This post is written in Binary Form.
When thinking about the relationship between numbers and music, often what first springs to mind are the proportional differences between the physical objects that generate musical intervals on which harmony and melody is so often based. The famous story of the blacksmith’s workshop and Pythagoras is sometimes used to illustrate this. Pythagoras was said to have been walking past a workshop where the sound of anvils of different weights were being struck, with each weight producing a different tone. Sounding together the result was consonant or stable. Upon investigation Pythagoras, it is said, realised that the pleasing aural result was accompanied by the pleasing integer proportion between the weights of the hammers, 6, 8, 9 and 12.
When one considers music in its own isolated discipline bubble, it might seem that it has a unique place beside mathematics (historically, music forms a pillar of the quadrivium alongside geometry, astronomy and athematic). However, perhaps the real power lies in the hands of numbers, and their ability to describe nature and her patterns.
“Numbers are a universal medium for the embedding of patterns of any sort, and that for that reason, statements seemingly about numbers alone can in fact encode statements about other universes of discourse”
Douglas Hofstadter, in his forward to Gödel’s Proof (Revised Edition)
Framed in this way, it is perhaps no surprise that so many composers and musicians are drawn to rigorously incorporate mathematical patterns into their art. The music of Iannis Xenakis is a great illustration of this mode of working and his book Formalized Music: Thought and Mathematics in Music, is essential reading. A slide of his work is almost an obligatory element of any presentation about these subjects.
Even when not being explicit in this intention many musicians and composers I have talked to feel that they are, at the very least, thinking in a ‘mathematical kinda way’.
The equation is balanced when it sounds good.
“Musical form is close to mathematics; not perhaps to mathematics itself but certainly to something like mathematical thinking and relationship.”
Above all music is inherently, deeply and strangely hierarchical.
Notated music of the western art tradition is basically a graph of frequency information quantised to the pitches of the chromatic scale over time. This is much more easily seen in piano roll notation, where the intervallic distances are not distorted by the staff or interrupted by pesky bar lines. From this perspective a C major scale, for example, is not a series of pitches, but rather a vector between to locations in ‘pitch space’.
I like Dmitri Tymoczko’s observation that in order to measure things we need a scale, and in the case of music the musical scale is a measurement of difference between pitches.
With musical notation reduced – or elevated – to lines on a graph, it begs the question what happens when the basic geometric transformations are applied.
By taking any melody and adding a constant to the Y value the musical material will start on a different pitch, but follow the exact contour as before. This transformation is a transposition and can be applied by experienced musicians in real-time. One could discuss here tuning systems at length, and how equal temperament ruined harmony: and why you should care, but maybe another day.
By reflecting about the Y Axis (in time), we retrograde the musical material. J.S. Bach used this to great effect in his musical offering, in which the reflected material is used to harmonise itself. If you were to write out this single melodic line on a piece of paper and half-twist it into a möbius loop, the same thing would result. There is a lovely video of this in action.
Haydn takes the retrograde ideas but establishes it in a more easily recognisable form in Symphony no. 47, the “Minuetto al Roverso”. This palindromic section is deliberately composed to accentuate the short motives (highlighted) asymmetrical structure. Haydn further embellishes the main motive with articulations and sharp dynamic contrast for the same reason. I believe that this section is successful in its transparency of process partly because this motive is about the same duration in time as our short term working memory (less than 6 seconds). We are able to imagine the retrograde at this level. The result is that the transformation being applied to the material is quite apparent to the audience without need of explanation.
Reflection on the X axis results in the inversion of pitches. This is a common procedure and produces the often-desirable effect of counterpoint between parts. The Christmas favorite Good King Wenceslas is a commonly sung in cannon or round. However, it is also possible to invert the melody to produce a different harmonic experience. Therefore the example below is the combination of two different geometric transformations; a reflection in the X axis and a translation along the X axis. (This example is courtesy of Scott Kim who describes it an accidental canon.)
Tafelmusik or Table Music is a short composition, in which the musical material has gone through a 180 Degree rotation. Now it is possible for two musicians to sit opposite each other and play from the other side.
Finally it is possible to scale a melody, either by intervallic augmentation, or by multiplying the rhythmic values by some value. For example, multiplying by 2 would half the perceived ‘speed’ of the music; a quaver would be come a crotchet etc. It is not necessary for the multiplication to be integers; a multiplier of 1.5 would create a dotted crotchet from a crotchet and so on. When performed against the untransformed sourse material the result is known as a prolation canon. A great example is Agnus Dei from Missa l’homme armé super voces musicales, by Josquin des Prez.
These examples demonstrate the fact that exact adherence to a mathematical process can be an effective approach to music composition. However in actual fact, these processes (although an integral component in the generation of musical material) is often much more fluid and subconscious in the mind of the composer or musician.
- Music and the Making of Modern Science by Peter Pesic
- Formalized Music by Iannis Xenakis
- A Geometry of music by Dmitri Tymockzo
- Gödel Escher Bach by Douglas Hofstadter
- How Equal Temperament Ruined Harmony by Ross Duffin