ATON is a hopefully evolving classification theory. It aspires to unify knowledge around numbers and prefers naive methods. Some of the older posts are wrong but I'll keep them for the sake of continuity.

Wednesday, February 14, 2007

An analysis of symmetry in BWV-999

This is my second and more detailed look at BWV-999, which is a lone prelude amongst the lute works by Bach. It seems to be the opening piece of all the lute works, which are 4 suites, prelude-fuge-allegro-998 which is a toughie and also very lovely, and a lone fugue which is also from the violine works. Apparently 999 is specifically marked to be for lute, and it also suits the 6 stringed modern guitar perfectly. It is also regarded to be one of the "kleine" keyboard preludes. It would be easy for the keyboard but not so for the guitar, with it's difficult chord changes and fast arpeggio. I will analyse all of it's 83 bars one by one, by looking at it from harmonic, melodic and fractal aspects, as much as I can with my limited musical knowledge. I hope to learn some more music through this analysis, as the piece is a relatively simple one. The main emphasis will be it's fractal structure. I am currently practising 999, 998 and the first suite.

The most outstanding aspect of a musical piece is it's measure, which is 3/4 here. 3/4 is a very special number with mythical, religious and mathematical significance. 999 is a uniform arpeggio piece with 12 sixteenths for each measure. 12 is also a magical number. Bach opens the keyboard prelude and fuges with an arpeggio prelude as well. A bit like a historical reference to the ancient harp. 999 opens with D-minor and closes with A-major.

Like most good musical pieces, it has {bass,mid-range,treble} symmetry. The mid-range is the spine of the piece, and bass and treble reflect each other and dance around it. I will use letters for notes and numbers for octaves, like D and 2D. Here is the first measure coded;
1. [(D,2D,2F,2A),(2F,2D,2F,2D),(A,2D,F,2D)]
The second measure is the same;
2. [(D,2D,2F,2A),(2F,2D,2F,2D),(A,2D,F,2D)]
The spine of the piece falls in the even numbers, or the alternate beats, if a measure is numbered from 1-12. All alternate beats are the same except the 4th sixteenth. The odd numbered notes make either side of the spine.

The first beat is the bass, and sets the dominant tone of the measure. It also completes the harmony started in the end of the previous measure. This bass linking is typical of most, if not all, musical pieces, and it comes from the linking of the scales. To see this, consider a 2 octave C-major scale;
When a 1 octave scale is played, it sounds complete only after adding the octave higher root (2C here). Thus we get 8 notes in a complete sounding octave, although there are only 7 different notes. When 2 octaves are played, it sounds complete with 15 notes. The middle note 2C both ends the previous octave and also starts the next. Therefore 2C has a dual, or linking, role.

The measure is divided into 3 quarters, each with 4 sixteenths. The 1st and 3rd quarters reflect each other around the mediator 2nd quarter. The 1st quarter(D,2D,2F,2A) is a typical 4 note chord with it's bass and 3 trebles (D minor here). The last 3 notes of the 1st q (2D,2F,2A) are reflected by the 1st and 3rd notes of the 3rd q ((A),2D,(F),2D), and the 1st note of the 1st q ((D),2D,2F,2A), in reverse order, half speed and in the previous octave (2D,2F,2A)-(A,X,F,X,D). The 2nd q acts as a mirror. But the symmetry is slightly distorted by the shifted position of the 1st chord. This creates tension and movement. Thus we get two chords reflecting each other and dancing around a spine, in a slightly imbalanced form. When you consider different tones for the chords in the rest of the piece, you get the fireworks.

The genius of Bach here shows itself in the complexity contained in the simplicity.

A measure is the smallest unit that contains all essential properties of a piece. In this sense, it corresponds to atoms of physics and cells of life. This is such a powerful metaphor. Universe and life are essentially musical forms.

To continue...

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