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). Play  Each step represents a frequency ratio of 2​1⁄53, or 22.6415 cents (Play ), an interval sometimes called the Holdrian comma.

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

53-EDO is a tuning of equal temperament in which the tempered perfect fifth is 701.89 cents wide, as shown in Figure 1.

The 53-EDO 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-EDO 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-EDO, 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. Ching 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 ​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 ​(353)⁄(284) = ​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 ≈ ​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.[4][5] This property of 53-EDO may have been known earlier; Isaac Newton's unpublished manuscripts suggest that he had been aware of it as early as 1664–1665.[6]

Music

In the 19th century, people began devising instruments in 53-EDO, 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. 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-EDO 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.[9][10][11] 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.).

Notation
Notation used in Ottoman classical music, where the tone is divided into 9 commas

Attempting to use standard notation, seven letter notes plus sharps or flats, can quickly become confusing. This is unlike the case with 19-EDO and 31-EDO 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. Ottoman classical music uses a notation of flats and sharps for the 9-comma tone.

In this article, diatonic notation will be used creating the following chromatic scale, where sharps and flats aren't enharmonic, only E and B are enharmonic with F and C. For the other notes, triple and quadruple sharps and flats aren't enharmonic.

C, C♯, C, C♯, C, D, D, D, D♭,

D, D♯, D, D♯, D, E, E, E, E♭,

E, E♯, E/F, F♭,

F, F♯, F, F♯, F, G, G, G, G♭,

G, G♯, G, G♯, G, A, A, A, A♭,

A, A♯, A, A♯, A, B, B, B, B♭,

B, B♯, B/C, C♭, C

Chords of 53 equal temperament

Since 53-EDO 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-F♭-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-EDO, 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-EDO 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.

Because 53-EDO 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).

Interval size

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 slightly tempered 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-EDO 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-EDO 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.

Size
(steps)		 Size
(cents)		 Interval name Just
ratio		 Just
(cents)		 Error
(cents)		 Limit
53 1200 perfect octave 2:1 1200 0 2
48 1086.79 classic major seventh 15:8 1088.27 −1.48 5
45 1018.87 just minor seventh 9:5 1017.60 +1.27 5
44 996.23 Pythagorean minor seventh 16:9 996.09 +0.14 3
43 973.59 harmonic seventh 7:4 968.83 +4.76 7
39 882.96 major sixth 5:3 884 −1.04 5
36 815.09 minor sixth 8:5 813.69 +1.40 5
31 701.89 perfect fifth 3:2 701.96 −0.07 3
27 611.32 Pythagorean augmented fourth 729:512 611.73 −0.41 3
26 588.68 diatonic tritone 45:32 590.22 −1.54 5
26 588.68 septimal tritone 7:5 582.51 +6.17 7
25 566.04 classic tritone 25:18 568.72 −2.68 5
24 543.40 undecimal tritone 11:8 551.32 −7.92 11
24 543.40 double diminished fifth 512:375 539.10 +4.30 5
24 543.40 undecimal augmented fourth 15:11 536.95 +6.45 11
23 520.76 acute fourth 27:20 519.55 +1.21 5
22 498.11 perfect fourth 4:3 498.04 +0.07 3
21 475.47 grave fourth 320:243 476.54 −1.07 5
21 475.47 septimal narrow fourth 21:16 470.78 +4.69 7
20 452.83 classic augmented third 125:96 456.99 −4.16 5
20 452.83 tridecimal augmented third 13:10 454.21 −1.38 13
19 430.19 septimal major third 9:7 435.08 −4.90 7
19 430.19 classic diminished fourth 32:25 427.37 +2.82 5
18 407.54 Pythagorean ditone 81:64 407.82 −0.28 3
17 384.91 just major third 5:4 386.31 −1.40 5
16 362.26 grave major third 100:81 364.80 −2.54 5
16 362.26 neutral third, tridecimal 16:13 359.47 +2.79 13
15 339.62 neutral third, undecimal 11:9 347.41 −7.79 11
15 339.62 acute minor third 243:200 337.15 +2.47 5
14 316.98 just minor third 6:5 315.64 +1.34 5
13 294.34 Pythagorean semiditone 32:27 294.13 +0.21 3
12 271.70 classic augmented second 75:64 274.58 −2.88 5
12 271.70 septimal minor third 7:6 266.87 +4.83 7
11 249.06 classic diminished third 144:125 244.97 +4.09 5
10 226.41 septimal whole tone 8:7 231.17 −4.76 7
10 226.41 diminished third 256:225 223.46 +2.95 5
9 203.77 whole tone, major tone 9:8 203.91 −0.14 3
8 181.13 whole tone, minor tone 10:9 182.40 −1.27 5
7 158.49 neutral second, greater undecimal 11:10 165.00 −6.51 11
7 158.49 grave whole tone 800:729 160.90 −2.41 5
7 158.49 neutral second, lesser undecimal 12:11 150.64 +7.85 11
6 135.85 major diatonic semitone 27:25 133.24 +2.61 5
5 113.21 Pythagorean major semitone 2187:2048 113.69 −0.48 3
5 113.21 just diatonic semitone 16:15 111.73 +1.48 5
4 90.57 major limma 135:128 92.18 −1.61 5
4 90.57 Pythagorean minor semitone 256:243 90.22 +0.34 3
3 67.92 just chromatic semitone 25:24 70.67 −2.75 5
2 45.28 just diesis 128:125 41.06 +4.22 5
1 22.64 syntonic comma 81:80 21.51 +1.14 5
0 0 perfect unison 1:1 0.00 0.00 1

Scale diagram

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

Interval (steps) 3 2 4 3 2 3 2 1 2 4 1 4 3 2 4 3 2 3 2 1 2
Interval (cents) 68 45 91 68 45 68 45 23 45 91 23 91 68 45 91 68 45 68 45 23 45
Note name C C♯ D D D♯ E E♭ E♯ F F F G G G♯ A A♭ A♯ B B♭ B♯ C C
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

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 Archived 2009-02-15 at the Wayback Machine. (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.
  11. MIDI modelled sounding of the 53EDO piece by J. Slavenski.
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