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. Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio. 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 Each step represents a frequency ratio of 21/53, 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 Holder's comma, and sometimes the Arabian comma, is a small 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)53) is very nearly equal to 31 octaves ((2 / 1)31). Jing Fang ( 78&ndash37 BC born Li Fang (李房 Courtesy name Junming (君明 was a Chinese music theorist, Mathematician The perfect fifth ( is the Musical interval between a note and the note seven Semitones above it on the musical scale He calculated this difference with six-digit accuracy to be 177147 / 176776. [1] Later the same observation was made by the mathematician and music theorist Nicholas Mercator (c. Nicholas ( Nikolaus) Mercator (c 1620 Eutin -1687 Versailles) also known by his Germanic name Kauffmann, was a 17th-century mathematician 1620-1687), who calculated this value precisely as (353 / 284), which is known as Mercator's Comma. 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. Pythagorean tuning is a system of Musical tuning in which the Frequency relationships of all intervals are based on the ratio 32.

After Mercator, William Holder published a treatise in 1694 which pointed out that 53 equal temperament also very closely approximates the just major third (to within 1. William Holder ( 1616 - January 24, 1698) was an English Music theorist of the 17th century A major third ( is one of two commonly occurring Musical intervals that span three Diatonic scale degrees the other being the Minor third. 4 cents), and consequently 53 equal temperament accommodates the intervals of 5-limit just intonation very well. In music theory limit can refer to a variety of methods used to characterize the harmonies found in a piece of music genre of music or by extension the harmonies that can be made with [2][3] 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. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements

## Comparison to other scales

Because a distance of 31 steps in this scale is almost precisely equal to a just perfect fifth, this scale can practically be considered a form of Pythagorean tuning that has been extended to 53 tones. In music just intonation is any Musical tuning in which the frequencies of Notes are related by Ratios of Whole numbers Any interval The perfect fifth ( is the Musical interval between a note and the note seven Semitones above it on the musical scale Pythagorean tuning is a system of Musical tuning in which the Frequency relationships of all intervals are based on the ratio 32. As such the intervals available can have the same properties as any Pythagorean tuning, such as fifths that are (practically) pure, major thirds that are wide from just (approximately 81/64 as opposed to the purer 5/4), and minor thirds that are conversely narrow (32/27 compared to 6/5).

However, unlike most Pythagorean forms of tuning, 53-TET contains additional intervals that are very close to just intonation. For instance, the interval of 17 steps is also a major third, but only 1. 4 cents narrower than the very pure just interval 5/4. 53-TET is very good as an approximation to any interval in 5-limit just intonation.

53-TET does not handle intervals involving the 7th or 11th overtones particularly well, especially when compared to the close matches it makes for the other intervals. All of these intervals fall close to the center of a single-step interval in 53-TET. By comparison, 31-TET has a much closer match to the 8:7 ratio and achieves a similar match to 7:6 with fewer divisions of the octave.

Unlike other tunings such as 19-TET and 31-TET, 53-TET is not suitable as an approximation for meantone temperament for various reasons. In music 19 equal temperament, called 19-TET 19- EDO, or 19-ET is the tempered scale derived by dividing the octave into 19 equally large steps In music 31 equal temperament, which can be abbreviated 31-tET 31- EDO, 31-ET is the tempered scale derived by dividing the Octave into 31 equal-sized Meantone temperament is a Musical temperament, which is a system of Musical tuning. The main problem is that a cycle of four fifths does not produce a near-just major third like most meantone temperaments, but instead produces the Pythagorean wide third (see above), making the primary advantage of choosing a meantone system (purer thirds) impossible using traditional western harmonic practice. As well, it does not have a suitable set of two different semitones which can sum to a "meantone" between 10/9 and 9/8. 53-TET approximates 10/9 and 9/8 very well, but nothing in between which is necessary for a meantone temperament.

 interval name size (steps) size (cents) just ratio just (cents) difference perfect fifth 31 701. The perfect fifth ( is the Musical interval between a note and the note seven Semitones above it on the musical scale 89 3:2 701. 96 0. 07 tritone 26 588. The tritone ( Tri - or three and tone) is a Musical interval that spans three whole tones. 68 7:5 582. 51 -6. 17 tritone 25 566. The tritone ( Tri - or three and tone) is a Musical interval that spans three whole tones. 04 18:13 563. 38 -2. 66 11th overtone 24 543. 4 11:8 551. 32 7. 92 15:11 ratio 24 543. 4 15:11 536. 95 -6. 45 perfect fourth 22 498. The perfect fourth () is a Musical interval which spans four scale degrees 11 4:3 498. 04 -0. 07 (13:10) ratio 20 452. 83 13:10 454. 21 1. 38 septimal major third 19 430. In Music, the septimal major third, also called the supermajor third (by Hermann Helmholtz) and sometimes Bohlen-Pierce third is 19 9:7 435. 08 4. 9 major third, Pythagorean 18 407. A major third ( is one of two commonly occurring Musical intervals that span three Diatonic scale degrees the other being the Minor third. Pythagorean tuning is a system of Musical tuning in which the Frequency relationships of all intervals are based on the ratio 32. 54 81:64 407. 82 0. 28 major third, just 17 384. In music just intonation is any Musical tuning in which the frequencies of Notes are related by Ratios of Whole numbers Any interval 91 5:4 386. 31 1. 4 (16:13) third 16 362. 26 16:13 359. 47 -2. 79 undecimal neutral third 15 339. A neutral third is a Musical interval between a Minor third and a Major third. 62 11:9 347. 41 7. 79 minor third, just 14 316. A minor third ( is the smaller of two commonly occurring musical intervals compounded of two steps of the Diatonic scale. 98 6:5 315. 64 -1. 34 minor third, Pythagorean 13 294. 34 32:27 294. 13 -0. 21 septimal minor third 12 271. In music the septimal minor third, also called the subminor third (by eg Helmholtz) is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies 70 7:6 266. 87 -4. 83 (15:13) ratio 11 249. 06 15:13 247. 74 -1. 32 septimal whole tone 10 226. In music the septimal whole tone is the musical interval exactly or approximately equal to a 8/7 ratio of frequencies 41 8:7 231. 17 4. 76 whole tone, major tone 9 203. A major second () also called a whole step or a whole tone, is a Musical interval that occurs between the first and second degrees of a 77 9:8 203. 91 0. 14 whole tone, minor tone 8 181. 13 10:9 182. 40 1. 27 neutral second, greater undecimal 7 158. A neutral second or medium second is a Musical interval between a Minor second and a Major second. 49 11:10 165. 00 6. 51 (13:12) second 6 135. 85 13:12 138. 57 2. 72 diatonic semitone, just 5 113. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the 21 16:15 111. 73 -1. 48 chromatic semitone, Pythagorean 5 113. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the 21 2187:2048 113. 69 0. 48 diatonic semitone, Pythagorean 4 90. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the 566 256:243 90. 225 -0. 34 chromatic semitone, just 3 67. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the 925 25:24 70. 672 2. 747

## 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. In Music, the schisma, also spelled skhisma, is the ratio between a Pythagorean comma and a Syntonic comma and equals 32805/32768 which is 1 In music theory the kleisma is an interval important to temperaments of the Bohlen-Pierce scale. 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. Regular temperament is any tempered system of Musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators In Music theory, a comma is a small or very small interval between two Enharmonic notes tuned in different ways was a Japanese physicist music theorist and inventor He graduated from Tokyo University in 1882 as a science student 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. In Music, the schismatic temperament is the result of tempering (musical tuning the Schisma of 3280532768 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. In music the ratio 225/224 is called the septimal kleisma ( Another name for it is the marvel comma, since the temperament tempering it out is sometimes called the Marvel 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.

## 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. Meantone temperament is a Musical temperament, which is a system of Musical 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. In Music, the schismatic temperament is the result of tempering (musical tuning the Schisma of 3280532768 to a unison 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. A seventh chord is a chord consisting of a triad plus a note forming an interval of a Seventh above the chord's root. Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones (multiples or "undertones" (divisors of a The utonal tetrad, the inversion of the otonal tetrad, is spelled C-D#-G-Gx. Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones (multiples or "undertones" (divisors of a

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. In music the ratio 225/224 is called the septimal kleisma ( Another name for it is the marvel comma, since the temperament tempering it out is sometimes called the Marvel So is the orwell tetrad, C-Fb-Dxx-Gx in its various inversions.

## Music

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[4] and the American tuner James Paul White[5]. Robert Holford Macdowall Bosanquet ( 31 July 1841 &ndash 7 August 1912) was an English scientist and music theorist and brother of Admiral Sir 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. The music of Turkey includes diverse elements ranging from Central Asian folk music and music from Ottoman Empire dominions such as Persian music, 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. Arabic music or Arab music ( Arabic: موسيقى عربية;) includes several genres and styles of Music ranging from Arabic classical A quarter tone is an interval about half as wide (aurally or logarithmically as a Semitone, which is half a Whole tone. 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.

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. Josip Štolcer-Slavenski ( May 11 1896 - November 30 1955) was a Croatian Composer. Robert Holford Macdowall Bosanquet ( 31 July 1841 &ndash 7 August 1912) was an English scientist and music theorist and brother of Admiral Sir [6][7]

In 2006 aforesaid 4 1/2-octave harmonium[8] was repaired by Phil & Pam Fluke[9] in England and now is playable.