## Monday, October 29, 2007

### Pythagoras Revisited

I wrote a brief post on a study of Pythagorean ratio rules and brain function, based solely upon the abstract of the article. Now I've read the article, and can answer some of the questions posed by me and others. Most importantly, the authors did not really test for Pythagorean ratio rules. Here is their justification:

For centuries after Pythagoras, his tuning system based on exact perfect consonances predominated. As Western music increased in complexity and range, however, slight modifications to the Pythagorean scale became necessary to preserve consistently tuned intervals across extremely large intervals (greater than one or two octaves) and small ones (half steps and intervals that are difficult to standardize using Pythagorean tuning). The difficulty arising from the increased range is apparent when one goes through 12 perfect fifths, for example, from the note C to a C seven octaves higher: the ratio of the harmonic to the fundamental starting tone is (3/2)^12=129.746. Going from a C to one seven octaves higher via the octave route, however, produces a tone with a frequency that has a ratio (2/1)^7 = 128 times higher than the starting tone. This small difference ultimately requires some temperament or modification of pure harmonic intervals to construct and tune instruments that can play pieces written with tones that span multiple octaves. Numerous fixes or temperaments for this problem have been devised over the centuries[1]. The one used almost universally today is known as equal temperament, inwhich the discrepancy of 1.746 is divided by narrowing each of the 12 previously perfect fifths in the seven-octave span, resulting in the 12 notes of the chromatic scale. Thus in equal temperament the fifths are no longer perfect, only close.

With this caveat of equal tempering – the temperament in which Western listeners are accustomed to hearing music – informing our search for neural correlates to the Pythagorean rules, we chose to study the neural activation pattern associated with hearing the perfect [sic] fifth (1.498:1), major sixth (1.682:1) and major seventh (1.888:1).

(The footnote cites Helmholtz. They couldn't find something a little more contemporary?) In addition, all of the "musician" participants were piano performance majors, which could bias certain brain responses to muscle memory activity. The purpose of this study was hidden from the participants, by throwing the intervals in after another listening test on sentences and progressions (probably like Steinbeis and Koelsch's experiment.) Overall, I believe the authors over-reached by claiming to test Pythagorean ratio rules. I believe they did find something about consonance and dissonance, but not specific to frequency ratios. Yet again, scientists really need to consult with theorists, so they don't make this kind of mistake. I think Tenney's A History of Consonance and Dissonance would be a great source of hypotheses to test.

Foss, AL. "Neural correlates of the Pythagorean ratio rules." Neuroreport 18/15 (October 2007) 1521-1525.