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 53 equal steps (equal frequency ratios) (). Each step represents a frequency ratio of   21 ∕ 53 , or 22.6415 cents (), an interval sometimes called the Holdrian comma. 53 TET is a tuning of equal temperament in which the tempered perfect fifth is 701.89 cents wide, as shown in Figure 1, and sequential pitches are separated by 22.642 cents. The 53-TET tuning equates to the unison, or tempers out, the intervals ⁠ 32 805 / 32 768 ⁠, known as the schisma, and ⁠ 15 625 / 15 552 ⁠, 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 TET 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 TET 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 closest to the 43rd note (counting from 0) and   243 ∕ 53 = 1.7548   is only 4.8 cents sharp from the harmonic 7th   = ⁠ 7 / 4 ⁠ in 53 TET, and using it for 7-limit harmony means that the septimal kleisma, the interval ⁠ 225 / 224 ⁠, is also tempered out.

History and use Theoretical interest in this division goes back to antiquity. Jing Fang (78–37 BCE), a Chinese music theorist, observed that a series of 53 just fifths ( [⁠ 3 / 2 ⁠]53 ) is very nearly equal to 31 octaves (231). He calculated this difference with six-digit accuracy to be ⁠ 177 147 / 176 776 ⁠. 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 ⁠ = ⁠ 19 383 245 667 680 019 896 796 723 / 19 342 813 113 834 066 795 298 816 ⁠, 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 ( ≈ 0.0682 cent ≈ ⁠1/ 315 ⁠ syntonic comma ≈ ⁠1/ 344 ⁠ pythagorean comma). Thus, 53 tone 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. 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 1664–1665.

Music In the 19th century, people began devising instruments in 53 TET, with an eye to their use in playing near-just 5-limit music. Such instruments were devised by R.H.M. Bosanquet and the American tuner J.P. White. Subsequently, the temperament has seen occasional use by composers in the West, and by the early 20th century, 53 TET had become the most common form of tuning in Ottoman classical music, replacing its older, unequal tuning. 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 TET should be used as the master scale for Arabic music. 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.

Furthermore, General Thompson worked in league with the London-based guitar maker Louis Panormo to produce the Enharmonic Guitar.

Notation

Standard Western notation, with seven letter-named notes plus sharps and flats, is not adequate to notate music in 53 TET. This is unlike the case with 19 TET and 31 TET where there is little ambiguity. The fact that it is not meantone adds additional problems. Specifically, the Pythagorean major third (ditone) and just major third are distinguished, as are the Pythagorean minor third (semiditone) and just minor third. The fact that the syntonic comma is not tempered out means that notes and intervals need to be defined more precisely. Ottoman classical music uses a notation of flats and sharps for the 9 comma tone. Furthermore, since 53 is not a multiple of 12, notes such as G♯ and A♭ are not enharmonically equivalent, nor are the corresponding key signatures. As a result, many key signatures will require the use of double sharps (such as G♯ major / E♯ minor), double flats (such as F♭ major / D♭ minor), or microtonal alterations. Extended pythagorean notation, using only sharps and flats, gives the following chromatic scale:

C, B♯, A♯, E, D♭, C♯, B, F, E, D, C, B♯, F, E♭, D♯, C♯, G, F♭, E, D, C/A, G, F, E♯, D♯, A, G♭, F♯, E, D/B, A, G, F, E♯, B, A♭, G♯, F♯, C, B, A, G, F/D, C, B♭, A♯, G♯, D, C♭, B, A, G/E, D, C The note names run out of letters, and quadruple sharps and flats can be required. As a result, a just major 3rd must be spelled as a diminished 4th. Ups and downs notation keeps the notes in order and also preserves the traditional meaning of sharp and flat. It uses up and down arrows, written as a caret or a lower-case "v", usually in a sans-serif font. One arrow equals one step of 53-TET. In note names, the arrows come first, to facilitate chord naming. The many enharmonic equivalences allow great freedom of spelling.

C, ^C, ^^C, vvC♯/vD♭, vC♯/D♭, C♯/^D♭, ^C♯/^^D♭, vvD, vD, D, ^D, ^^D, vvD♯/vE♭, vD♯/E♭, D♯/^E♭, ^D♯/^^E♭, vvE, vE, E, ^E, ^^E/vvF, vF, F, ^F, ^^F, vvF♯/vG♭, vF♯/G♭, F♯/^G♭, ^F♯/^^G♭, vvG, vG, G, ^G, ^^G, vvG♯/vA♭, vG♯/A♭, G♯/^A♭, ^G♯/^^A♭, vvA, vA, A, ^A, ^^A, vvA♯/vB♭, vA♯/B♭, A♯/^B♭, ^A♯/^^B♭, vvB, vB, B, ^B, ^^B/vvC, vC, C

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