Hindemith chose Bach’s F minor 3 part invention for analysis to demonstrate his theory of composition in “The craft of musical composition”. It is a fascinating piece – one of Bach’s most chromatic and expressive pieces of keyboard writing, and rather like other extraordinary musical moments (like Wagner’s musical orgasm in Tristan), it raises the question “What is going on?”. I like Hindemith’s theory very much (although not as much as I like his music!), but his analysis sent me on my own analytical journey through the lens of information theory.
What happens in music, I believe, is the unfolding of a structure where multiple constraints are interwoven and overlaid. Information theory can provide some insight into this (as is discussed in a very recent paper from Loet Leydesdorff, myself and Inga Ivanova in the Journal of the Association for Information Science and Technology: https://onlinelibrary.wiley.com/doi/full/10.1002/asi.24052), and particularly the meaningfulness of the communication.
When considering music from the perspective of information theory, there are three fundamental problems to be overcome:
- Music has no object of reference. So how is meaning communicated without reference?
- Music emerges over time, producing novelty and unfolding a diachronic structure which appears to be linked to its synchronic structure. For this reason, music is not ergodic, unlike the use of letters in a language: its entropy over one period of time is not the same as its entropy over a different period of time.
- Music’s unfolding novelty is not arbitrary: novelty in music appears to be a symmetry-breaking process similar to that found in epigenesis where both synchronic and diachronic symmetries gradually define structure
The first page of Bach’s music looks like this:
Here’s a performance:
The piece is fugal, and obviously in three parts, there is a very bare texture, and this bareness seems to contribute to the expressiveness of the music. However, there is a harmonic structure which is articulated throughout the piece, and a reduction of the harmonic written as chords per beat, looks something like this:
This kind of harmonic reduction is very common in music analysis as a method for getting at the “deep structure” of music (particularly in Schenker). It is typical of Bach’s music that the harmonic reduction is very much like a chorale (hymn). In trying to understand how Bach’s music works, we can start by asking about the relation between the harmonic reduction and the finished piece.
At first glance, from an information theory perspective, the block chords of the reduction seem to remove a considerable amount of entropy which exists in the movement of parts in the original. It does this by compressing the variety into single “beats”, which taken as an entirety have an entropy of 1. However, the variety compression makes more apparent the shifting harmonies. Written in chord symbols, this is an extended I (tonic) – V (dominant) – I (tonic) movement, interspersed with diminished chords (which are harmonically ambiguous) and a oscillation between major and minor chords. But if one was to calculate the entropy of the harmony, it wouldn’t be that great.