LIMITLESS COMBINATORICS

ERIC PETTINE postulates
'No Shortage of Melodies Anytime Soon'
and offers some real hope for cynical
musicians who think they've heard it all before

In his 1997 book, How The Mind Works, Steven Pinker describes combinatorics, which highlight the seemingly endless cache of a person's thoughts and subsequent actions. Pinker's posits that if you combine a few elements and a few rules, you can generate a virtually infinite number of ideas. For example, in manipulating sentence structures of twenty or less words with ten choices for each word of the sentence, the number of possible sentences extends exponentially into the realm of a hundred million trillion different sentences. Pinker then relates this concept to the musical sphere, claiming:

We are unlikely to have a melody shortage anytime soon because music is combinatorial: if each note of a melody can be selected from, say, eight notes on average, there are:

64 pairs of notes,

512 motifs of three notes,

4,096 phrases of four notes,

and so on, multiplying out to millions and trillions of musical pieces.

Appying similar calculations to harmony, it also appears that while perhaps not as infinite as melody, the harmonic structures which often define and shape melodies are also multitudinous. Author Duane Shinn, in his article, How Many Chords Are There Anyway? stated that there are a total of 48 possible triads involving major and minor thirds, the common building blocks of standard practice tonal harmony (e.g. twelve major, minor, diminished and augmented triads). Additionally, there are a total of 84 different standard practice chords comprising four notes (eg twelve each of diminished 7th chords, major 6th chords, minor 6th chords, dominant 7th chords, major 7th chords, minor 7th chords, and half-diminished chords). Then there are twelve each of larger collections and other alterations which frequently turn up in modern jazz harmony: 9th chords, flat 9th chords, 9th/major 7th chords, 9th/minor 7th chords, 11th chords, 13th chords, suspensions, flat 5th chords, flat 5th major 7th chords, and on and on. Plus, each of these chords can be inverted, combined into polychords and voiced in numerous ways.

While the data enumerated by Pinker and Shinn is very cogent, and Pinker's observations are revelatory about human mental and creative possibilities, hope for an infinite supply of tonal resources quickly turns to cynicism once you realize that none of this data mentions (nor could really mention) the quality or the kind of melodies and sentences that are possible. Nor can this data determine how many of these melodies/sentences would have real and personal meaning to someone, convey a universal emotion, or even make musical/literary sense -- all elements which are essential to both great melody and great sentence-making.

Marvin Minsky has pondered this same terrain in his article, Music, Mind and Meaning. In the section entitled 'The Syntactic Theories of Music' he states that:

We like certain tunes (melodies) because they have certain structural features and they resemble other tunes that we like (the songs, carols, rhymes, and hymns we liked in childhood.

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