What if we would divide the length of notes not in rational ratios, but in the golden ratio? If we, instead of halving and quartering notes, divided them in the ratio 1:φ where φ=1.6180339887...?
Using the basis φ one can build a place-value number system that also contains the whole numbers:
- 1 = φ0 = 1φ
- 2 = φ1 + φ-2 = 10.01φ
- 3 = φ2 + φ-2 = 100.01φ
- 4 = φ2 + φ0 + φ-2 = 101.01φ
Each number consists of tones for the ones in the phinary representation whose pitch is calculated, if I recall correctly, from the place-value. The length of the preceding pause is proportional to the place-value. The lengths are ordered decreasingly, so that the big powers of φ come first with great gaps between them, and then, towards the end, the negative powers of φ in rapid succession. That leads to regular accumulations of tones, one per natural number.
A little less systematically, the following little piece has come about:
Here I distributed a melody over two
measures of equal length, which are iteratedly subdivided in the ratio 1:φ or φ:1.
The notes all have lengths proportional to powers of φ.
A steadily accelerating (or decelerating) sequence of notes, where each note is shorter (or longer) than the preceding one by a factor of φ,
also fits in this length scheme, as φ0 + φ-1 + φ-2 + φ-3 + φ-4 + ...= φ2.
Hence the following melody, consisting of three
measures, the first and last of which have equal length and contain an accelerating and a decelerating
tone sequence respectively, whereas the middle measure is shorter by a factor of φ and contains an interlude: