In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered 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. It has been suggested that this article or section be merged into Equal temperament. ... In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. ... 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. ...

History

Theoretical interest in this division goes back to antiquity. Ching Fang (78-37BC), a Chinese music theorist, observed that a series of 53 just fifths (3 / 2) is very nearly equal to 31 octaves. He calculated this difference with six-digit accuracy to be 177147 / 176776. Later the same observation was made by the mathematician and music theorist Nicholas Mercator (c. 1620-1687), who calculated this value precisely as (3⁵³ / 2⁸⁴), which is known as Mercator's Comma. Ching Fang (78-37BC) was a music theorist, most known for being the first to notice how closely a succession of 53 just fifths approximates 31 octaves. ... The perfect fifth or diapente is the interval between the first note (the root or tonic) and the fifth note in a major scale. ... Nicholas (Nikolaus) Mercator (c. ...

Mercator's Comma is of such small value to begin with (~3.615 cents), but 53 equal temperament flattens each fifth by only 1 / 53 of that comma. Thus, 53 equal temperament is for all practical purposes equivalent to an extended pythagorean tuning. After Mercator, William Holder pointed out that 53 equal temperament also very closely approximates the just major third (to within 1.4 cents). Consequently, 53 equal temperament represents the intervals of 5-limit just intonation with great accuracy. Isaac Newton's unpublished manuscripts show that he was also aware of this. Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... William Holder (1616-1698) was an English music theorist of the 17th century. ... A major third is the larger of two commonly occurring musical intervals that span three diatonic scale degrees. ... 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, FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, alchemist, and natural philosopher, regarded by many as the greatest figure in the history of science. ...

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. ... Shohé Tanaka is a Japanese physicist,music theorist. ... 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 nineteenth 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 borne 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. ... History (Timeline and Samples) Genres: Alternative - Classical - Dance - Folk - Hip hop - Jazz - Military - Ottoman - Pop - Religious - Rock Music awards Kral - MÜ-YAP - MGD Charts Powerturk 40 - Kral 20 Annual festivals Istanbul International Music Festival - Istanbul International Jazz Festival - Ankara IMF - Izmir European Jazz Festival Media Bant magazine - Mix! - Adante - BlueJean... A quarter tone is an interval half as wide (aurally, or logarithmically) as a semitone, which is half a whole tone. ...

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

McClain, Ernest, Chinese Cyclic Tunings in Late Antiquity, Ethnomusicology Vol. 23 No. 2, 1979. pp. 205-224.