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# 53 equal temperament

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 21/53, or 22.6415 cents, an interval sometimes called the Holdrian comma.

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 ($\left(3/2\right)^\left\{53\right\}$) is very nearly equal to 31 octaves ($\left(2/1\right)^\left\{31\right\}$). He calculated this difference with six-digit accuracy to be $177147 / 176776$. [McClain, Ernest and Ming Shui Hung. "Chinese Cyclic Tunings in Late Antiquity", Ethnomusicology Vol. 23 No. 2, 1979. pp. 205-224.] Later the same observation was made by the mathematician and music theorist Nicholas Mercator (c. 1620-1687), who calculated this value precisely as $\left( 3^\left\{53\right\} / 2^\left\{84\right\} \right)$, 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.

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 1.4 cents), and consequently 53 equal temperament accommodates the intervals of 5-limit just intonation very well. [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and Music Theory", The Edwin Mellen Press, 2002] This property of 53-TET may have been known earlier; Isaac Newton's unpublished manuscripts suggest that he had been aware of it as early as 1664Fact|date=August 2007.

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. 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. It does, however, like 34-TET (which also poorly matches the 7th harmonic), provide a fine match to the 13th harmonic, resulting in various neutral, supra-major and sub-minor intervals not found in many other temperaments.

Unlike other tunings such as 19-TET and 31-TET, 53-TET is not suitable as an approximation for meantone temperament for various reasons. 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.

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 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.

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.

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.

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 [Helmholtz, L. F., and Ellis, Alexander, "On the Sensations of Tone", second English edition, Dover Publications, 1954. Pp. 328-329.] and the American tuner James Paul White [Ibid. Page 329.] . 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 IlericiFact|date=July 2007. 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 musicFact|date=July 2007. 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". [ [http://commons.wikimedia.org/wiki/
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In 2006 aforesaid 4 1/2-octave harmonium [ [http://tardis.dl.ac.uk/FreeReed/organ_book/node16.html#SECTION000162000000000000000 Bosanquet's 53EDO Enharmonic Harmonium] . Allan, R. J. The Free-Reed Organ in England. (2004)] was repaired by Phil & Pam Fluke [ [http://tardis.dl.ac.uk/FreeReed/organ_book/node3.html#SECTION00034000000000000000 Phil & Pam Fluke] ] in England and now is playable.

* [http://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html Whisper Song in 53EDO by Prent Rodgers]
* [http://www.anaphoria.com/hanson.PDF PDF file: Larry Hanson. Development of a 53EDO Keyboard Layout]
* [http://sonantometry.blogspot.com/2007_05_01_archive.html Tonal Functions as 53EDO grades]

References

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