Syntonic comma

Syntonic Comma

The syntonic comma is a small musical interval that is the difference between two different tuning systems: just intonation and Pythagorean tuning. It is an important concept in the study of musical temperament and intonation.

Syntonic comma on C

Definition

The syntonic comma is defined as the difference between a justly intonated major third and a Pythagorean major third. In terms of frequency ratios, the just major third is 5:4, while the Pythagorean major third is 81:64. The syntonic comma is therefore the ratio 81:80.

Calculation

Mathematically, the syntonic comma can be calculated as follows:

\[ \text{Syntonic comma} = \frac{81}{80} = \frac{(3/2)^4}{(5/4)} \]

This ratio represents a very small interval, approximately 21.51 cents, which is slightly larger than a quarter tone.

Musical Implications

The presence of the syntonic comma has significant implications for musical tuning systems. In just intonation, intervals are tuned to simple whole number ratios, which results in pure-sounding harmonies. However, when using Pythagorean tuning, which is based on stacking perfect fifths, the syntonic comma arises as a discrepancy.

Just perfect fifth on D

In practice, this means that if one tunes a series of perfect fifths starting from a given note, the resulting major third will be slightly sharper than the just major third. This discrepancy must be addressed in temperament systems, such as equal temperament, which distribute the syntonic comma across multiple intervals to achieve a more uniform tuning.

Audio Examples

• File:Syntonic comma on C.mid
• File:Just perfect fifth on D.mid
• File:Major tone on C.mid

Visual Representations

Syntonic comma on C in Helmholtz-Ellis notation

The syntonic comma can be visually represented in various musical notations. For example, in Helmholtz-Ellis notation, specific symbols are used to indicate microtonal adjustments, such as the syntonic comma.

Related Concepts

• Pythagorean comma
• Just intonation
• Equal temperament
• Musical temperament

Related Pages

• Pythagorean tuning
• Just intonation
• Musical temperament
• Cents (music)