves two parallel lines through same hidden unit space as the one on the this space; patterns that fall between the two left. In this version of the space two parallel lines turn the output unit on. The lines are lines have been added to capture the major very close together, because an output val- seventh tetrachords (the triangles). Note ue unit is sensitive to a very narrow range of that this is the only orientation of these two net inputs. parallel lines that results in all of the triangles falling between them. However, this The hidden unit space on the left side of positioning of the two lines does not sepa- Figure 7-26 illustrates the parallel cuts that rate the major seventh tetrachords from all can be made through this space by three of of the other types: notice that dominant sev- the output units. Each of these three pairs enth and minor seventh flat fifth chords also of cuts separates one type of tetrachord fall between these two lines. from all three of the other types, permitting the output unit that makes these cuts to cor- This suggests that the functional role of rectly classify the chords. The three cuts Hidden Unit 2 in the multilayer perceptron is illustrated on the left of Figure 7-26 demon- to arrange the major seventh tetrachords in strate that this two-dimensional space ar- a pattern that permits them to be separated ranges input patterns that would permit the from the other chords that they cannot be network to correctly classify all of the minor separated from when Hidden Unit 2 is ab- seventh, dominant seventh, and minor sev- sent. Looking back at Figure 7-25, this enth flat fifth tetrachords. seems to be exactly what the Hidden Unit 2 dimension is adding to the hidden unit The problem with this two-dimensional space. That dimension appears to capture a hidden unit space, though, is that it does not handful of major seventh tetrachords and

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 25 pulls them towards the back of the cube. provided below in Table 7-6. This observa- This permits these chords to be arranged tion is important because we noted above along a plane, and also permits an output that they key function of Hidden Unit 2 is to unit to define a decision region that only enable this type of chord to be classified; captures these patterns. The other effect of major seventh chords are the only chords Hidden Unit 2 is that it also draws a handful that cannot be correctly separated in the of minor seventh flat fifth tetrachords to the two-dimensional pattern space. back of the cube. This suggests that these input patterns possess some musical prop- Third, the four tetrachords that produce erty that is being used by the hidden unit to high activity in Hidden Unit 2 are all minor pull the major seventh tetrachords to the seventh flat fifth chords that are nearly iden- back. tical to the four major seventh chords that produce moderate activity in this same unit. This functional account of Hidden Unit They are nearly identical in several respects: 2’s role in the network is confirmed by exam- they share three pitch-classes with one of ining the subset of input patterns that pro- the major seventh tetrachords, the remain- duce higher Hidden Unit 2 activity in the ing pitch-class is only a minor second away context of the connection weights that were from the fourth pitch-class, and the connec- provided earlier in Figures 7-23 and 7-24. tion weight associated with the fourth pitch- class has a similar value to the connection First, Hidden Unit 2 generates moderate weight associated with the fourth (i.e. the to high activity to only eight different tetra- dissimilar) pitch-class in the major seventh chords. This confirms our observation chord. The properties of the four minor sev- above that the role of Hidden Unit 2 is to enth flat fifth tetrachords are provided in Ta- only move a small number of input patterns ble 7-7. The grey cells in Tables 7-6 and 7- to permit their correct detection. 7 indicate the single difference, in either a pitch-class or a connection weight, between Second, four of the tetrachords that pro- a major seventh tetrachord and its similar duce moderate activity in Hidden Unit 2 are minor seventh flat fifth tetrachord. major seventh chords whose properties are

Chord Notes Weights Net H2 Similar A#maj7 A A# D F 0.92 2.69 -0.39 -3.77 -0.55 0.49 Bm7♭5 Gmaj7 B D F# G 3.03 -0.39 -2.19 -0.05 0.40 0.49 G#m7♭5 C#maj7 C C# F G# 4.36 -0.75 -3.77 -0.40 -0.55 0.48 Dm7♭5 Emaj7 B D# E G# 3.03 0.95 -4.15 -0.40 -0.56 0.47 Fm7♭5 Table 7-6. The four major seventh tetrachords that produce moderate activity in Hidden Unit 2. Grey cells indicate the one difference between each of these chords and the minor seventh flat fifth tetrachord to which they are similar, and whose properties are provided in Table 7-7. The ‘Net’ column provides the net input produced for Hidden Unit 2, and the ‘H2’ column provides Hidden Unit 2 activity.

Chord Notes Weights Net H2 Similar Bm7♭5 A B D F 0.92 3.03 -0.39 -3.77 -0.18 0.96 A#maj7 G#m7♭5 B D F# G# 3.03 -0.39 -2.19 -0.40 -0.19 0.96 Gmaj7 Dm7♭5 C D F G# 4.36 -0.39 -3.77 -0.40 0.06 0.95 C#maj7 Fm7♭5 B D# F G# 3.03 0.95 -3.77 -0.40 -0.20 0.95 Emaj7 Table 7-7. The four minor seventh flat fifth tetrachords that produce moderate activity in Hidden Unit 2. Grey cells indicate the one difference between each of these chords and the major seventh tetrachord to which they are similar, and whose properties are provided in Table 7-6. The ‘Net’ column provides the net input produced for Hidden Unit 2, and the ‘H2’ column provides Hidden Unit 2 activity.

The high specificity of the connection First, these connection weights capture weights that were presented in Figures 7-23 very specific relationships (and not general and 7-24 now make perfect sense in light of musical properties) because all Hidden Unit the properties detailed in the two tables 2 really has to do is move four major sev- above. enth tetrachords away from the others in the three-dimensional hidden unit space. If this

© Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 26 unit was sensitive to more general proper- Third, the four minor seventh flat fifth tet- ties, it would affect the position of a larger rachords that produce high activity in Hidden number of tetrachords. Unit 2 do so because they are each nearly identical to one of the major seventh chords Second, the specific balancing that was that this unit moves. They differ in only one observed in Figure 7-24 nicely accomplishes pitch-class, and this difference is only a the main task of Hidden Unit 2. The A#maj7 semitone. This in turn means that one struc- chord produces moderate activity in this unit tural property of the weights in Figure 7-23 is because the weights from A and from D that pairs of pitch-classes a minor second roughly balance one another, as do the apart ([A#, B], [C#, D], [E, F] and [G, G#]) weights from A# and F. A similar rough bal- must have similar weight values. An inspec- ancing of pairs of pitch-classes is true for tion of this figure indicates that this property Emaj7. The remaining two major seventh is indeed apparent. chords balance one extreme positive weight (C for C#maj7, B for Gmaj7) with a combination of three negative weights.

7.5 Summary and Implications 7.5.1 Summary unit to correctly classify all of these tetrachords. Section 7.4 provided a detailed interpretation of the internal structure of a multilayer To relate this interpretation to material perceptron that learned to classify tetra- covered in Chapter 6, these three hidden chords into four different points. There are units provide another example of coarse three key elements of this interpretation. coding. None of the hidden units detect a specific property that is consistent with only First, both Hidden Units 1 and 3 organize one type of tetrachord: two or more different pitch-classes into equivalence classes types of tetrachords can produce high activi- based upon circles of intervals. Hidden Unit ty in any of the hidden units. However, 1 assigns the same weight values to two when the activities produced by an input pitch-classes that belong to the same circle pattern in the hidden units are considered of tritones. Hidden Unit 3 assigns the same simultaneously, the output units can deter- weight values to three pitch-classes that mine the input pattern’s type. belong to the same circle of minor thirds. 7.5.2 Implications Second, the particular weight values that both Hidden Units 1 and 3 assign to each The connection weights for the two gen- equivalence class are very systematic. eralist hidden units (Hidden Units 1 and 3) Their values permit pitch-classes separated have one interesting implication to keep in by other musical intervals to be balanced, mind for later network interpretations that increasing hidden unit activation. involve strange circles. Both of these hidden units use their weights to organize pitch- Taken together, these two above obser- classes by more than one musical interval. vations indicate that Hidden Units 1 and 3 The reason that this is possible is because detect general musical properties that permit some of the strange circles that were intro- varied and useful responses to pitch-class duced beginning in Section 7.1.5 are hierar- combinations that are either part of, or not chically related to other strange circles. part of, specific tetrachord structures. In- deed Figure 7-26 indicates that these two For example, each of the circles of major hidden units alone are capable of supporting 2nds contains the pitch-classes that belong to the correct identification of all of the mem- two of the circles of major 3rds. Similarly bers of three of the four types of tetrachords. each of the three circles of major 3rds contains the notes that belong to two of the cir- Third, Hidden Unit 2 detects very specific cles of tritones. These hierarchical relation- properties (i.e. properties related to a small ships permit one set of connection weights number of individual chords, and not to a to organize input pitch-classes in complex larger set or chord types) that serve to ar- ways. For example, one hidden unit could range the major seventh chords in hidden use the sign of the connection weight to unit space in such a way that they can be separate pitch-classes into the two circles of correctly identified. The specific properties major 2nds. However, variations in magni- detected by this hidden unit also capture tudes of these same weights can simultane- four different minor seventh flat fifth chords; ously be used to organize the pitch-classes there is a strong musical relationship be- into circles of tritones because of a hierar- tween the properties of these chords and the chical relationship between the two. In properties of the major seventh chords that Chapter 8 we explore a more complex net- Hidden Unit 2 segregates. work trained on an elaboration of the tetrachord task, and discover that many of its In sum, this multilayer perceptron uses hidden units exploit the hierarchical relation- two generalist hidden units (that employ cir- ships amongst strange circles in their repre- cles of intervals) and one specialist hidden sentation of input pitch-classes.

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