Monday, May 25, 2015

Coda: Our New Music

As described in this previous post, the  text below is a draft of one of several "interludes" to be included in a book that I am working on concerned with music and artificial neural networks.  This particular post is the Coda for the book; the interlude that comes at the end of the main text.

Figure C-1. Four key notes for the song “How Dry I Am” used by Bernstein to illustrate the infinite variety of music.
The tonality of Western music arises from its exclusive use of twelve pitch-classes.  In spite of being constrained by this sparse set of basic musical elements, composer Leonard Bernstein argues that Western music is infinite in its variety (Bernstein, 1966).  He observes that if one considers the twelve pitch-classes in a single range, and computes their possible melodic combinations, the result is 1,302,061,344.  If one extends this approach to consider both melodic and harmonic combinations of these elements the result is 127 googols, where a googol is a digit followed by 100 zeroes.  “The realm of music is an infinity into which the composer’s mind goes wandering” (Bernstein, 1966, p. 34).

Bernstein (1966) explores this theme with a particular example, the four note melody that starts the song “How Dry I Am”.  These four notes are provided in Figure C-1.  He notes the importance of this pattern, and the variations of musical effects that it can produce, by noting its presence in a huge range of compositions that begins with a French folk song and ends with the final movement of Shostakovich’s Fifth Symphony.  Bernstein ends his discussion by proposing a variation of Figure C-1 that depicts “a motto of man’s infinite variety” (Bernstein, 1966, pp. 46-47).

Early in this book we saw another example of variety from small numbers of elements.  The musical signal composed by John Williams for Steven Spielberg’s 1977 movie Close Encounters of the Third Kind (Figure O-1) was selected from a sample of 350 five-note compositions created by Williams.  Had Williams composed all possible five-note melodies the movie’s signal would have been selected from among about 134,000 possibilities.  The calculation of possibilities is conservative because it fails to take into account rhythmic variations; not all of the notes in Williams’ signal have the same duration.

The infinite possibilities of Western tonal music are reflected in music’s constant evolution.  American composer Aaron Copland wrote Our New Music (Copland, 1941) to explain the circumstances that had led to modern classical music.  His goal was to alleviate his readers’ bewilderment with modern music.  “Being unaware of the separate steps that brought about these revolutionary changes, they are naturally at a loss to understand the end result” (Copland, 1941, p. v).  He traced modern music’s development as a move away from a century of Germanic musical influences.  This move begins with explorations of folk music in the late 19th century, proceeds through explorations of new views of harmony, rhythm, and tonality.  Copland argues that it ends by coming full circle, in Stravinsky’s compositions of the late 1920s and early 1930s, and returning to melodic forms from the 18th century.

The infinite possibilities of Western tonal music make it nearly impossible to predict its future too.  In the early 1940s one could analyze existing modern music and describe a neoclassicism that had roots in the 18th century (Copland, 1941).  However, Copland’s analysis of modern American music does not even hint of the radical developments that would flourish there beginning in the 1960s with, for example, the invention of mimimalism (Glass, 1987; Griffiths, 1994, 1995; Hartog, 1957; Nyman, 1999; Pleasants, 1955; Potter, 2000; Reich, 1974, 2002).

Western tonal music has infinite variety and unpredictability.  However, it is neither accidental nor unsystematic.  When a composer’s mind goes wandering into the infinite musical realm, it does not randomly move from one musical entity to another.  Its search through this realm is guided by new ideas concerning musical structure – new notions of melody, harmony, rhythm and the like – in short, new music theory.  Rather than being “dusty abstract rules of form and harmonic structure” (Bernstein, 1966, p. 24), music theory itself seems both vast and dynamic.  When violent upheaval is heard in classical music, its root cause must be changing conceptions of music’s structure.

Does musical theory itself exhibit infinite variety?  I have no idea.  However, historical examinations reveal enormous changes in basic ideas, such as whether different inversions of a chord are the same chord, or what is the root note of a major or minor triad (Damschroder, 2008; Rehding, 2003; Riemann, 1895).  We saw in Chapter 1 that the psychophysical study of music that began in the late 19th century faced the tension between the physics of sound and individual differences in aesthetics that permitted just intonation to be replaced by equal temperament (Hui, 2013).

As well, evolving notions of consonance have permitted new musical intervals to become accepted in music.  The dissonance of the flattened seventh note led Helmholtz to reject its use in his advice to composers (Helmholtz & Ellis, 1863/1954); now it is definitive to the blues and plays a central role in Gershwin’s classic Rhapsody In Blue (Adams, 2008).  Later, seasoned jazz musicians who were completely comfortable with the flattened seventh were jarred and puzzled by the flattened fifth interval when it was introduced to jazz via bebop (Kelley, 2009).

Clearly there is no single, unified theory of music.  A multitude of music theories have existed; many different theories can exist at the same time; new theories can be invented or discovered.  One approach to composing innovative music involves taking a new musical theory an examining the compositions that it can pick out of the infinite realm of music.  Where might one find a new musical theory to exploit in this fashion?

There are many, many possible answers to this question.  One reading of the current book suggests one: train an artificial neural network to map some musical inputs to some other musical outputs.  The kind of training that we have seen in preceding chapters informs networks about their progress, but does not inform them how to construct the mapping.  As a result, these networks can discover new musical regularities or ideas for performing the mapping.  We have seen many instances of this in the current book, even when networks are trained on basic, traditional musical tasks.

Crucially, for a network to deliver a new musical theory its internal structure must be explored.  Artificial neural networks can only inform the study of music if we first reject the romanticism that characterizes much of connectionist cognitive science.



Saturday, May 16, 2015

Coltrane Changes on the Ukulele

As described in this previous post, the  text below is a draft of one of several "interludes" to be included in a book that I am working on concerned with music and artificial neural networks.

Figure I-14. The most basic ukulele chords for the ii-V-I progression in the key of C major.  See text for details.
When investigating musical problems with artificial neural networks, I find it useful to hear the stimuli on a musical instrument.  While I have spent a lot time working out stimulus patterns on the keyboard of my piano, these days my instrument of choice is the ukulele.  In this interlude I will provide the chords that I use to play the Coltrane changes in the key of C major, developing this chord structure from variations of the ii-V-I progression.
Figure I-14 provides the three chords that define the simplest version of the ii-V-I progression on the ukulele.  Each chord diagram illustrates the four strings of the ukulele, and the dots on the diagram indicate the fret at which each particular string is depressed.  The three chords that are presented are based on the assumption that the progression involves a sequence of three triads (D minor, G major, and C major).  That is, if one creates these three triads using only the notes available in the C major scale, then one of these chords is necessarily minor, while the other two are major (see the discussion of Figure 7-15).
The ii-V-I progression is a staple in jazz, and using triads doesn’t provide the jazziest sound.  Jazz musicians are more likely to extend the triads used to create the Figure I-14 chords to create tetrachords.  This extension, which involves adding an additional note to each chord from the C major scale, was also illustrated earlier in Figure 7-15.  Figure I-15 illustrates how one would play the C major tetrachords for the ii-V-I on the ukulele.  Note that the D minor has now become a D minor seventh, the G major has become a G dominant seventh, and the C major has become a C major seventh.
Figure I-15. The ukulele tetrachords for the ii-V-I progression in the key of C major.  See text for details.
All of the chords illustrated in Figures I-14 and I-15 are called open position chords.  This is because at least one ukulele string in the chord is open; that is, it is not pressed down by a finger.  The advantage of open position chords is that they generally are easier to play.  The disadvantage of such chords is that they are special in the sense that they cannot be moved up or down along the ukulele fret board to play the same type of chord in a different key.  This makes these chords different from the closed form chords that were the topic of the previous interlude “Ukulele Chords and Perceptrons”.  Our next move is to transform the Figure I-15 progression into one that uses closed position chords.
In order to perform this transformation, we will use two different tricks.  The first is to replace the Dm7 and Cmaj7 chords with alternative fingerings that can be found in a decent ukulele chord book (Johnson, 2005).  For these two chords we pick two fingerings that are related; both are barre chords that involve pressing the index finger down across all the strings at the fifth fret.
The second trick is to take advantage of chord substitution.  In general, jazz musicians see a chord’s name as an indicator of potential chords.  For instance, when such a musician sees that a G7 is the next chord, they would feel perfectly comfortable with substituting a different, but related, chord.  Chord substitution conventions permit G7 to be replaced, for example, with G9 or with G13 in order to add musical variety.  We will choose the G9 chord because the barre form of this chord places it in a similar position on the fret board to the other closed form chords in the ii-V-I, as is illustrated in Figure I-16.
Figure I-16. Closed form ukulele chords for the ii-V-I progression in the key of C major.  See text for details.
There are three important points to make about Figure I-16.  First, the particular chord choices that it illustrates begin quite a bit further down the fret board (at either fret 5 or 4) than was the case in Figures I-14 or I-16.  Each chord diagram has a number on the left indicating the starting fret, and the chord diagrams have been extended more than is typical to show where on the ukulele each chord is being played.
Second, the G9 chord that is illustrated is not likely to be found in many ukulele chord dictionaries.  This is because this form of the chord does not include the root note G.  Instead, it uses the other four pitches that are part of G9.  These four pitches actually define a minor seventh (flat fifth) chord in a different key.  G9 as illustrated in Figure I-16 is also Bm75.
Third, because these three chords are all in closed form one can move these patterns up or down the fret board to play the ii-V-I progression in a different key.  For instance if one uses the same chord patterns illustrated in Figure I-16, but moves each upwards a fret (towards the top of the page), then the result is the ii-V-I progression in the key of B major.
As detailed in Chapter 9, the Coltrane changes elaborate the ii-V-I progression by using the same three chords in Figure I-16, but also adds four additional chords that serve as lead ins.  We can create the Coltrane changes by adding these four chords to Figure I-16, attempting to choose closed form chords that minimize movements along the fret board.
The Coltrane changes for the ukulele in key of C major are presented in Figure I-17.  Each chord is a barre chord, meaning that this figure defines chord patterns that can be shifted to different fret board positions to generate the Coltrane changes in a different key.  For example, shifting each chord a fret downwards (towards the bottom of the page) produces this progression in the key of C# major.
Figure I-17.  Closed form ukulele chords for the Coltrane changes in the key of C major.
Creating the chord patterns in Figure I-17 serves the primary purpose of permitting me to play the Coltrane changes on the ukulele.  However, this set of chord diagrams suggests other uses.  One of the themes in Chapter 9 was exploring different encodings of jazz progressions for networks.  One could imagine adapting the encoding developed in the interlude that preceded Chapter 9 to present the jazz progressions to networks as sequences of ukulele chords.  What effect might this representation have on network complexity?

Similarly, Figure I-17 raises the possibility of generating alternative versions of the Coltrane changes for ukulele.  For instance, might easier chord fingerings emerge if one explores chord substitutions for the other dominant seventh chords in the figure?