53 equal temperament
Each step represents a frequency ratio of 21 ∕ 53 , or 22.6415 cents (Playⓘ), an interval sometimes called the Holdrian comma.
These are both 5 limit intervals, involving only the primes 2, 3, and 5 in their factorization, and the fact that 53 TET tempers out both characterizes it completely as a 5 limit temperament: It is the only regular temperament tempering out both of these intervals, or commas, a fact which seems to have first been recognized by Japanese music theorist Shohé Tanaka.
Jing Fang (78–37 BCE), a Chinese music theorist, observed that a series of 53 just fifths ( [ 3 / 2 ]53 ) is very nearly equal to 31 octaves (231).
Thus, 53 tone equal temperament is for all practical purposes equivalent to an extended Pythagorean tuning.
[5][6] This property of 53 TET may have been known earlier; Isaac Newton's unpublished manuscripts suggest that he had been aware of it as early as 1664–1665.
[7] In the 19th century, people began devising instruments in 53 TET, with an eye to their use in playing near-just 5-limit music.
[citation needed] Croatian composer Josip Štolcer-Slavenski wrote one piece, which has never been published, which uses Bosanquet's Enharmonium during its first movement, entitled Music for Natur-ton-system.
[12] Attempting to use standard notation, seven-letter notes plus sharps or flats, can quickly become confusing.
The fact that the syntonic comma is not tempered out means that notes and intervals need to be defined more precisely.
Ottoman classical music uses a notation of flats and sharps for the 9 comma tone.
Ups and downs notation[13] keeps the notes in order and also preserves the traditional meaning of sharp and flat.
It uses up and down arrows, written as a caret or a lower-case "v", usually in a sans-serif font.
Since 53-TET is a Pythagorean system, with nearly pure fifths, justly-intonated major and minor triads cannot be spelled in the same manner as in a meantone tuning.
The 11th harmonic and intervals involving it are less closely matched, as illustrated by the undecimal neutral seconds and thirds in the table below.
The origin of Holder's comma resides in the fact that the Ancient Greeks (or at least to the Roman Boethius[b]) 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.
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 to 55 TET.
The Holdrian comma has been employed mainly in Ottoman/Turkish music theory by Kemal Ilerici, and by the Turkish composer Erol Sayan.
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,[e] 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.
