53 equal temperament

File:Syntonic tuning continuum.svg

Figure 1: 53 TET on the syntonic temperament's tuning continuum at 701.89 cents, from Template:Harvp

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) (Template:ErrorTemplate:Category handler). Each step represents a frequency ratio of   2^1 ∕ 53 , or 22.6415 cents (Template:ErrorTemplate:Category handler), 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 Template:Sfrac, known as the schisma, and Template:Sfrac, 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 Template:Sfrac is closest to the 43rd note (counting from 0) and   2^{43 ∕ 53} = 1.7548   is only 4.8 cents sharp from the harmonic 7th   = Template:Sfrac in 53 TET, and using it for 7-limit harmony means that the septimal kleisma, the interval Template:Sfrac, 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 ( [Template:Sfrac]⁵³ ) is very nearly equal to 31 octaves (2³¹). He calculated this difference with six-digit accuracy to be Template:Sfrac. Later the same observation was made by the mathematician and music theorist Nicholas Mercator (c. Template:Trim), who calculated this value precisely as Template:Sfrac = Template:Sfrac, 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 Template:Sfrac of that comma ( ≈ 0.0682 cent ≈ Template:Sfrac syntonic comma ≈ Template:Sfrac 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. BosanquetTemplate:Rp and the American tuner J.P. White.Template:Rp 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.Template:Efn

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

Notation

File:Koma gosterim.png

Notation used in Ottoman classical music, where the whole notes are 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 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 (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♯double sharp, Etriple flat, D♭, C♯, Bdouble sharp, Ftriple flat, Edouble flat,
• D, Cdouble sharp, B♯double sharp, Fdouble flat, E♭, D♯, C♯double sharp, Gtriple flat, F♭,
• E, Ddouble sharp, Cdouble sharpdouble sharp/Adouble flatdouble flat, Gdouble flat,
• F, E♯, D♯double sharp, Atriple flat, G♭, F♯, Edouble sharp, Ddouble sharpdouble sharp/Bdouble flatdouble flat, Adouble flat,
• G, Fdouble sharp, E♯double sharp, Btriple flat, A♭, G♯, F♯double sharp, Ctriple flat, Bdouble flat,
• A, Gdouble sharp, Fdouble sharpdouble sharp/Ddouble flatdouble flat, Cdouble flat, B♭, A♯, G♯double sharp, Dtriple flat, C♭,
• B, Adouble sharp, Gdouble sharpdouble sharp/Edouble flatdouble flat, Ddouble flat, 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 (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-Cdouble flat, and C-F♭-G-A♯ is still another seventh chord. The utonal tetrad, the inversion of the otonal tetrad, is spelled C-D♯-G-Gdouble sharp.

Further septimal chords are the diminished triad, having the two forms C-D♯-G♭ and C-Fdouble flat-G♭, the subminor triad, C-Fdouble flat-G, the supermajor triad C-Ddouble sharp-G, and corresponding tetrads C-Fdouble flat-G-Bdouble flat and C-Ddouble sharp-G-A♯. Since 53-TET tempers out the septimal kleisma, the septimal kleisma augmented triad C-F♭-Btriple flat in its various inversions is also a chord of the system. So is the Orwell tetrad, C-F♭-Ddouble sharpdouble sharp-Gdouble sharp 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 Template:Sfrac 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

File:53ed2.svg

7-Limit just intonation intervals approximated in 53 TET

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 Template:Sfrac opposed to the purer Template:Sfrac, and minor thirds that are conversely narrow (Template:Sfrac compared to Template:Sfrac).

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 Template:Sfrac. 53 TET is very good as an approximation to any interval in 5 limit just intonation. Similarly, the pure just interval Template:Sfrac 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 (43 steps are 4.8 cents sharp for [[harmonic seventh|Template:Sfrac]]), but all such intervals are still quite closely matched with the highest deviation being the Template:Sfrac 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 (steps)	Size (cents)	Interval name	Nearest Just ratio	Just (cents)	Error (cents)	Limit
53	1200	perfect octave	Template:Sfrac	1200	0	2
52	1177.36	grave octave	Template:Sfrac	1178.49	−1.14	5
51	1154.72	just augmented seventh	Template:Sfrac	1158.94	−4.22	5
50	1132.08	diminished octave	Template:Sfrac	1129.33	+2.75	5
48	1086.79	just major seventh	Template:Sfrac	1088.27	−1.48	5
45	1018.87	just minor seventh	Template:Sfrac	1017.60	+1.27	5
44	996.23	Pythagorean minor seventh	Template:Sfrac	996.09	+0.14	3
43	973.59	accute augmented sixth	Template:Sfrac	976.54	−2.95	5
43	973.59	harmonic seventh	Template:Sfrac	968.83	+4.76	7
43	973.59	accute diminished seventh	Template:Sfrac	968.43	+5.15	5
42	950.94	just augmented sixth	Template:Sfrac	955.03	−4.09	5
42	950.94	just diminished seventh	Template:Sfrac	946.92	+4.02	5
39	883.02	major sixth	Template:Sfrac	884.36	−1.34	5
37	837.73	tridecimal neutral sixth	Template:Sfrac	840.53	−2.8	13
36	815.09	minor sixth	Template:Sfrac	813.69	+1.40	5
31	701.89	perfect fifth	Template:Sfrac	701.96	−0.07	3
30	679.25	grave fifth	Template:Sfrac	680.45	−1.21	5
28	633.96	just diminished fifth (greater tritone)	Template:Sfrac	631.28	+2.68	5
27	611.32	Pythagorean augmented fourth	Template:Sfrac	611.73	−0.41	3
27	611.32	greater ‘classic’ tritone	Template:Sfrac	609.78	+1.54	5
26	588.68	lesser ‘classic’ tritone	Template:Sfrac	590.22	−1.54	5
26	588.68	septimal tritone	Template:Sfrac	582.51	+6.17	7
25	566.04	just augmented fourth (lesser tritone)	Template:Sfrac	568.72	−2.68	5
24	543.40	undecimal major fourth	Template:Sfrac	551.32	−7.92	11
24	543.40	double diminished fifth	Template:Sfrac	539.10	+4.30	5
24	543.40	undecimal augmented fourth	Template:Sfrac	536.95	+6.45	11
23	520.76	acute fourth	Template:Sfrac	519.55	+1.21	5
22	498.11	perfect fourth	Template:Sfrac	498.04	+0.07	3
21	475.47	grave fourth	Template:Sfrac	476.54	−1.07	5
21	475.47	septimal narrow fourth	Template:Sfrac	470.78	+4.69	7
20	452.83	just augmented third	Template:Sfrac	456.99	−4.16	5
20	452.83	tridecimal augmented third	Template:Sfrac	454.21	−1.38	13
19	430.19	septimal major third	Template:Sfrac	435.08	−4.90	7
19	430.19	just diminished fourth	Template:Sfrac	427.37	+2.82	5
18	407.54	Pythagorean ditone	Template:Sfrac	407.82	−0.28	3
17	384.91	just major third	Template:Sfrac	386.31	−1.40	5
16	362.26	grave major third	Template:Sfrac	364.80	−2.54	5
16	362.26	neutral third, tridecimal	Template:Sfrac	359.47	+2.79	13
15	339.62	neutral third, undecimal	Template:Sfrac	347.41	−7.79	11
15	339.62	acute minor third	Template:Sfrac	337.15	+2.47	5
14	316.98	just minor third	Template:Sfrac	315.64	+1.34	5
13	294.34	Pythagorean semiditone	Template:Sfrac	294.13	+0.21	3
12	271.70	just augmented second	Template:Sfrac	274.58	−2.88	5
12	271.70	septimal minor third	Template:Sfrac	266.87	+4.83	7
11	249.06	just diminished third	Template:Sfrac	244.97	+4.09	5
10	226.41	septimal whole tone	Template:Sfrac	231.17	−4.76	7
10	226.41	diminished third	Template:Sfrac	223.46	+2.95	5
9	203.77	whole tone, major tone, greater tone, just second	Template:Sfrac	203.91	−0.14	3
8	181.13	grave whole tone, minor tone, lesser tone, grave second	Template:Sfrac	182.40	−1.27	5
7	158.49	neutral second, greater undecimal	Template:Sfrac	165.00	−6.51	11
7	158.49	doubly grave whole tone	Template:Sfrac	160.90	−2.41	5
7	158.49	neutral second, lesser undecimal	Template:Sfrac	150.64	+7.85	11
6	135.85	accute diatonic semitone	Template:Sfrac	133.24	+2.61	5
5	113.21	greater Pythagorean semitone	Template:Sfrac	113.69	−0.48	3
5	113.21	just diatonic semitone, just minor second	Template:Sfrac	111.73	+1.48	5
4	90.57	major limma	Template:Sfrac	92.18	−1.61	5
4	90.57	lesser Pythagorean semitone	Template:Sfrac	90.22	+0.34	3
3	67.92	just chromatic semitone	Template:Sfrac	70.67	−2.75	5
3	67.92	greater diesis	Template:Sfrac	62.57	+5.35	5
2	45.28	just diesis	Template:Sfrac	41.06	+4.22	5
1	22.64	syntonic comma	Template:Sfrac	21.51	+1.14	5
0	0	perfect unison	Template:Sfrac	0	0	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 (Pythagorean notation)	C		Etriple flat		C♯		D		Fdouble flat		D♯		F♭		Ddouble sharp		Cdouble sharpdouble sharp/Adouble flatdouble flat		F		G♭		F♯		G		Btriple flat		GTemplate:Sharp		Bdouble flat		Cdouble flat		ATemplate:Sharp		C♭		Adouble sharp		Gdouble sharpdouble sharp/Edouble flatdouble flat		C
Note name (ups and downs notation)	C		vvCTemplate:Sharp/vDTemplate:Flat		CTemplate:Sharp/^DTemplate:Flat		D		vvDTemplate:Sharp/vETemplate:Flat		DTemplate:Sharp/^ETemplate:Flat		vE		^E		^^E/vvF		F		vFTemplate:Sharp/GTemplate:Flat		FTemplate:Sharp/^GTemplate:Flat		G		vvGTemplate:Sharp/vATemplate:Flat		GTemplate:Sharp/^ATemplate:Flat		vA		vvATemplate:Sharp/vBTemplate:Flat		ATemplate:Sharp/^BTemplate:Flat		vB		^B		^^B/vvC		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

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 22.6415 cents, equal to one step of 53 equal temperament, or $\sqrt[53]{2}$ (Template:ErrorTemplate:Category handler). 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 (21.51 cents) (Template:ErrorTemplate:Category handler), 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 BoethiusTemplate:Efn) 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 5 tones + 2 diatonic semitones,   5 × 9 + 2 × 4 = 53 equal commas. Holder attributes the division of the octave in 53 equal parts to Nicholas Mercator,Template:Efn who himself had proposed that Template:Sfrac part 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.Template:Efn One of these intervals was first described by Jing Fang in 45 Template:Sc. Mercator applied logarithms to determine that $\sqrt[55]{2}$ (≈ 21.8182 cents) was nearly equivalent to a syntonic comma of ≈ 21.5063 cents (a feature of the prevalent meantone temperament of the time). He also considered that an "artificial comma" of $\sqrt[53]{2}$ might be useful, because 31 octaves could be practically approximated by a cycle of 53 just fifths. Holder, for whom the Holdrian comma is named, favored this latter unit because the intervals of 53 equal 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.

Name of interval	Commas	Cents	Symbol
Koma	1	22.64	F
Bakiye	4	90.57	B
Küçük Mücennep	5	113.21	S
Büyük Mücennep	8	181.13	K
Tanini	9	203.77	T
Artık Aralık (12)	12	271.70	A (12)
Artık Aralık (13)	13	294.34	A (13)

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:

File:Makam aeu rast wp.png

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

File:Makam aeu nihavent wp.png

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 9 commas.

External links

• — Tonal functions as 53 TET grades.