Chance Music

When a composer or songwriter finishes a new piece of music, typically the result is a fixed and defined work that can be reproduced consistently. Classically, a set of score dictates what notes to play, when to play them, for how long, how loudly, and so on. Even in jazz, a quintessentially improvisational form of performance, a fixed melody known as the “head” is played (usually at the start and end of the piece), and a lead sheet specifies the chord changes through which the music cycles. Recordings capture performances and multitrack constructions with excellent fidelity. In all these cases, there is some degree of chance, relying on the skills and inclinations of the performers or the capabilities and settings of the equipment involved, as well as the acoustic properties of the listening environment; however, some composers incorporate chance into the very structure of the music itself.

At least as far back as the late 15th Century, games existed in which short fragments of music were selected and arranged according to the results of rolling dice. A manuscript written by Wolfgang Amadeus Mozart in 1787 consists of several two-bar segments of score with various marks indicating some system was used to piece them together, and one example is even included, though the precise rules were not recorded (or have not survived) and attempts to deduce them remain inconclusive. Several other varieties of "Musikalisches Würfelspiele" (musical dice games) were credited by their publishers to Mozart, though nearly all of these are almost certainly inauthentic; one such game was produced by Nikolaus Simrock in Berlin, who was Mozart's own publisher, though its attribution to Mozart has never been confirmed.

In the early 20th Century, Marcel Duchamp used chance in the composition of two musical pieces, while Charles Ives and Henry Cowell produced pieces that required performers to arrange fragmentary music. Soon, many other composers began experimenting with chance in various ways, including Karlheinz Stockhausen, Pierre Boulez, and John Cage. “Water Walk,” a piece Cage performed on the show I've Got A Secret in January of 1960 had to be modified for television, as the composition originally required five radios turned on and off at specific instances, incorporating by chance whatever might be on the air at the time of performance, but a dispute between two labor unions on the set of the show meant the radios could not be used; as a solution, Cage instead opted to smack the top of each radio at the designated moments to signify turning them on, and to signify turning them off, he knocked them to the floor.

Developments in computing technology allow for the production of highly variable music. For example, Brian Eno's 1996 CD-ROM album Generative Music 1 is a collection of twelve compositions where each conforms to its own complex parameters but never plays out quite the same way twice. More recently, Eno also composed the elements used in the procedural music system for the videogame Spore, yielding a soundtrack that never repeats. Many programs for creating different varieties of chance music are available, both commercially and as free and open source software, with some offering the user little control while others provide extensive options.

Though it may seem easier for a composer to relinquish decision-making to chance, this is not usually the case; in fact, it's often far simpler to write a traditional fixed piece than to devise a system of parameters within which the results are bound to produce the sort of effect a composer wishes to convey. Not only does the incorporation of chance allow a musician to emphasize the broader shape of a piece rather than place too much significance in the specifics of one possible implementation of an idea, it can also offer the listener a fresher experience each time through.

I've conducted several experiments with different types of chance music, and I'd like to offer a brief example. “Inganok” is one selected result of a C++ program I wrote in 2008 to produce highly variable Csound scores, a technique I've since been revisiting and revising with Python. Of course, this simplistic mood piece reflects more on my own tastes and abilities than on the usefulness of chance music in the hands of more skilled musicians!

Just Intonation

Often the first step a musician might take away from the familiar environs of the standard Western twelve-tone equal temperament is to explore a method of tuning known as just intonation.

In an equal temperament, the distance between every pair of neighboring pitches is exactly the same. This is very useful for practices such as transposition and modulation, since the relationships between notes don't change as they are moved up or down as a group.

However, today's standard is the result of numerous compromises made throughout history. For thousands of years, until only about three centuries ago in Western culture, tuning was done with pitches being related to one another by specific ratios. The smaller the numbers in the ratio, the more consonant (smooth, pleasant, at rest) the interval sounds, as it is physically easier for the human ear to identify its harmonic properties.

The mathematical procedure to apply these ratios is relatively straightforward: a basic pitch is chosen, and the frequency of that pitch is multiplied by the desired ratio to arrive at the second pitch. For example, standard concert tuning is known as A440, because the A above Middle C is set at 440 Hz (Hertz, a unit that measures vibrations per second). If we take this as our basic pitch and want to find the pitch at a ratio of 3:2 above it, we'd have 440*(3/2) = 660 Hz.

RATIO	INTERVAL
1:1	Unison
9:8	Tonus (Whole Tone)
4:3	Diatessaron (Perfect Fourth)
3:2	Diapente (Perfect Fifth)
2:1	Octave

The simplest ratio, 1:1, is two instances of the same pitch, and is therefore appropriately termed a unison. The next-simplest ratio, 2:1, is one pitch exactly double the frequency of the other, which for historical reasons is called an octave ("oct-" being a prefix meaning "eight", the octave is the eighth note at the end of a heptatonic scale, a return to the functional position of the tonic; in other words, the distance between the first and last "do" of "do re mi…").

The next-simplest ratio, 3:2, was named the diapente ("dia-" translated variously as "by/from/through/of", "pente" meaning "five") and is the fifth step of "do re mi…", namely "sol". Its inversion is the diatessaron ("by/from/through/of four"), ratio 4:3, the "fa" in "do re mi…". The distance between the diapente and the diatessaron was called the tonus, ratio 9:8, what is today called a whole tone; a whole tone up from "do" is "re".

These intervals all exist in what is known as Pythagorean tuning, an ancient Greek system in which the ratios involved exhibit no prime factors greater than 3; for example, 9:8 factors to (3*3):(2*2*2). This system can also be called 3-limit just intonation, and it's the most fundamental of all musical tunings beyond the basic unison and octave. It's easy to tune because the octave and pure perfect fifth are simple to hear, and a Pythagorean scale is achieved by stacking pure fifths and collapsing the results into the span of an octave.

However, limiting eligible ratios to the prime factor of 3 can yield some relatively complicated intervals: for example, a Pythagorean major third (more properly called a ditone, a stack of two whole tones) is the ratio 81:64, and the minor second (or semitone) could be either 256:243 (called the Pythagorean limma, or diatonic semitone) or 2187:2048 (the Pythagorean apotome, or chromatic semitone). Such complicated intervals tend to sound dissonant (harsh), as they are more difficult for the human ear to process.

A far simpler system can be constructed by raising the prime limit to 5, thus allowing ratios like 5:4 to serve as the major third, 6:5 the minor third, and 16:15 the minor second.

RATIO	CENTS	INTERVAL
1:1	0	Unison
16:15	112	Minor Second
9:8	204	Major Second
6:5	316	Minor Third
5:4	386	Major Third
4:3	498	Perfect Fourth
3:2	702	Perfect Fifth
8:5	814	Minor Sixth
5:3	884	Major Sixth
9:5	1018	Minor Seventh
15:8	1088	Major Seventh
2:1	1200	Octave

Notice that the most basic Pythagorean intervals are retained in this tuning, as they are simpler than ratios involving the prime factor 5; for example, 9:8 is simpler than 10:9, which would otherwise be the simplest ratio for a major second.

Notice also the measurements given in the unit of cents. This common measurement scheme is an extremely convenient means to compare the sizes of any intervals, and is based on 1200 cents per octave. Dividing the octave into twelve equal tones, as does the common Western standard tuning, yields one tone every 100 cents: 100 is the minor second, 200 is the major second, 300 is the minor third, and so on. Understanding this, and given the chart above, it becomes clear how impure the intervals of standard tuning are; for example, a minor third in twelve-tone equal temperament is 16 cents (about a sixth of a semitone) flat from the pure version of the same interval, while a major third is 14 cents sharp, and the same is also true (but reversing the sharp and the flat) for their inversions, the sixths.

The pure intervals have been tempered to equal widths for the sake of conveniences like unhindered transposition and modulation. However, the cost of this compromise is the loss of the distinctive characters of each key and their modes, and the harmonic purity of chords.

5-limit just intonation is perhaps the most welcoming first step out of equal temperament, as a great deal of existing traditional and contemporary popular music can be performed in this tuning and (due to the purity of the intervals) end up sounding sweeter, literally more harmonic, than it would in standard twelve-tone tuning.

On the other hand, there are serious dangers inherent in just intonation: beware, for outside the comfortable conveniences of equal temperament, there be wolves!

A wolf interval is so named because it is so dissonant (far from pure) that it "howls". The size of a fifth built on most degrees of a just-intoned scale will be pure (there are 702 cents between the just minor second and just minor sixth, for example), but certain fifths are monstrously wrong: the distance between a just major second and just major sixth is only about 680 cents, almost a quarter of a semitone flat from pure!

Taming these wolves was another cause for the compromise of equal temperament. Without tempering, the second chord of both the major and minor scales would be practically unusable for the vast majority of purposes, and having access to this chord is certainly very useful. This is one excellent reason that equal temperament started to find favor about three centuries ago, and has since become the de facto standard tuning of practically all Western music.

However, for music which cleverly evades those rare wolf intervals, and which does not have a great need for complete freedom of transposition or modulation, just intonation offers a sweetness of sound unparalleled by standard tuning. In fact, transposition and modulation are still perfectly useful techniques within just intonation, but certain modes work better together than others, as they each have a unique arrangement of intervals rather than the uniform smear of equal temperament.

For the exceptionally adventurous musician, just intonation can even be extended beyond the 5-limit into higher primes. Septimal just intonation incorporates ratios involving the prime factor of 7, most readily recognized in the "barbershop seventh" of 7:4 (so called due to its ubiquity in the purely harmonized performances of barbershop quartets, more formally known as the "harmonic seventh", "subminor seventh", or "septimal minor seventh", and sometimes as the "perfect seventh" or "pure seventh"). Even higher primes have seen occasional use by the arguably insane (such as the good folks of the Alternate Tunings group on Yahoo! and the Xenharmonic Alliance on Facebook), though a highly trained ear is required to discern such intervals. Sometimes, to achieve a particularly unusual tuning, certain low primes (even 3 and 5) are actually excluded!

With modern technology, just intonation has become widely accessible. Even an inexpensive synthesizer may have a retuning capability, often with a list of presets including basic just intonation. Many acoustic instruments support just intonation natively, though fretted instruments like guitars may require modification (the frets must be bent into carefully calculated squiggles). In most professional multitracking software, MIDI tracks can be retuned instantly, lending the purity of just intonation to a song that was originally composed in equal temperament and giving it a little extra shine. Give it a try!

Heptatonia

The foundation of the Western musical tradition is the diatonic scale, known best by its "major" and "minor" modes, though there are five other modes of this scale used somewhat less often. The diatonic scale is a specific pattern of intervals, called whole- and half-steps (or more formally, tones and semitones), arranged in a circle. Any of the seven degrees of this circle can be used as a starting position, becoming the tonic of a particular mode, the pattern of which is formed by going around the circle until returning to the tonic.

On a piano, starting on a particular white key and playing up along the next seven white keys yields one of the modes, with a "whole-step" being whenever a black key is skipped and a "half-step" being wherever there is no black key. However, the pattern itself is what defines a mode, and the pattern could be started from any key, skipping or landing on both white and black keys as the pattern dictates; the "white keys only" method simply provides one convenient example of each mode to understand and remember.

PATTERN	2367	MODE	WHITE KEY
WWHWWWH	MMMM	Ionian (major scale)	C
WHWWWHW	MmMm	Dorian	D
HWWWHWW	mmmm	Phrygian	E
WWWHWWH	MMMM+#4	Lydian	F
WWHWWHW	MMMm	Mixolydian	G
WHWWHWW	Mmmm	Aeolian (natural minor scale)	A
HWWHWWW	mmmm+b5	Locrian	B

In this table, the "2367" column gives the patterns of major (M) and minor (m) seconds, thirds, sixths, and sevenths for each mode; the fourth and fifth degrees are assumed perfect, though the Lydian mode has an augmented fourth and the Locrian mode has a diminished fifth, as noted. This view provides an easy way to construct the modes from any starting key, and to directly compare their properties.

Notice that each sequence in the "Pattern" column features the same sequence as its neighbors above and below, but with the interval at one end (either a Whole- or Half-step) moved to the other end; the sequences at the top and bottom of the list are likewise related. This reflects the circulating nature of the diatonic scale.

While the "major" and "natural minor" modes have been most commonly used, all seven modes of the diatonic scale have been employed throughout history and are still in use today. There are, however, two other circulating seven-tone scales that can be built out of the standard twelve-tone system using only whole- and half-steps, and these scales remain considerably more obscure. In this context, the diatonic scale is known as Heptatonia Prima, in contrast to Heptatonia Secunda and Heptatonia Tertia.

HEPTATONIA SECUNDA

PATTERN	2367
WWHWHWW	MMmm
WHWHWWW	Mmmm+b5
HWHWWWW	mmmm+b4+b5
WHWWWWH	MmMM
HWWWWHW	mmMm
WWWWHWH	MMMM+#4+#5
WWWHWHW	MMMm+#4

HEPTATONIA TERTIA

PATTERN	2367
HWWWWWH	mmMM
WWWWWHH	MM#M+#4+#5
WWWWHHW	MMMm+#4+#5
WWWHHWW	MMmm+#4
WWHHWWW	MMmm+b5
WHHWWWW	Mmmm+b4+b5
HHWWWWW	mbmm+b4+b5

A quick glance at these tables can give a good idea why these patterns haven't often been explored: they exhibit many more augmented and diminished intervals (and resulting chords) than Heptatonia Prima, which makes working with them harmonically considerably more difficult. Tertia even has an augmented sixth in one mode, and (even worse for the tonic chord) a diminished third in its inversion! These are certainly awkward in comparison to Prima.

However, there are clearly a few modes that are relatively straightforward. For example, the MmMM mode of Secunda is also known as the "melodic minor" scale, and has been used regularly. Its inversion, the mmMm mode, is equally stable, though both these modes do feature some augmented and diminished chords, and therefore some skill is required to handle them well. Similarly, the mmMM mode of Tertia features both augmented and diminished chords, but has seen use as the "Neapolitan major" scale.

An exploration of even the most complicated modes could yield interesting results in the hands of an adventurous composer. It is a challenge not often undertaken, and many distinctive works may yet be produced!

Writing in Locrian

In modern Western music, the Locrian mode is built from the seventh degree of the major scale (at a piano, you could start on B and play only the white keys). In this mode, the tonic triad is diminished. This makes it very difficult (or even, according to traditional phrasing practice, impossible) to write in this mode without the result sounding like it's actually in a different key.

The tonic must be continually reasserted as the center, which is challenging because a diminished chord is quite dissonant and does not sound like a place of rest. This chord can be made more manageable, though less distinctly Locrian, by omitting its fifth degree and adding a minor seventh; another option is to "cheat" by raising the fifth degree chromatically only for the tonic chord. In either case, to define a Locrian identity, the modified tonic chord would require balance from the iii and V chords in proper diatonic form, where the true fifth degree of the mode can be voiced more comfortably.

If the fifth degree is raised permanently, the mode becomes Phrygian. This change makes the iii chord major, but more importantly, the V chord becomes diminished, which can be nearly as troublesome as a diminished tonic chord. In both Locrian and Phrygian, the second, third, sixth, and seventh degrees are all minor; they can sound even darker than the Aeolian mode, otherwise known as the natural minor scale, which has a major second. If the pattern of whole- and half-steps in the Locrian mode (HWWHWWW) is reversed, the result is the Lydian mode, which is completely major but features an augmented fourth.

Due to the difficulties inherent in working with the Locrian mode, it has always been extremely uncommon, though a few composers have made use of it. Claude Debussy exhibits the mode in three passages of his last orchestral work, Jeux. It can be done!