Syntonic comma

A small musical interval in just intonation

Syntonic comma

The syntonic comma is a small musical interval that is significant in the study of just intonation and musical temperament. It is the difference between a justly intonated major third and a Pythagorean third. This interval is crucial in understanding the tuning systems used in Western music.

Definition

In terms of frequency ratios, the syntonic comma is the difference between the ratio of a justly tuned major third (5:4) and a Pythagorean major third (81:64). Mathematically, it can be expressed as:

\[ \text{Syntonic comma} = \frac{81}{80} \]

This ratio, 81:80, represents a very small interval, approximately 21.51 cents, which is slightly larger than a Pythagorean comma.

Historical Context

The concept of the syntonic comma has been known since ancient times and was extensively studied by Pythagoras and later by Ptolemy. It became particularly important during the Renaissance when musicians and theorists sought to reconcile the mathematical purity of just intonation with the practical needs of musical performance.

Musical Implications

The presence of the syntonic comma in tuning systems means that it is impossible to perfectly tune all intervals in a scale using just intonation. This led to the development of various temperament systems, such as meantone temperament, which temper the syntonic comma to achieve a more usable tuning across all keys.

Meantone Temperament

In meantone temperament, the syntonic comma is distributed across several intervals, allowing for more consonant thirds at the expense of slightly less pure fifths. This system was widely used during the Baroque period and is still of interest to musicians and musicologists today.

Mathematical Properties

The syntonic comma is a prime example of how small discrepancies in tuning can accumulate to create noticeable differences in musical performance. It is one of several commas that arise in the study of musical acoustics and tuning theory.

Visual Representation

File:Just vs Pythagorean.svg

Comparison of just and Pythagorean tuning. The syntonic comma is the small difference between the two tuning systems.

The diagram above illustrates the difference between just and Pythagorean tuning, highlighting the syntonic comma as the small interval that separates them.

Related Pages

• Just intonation
• Pythagorean tuning
• Musical temperament
• Meantone temperament
• Comma (music)