Enharmonic comma
Overview
The enharmonic comma, also known as the ditonic comma, is a small interval in music theory, specifically in the field of tuning and temperament. It is defined as the difference between twelve justly tuned perfect fifths and seven octaves Octave. In terms of frequency ratio, it is approximately 1.01364:1, or about 21.51 cents Cent.
Mathematical Explanation
The enharmonic comma can be calculated using the formula for the frequency ratio of a perfect fifth, which is 3:2, and the formula for the frequency ratio of an octave, which is 2:1. If we multiply the frequency ratio of a perfect fifth by itself twelve times, and then divide by the frequency ratio of an octave raised to the seventh power, we get the frequency ratio of the enharmonic comma.
To put it in mathematical terms: (3/2)^12 / (2)^7 ≈ 1.01364
This calculation results in a number slightly greater than 1, which represents the enharmonic comma. This number is the factor by which a note will be sharp or flat if it is tuned using twelve perfect fifths instead of seven octaves.
Historical Context
The concept of the enharmonic comma has been known since the time of ancient Greek music theorists, who recognized that a series of perfect fifths will not align perfectly with a series of octaves. This discrepancy was a source of much debate and controversy among ancient theorists, and it continues to be a topic of discussion among modern music theorists and musicians.
Significance in Tuning Systems
The enharmonic comma plays a significant role in the design of musical tuning systems. In systems that use equal temperament, such as the standard tuning system for modern Western music, the enharmonic comma is distributed evenly among all twelve notes in the octave, making each note slightly out of tune with the justly tuned version of that note. This is done in order to make the tuning system usable in all keys.
In other tuning systems, such as just intonation or various types of meantone temperament, the enharmonic comma is dealt with in different ways. In just intonation, for example, the comma is avoided by only using a limited number of keys. In meantone temperaments, the comma is distributed unevenly among the notes, resulting in some keys sounding more in tune than others.