# 53 equal temperament

Figure 1: 53-TET on the syntonic temperament’s tuning continuum at 701.89, from (Milne et al. 2007).[1]

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 the syntonic temperament in which the tempered perfect fifth is 701.89 cents wide, as shown in Figure 1.

## History

Theoretical interest in this division goes back to antiquity. Ching Fang (78–37 BC), a Chinese music theorist, observed that a series of 53 just fifths (${\displaystyle (3/2)^{53}}$) is very nearly equal to 31 octaves (${\displaystyle (2/1)^{31}}$). He calculated this difference with six-digit accuracy to be ${\displaystyle 177147/176776}$.[2] Later the same observation was made by the mathematician and music theorist Nicholas Mercator (c. 1620–1687), who calculated this value precisely as ${\displaystyle (3^{53}/2^{84}=19383245667680019896796723/19342813113834066795298816)}$, which is known as Mercator's comma.[3] 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). 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of 5-limit just intonation very well.[4][5] This property of 53-TET may have been known earlier; Isaac Newton's unpublished manuscripts suggest that he had been aware of it as early as 1664–65.[6]

## Scale diagram

 Interval (steps) 3 2 4 3 2 3 2 1 2 4 1 4 3 2 3 4 2 3 2 1 2 Interval (cents) 68 45 91 68 45 68 45 23 45 91 23 91 68 45 68 91 45 68 45 23 45 Note name C0 CTemplate:Music-3 DTemplate:Music1 D0 DTemplate:Music-3 ETemplate:Music1 E-1 FTemplate:Music3 ETemplate:Music-4 F0 FTemplate:Music-2 GTemplate:Music2 G0 GTemplate:Music-3 ATemplate:Music1 A-1 ATemplate:Music-3 BTemplate:Music1 B-1 CTemplate:Music3 BTemplate:Music-4 C0 Note (cents) 0 68 113 204 272 317 385 430 453 498 589 611 702 770 815 883 974 1018 1087 1132 1155 1200 Note (steps) 0 3 5 9 12 14 17 19 20 22 26 27 31 34 36 39 43 45 48 50 51 53

## Comparison to other scales

Because a distance of 31 steps in this scale is almost precisely equal to a just perfect fifth, in theory this scale can 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 (about 81/64 opposed to the purer 5/4), and minor thirds that are conversely narrow (32/27 compared to 6/5).

However, 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.

The matches to the just intervals involving the 7th harmonic are slightly less close, but all such intervals are still matched with the highest deviation being the 7:5 tritone. The 11th harmonic and intervals involving it are less closely matched, as illustrated with the undecimal neutral seconds and thirds in the table below.

 interval name size (steps) size (cents) just ratio just (cents) error harmonic seventh 43 973.59 7:4 968.83 +4.76 perfect fifth 31 701.89 3:2 701.96 −0.07 diatonic tritone 26 588.68 45:32 590.22 −1.54 septimal tritone 26 588.68 7:5 582.51 +6.17 classic tritone 25 566.04 25:18 568.72 −2.68 undecimal tritone 24 543.40 11:8 551.32 −7.92 double diminished fifth 24 543.40 512:375 539.10 +4.30 undecimal augmented fourth 24 543.40 15:11 536.95 +6.45 acute fourth 23 520.76 27:20 519.55 +1.21 perfect fourth 22 498.11 4:3 498.04 +0.07 grave fourth 21 475.47 320:243 476.54 −1.07 septimal narrow fourth 21 475.47 21:16 470.78 +4.69 classic augmented third 20 452.83 125:96 456.99 −4.16 tridecimal augmented third 20 452.83 13:10 454.21 −1.38 septimal major third 19 430.19 9:7 435.08 −4.90 classic diminished fourth 19 430.19 32:25 427.37 +2.82 Pythagorean ditone 18 407.54 81:64 407.82 −0.28 just major third 17 384.91 5:4 386.31 −1.40 neutral third, tridecimal 16 362.26 16:13 359.47 +2.79 grave major third 16 362.26 100:81 364.80 −2.54 neutral third, undecimal 15 339.62 11:9 347.41 −7.79 acute minor third 15 339.62 243:200 337.15 +2.47 just minor third 14 316.98 6:5 315.64 +1.34 semiditone 13 294.34 32:27 294.13 +0.21 classic augmented second 12 271.70 75:64 274.58 −2.88 septimal minor third 12 271.70 7:6 266.87 +4.83 classic diminished third 11 249.06 144:125 244.97 +4.09 septimal whole tone 10 226.41 8:7 231.17 −4.76 diminished third 10 226.41 256:225 223.46 +2.95 whole tone, major tone 9 203.77 9:8 203.91 −0.14 whole tone, minor tone 8 181.13 10:9 182.40 −1.27 neutral second, greater undecimal 7 158.49 11:10 165.00 −6.51 neutral second, grave whole tone 7 158.49 800:729 160.90 −2.41 neutral second, lesser undecimal 7 158.49 12:11 150.64 +7.85 neutral second, large limma 6 135.85 27:25 133.24 +2.61 Pythagorean major semitone 5 113.21 2187:2048 113.69 −0.48 just diatonic semitone 5 113.21 16:15 111.73 +1.48 major limma 4 90.57 135:128 92.18 −1.61 Pythagorean minor semitone 4 90.57 256:243 90.22 +0.34 just chromatic semitone 3 67.92 25:24 70.67 −2.75 just diesis 2 45.28 128:125 41.06 +4.22 syntonic comma 1 22.64 81:80 21.51 +1.14

## 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.

## Notation

Attempting to use standard notation, seven letter notes plus sharps or flats, can quickly become confusing. This is unlike the case with 19-ET and 31-ET where there is little ambiguity. By not being meantone, it adds some problems that require more attention. Specifically, the major third is different from a ditone, two tones, each of which is two fifths minus an octave. Likewise, the minor third is different from a semiditone. The fact that the syntonic comma is not tempered out means that notes and intervals need to be defined more precisely.

## 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-FTemplate:Music-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-DTemplate:Music-G. In 53-ET the dominant seventh chord would be spelled C-FTemplate:Music-G-BTemplate:Music, but the otonal tetrad is C-FTemplate:Music-G-CTemplate:Music, and C-FTemplate:Music-G-ATemplate:Music is still another seventh chord. The utonal tetrad, the inversion of the otonal tetrad, is spelled C-DTemplate:Music-G-GTemplate:Music.

Further septimal chords are the diminished triad, having the two forms C-DTemplate:Music-GTemplate:Music and C-FTemplate:Music-GTemplate:Music, the subminor triad, C-FTemplate:Music-G, the supermajor triad C-DTemplate:Music-G, and corresponding tetrads C-FTemplate:Music-G-BTemplate:Music and C-DTemplate:Music-G-ATemplate:Music. Since 53-ET tempers out the septimal kleisma, the septimal kleisma augmented triad C-FTemplate:Music-BTemplate:Music in its various inversions is also a chord of the system. So is the orwell tetrad, C-FTemplate:Music-DTemplate:MusicTemplate:Music-GTemplate:Music in its various inversions.

Because 53-TET is compatible with both the schismatic temperament and the syntonic temperament, it can be used as a pivot tuning in a temperament modulation (a musical effect enabled by dynamic tonality).

## Music

In the 19th 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[7] and the American tuner James Paul White.[8] 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{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}. 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{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}.

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.[9][10] Furthermore, General Thompson worked in league with the London-based guitar maker Louis Panormo to produce the Enharmonic Guitar (see: James Westbrook,‘General Thompson’s Enharmonic Guitar’, Soundboard: XXXVIII: 4, pp. 45–52.).

## References

1. Milne, A., Sethares, W.A. and Plamondon, J.,"Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings Across a Tuning Continuum", Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.
2. McClain, Ernest and Ming Shui Hung. Chinese Cyclic Tunings in Late Antiquity, Ethnomusicology Vol. 23 No. 2, 1979. pp. 205–224.
3. Monzo, Joe (2005). "Mercator's Comma", Tonalsoft.
4. Holder, William, Treatise on the Natural Grounds and Principles of Harmony, facsimile of the 1694 London edition, Broude Brothers, 1967
5. Stanley, Jerome, William Holder and His Position in Seventeenth-Century Philosophy and Music Theory, The Edwin Mellen Press, 2002
6. Barbieri, Patrizio. Enharmonic instruments and music, 1470–1900. (2008) Latina, Il Levante Libreria Editrice, p. 350.
7. Helmholtz, L. F., and Ellis, Alexander, On the Sensations of Tone, second English edition, Dover Publications, 1954. pp.328–329.
8. Helmholtz, L. F., and Ellis, Alexander, On the Sensations of Tone, second English edition, Dover Publications, 1954. p.329.
9. Facsimile of the 53EDO piece preface by J. Slavenski.
10. Facsimile of the 53EDO piece title page by J. Slavenski.