# Holdrian comma

In music theory and musical tuning the Holdrian comma, also called Holder's comma, and sometimes the Arabian comma,[1] is a small musical interval of approximately 22.6415 cents,[1] 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.

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.[1]

## 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 sixth-comma meantone temperament tuning system 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 just intonation 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:

        c d e f g a b c'
commas:  9 8 5 9 9 8 5


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

        c d e♭ f g a♭ b♭ c'
commas:  9 4  9 9 4  9 9


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.[1]

## References

1. ^ a b c d Habib Hassan Touma (1996). The Music of the Arabs, p.23. trans. Laurie Schwartz. Portland, Oregon: Amadeus Press. ISBN 0-931340-88-8.