Holdrian comma

In music theory and musical tuning the Holdrian comma, also called Holder's comma, and sometimes the Arabian comma, is a small musical interval of approximately 22.6415 cents, equal to one step of 53 equal temperament, or $$\sqrt[53]{2}$$. 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 (21.51 cents), 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 (or at least Boethius ) 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 Nicolas Mercator, who would have named the 1/53 part of the octave the "Artificial 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's comma and the Holdrian comma
Mercator applied logarithms to determine that $$\sqrt[55]{2}$$ (≈ 21.8182 cents) was nearly equivalent to a syntonic comma of ≈ 21.5063 cents (a feature of the prevalent meantone temperament of the time). He also considered that an "artificial comma" of $$\sqrt[53]{2}$$ might be useful, because 31 octaves 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 Pythagorean tuning than that of 55. Thus Mercator's comma and the Holdrian comma are two distinct but related intervals.

Arabian comma
The name "Arabian comma" may be inaccurate; the 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 makam rast (similar to the Western major scale) may be considered in terms of Holdrian commas: (In common Arabic and Turkish practice, the third note e in rast is lower than in this theory, almost exactly halfway between western major and minor thirds, i.e. closer to 6,5 commas above d and 6,5 below f, the third c-e often referred to as a "neutral third" by musicologists.)

while in contrast, the makam nihavend (similar to the Western minor scale): 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.