CHAPTER 9: EXPLORING THE II-V-I CHORD PROGRESSION The purpose of this chapter is to explore an important source of tonality in Western music, a musically related sequence of chords called a chord progression. The chapter begins by discussing chord progressions in the context of establishing tonality. It then describes an important chord progression in jazz called the ii-V-I progression. The goal of this chapter is to train a number of different networks on this progression; when provided one chord, the network is trained to respond with the next chord in the progression that should be played. While all of the networks in the chapter are trained on the same jazz progression problem, different networks use different codes to represent input and output chords. The question of interest is whether the choice of encoding has any effect on the ease of discovering a solution to the progression problem. We present results that show that this is definitely true: when an abstract encoding is employed, a multilayer perceptron with many hidden units is required to learn the chord progression. In contrast, when different encodings are used the same problem can be learned by a perceptron that has no hidden units at all. We demonstrate a practical implication of this by demonstrating that a simple network can be easily interpreted. We end the chapter by pointing out that other factors might influence encoding selection; depending on the goals of a simulation, one may not always be seeking the encoding that leads to the simplest network solution. 9.1 Tonality and Chord Progressions ............................... 2 9.2 The ii-V-I Progression .................................................. 5 9.3 The Importance of Encodings ..................................... 8 9.4 Four Encodings of the ii-V-I Problem ....................... 10 9.5 Simulations With Pitch-Class Encoding .................. 15 9.6 Simulations Using Pitch Encodings of Root Forms 17 9.7 Simulations Using Pitch Encodings of Inverted Forms ............................................................................... 18 9.8 Simulations Using Lead Sheet Encodings ............... 19 9.9 Interpreting A Lead Sheet Perceptron ...................... 20 9.10 A Progression of Progressions .............................. 26 9.11 Summary and Implications ..................................... 31 9.12 References ............................................................... 34 © Michael R. W. Dawson 2014 Chapter 9 Exploring the ii-V-I Progression 2 9.1 Tonality and Chord Progressions Tonality is a central characteristic of of balanced support of the tonic, like two Western music (Piston, 1962; Schoenberg, equidistant weights on either side of a 1969). Tonality is the sense that a fulcrum” (Piston, 1962, p. 31). Alternatively, composition belongs to a particular musical some tones are intrinsically unstable in a key, and has been a topic central to many of particular musical key; their presence the earlier parts of this book, including produces a musical tension that demands Chapter 3 on identifying scale tonics, resolution in the form of hearing a more Chapter 4 on identifying scale modes, and stable musical entity. Chapter 5 on key finding. Establishing tonality, then, also requires In what is known as the era of ‘common communicating and exploiting the various practice’ in Western classical music, which relationships amongst tones in a musical spans the 18th and 19th centuries, key. “It is more a process of setting forth the establishing a musical key almost always organized relationship of these tones to one meant establishing a major or minor key of among them which is to be the tonal center” the type that we have already encountered (Piston, 1962, p. 31). In common practice in previous chapters. Although other modes the process of establishing tonality and its can and have been used in Western structure is accomplished by using harmony. compositions, they are essentially ignored by common practice’s focus on the ‘major- The basic element of harmony is the minor system’ of music. “We are so imbued musical interval, the simultaneous presence with this tradition that we tend to interpret of two tones a specific musical distance music based on other modes as being in apart. Chords involve the presentation of either major or minor, usually with somewhat more than two simultaneous tones, and unsatisfactory results” (Piston, 1962, p. 30). therefore the presence of more than one musical interval. Earlier in Chapter 7 we In our earlier chapters on the modality saw how one could construct particular and tonics of musical scales we noted that chords (triads and tetrachords) on each of one could define a musical key by the degrees of a musical scale (Figure 7- constraining pitch-class choice: specifically, 15). Accounts of harmony often begin by by only using tones that belonged to a considering the use of, and the relationship particular scale. For instance, to set a between, triads (Piston, 1962; Schoenberg, composition in the key of C major a 1969). composer would only select its tones from the C major scale, and not employ tones Just as the presence of a single tone that do not belong to this scale. cannot by itself establish a musical key, the occurrence of a triad in isolation cannot However, establishing tonality is more establish tonality. “A triad standing alone is complex than merely restricting the use of entirely indefinite in its harmonic meaning; it particular tones. The tones that define a may be the tonic of one tonality or one musical scale have an organized degree of several others” (Schoenberg, relationship to one another, relationships 1969, p. 1). In order for tonality to be that even listeners with no musical training established, a succession of triads must be are aware of (Krumhansl, 1990). Because presented. The succession must be of these relationships, different tones in a structured so that there is the relationship scale – typically identified by their scale from one triad to the next takes the listener degree (i.e. the Roman numerals used to an intended goal. Such a structured earlier in Figures 4-11 and 7-15), such as I succession of chords is called a chord for the tonic, IV for the subdominant, and V progression. In jazz a chord progression is for the dominant – have specific tonal often called ‘the changes’. functions. There are two strongly related aspects to For example, “dominant and creating a chord progression: establishing a subdominant seem to given an impression sequence of chord roots, and defining the © Michael R. W. Dawson 2014 Chapter 9 Exploring the ii-V-I Progression 3 structure of each chord to establish an imperfect cadence or a half cadence appropriate voice leading. Let us briefly ends on a V chord, and can be preceded by consider these two aspects in turn. any of a number of different chords (e.g. IV or I). The tension produced by ending on Defining a particular succession of the V chord provides a clear signal that chords requires considering the succession further music is coming. It is “like a comma, of chord roots independently of the form (the indicating a partial stop in an unfinished inversion) of each chord. “Chord succession statement” (Piston, 1962, p. 60). can be reduced to root succession (or root progression), which in turn can be translated Root progressions can also be used to into Roman numerals representing a perform other functions, such as modulating succession of scale degrees” (Piston, 1962, from one musical key to another p. 18). Piston notes that common practice (Schoenberg, 1969). We saw earlier in reveals a set of typical root progressions Chapter 4 that different musical scales are which are summarized in Table 9-1. Each similar to one another because they share row of this table provides the root of the many tones. As a result the same chord can current chord in a progression, and notes be found in more than one musical key; the root of the next chord that typically these are called common chords. follows, that sometimes follows, and that less often follows. For instance its first row For instance, A minor is a common chord can be interpreted as “If the current chord found in both the key of C major (where it is has I as its root, it is typically followed by a the built on the VI scale degree) and in the IV or by a V chord, it is sometimes followed key of G major (where it is built on the II by a VI chord, and it is less often followed by scale degree). One can therefore use A a II or III chord.” minor as a pivot chord in a cadence that modulates the key of a composition from C Less Root Of Typically Sometimes major to G major. Often Current Followed Followed Followed Chord By By By Root successions are only aspect of a I IV or V VI II or III chord progression. A second strongly II V VI I, III, or IV related aspect is voice leading (see the III VI IV II or V earlier discussion of this term in Section IV V I or II III or VI 4.5.3). In choral music different voices V I VI or IV III or II perform the component notes of each chord VI II or V III or IV I VII III in a progression. In addition to defining the Table 9-1. The usual progression of chord roots in succession of chord roots in this common practice. See text for details. progression, a composer of choral music must also decide which voice is to move Why do the root progressions from one tone in the first chord to another summarized in Table 9-1 emerge from tone in the second. common practice? The reason is that the relations amongst chords with these roots in Common practice adopts principles that terms of a particular tonal center are such lead to efficient voice leading, which that they instill a particular musical direction attempts to minimize the musical distance to a listener.