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 . Each step represents a frequency ratio of 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 known as the schisma, and 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, tempering out the kleisma.
The interval of is closest to the 43rd note and is only 4.8 cents sharp from the harmonic 7th in 53 TET, 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, a Chinese music theorist, observed that a series of 53 just fifths is very nearly equal to 31 octaves. 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 53 equal temperament flattens each fifth by only of that comma, 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 This property of 53 TET may have been known earlier; Isaac Newton's unpublished manuscripts suggest that he had been aware of it as early as 1664–1665.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

Attempting to use standard notation, seven-letter notes plus sharps or flats, can quickly become confusing. This is unlike the case with 19 TET and 31 TET where there is little ambiguity. By not being meantone, it adds some problems that require more attention. Specifically, the Pythagorean major third and just major third are distinguished, as are the Pythagorean minor third 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, double flats, 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

Unfortunately, the notes run out of letter-order, and up to quadruple sharps and flats are 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, 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 53 TET, 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.

Major triad: C-vE-G
Minor triad: C-^E-G
Dominant 7th: C-vE-G-B
Otonal tetrad: C-vE-G-vB
Utonal tetrad: C-^E-G-^A
Diminished triad: C-^E-G
Diminished triad: C-vE-G
Subminor triad: C-vE-G
Supermajor triad: C-^E-G
Subminor tetrad: C-vE-G-vA
Supermajor tetrad: C-^E-G-^B
Augmented triad: C-vE-vvG
Orwell triad: C-vE-vvG-^A
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 pure, major thirds that are wide from just.
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 53 TET is very good as an approximation to any interval in 5 limit just intonation. Similarly, the pure just interval is only 1.3 cents wider than 14 steps in 53 TET.
The matches to the just intervals involving the 7th harmonic are slightly less close, but all such intervals are still quite closely matched with the highest deviation being the tritone. The 11th harmonic 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.

Size	Size	Interval name	Nearest Just ratio	Just	Error	Limit
53	1200	perfect octave		1200	0	2
52	1177.36	grave octave		1178.49	−1.14	5
51	1154.72	just augmented seventh		1158.94	−4.22	5
50	1132.08	diminished octave		1129.33	+2.75	5
48	1086.79	just major seventh		1088.27	−1.48	5
45	1018.87	just minor seventh		1017.60	+1.27	5
44	996.23	Pythagorean minor seventh		996.09	+0.14	3
43	973.59	acute augmented sixth		976.54	−2.95	5
43	973.59	harmonic seventh		968.83	+4.76	7
43	973.59	acute diminished seventh		968.43	+5.15	5
42	950.94	just augmented sixth		955.03	−4.09	5
42	950.94	just diminished seventh		946.92	+4.02	5
41	928.30	septimal major sixth		933.13	−4.83	7
40	905.66	Pythagorean major sixth		905.87	−0.21	3
39	883.02	major sixth		884.36	−1.34	5
37	837.73	tridecimal neutral sixth		840.53	−2.8	13
36	815.09	minor sixth		813.69	+1.40	5
31	701.89	perfect fifth		701.96	−0.07	3
30	679.25	grave fifth		680.45	−1.21	5
28	633.96	just diminished fifth		631.28	+2.68	5
27	611.32	Pythagorean augmented fourth		611.73	−0.41	3
27	611.32	greater ‘classic’ tritone		609.78	+1.54	5
26	588.68	lesser ‘classic’ tritone		590.22	−1.54	5
26	588.68	septimal tritone		582.51	+6.17	7
25	566.04	just augmented fourth		568.72	−2.68	5
24	543.40	undecimal major fourth		551.32	−7.92	11
24	543.40	double diminished fifth		539.10	+4.30	5
24	543.40	undecimal augmented fourth		536.95	+6.45	11
23	520.76	acute fourth		519.55	+1.21	5
22	498.11	perfect fourth		498.04	+0.07	3
21	475.47	grave fourth		476.54	−1.07	5
21	475.47	septimal narrow fourth		470.78	+4.69	7
20	452.83	just augmented third		456.99	−4.16	5
20	452.83	tridecimal augmented third		454.21	−1.38	13
19	430.19	septimal major third		435.08	−4.90	7
19	430.19	just diminished fourth		427.37	+2.82	5
18	407.54	Pythagorean ditone		407.82	−0.28	3
17	384.91	just major third		386.31	−1.40	5
16	362.26	grave major third		364.80	−2.54	5
16	362.26	neutral third, tridecimal		359.47	+2.79	13
15	339.62	neutral third, undecimal		347.41	−7.79	11
15	339.62	acute minor third		337.15	+2.47	5
14	316.98	just minor third		315.64	+1.34	5
13	294.34	Pythagorean semiditone		294.13	+0.21	3
12	271.70	just augmented second		274.58	−2.88	5
12	271.70	septimal minor third		266.87	+4.83	7
11	249.06	just diminished third		244.97	+4.09	5
10	226.41	septimal whole tone		231.17	−4.76	7
10	226.41	diminished third		223.46	+2.95	5
9	203.77	whole tone, major tone, greater tone, just second		203.91	−0.14	3
8	181.13	grave whole tone, minor tone, lesser tone,		182.40	−1.27	5
7	158.49	neutral second, greater undecimal		165.00	−6.51	11
7	158.49	doubly grave whole tone		160.90	−2.41	5
7	158.49	neutral second, lesser undecimal		150.64	+7.85	11
6	135.85	acute diatonic semitone		133.24	+2.61	5
5	113.21	greater Pythagorean semitone		113.69	−0.48	3
5	113.21	just diatonic semitone, just minor second		111.73	+1.48	5
4	90.57	major limma		92.18	−1.61	5
4	90.57	lesser Pythagorean semitone		90.22	+0.34	3
3	67.92	just chromatic semitone		70.67	−2.75	5
3	67.92	greater diesis		62.57	+5.35	5
2	45.28	septimal diesis		48.77	−3.49	7
2	45.28	just diesis		41.06	+4.22	5
1	22.64	syntonic comma		21.51	+1.14	5
0	0	perfect unison		0	0	1

53 equal temperament

History and use

Music

Notation

Chords of 53 equal temperament

Interval size