53 equal temperament

Figure 1: 53 TET on the syntonic temperament's tuning continuum at 701.89 cents, from Milne, Sethares & Plamondon (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). Play^ⓘ Each step represents a frequency ratio of   2^1 ∕ 53 , or 22.6415 cents (Play^ⓘ), 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 ⁠ 32 805 / 32 768 ⁠, known as the schisma, and ⁠ 15 625 / 15 552 ⁠, 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 ⁠ 7 / 4 ⁠ 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   = ⁠ 7 / 4 ⁠ in 53 TET, 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. Jing 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 ⁠ 177 147 / 176 776 ⁠.^[2][3] 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 ⁠ = ⁠ 19 383 245 667 680 019 896 796 723 / 19 342 813 113 834 066 795 298 816 ⁠,^{[verification needed]} which is known as Mercator's comma.^[4] 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.^[5][6] 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.^[7]

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^{[8](p 328–329)} and the American tuner J.P. White.^[8](p 329) 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.^{[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][11][a]}

Furthermore, General Thompson worked in league with the London-based guitar maker Louis Panormo to produce the Enharmonic Guitar.^[12]

Notation

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♯, 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^[13] 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 53 TET,^[14] it provides precise chord names too. The pythagorean minor chord with a ⁠ 32 / 27 ⁠ 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

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 ⁠ 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. Similarly, the pure just interval ⁠ 6 / 5 ⁠ 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 ⁠ 7 / 4 ⁠), but all such intervals are still quite closely matched with the highest deviation being the ⁠ 7 / 5 ⁠ 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	⁠ 2 / 1 ⁠	1200	0	2
52	1177.36	grave octave	⁠ 160 / 81 ⁠	1178.49	−1.14	5
51	1154.72	just augmented seventh	⁠ 125 / 64 ⁠	1158.94	−4.22	5
50	1132.08	diminished octave	⁠ 48 / 25 ⁠	1129.33	+2.75	5
48	1086.79	just 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	accute augmented sixth	⁠ 225 / 128 ⁠	976.54	−2.95	5
43	973.59	harmonic seventh	⁠ 7 / 4 ⁠	968.83	+4.76	7
43	973.59	accute diminished seventh	⁠ 17 496 / 10 000 ⁠	968.43	+5.15	5
42	950.94	just augmented sixth	⁠ 125 / 72 ⁠	955.03	−4.09	5
42	950.94	just diminished seventh	⁠ 216 / 125 ⁠	946.92	+4.02	5
39	883.02	major sixth	⁠ 5 / 3 ⁠	884.36	−1.34	5
37	837.73	tridecimal neutral sixth	⁠ 13 / 8 ⁠	840.53	−2.8	13
36	815.09	minor sixth	⁠ 8 / 5 ⁠	813.69	+1.40	5
31	701.89	perfect fifth	⁠ 3 / 2 ⁠	701.96	−0.07	3
30	679.25	grave fifth	⁠ 40 / 27 ⁠	680.45	−1.21	5
28	633.96	just diminished fifth (greater tritone)	⁠ 36 / 25 ⁠	631.28	+2.68	5
27	611.32	Pythagorean augmented fourth	⁠ 729 / 512 ⁠	611.73	−0.41	3
27	611.32	greater ‘classic’ tritone	⁠ 64 / 45 ⁠	609.78	+1.54	5
26	588.68	lesser ‘classic’ tritone	⁠ 45 / 32 ⁠	590.22	−1.54	5
26	588.68	septimal tritone	⁠ 7 / 5 ⁠	582.51	+6.17	7
25	566.04	just augmented fourth (lesser tritone)	⁠ 25 / 18 ⁠	568.72	−2.68	5
24	543.40	undecimal major fourth	⁠ 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	just 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	just 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	just 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	just 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, greater tone, just second	⁠ 9 / 8 ⁠	203.91	−0.14	3
8	181.13	grave whole tone, minor tone, lesser tone, grave second	⁠ 10 / 9 ⁠	182.40	−1.27	5
7	158.49	neutral second, greater undecimal	⁠ 11 / 10 ⁠	165.00	−6.51	11
7	158.49	doubly 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	accute diatonic semitone	⁠ 27 / 25 ⁠	133.24	+2.61	5
5	113.21	greater Pythagorean semitone	⁠ 2 187 / 2 048 ⁠	113.69	−0.48	3
5	113.21	just diatonic semitone, just minor second	⁠ 16 / 15 ⁠	111.73	+1.48	5
4	90.57	major limma	⁠ 135 / 128 ⁠	92.18	−1.61	5
4	90.57	lesser Pythagorean semitone	⁠ 256 / 243 ⁠	90.22	+0.34	3
3	67.92	just chromatic semitone	⁠ 25 / 24 ⁠	70.67	−2.75	5
3	67.92	greater diesis	⁠ 648 / 625 ⁠	62.57	+5.35	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	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		E		C♯		D		F		D♯		F♭		D		C/A		F		G♭		F♯		G		B		G♯		B		C		A♯		C♭		A		G/E		C
Note name (ups and downs notation)	C		vvC♯/vD♭		C♯/^D♭		D		vvD♯/vE♭		D♯/^E♭		vE		^E		^^E/vvF		F		vF♯/G♭		F♯/^G♭		G		vvG♯/vA♭		G♯/^A♭		vA		vvA♯/vB♭		A♯/^B♭		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,^[15] is a small musical interval of approximately 22.6415 cents,^[15] equal to one step of 53 equal temperament, or ${\displaystyle \ {\sqrt[{53}]{2\;}}\ }$ (play^ⓘ). 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)(play^ⓘ), 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^[b]) 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^[18] attributes the division of the octave in 53 equal parts to Nicholas Mercator,^[c] who himself had proposed that ⁠1/ 53 ⁠ 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.^[d] One of these intervals was first described by Jing Fang in 45 BCE.^[15] Mercator applied logarithms to determine that ${\displaystyle \ {\sqrt[{55}]{2\;}}\ }$ (≈ 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 ${\displaystyle \ {\sqrt[{53}]{2\;}}\ }$ 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 Holder koması 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:

where denotes a Holdrian comma flat,^[e] 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 9 commas.^[15]

Notes

1. "Croatian composer Josip Štolcer-Slavenski wrote one piece,^[9][10] which has never been published, which uses Bosanquet's Enharmonium during its first movement, entitled Music for Natur-ton-system".^[11]
2. According to Boethius, Pythagoras' disciple Philolaus of Croton would have said that the tone consisted in two diatonic semitones and a comma; the diatonic semitone consisted in two diaschismata, each formed of two commas.^[16][17]
3. "The late Nicholas Mercator, a Modest Person, and a Learned and Judicious Mathematician, in a Manuscript of his, of which I have had a Sight."^[18]
4. Holder (1731) writes that Marin Mersenne had calculated 58 ⁠ 1 / 4 ⁠ commas in the octave; Mercator "working by the logarithms, finds out but 55, and a little more."^[18]
5. In common Arabic and Turkish practice, the third note e and the seventh note b in Rast are even lower than in this theory, almost exactly halfway between western major and minor thirds above c and g, i.e. closer to 6.5 commas (three-quarter tone) above d or a and 6.5 below f or c, the thirds c–e and g–b often referred to as a "neutral thirds" by musicologists.

References

1. Milne, Andrew; Sethares, William; Plamondon, James (2007). "Isomorphic controllers and dynamic tuning: Invariant fingering over a tuning continuum". Computer Music Journal. 31 (4): 15–32. doi:10.1162/comj.2007.31.4.15. S2CID 27906745 – via mitpressjournals.org.
2. McClain, Ernest; Hung, Ming Shui (1979). "Chinese cyclic tunings in late antiquity". Ethnomusicology. 23 (2): 205–224.
3. "後漢書/卷91 - 维基文库，自由的图书馆" [Book of the Later Han Dynasty / Volume 91 - Wikisource, the free library]. zh.wikisource.org (in Chinese). Retrieved 2022-06-23.
4. Monzo, Joe (2005). "Mercator's comma". Tonalsoft.
5. Holder (1967)
6. Stanley, Jerome (2002). William Holder and His Position in Seventeenth-Century Philosophy and Music Theory. The Edwin Mellen Press. — see also Holder (1967)
7. Barbieri, Patrizio (2008). Enharmonic Instruments and Music, 1470–1900. Latina, Il Levante Libreria Editrice. p. 350. Archived from the original on 2009-02-15.
8. von Helmholtz, H.L.F. (1954). Ellis, Alexander (ed.). On the Sensations of Tone (2nd English ed.). Dover Publications. pp. 328–329.
9. Slavencki, Josip (21 June 2007a). "Preface". 53 EDO piece (manuscript). Belgrade, Serbia: The Faculty of Music – via Wikimedia Commons.
10. Slavencki, Josip (21 June 2007b). "Title". 53 EDO movement (manuscript). Belgrade, Serbia: The Faculty of Music – via Wikimedia Commons.
11. Slavenski, Josip (February 2018). Khramov, Mykhaylo (ed.). "Music Natural 53e6v". 53 EDO movement. soundcloud.com – via soundcloud.com.
    Khramov, Mykhaylo, ed. (February 2018). "Link to ZIP with materials". 53 EDO movement – via Google Drive.
12. Westbrook, James (2012). "General Thompson's enharmonic guitar". Soundboard. Vol. 38, no. 4. pp. 45–52.
13. "Ups and downs notation". Xenharmonic Wiki (en.xen.wiki). Retrieved 19 August 2024.
14. "53edo intervals". Xenharmonic Wiki (en.xen.wiki). Retrieved 19 August 2024.
15. Touma, H.H. (1996). The Music of the Arabs. Translated by Schwartz, Laurie. Portland, OR: Amadeus Press. p. 23. ISBN 0-931340-88-8.
16. Boethius, Anicius Manlius Severinus. De institutione musica. Book 3, Chapter 8.
17. Barbour, J.M. (1951). Tuning and Temperament: A historical survey. p. 123.
18. Holder, W. (1731). A Treatise of the Natural Grounds, and Principles of Harmony (3rd ed.). London, UK. p. 79.

• Holder, William (1967) [1694]. A Treatise on the Natural Grounds, and Principles of Harmony (facsimile ed.). New York, NY: Broude Brothers. pp. 103–106.

External links

• Rodgers, Prent (May 2007). "Whisper song in 53 EDO". Bumper Music (podcast) (slower ed.).
• Hanson, Larry (1989). "Development of a 53 tone keyboard layout" (PDF). Xenharmonicon. XII. Hanover, NH: Frog Peak Music: 68–85. Retrieved 4 January 2021 – via Anaphoria.com.
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