In music theory and musical tuning the Holdrian comma, also called Holder's comma, and rarely the Arabian comma, is a small musical interval of approximately 22.6415 cents, equal to one step of 53 equal temperament, or . The name comma is misleading, since this interval is an irrational number and does not describe the compromise between intervals of any tuning system; it assumes this name because it is an approximation of the syntonic comma, which was widely used as a measurement of tuning in William Holder's time. The origin of Holder's comma resides in the fact that the Ancient Greeks believed that in the Pythagorean tuning the tone could be divided in nine commas, four of which forming the diatonic semitone and five the chromatic semitone. If all these commas are exactly of the same size, there results an octave of 5 tones + 2 diatonic semitones, 5 × 9 + 2 × 4 = 53 equal commas. Holder attributes the division of the octave in 53 equal parts to Nicholas Mercator, who would have named the 1/53 part of the octave the "artificial comma".
Mercator's comma and the Holdrian comma
Mercator's comma is a name often used for a closely related interval because of its association with Nicholas Mercator. One of these intervals was first described by Ching-Fang in 45 BCE. Mercator applied logarithms to determine that was nearly equivalent to a syntonic comma of ≈ 21.5063 cents. He also considered that an "artificial comma" of might be useful, because 31octaves could be practically approximated by a cycle of 53 just fifths. William Holder, for whom the Holdrian comma is named, favored this latter unit because the intervals of 53 equal temperament are closer to just intonation than that of 55. Thus Mercator's comma and the Holdrian comma are two distinct but related intervals.
The Holdrian comma has been employed mainly in Turkish music theory by Kemal Ilerici, and by the Turkish composer Erol Sayan. The name of this comma is Holder koması in Turkish. For instance, the Rast makam may be considered in terms of Holdrian commas: , where denotes a Holdrian comma flat, while in contrast, the Nihavend makam : , where denotes a five-comma flat, has medium seconds between d–e, e–f, g–a, a–b, and b–c', a medium second being somewhere in between 8 and 9 commas.
