Holdrian comma

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].

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 name "comma", however, is technically misleading, since this interval is an irrational number and it does not describe a compromise between intervals of any tuning system. 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 fact that 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 diatonic semitones and a comma; the diatonic semitone consisted in two diaschismata, each formed of two commas.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 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".

See also

• Mercator's comma
• Historical temperaments