Syntonic comma

Syntonic Comma (also known as the Didymus comma or meantone comma) is a small musical interval, specifically a type of comma, that is mathematically represented as the frequency ratio 81:80. This interval, approximately equal to 21.51 cents, plays a significant role in the theory and practice of tuning systems, particularly in the context of Just Intonation and Meantone temperament.

Definition

The syntonic comma arises in musical tuning systems when comparing the sizes of major thirds. In Pythagorean tuning, which is based on a stack of perfect fifths, the major third (from C to E, for example) is slightly larger than the sum of two whole tones (from C to D and D to E). This discrepancy is due to the fact that the Pythagorean major third is derived from four perfect fifths (C-G-D-A-E), resulting in a ratio of 81:64, whereas the just major third, more consonant to the human ear, has a simpler ratio of 5:4 (80:64). The difference between these two intervals is the syntonic comma.

Historical Context

The syntonic comma has been a subject of interest since the ancient Greek theorists, but it was particularly noted by Didymus and Ptolemy in their explorations of musical tuning. The awareness of this interval influenced the development of various tuning systems throughout history, including meantone temperament, which attempts to temper the syntonic comma in order to achieve more consonant thirds.

Impact on Tuning Systems

In the context of Meantone temperament, the syntonic comma is tempered, meaning it is distributed among several intervals to reduce its dissonance. This is achieved by narrowing the fifths slightly, which in turn makes the major thirds closer to the pure ratio of 5:4. This compromise allows for more consonant harmonies in a wider range of keys than Pythagorean tuning, though it introduces slight impurities in other intervals.

Just Intonation also deals with the syntonic comma, but rather than tempering it, this system embraces the pure intervals, including the pure major third. This results in a tuning system that can be highly consonant in certain keys but may lead to significant dissonances in others, due to the accumulation of syntonic commas.

Mathematical Description

Mathematically, the syntonic comma can be described as the difference between two kinds of major thirds: the Pythagorean major third (81:64) and the just major third (5:4). This difference is calculated as the ratio 81:80. When expressed in cents, a logarithmic measure of intervals, the syntonic comma is approximately 21.51 cents.

Conclusion

The syntonic comma is a fundamental concept in the study of musical tuning, illustrating the compromises and choices made in the development of musical scales and temperaments. Its existence highlights the complexity of achieving consonance within a musical system and the intricate balance between mathematical ratios and perceptual qualities of sound.