53 equal temperament
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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.
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[edit] 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.
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 Isaac Newton's unpublished manuscripts show that he was also aware of this.
[edit] 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.
[edit] 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.
[edit] 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.
[edit] 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.
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| Pythagorean · Just intonation · Harry Partch's 43-tone scale | |||||
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| Well temperament | |||||
