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 to- gether in circles where each adjacent note in the circle is separated by a particular musical inter- val. 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 © Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 2 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 re- vealed 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 “pi- ano keyboard’ encoding was used to pre- sent different inversions of the triads, pitch- class equivalences involving different musi- cal 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 relation- ships. 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 © Michael R. W. Dawson 2014 Chapter 7 Strange Circles and Tetrachords 3 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 differ- ent 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 indi- cates 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 pitch- es. See text for details piano keys) or a minor sixth apart (eight pi- ano 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.