## Tuesday, June 29, 2004

### Sometimes 8/13 is just 8/13

Number freaks love to find special proportions in nature and art, suggesting that these ratios have extra aesthetic value. One of the most popular is the Golden Section, especially in discussions of temporal ratios in music. The Golden Section started out as a geometric division of a line segment, such that the ratio of the smaller line segment to the larger line segment is equal to the ratio of the larger line segment to the whole line segment. In fancy algebraic terms, a/b = b/(a+b). This is related to the Fibonacci Series 1, 1, 2, 3, 5, 8, 13, 21, 34, ... in that any three sequential numbers from this series obey the Golden Section, approximately. So 5/8 is about equal to 8/13.

These ratios are not expressed in pitches well, but theorists find the Golden Section as a rhythmic/metric phenomenon all over the place in the music of Bach, Bartók, Schoenberg, Crumb, and others. Bartók did write about his admiration for the Golden Section, and Bach did enjoy inserting number games in his music. So finding and writing about the existence of the Golden ratio in works by these composers has some relevance. But in other works where there is no evidence of intentionality by the composer, the theorist needs to rely upon the perceptual salience of these durational ratios, usually in works that last 5-15 minutes. Can you accurately judge and compare three minutes versus five minutes, without using a clock? If not, there is no validity to the claim that the ratio is aesthetically significant. These findings become even more dicey when theorists add a safety margin, allowing ratios that are "close" to the golden mean. If a significant chord occurs on measure 28, but the Golden Section is really at measure 27, well, that is worth writing about! And theorists are willing to shift between units of time measured in measures, beats, or seconds, often without specifying why a particular unit is appropriate for the given piece.

Here are two things Schoenberg had to say about the Golden Section:
I don’t believe in the golden section. At least, I don’t believe that it is the single formal principle for our sense of beauty; rather at most one among many, among countless many.
Harmonielehre
All form-making, all conscious formmaking, is connected with some kind of mathematics, or geometry, or with the golden section or suchlike. But only unconscious form-making, which sets up the equation “form – outward shape,” really creates forms; that alone brings forth prototypes which are imitated by unoriginal people and become formulas.
Letter to Kandinsky