Collection V

Convergence to the Golden Ratio no. 7

Casey Mongoven

November 23, 2007

description of piece: Based on a fundamental frequency of 33 Hz, the speakers play triangle waves at pitches derived from the harmonic series - the Fibonacci-number partials are heard. The left channel plays pitches having a fundamental frequency of the partials 1,1,2,3,5,8,13,21,34,55,89,144,233; the right speaker plays pitches having a fundamental frequency of the partials 1,2,3,5,8,13,21,34,55,89,144,233,377 at the same time (see list of pitches below). Again, all these partials are based on a fundamental frequency of 33 Hz, which represents 1 in the Fibonacci sequence. The convergence of the Fibonacci-number partials in the harmonic series to the golden ratio is thus illustrated through the ratio of the frequencies of the left and right speakers. The duration of each interval is determined by the convergence to the golden ratio as well, with .82 seconds representing the larger golden ratio value 1.618033989... .
description of sequence: Fibonacci sequence: a(1) = 1, a(2) = 1, a(n) = a(n - 1) + a(n - 2) for n > 2
left channel offset: 1
right channel offset: 2
number of members used: 14 (13 per channel)
number of channels: 2
piece length: 10.4986 seconds
note values in seconds in order heard: .5068, 1.0136, .7602, .8446, .8109, .8235, .8187, .8205, .8198, .8201, .82, .82, .82
temperament: based on harmonic series
lowest frequency: 33 Hz highest frequency: 12441 Hz
number of unique frequencies: 13
synthesis technique: classic waves
wave: triangle
dynamic: mp
attack: .008
release: .008
synthesis language used: Csound

integer (partial)		frequency (of triangle wave)

1		33
2		66
3		99
5		165
8		264
13		429
21		693
34		1122
55		1815
89		2937
144		4752
233		7689
377		12441