53 equal temperament on arvau

Wikipedia / 53 equal temperament

In music, 53equal temperament, called 53TET, 53EDO, or 53ET, is the tempered scale derived by dividing the octave into 53equal steps (equal frequency ratios) (). Each step represents a frequency ratio of or 22.6415cents (), an interval sometimes called the Holdrian comma.

53TET is a tuning of equal temperament in which the tempered perfect fifth is 701.89cents wide, as shown in Figure1, and sequential pitches are separated by 22.642cents.

The 53-TET tuning equates to the unison, or tempers out, the intervals known as the schisma, and known as the kleisma. These are both 5limit intervals, involving only the primes 2, 3, and 5 in their factorization, and the fact that 53TET tempers out both characterizes it completely as a 5limit temperament: It is the only regular temperament tempering out both of these intervals, or commass, a fact which seems to have first been recognized by Japanese music theorist Shohé Tanaka. Because it tempers these out, 53TET can be used for both schismatic temperament, tempering out the schisma, and Hanson temperament (also called kleismic), tempering out the kleisma.

The interval of is closest to the 43rd note (counting from 0) and is only 4.8cents sharp from the harmonic 7th in 53TET, and using it for 7-limit harmony means that the septimal kleisma, the interval , is also tempered out.

History and use

Theoretical interest in this division goes back to antiquity. Jing Fang (78–37BCE), a Chinese music theorist, observed that a series of 53just fifths is very nearly equal to 31octaves (). He calculated this difference with six-digit accuracy to be . Later the same observation was made by the mathematician and music theorist Nicholas Mercator (), who calculated this value precisely as which is known as Mercator's comma. Mercator's comma is of such small value to begin with but 53equal temperament flattens each fifth by only of that comma ( Thus, 53tone 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 53equal temperament also very closely approximates the just major third (to within 1.4cents), and consequently 53equal temperament accommodates the intervals of 5 limit just intonation very well. This property of 53TET may have been known earlier; Isaac Newton's unpublished manuscripts suggest that he had been aware of it as early as 1664–1665.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 19thcentury, people began devising instruments in 53TET, 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 20thcentury, 53TET 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 24parts a 24note scale in 53TET 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 9comma 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

Chords of 53 equal temperament

Since 53-TET is a Pythagorean system, with nearly pure fifths, justly-intonated major and minor triads cannot be spelled in the same manner as in a meantone tuning. Instead, the major triads are chords like C-F-G (using the Pythagorean-based notation), 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-TET, the dominant seventh chord would be spelled C-F-G-B, but the otonal tetrad is C-F-G-C, and C-F-G-A is still another seventh chord. The utonal tetrad, the inversion of the otonal tetrad, is spelled C-D-G-G.

Further septimal chords are the diminished triad, having the two forms C-D-G and C-F-G, the subminor triad, C-F-G, the supermajor triad C-D-G, and corresponding tetrads C-F-G-B and C-D-G-A. Since 53-TET tempers out the septimal kleisma, the septimal kleisma augmented triad C-F-B in its various inversions is also a chord of the system. So is the Orwell tetrad, C-F-D-G in its various inversions.

Ups and downs notation permits more conventional spellings. Since it also names the intervals of 53TET, it provides precise chord names too. The pythagorean minor chord with a third is still named Cm and still spelled C–E–G. But the 5-limit upminor chord uses the upminor 3rd 6/5 and is spelled C–^E–G. This chord is named C^m. Compare with ^Cm (^C–^E–^G).

Major triad: C-vE-G (downmajor)
Minor triad: C-^E-G (upminor)
Dominant 7th: C-vE-G-B (down add-7)
Otonal tetrad: C-vE-G-vB (down7)
Utonal tetrad: C-^E-G-^A (upminor6)
Diminished triad: C-^E-G (updim)
Diminished triad: C-vE-G (downdim)
Subminor triad: C-vE-G (downminor)
Supermajor triad: C-^E-G (upmajor)
Subminor tetrad: C-vE-G-vA (downminor6)
Supermajor tetrad: C-^E-G-^B (up7)
Augmented triad: C-vE-vvG (downaug dud-5)
Orwell triad: C-vE-vvG-^A (downmajor dud-5 up6)

Interval size

Because a distance of 31steps in this scale is almost precisely equal to a just perfect fifth, in theory this scale can be considered a slightly tempered form of Pythagorean tuning that has been extended to 53tones. 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 (about opposed to the purer and minor thirds that are conversely narrow ( compared to ).

However, 53TET contains additional intervals that are very close to just intonation. For instance, the interval of 17steps is also a major third, but only 1.4cents narrower than the very pure just interval 53TET is very good as an approximation to any interval in 5limit just intonation. Similarly, the pure just interval is only 1.3cents wider than 14steps in 53TET.

The matches to the just intervals involving the 7thharmonic are slightly less close (43steps are 4.8cents sharp for  7 4), but all such intervals are still quite closely matched with the highest deviation being the tritone. The 11thharmonic and intervals involving it are less closely matched, as illustrated by the undecimal neutral seconds and thirds in the table below. 7-limit ratios are colored light gray, and 11- and 13-limit ratios are colored dark gray.

Scale diagram

The following are 21 of the 53 notes in the chromatic scale. The rest can easily be added.

Holdrian comma

In music theory and musical tuning the Holdrian comma, also called Holder's comma, and rarely the Arabian comma, is a small musical interval of approximately 22.6415cents, equal to one step of 53 (). The name "comma", however, is technically misleading, since this interval is an irrational number and it does not describe a compromise between intervals of any tuning system. The interval gets the name "comma" because it is a close approximation of several commas, most notably the syntonic comma (21.51cents) (), which was widely used as a unit of tonal measurement during Holder's time.

The origin of Holder's comma resides in the fact that the Ancient Greeks (or at least to the Roman Boethius) believed that in the Pythagorean tuning the tone could be divided in nine commas, four of which forming the diatonic semitone and five the chromatic semitone. If all these commas are exactly of the same size, there results an octave of semitones, commas. Holder attributes the division of the octave in 53equal parts to Nicholas Mercator, who himself had proposed that of the octave be named the "artificial comma". === Mercator's comma and the Holdrian comma === Mercator's comma is a name often used for a closely related interval because of its association with Nicholas Mercator. One of these intervals was first described by Jing Fang in (≈ 21.8182cents) was nearly equivalent to a syntonic comma of ≈ 21.5063cents (a feature of the prevalent meantone temperament might be useful, because 31octaves could be practically approximated by a cycle of 53just fifths. Holder, for whom the Holdrian comma is named, favored this latter unit because the intervals of 53equal temperament are closer to just intonation than to 55 TET. Thus Mercator's comma and the Holdrian comma are two distinct but nearly equal intervals.

Use in Turkish makam theory

The Holdrian comma has been employed mainly in Ottoman/Turkish music theory by Kemal Ilerici, and by the Turkish composer Erol Sayan. The name of this comma is in Turkish.For instance, the Rast makam (similar to the Western major scale, or more precisely to the justly-tuned major scale) may be considered in terms of Holdrian commas:

where denotes a Holdrian comma flat, while in contrast, the Nihavend makam (similar to the Western minor scale):

where denotes a five-comma flat, has medium seconds between d–e, e–f, g–a, a–b, and b–c, a medium second being somewhere in between 8 and 9commas.