Mercator's comma is a name often used for a closely related interval because of its association with Nicholas Mercator.^[5] One of these intervals was first described by Ching-Fang in 45 BCE.^[1] Mercator applied logarithms to determine that ${\displaystyle {\sqrt[{55}]{2}}}$  (≈ 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 ${\displaystyle {\sqrt[{53}]{2}}}$  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.

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 (similar to the Western major scale, or more precisely to the justly-tuned major scale) may be considered in terms of Holdrian commas:

 , where   denotes a Holdrian comma flat^∗,

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

 , 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.^[1]

^∗ In common Arabic and Turkish practice, the third note e  and the seventh note b  in Rast are even lower than in this theory, almost exactly halfway between western major and minor thirds above c and g, i.e. closer to 6.5 commas (three-quarter tone) above d or a and 6.5 below f or c, the thirds c–e  and g–b  often referred to as a "neutral thirds" by musicologists.

• Holder, William, A Treatise on the Natural Grounds, and Principles of Harmony, facsimile of the 1694 edition, Broude Brothers, New York, 1967. (Original pp. 103–106.)