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Holdrian comma

## Summary

In music theory and musical tuning the Holdrian comma, also called Holder's comma, and rarely the Arabian comma,[1] is a small musical interval of approximately 22.6415 cents,[1] equal to one step of 53 equal temperament, or ${\displaystyle {\sqrt[{53}]{2}}}$ ( ). 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[2]) 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[3] attributes the division of the octave in 53 equal parts to Nicholas Mercator,[4] 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.[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.

## Use in Ottoman music

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]

## Notes

^∗ 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.

## 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.
2. ^ A. M. S. Boethius, De institutione musica, Book 3, Chap. 8. According to Boethius, Pythagoras' disciple Philolaos would have said that the tone consisted in two diatonic semitones and a comma; the diatonic semitone consisted in two diaschismata, each formed of two commas. See J. Murray Barbour, Tuning and Temperament: A Historical Survey, 1951, p. 123
3. ^ W. Holder, A Treatise of the Natural Grounds, and Principles of Harmony, London, 3d edition, 1731, p. 79.
4. ^ "The late Nicholas Mercator, a Modest Person, and a Learned and Judicious Mathematician, in a Manuscript of his, of which I have had a Sight."
5. ^ W. Holder, A Treatise..., ibid., writes that Mersenne had calculated 58¼ commas in the octave; Mercator "working by the Logarithms, finds out but 55, and a little more."