In music, 53 equal temperament, called 53-tet, 53-edo, or 53-et, is the scale derived by dividing the octave into fifty-three equally large steps. Each step represents a frequency ratio of 2^1/53, or 22.6415 cents, an interval sometimes called the Holdrian comma. The cent is a logarithmic unit of measure used for musical intervals. ... The Holdrian comma, also called Holders comma or the Arabian comma, is a musical interval of 22. ...

Theoretical interest in this division goes back to antiquity. Jing Fang (also spelled Ching Fang and King Fang), a Chinese music theorist, observed around 40 BC that the fifths very nearly close into a circle of 53 fifths, making up 31 octaves. The same obsevation was eventually made in the west by the mathematician and music theorist Nicholas Mercator. This is due to one of the most salient features of this temperament, which is that the fifths are very nearly the pure 3/2 ratio of just intonation, being a mere 0.07 cents flat. While it does not represent major thirds to anything like this degree of accuracy, major thirds are still only 1.4 cents flat; consequently, 53 equal temperament represents the intervals of 5-limit just intonation with great accuracy. This fact seems to have first been published by the seventeenth century music theorist William Holder, though Isaac Newton's unpublished manuscripts show he was aware of it also. Nicholas (Nikolaus) Mercator (c. ... In music, Just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by whole number ratios; that is, by positive rational numbers. ... Just intonation tunings and scales can be described by giving an upper bound on the complexity of the harmonies admitted by the tuning or scale. ... Sir Isaac Newton, PRS, (4 January [O.S. 25 December 1642] 1643 – 31 March [O.S. 20 March] 1727) was an English physicist, mathematician, astronomer, alchemist, inventor and natural philosopher who is generally regarded as one of the most influential scientists in history. ...

Theoretical properties

The 53-et tuning equates to the unison, or tempers out, the intervals 32805/32768, known as the schisma, and 15625/15552, known as the kleisma. These are both 5-limit intervals, involving only the primes 2, 3 and 5 in their factorization, and the fact that 53-et 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. Because it tempers these out, 53-et can be used for both schismatic temperament, tempering out the schisma, and hanson temperament (also called kleismic), tempering out the kleisma. The schisma, also spelled skhisma, is the ratio between a Pythagorean comma and a syntonic comma and equals 32805/32768, which is 1. ... Regular temperament is a system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. ... In music theory, a comma is a small or very small interval between two enharmonic notes tuned in different ways. ... In music, schismatic temperament is the temperament which results from tempering the schisma of 32805:32768 to a unison. ...

The interval of 7/4 is 4.8 cents sharp in 53-et, and using it for 7-limit harmony means that the septimal kleisma, the interval 225/224, is also tempered out. So is the interval 1728/1715, sometimes called the orwell comma. As a consequence, 53-et supports various 7-limit temperaments, some of which have recently been named orwell, garibaldi, and catakleismic. In music, the ratio 225/224 is called the septimal kleisma. ...

Chords of 53 equal temperament

Standard musical notation can be used to denote 53 equal temperament; however, since it is a Pythagorean system, with nearly pure fifths, major and minor triads cannot be spelled in the same manner as in a meantone tuning. Instead, the major triads are chords like C-Fb-G, where the major third is a diminished fourth; this is the defining characteristic of schismatic temperament. Likewise, the minor triads are chords like C-D#-G. In 53-et the dominant seventh chord would be spelled C-Fb-G-Bb, but the otonal tetrad is C-Fb-G-Cbb, and C-Fb-G-A# is still another seventh chord. The utonal tetrad, the inversion of the otonal tetrad, is spelled C-D#-G-Gx. Meantone temperament is a system of musical tuning. ... In music, schismatic temperament is the temperament which results from tempering the schisma of 32805:32768 to a unison. ... A seventh chord is a chord or triad which has a note the seventh above the tonic in it. ... Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones (multiples) or undertones (divisors) of a given fixed tone. ... Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones (multiples) or undertones (divisors) of a given fixed tone. ...

Further septimal chords are the diminished triad, having the two forms C-D#-Gb and C-Fbb-Gb, the subminor triad, C-Fbb-G, the supermajor triad C-Dx-G, and corresponding tetrads C-Fbb-G-Bbb and C-Dx-G-A#. Since 53-et tempers out the septimal kleisma, the septimal kleisma augmented triad C-Fb-Bbb in its various inversions is also a chord of the system. So is the orwell tetrad, C-Fb-Dxx-Gx in its various inversions. In music, the ratio 225/224 is called the septimal kleisma. ...

Music in 53 equal temperament

In the ninteenth century, people began devising instruments in 53-et, with an eye to their use in playing near-just 5-limit music. Such instruments were devised by RHM Bosanquet and the American tuner James Paul White. Subsequently the temperament has seen occasional use by composers in the west, and has been used in Turkish music as well; the Turkish composer Erol Sayan has employed it, following theoretical use of it by Turkish music theorist Kemal Ilerici. Arabic music, which for the most part bases its theory on quartertones, has also made some use of it; the Syrian violinist and music theorist Twfiq Al-Sabagh proposed that instead of an equal division of the octave into 24 parts a 24-note scale in 53-et should be used as the master scale for Arabic music. It should also be bourne in mind that any music in 5-limit just intonation, or the temperaments supported by 53-et such as schismatic, can be performed in 53-et as well. Robert Holford Macdowall Bosanquet (1841-1913) was an English scientist and music theorist, brother of the philosopher Bernard Bosanquet. ... Turkey is a country on the eastern shore of the Mediterranean Sea, and is a crossroads of cultures from across Europe, North Africa, the Middle East, the Caucasus and South and Central Asia. ... A quarter tone is an interval half as wide (aurally, or logarithmically) as a semitone, which is half a whole tone. ...

Musical examples

• Melancholic by Aaron Krister Johnson

• A composition based on a folk melody of the Shiraz region, in shur-dastgah by Mohajeri Shahin

• An experiment in Iranian homayun and chahargah modes by Mohajeri Shahin

• Trio for Clarinet, English Horn and Banjo by Gene Ward Smith

• Pachelbel's canon in 53-et

References

Helmholtz, L. F., and Ellis, Alexander, On the Sensations of Tone, second English edition, Dover Publications, 1954

Holder, William, Treatise on the Natural Grounds and Principles of Harmony, facimile of the 1694 London edition, Broude Brothers, 1967

Stanley, Jerome, William Holder and His Position in Seventeenth-Century Philosophy and Music Theory, The Edwin Mellen Press, 2002