Holdrian comma

Interval information

Expression	[math]\displaystyle{ 2^{1/53} }[/math]
Size in cents	22.6415¢
Names	Holdrian comma, Holder's comma, Arabian comma, 1 step of 53edo
Special properties	reduced
Comma size	small
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The Holdrian comma, also called Holder's comma, rarely the Arabian comma, is a small interval of approximately 22.6415 cents, equal to exactly one step of 53edo, or [math]\displaystyle{ \ \sqrt[53]{2\;}\ }[/math].

The name comma describes its size and does not describe a compromise between intervals of any tuning system, since this interval is an irrational number. The interval gets the name comma because it is a close approximation of several commas, most notably the syntonic comma (21.51 cents), which was widely used as a unit of tonal measurement during William Holder's time.

Historical origin

The origin of Holder's comma resides in the Ancient Greeks (or at least to the Roman Anicius Manlius Severinus Boethius). According to Boethius, Pythagoras' disciple Philolaus of Croton would have said that the tone consisted in two Pythagorean diatonic semitones and a comma (the Pythagorean diatonic semitone consisted in two diaschismata, each formed of two commas.) and believed that in the Pythagorean tuning the tone could be divided in nine commas, four of which forming the Pythagorean diatonic semitone and five the Pythagorean 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: "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." who himself had proposed that 1/53 of the octave be named the "artificial comma".

Mercator's comma, Mercator's old comma, and the Holdrian comma

Mercator's old comma is a name sometimes used for a closely related interval because of its association with Nicholas Mercator.

Holder 1731 writes that Marin Mersenne had calculated 58¹/₄s in the octave; Mercator "working by the logarithms, finds out but 55, and a little more."

One of these intervals was first described by Jing Fang in 45 BCE. Mercator applied logarithms to determine that [math]\displaystyle{ \ \sqrt[55]{2\;}\ }[/math] (≈ 21.8182 cents), exactly one step of 55edo, 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 [math]\displaystyle{ \ \sqrt[53]{2\;}\ }[/math] 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 53edo are closer to just intonation than to 55edo. Thus Mercator's old comma and the Holdrian comma are two distinct but nearly equal intervals.

There is another comma named Mercator's comma which receives much more usage in modern musical tuning. It a small comma of 3.615 cents which is the amount by which 53 perfect fifths exceed 31 octaves, in other words (3/2)⁵³/2³¹. It has its own dedicated article.

See also

• Historical temperaments
• Interval size measure: both the Holdrian comma and Mercator's old comma are examples of this