The Pythagorean comma

It is interesting to note that a sequence of ever higher 3:2 "fifths" never produces a frequency ratio that can be reduced to the ratio 2:1.{1} For example,

Fig. 1. Calculating the Pythagorean comma: 12 3:2 fifths vs. 7 2:1 octaves, 129.75:128

													2:1-related with F3?
Pitch	F3	C4	G4	D5	A4	E6	B7	F#7	C#8	G#8	D#9	A#9	(E#10)
No. of 3:2s		1	2	3	4	5	6	7	8	9	10	11	12

Calculating from F₃ up 12 fifths: (3/2)¹² = 531441/4096 and produces the pitch E#₁₀which is 129.75 times higher than the original starting point. By comparison, 7 octaves above F₃,(2/1)⁷ = 128/1 produces a tone 128 times higher. From a musical point of view, it would be nice if these two numbers were the same. Of course, they are not. This type of discrepancy is called a comma. A large number of such discrepancies arise in tuning theory. As theorists discovered various commas, they usually them assigned a unique name. For example, and the discrepancy discussed above which arose from the Pythagorean tuning process scale is called the Pythagorean comma. Its size may alternatevly expressed as the ratio 531441/8092, 129.75:128, 23 cents, or about one-fourth of a semitone.

Hear the Pythagorean Comma

Fig. 2. Twelve ascending 3:2 fifths above C₂ vs. seven 2:1 octaves above C₂ does not produce the same pitch.

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Updated: September 24, 2002

Reginald Bain | University of South Carolina | School of Music | Disclaimer
http://www.music.sc.edu/fs/bain/atmi02/
A Web-based Multimedia Approach to the Harmonic Series: The Pythagorean Comma

© 1997-2002 Reginald Bain. All rights reserved.

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