Saturday, 17 May 2014

Systematic methods in music and their application to Visual Art

My intention here is to address a number of musical forms, including serialism, but as yet this post is incomplete. Also, the audio player is not performing/displaying correctly in some browsers. The large gap above the audio player is to allow it to render more effectively on my smart phone.

Music may be the most systematic of the art forms, although dance must come a close second.

I am, of course, mostly interested in music which has been composed, and for which a formal description (a score) has been written by a composer.

For example, here is a piece of Bach: the first two measures of the Prelude in C major.

Each bar follows a particular pattern, which I will explain below, but even if you are unfamiliar with reading music you will surely see the symmetry in the two halves of the above image. In the Prelude, Bach uses the same pattern in the first 33 of the 35 bars.

Here is a rendering of the Prelude as written by Bach [first 15 measures]. Perhaps you can hear the pattern

Bach's original - r r  a b c  a b c

The pattern in the top line of each measure is

r r  a b c  a b c    r r  a b c  a b c

That is, two rests followed by the phrase "a b c" twice, making the first half of each measure. The second half of each measure is identical to the first half.

Although the patterns of the two measures are identical, the actual notes chosen by Bach differ between the measures. In the first measure they are G C E and in the second they are A D F. All 33 pattern-identical measures have "a b c" substituted by such chords, and it is the choice of these chords which give the music it's brilliance. It's similar to choosing the colours to use in an abstract painting.

The bottom line (the left hand) also plays a pattern which is repeated in the same fashion over 33 bars. It's analysis is similar to the analysis I am making of the top line, so I won't go into details here.

I alluded above to a possible relationship between patterns in music and visual art, when I said that the composer's choice of chords could be likened to the artist's choice of colours. Klee imagined such a relationship and even made paintings where the composition of colours was made by overpainting tones rather like the effect we get on the piano by sounding notes simultaneously.

Here for example is the abcabc sequence, this time for the first eleven bars, as a colour chart.

The colours are dictated by the chords that Bach chose (to arpeggiate), coded according to the following scale, which shows the three octaves C3 to C6.

The actual hues chosen are dictated by the degree on the scale (C is 0, D is 1, ..., B is 6) and the brightness is chosen by the octave. So the first line of the pattern for Bach's composition is  G4 C5 E5 G4 C5 E5, which is blue, red, green, blue, red, green from the scale.

It's interesting that Bach's Prelude is so rich, given how simple the pattern is.

What would happen if we changed this pattern a little? Would it sound like Bach? First, let's simply reverse the order of the three note arpeggio "a b c" and replace it throughout with "c b a"

pattern is r r  c b a  c b a

That sound's pretty good to me. I've left the bottom line exactly as Bach had it and left all the chord choices the same. All the pianist is asked to do is to play that three note arpeggio 4-3-2 rather than 2-3-4. So we have a novel "Prelude".

With this simple kind of transformation, we can create countless "Preludes". Even if we restrict ourselves to just juggling those three notes, there are four more patterns to be used.

We can go further. The pattern "r r a b c a b c" is eight sixteenth-notes long. Suppose we replace this with the pattern "aa b c aa b c" where "aa" is an eighth-note played for " r a" then again we have a phrase that is eight sixteenth-notes long (2+1+1+2+1+1), but now we have a different Rhythm.

pattern is aa b c  aa b c

Not a Prelude perhaps?  But interesting, nevertheless.

What is the general lesson we learn here?

Simple patterns can generate complex constructions. In the case of music, they can often be very beautiful. But, of course, we could make patterns that were not so beautiful. So far, I have relied on Bach to choose the chords and so I have inherited his wisdom. If I had chosen the chords myself, I may not have been so effective.

Which brings me to drumming.

The basic rock beat for a drummer is something like


where K is the bass (kick) drum operated by the foot, H is the cymbal (hat) and S is the snare drum. See the pattern? Here it is at 120 beats.

pattern is KHSH

It's simple enough to create a variation on that.

pattern is KHHS
And indeed to combine them.

pattern is KHSH KHHS

In practice a drummer will have many variations of this sort and will choose to play them according to the requirements of the performance.

Whether the patterns are written down this explicitly or not, the drummer will be performing according to patterns of this sort.

For example, suppose we identify the following six patterns


then we can play the following changes

1 2 3 4 5 6
2 6 4 3 1 5
3 5 2 1 6 4
4 6 1 2 5 3
5 1 3 4 6 2
6 5 4 3 2 1

which sounds like this

pattern is 123456 etc.

This last drumming pattern, as a visual looks like this.

The cymbal (H) is red. The top line begins KHSH KHHS KHKH ... What is interesting here is the visual effect. You can "see" the pattern that you can hear in the audio. Of course by drumming standards, this is a trivial example. Think, for example, of some of Steve Reich's compositions where one set of drums is phased against another. Nevertheless, this method of systematic composition has some merits and looking at the visual has enabled me to revise the sequence.

A drumming sequence like this was added to the bachPrelude revised to "cbacba" as described above and used as the backing track for the video "Something Happened" (see

Which, ultimately, brings me to serialism. Serialism is a style of musical composition which is based on patterns. It began (I believe) with the adoption of the 12-tone scale by some composers (Schoenberg, for example) early in the 20th century. The classical methods of composition needed something additional in order to organise the sequencing and concurrency of sounds; something to replace chords and intervals and chord sequences.

Simple serial composition comprises selecting a tone-row, placing all the 12 tones of an octave in some order. From this various rearrangements are constructed, in particular reversals and inversions of the sequence. In fact, if my memory serves me right, there was a scheme for constructing 48 rows in this way. Composition then consisted of ordering these, to be played sequentially or concurrently and assigning durations to each note. This was not a trivial task. The successful composers (not least Schoenberg) were accomplished classical composers who branched out into this new form.

I'm just going to take the basic idea of pattern-driven composition to describe how I constructed a work that became the soundtrack of the video "1000 Works of Systems Art" (see

The basic design was simple, and is shown in the following two extracts from my notebook.

There are four blocks labelled rr, ff, rf and fr, for rising-rise, falling-fall, falling-rise and rising-fall respectively. The pianist is expected to start on any chord at 0 and then work along the rows of each block going up a degree if rising, or down a degree if falling. So for the first block, the pianist plays the chords 0,1,2,3,4,1,2,3,4,5,2,3,4,5,6, ... and so on. In traditional notation, this looks like this (top line)

The names rising-rise etc. helped me construct the work. If you look at the first block of numbers (rising -rise) you can see that the chords are rising in each line and that each line rises above its predecessor. Rising-fall on the other hand has falling chords on each line and rises from line to line. These constructions give me four blocks to play with.

Here is the pattern as a visual, using the same colour coding as above.

The traditional notation shows a choice of rhythm for the work, chosen to synchronise with the video. This gives me 8 (eighth) notes to the bar, whereas I have only 5 in the sequence. An alternative choice of rhythm might be quarter, quarter, eighth, eighth, quarter which would map the 5 "syllables" onto the measure exactly. Shuffling the eighths and quarters so they add up to one gives a number of variations. This actually sounds quite good, or no worse than the version actually used in the video. Here are three of them

Where next? The Visual Art illustrated here is trivial, although I have made more complex images by systematic means, as earlier posts on this blog will attest. Following the example of being able to map the same sequences systematically onto visuals and onto music is a topic worthy of further investigation. I am intrigued by the super-synchronisation achieved in "1000 Works" and with the lyrical effect of combining the third derived Prelude with moving images in "Water, Bamboo, Water" (see The next step for me is combining these two ideas.