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  • Peter Ivan Edwards

Uniform Waves: Part 2, in which random melodies are minted

In the previous post, the difference between random permutation and uniform distribution was explained. By the end of it, you might have asked yourself, "But what does this have to do with music?" A fitting question given the theme of this blog. Put simply, different selection scenarios can yield different musical experiences. The musical experience resulting from random permutation being applied to musical elements like pitch and note duration can be distinguished from that resulting from a uniform distribution applied to those same elements. Below is an example. Two pitch series, each including 48 notes, utilize the 12 semitones in the middle C register, but the first employs random permutations, while the second results from uniform distribution.

The differences aren't striking, but they are noticeable. Looking closely, for instance, one sees within the first 2 beats of the uniform distribution results, which start in bar 4, that the F is repeated twice and the G-sharp is repeated 3 times. This is impossible in a random permutation because all notes must be selected once before a note can be repeated. Adjacent note repetition only happens when the last note of one 12-note series is the same as the first note of the next series. Does the uniform distribution note series not sound random? No, it certainly does, but it behaves differently.

In part, the uniform distribution results sound random because there are 48 of them and only 12 values to choose from. In such a situation, there are plenty of chances for all notes to sound at least once. But looking at the number of appearances for each note within the uniform distribution results reveals another way in which they differ from those of random permutation. With the random permutation results, there are exactly 4 appearances of each note. But the uniform distribution results break down as follows:

The pitches Bb and D only occur twice, while C# and F# occur 7 times. When we apply uniform distribution to select only a handful of pitches, the sense of randomness can disappear. Below are 5 melodies generated using uniform randomness. They sound, well, really melodic.

The melodic potential of these random note series becomes even more noticeable when they are harmonized. Granted, the harmonizations are very chromatic, but they don't include anything that a European composer in the late 19th century wouldn't have used. Take a listen below.

Here are the uniform distribution results from the previous post [5, 3, 5, 6, 6, 2, 3, 4, 1, 1] played as a melody with chromatic mediant harmonization. With these examples we are a far cry from the 12-note Schoenberg-esque harmonic world that permeated Shuffled.

A final example using the series [8, 5, 4, 5, 1, 9].

The first 4 pitches suggest an f-minor chord; the last 2 suggest A-major. These chords have a chromatic mediant relationship: their roots are a third apart - that's the mediant part of the term chromatic mediant - but they are not diatonically related, that is, an f-minor chord isn't one of the chords in the key of A major and an A-major chord isn't one in the key of f minor. The chromatic mediant progression is part of Romantic musical language - it is all over Liszt's music as well as many other composers of his and later generations - and a particularly beloved device in Hollywood. Hence, we will fashion a Harry Potter-esque soundtrack from our randomly generated melody.

Via transposition and repetition, the chromatic mediant progression can be continued:

The harmonic progression here is f-A-c-E-g-B-d-F#. It ascends by thirds, alternating between minor and major triads, but these triads are always chromatically related. When you orchestrate the hell out of it, placing the melody in 5 octaves in the winds, accompanying it with middle register, tremoloing strings and swells in the trumpets and Wagner tubas - they're so neglected - you get a transitional passage fitting for the realm of Hogswarts School of Witchcraft and Wizardry.

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