53edo: Difference between revisions

← 52edo 53edo 54edo →
Prime factorization	53 (prime)
Step size	22.6415 ¢
Fifth	31\53 (701.887 ¢); (convergent)
Semitones (A1:m2)	5:4 (113.2 ¢ : 90.57 ¢)
Consistency limit	9
Distinct consistency limit	9

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53 equal temperament

53 equal divisions of the octave (abbreviated 53edo or 53ed2), also called 53-tone equal temperament (53tet) or 53 equal temperament (53et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 53 equal parts of about 22.6 ¢ each. Each step represents a frequency ratio of 2^1/53, or the 53rd root of 2.

Theory

53edo is notable as an excellent 5-limit system, a fact apparently first noted by Isaac Newton^[1]. It is the seventh strict zeta edo. In the opinion of some, 53edo is the first equal division to deal adequately with the 13-limit, while others award that distinction to 41edo or 46edo. Like 41 and 46, 53 is distinctly consistent in the 9-odd-limit (and if we exclude the most damaged interval pair, 7/5 and 10/7, is consistent to distance 2), but among them, 53 is the first that finds the interseptimal intervals 15/13 and 13/10 distinctly from adjacent septimal intervals 8/7 and 7/6, and 9/7 and 21/16, respectively, which is essential to its 13-limit credibility. It also avoids equating 11/9 with 16/13, so that the former is tuned very flat to equate it with a slightly flat ~39/32 – a feature shared by 46edo. It is almost consistent to the entire 15-odd-limit, with the only inconsistency occurring at 14/11 (and its octave complement), which is mapped inconsistently sharp and equated with 9/7, but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the 27-odd-limit. It shines however in the 2.3.5.19 and 2.3.5.13 subgroups, where it offers excellent approximations with decent complexity.

53edo has also found a certain dissemination as an edo tuning for Arabic, Turkish, and Persian music. It can also be used as an extended Pythagorean tuning, since its fifths are indistinguishable from just in most contexts.

53edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the step's role as a "comma" comes from it being an approximation of the Pythagorean comma and syntonic comma.

Prime harmonics

Approximation of prime harmonics in 53edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.07	-1.41	+4.76	-7.92	-2.79	+8.25	-3.17	+5.69	-10.71	+9.68
Error	Relative (%)	+0.0	-0.3	-6.2	+21.0	-35.0	-12.3	+36.4	-14.0	+25.1	-47.3	+42.8
Steps (reduced)		53 (0)	84 (31)	123 (17)	149 (43)	183 (24)	196 (37)	217 (5)	225 (13)	240 (28)	257 (45)	263 (51)

Approximation of prime harmonics in 53edo (continued)
Harmonic		37	41	43	47	53	59	61	67	71	73	79	83
Error	Absolute (¢)	-2.29	+1.13	+9.24	-8.90	+9.51	+4.98	-7.45	+11.26	+1.44	-1.37	-2.27	+2.78
Error	Relative (%)	-10.1	+5.0	+40.8	-39.3	+42.0	+22.0	-32.9	+49.7	+6.3	-6.1	-10.0	+12.3
Steps (reduced)		276 (11)	284 (19)	288 (23)	294 (29)	304 (39)	312 (47)	314 (49)	322 (4)	326 (8)	328 (10)	334 (16)	338 (20)

See #Approximation to JI for details and a more in-depth discussion on the higher harmonics.

As a tuning of other temperaments

As an equal temperament, 53et notably tempers out Mercator's comma (3⁵³/2⁸⁴), the schisma (32805/32768), kleisma (15625/15552), and amity comma (1600000/1594323). In the 7-limit it tempers out the marvel comma (225/224) for which it is a relatively efficient tuning, orwellisma (1728/1715), gariboh comma (3125/3087), and ragisma (4375/4374). In the 11-limit, it tempers out 99/98 and 121/120 (in addition to their difference, 540/539), and is the optimal patent val for big brother temperament, which tempers out both, as well as 11-limit orwell temperament, which also tempers out the 11-limit commas 176/175 and 385/384. In the 13-limit, it tempers out 169/168, 275/273, 325/324, 625/624, 676/675, 1001/1000, 2080/2079, and 4096/4095, and gives the optimal patent val for athene temperament.

Subsets and supersets

53edo is the 16th prime edo, following 47edo and coming before 59edo.

Many of its multiples such as 159edo, 212edo, 742edo, 901edo and the zeta 954edo have good consistency limits and are each of their own interest. The mercator family comprises rank-2 temperaments with 1/53-octave periods.

Intervals

#	Cents	Approximate ratios^{[note 1]}	Ups and downs notation (EUs: v⁵A1 and ^d2)			Solfeges
0	0.0	1/1	P1	unison	D	Da	Do
1	22.6	50/49, 64/63, 81/80	^1	up unison	^D	Du	Di
2	45.3	33/32, 36/35, 49/48, 128/125	^^1, vvm2	dup unison, dudminor 2nd	^^D, vvEb	Di / Fre	Daw
3	67.9	22/21, 25/24, 26/25, 27/26, 28/27	vvA1, vm2	dudaug 1sn, downminor 2nd	vvD#, vEb	Fro	Ro
4	90.6	19/18, 20/19, 21/20, 256/243	vA1, m2	downaug 1sn, minor 2nd	vD#, Eb	Fra	Rih
5	113.2	15/14, 16/15	A1, ^m2	aug 1sn, upminor 2nd	D#, ^Eb	Fru	Ra
6	135.8	13/12, 14/13, 27/25	^^m2	dupminor 2nd	^^Eb	Fri	Ru
7	158.5	11/10, 12/11, 35/32, 57/52, 800/729	vvM2	dudmajor 2nd	vvE	Re	Ruh
8	181.1	10/9	vM2	downmajor 2nd	vE	Ro	Reh
9	203.8	9/8	M2	major 2nd	E	Ra	Re
10	226.4	8/7, 256/225	^M2	upmajor 2nd	^E	Ru	Ri
11	249.1	15/13, 22/19, 125/108, 144/125	^^M2, vvm3	dupmajor 2nd, dudminor 3rd	^^E, vvF	Ri / Ne	Raw
12	271.7	7/6, 75/64	vm3	downminor 3rd	vF	No	Ma
13	294.3	13/11, 19/16, 32/27	m3	minor 3rd	F	Na	Meh
14	317.0	6/5	^m3	upminor 3rd	^F	Nu	Me
15	339.6	11/9, 243/200	^^m3	dupminor 3rd	^^F	Ni	Mu
16	362.3	16/13, 100/81	vvM3	dudmajor 3rd	vvF#	Me	Muh
17	384.9	5/4	vM3	downmajor 3rd	vF#	Mo	Mi
18	407.5	19/15, 24/19, 81/64	M3	major 3rd	F#	Ma	Maa
19	430.2	9/7, 14/11	^M3	upmajor 3rd	^F#	Mu	Mo
20	452.8	13/10, 125/96, 162/125	^^M3, vv4	dupmajor 3rd, dud 4th	^^F#, vvG	Mi / Fe	Maw
21	475.5	21/16, 25/19, 320/243, 675/512	v4	down 4th	vG	Fo	Fe
22	498.1	4/3	P4	perfect 4th	G	Fa	Fa
23	520.8	19/14, 27/20	^4	up 4th	^G	Fu	Fih
24	543.4	11/8, 15/11, 26/19	^^4	dup 4th	^^G	Fi / She	Fu
25	566.0	18/13	vvA4, vd5	dudaug 4th, downdim 5th	vvG#, vAb	Pe / Sho	Fuh
26	588.7	7/5, 45/32	vA4, d5	downaug 4th, dim 5th	vG#, Ab	Po / Sha	Fi
27	611.3	10/7, 64/45	A4, ^d5	aug 4th, updim 5th	G#, ^Ab	Pa / Shu	Se
28	634.0	13/9	^A4, ^^d5	upaug 4th, dupdim 5th	^G#, ^^Ab	Pu / Shi	Suh
29	656.6	16/11, 19/13, 22/15	vv5	dud 5th	vvA	Pi / Se	Su
30	679.2	28/19, 40/27	v5	down 5th	vA	So	Sih
31	701.9	3/2	P5	perfect 5th	A	Sa	Sol
32	724.5	32/21, 38/25, 243/160, 1024/675	^5	up 5th	^A	Su	Si
33	747.2	20/13, 125/81, 192/125	^^5, vvm6	dup 5th, dudminor 6th	^^A, vvBb	Si / Fle	Saw
34	769.8	11/7, 14/9, 25/16	vm6	downminor 6th	vBb	Flo	Lo
35	792.5	19/12, 30/19, 128/81	m6	minor 6th	Bb	Fla	Leh
36	815.1	8/5	^m6	upminor 6th	^Bb	Flu	Le
37	837.7	13/8, 81/50	^^m6	dupminor 6th	^^Bb	Fli	Lu
38	860.4	18/11, 400/243	vvM6	dudmajor 6th	vvB	Le	Luh
39	883.0	5/3	vM6	downmajor 6th	vB	Lo	La
40	905.7	22/13, 27/16, 32/19	M6	major 6th	B	La	Laa
41	928.3	12/7	^M6	upmajor 6th	^B	Lu	Li
42	950.9	19/11, 26/15, 125/72, 216/125	^^M6, vvm7	dupmajor 6th, dudminor 7th	^^B, vvC	Li / The	Law
43	973.6	7/4	vm7	downminor 7th	vC	Tho	Ta
44	996.2	16/9	m7	minor 7th	C	Tha	Teh
45	1018.9	9/5	^m7	upminor 7th	^C	Thu	Te
46	1041.5	11/6, 20/11, 64/35, 729/400	^^m7	dupminor 7th	^^C	Thi	Tu
47	1064.2	13/7, 24/13, 50/27	vvM7	dudmajor 7th	vvC#	Te	Tuh
48	1086.8	15/8	vM7	downmajor 7th	vC#	To	Ti
49	1109.4	19/10, 36/19, 40/21, 243/128	M7	major 7th	C#	Ta	Tih
50	1132.1	21/11, 25/13, 27/14, 52/27, 48/25	^M7	upmajor 7th	^C#	Tu	To
51	1154.7	35/18, 64/33, 96/49, 125/64	^^M7, vv8	dupmajor 7th, dud 8ve	^^C#, vvD	Ti / De	Taw
52	1177.4	49/25, 63/32, 160/81	v8	down 8ve	vD	Do	Da
53	1200.0	2/1	P8	perfect 8ve	D	Da	Do

Interval quality and chord names in color notation

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

Quality	Color	Monzo format	Examples
downminor	zo	(a, b, 0, 1)	7/6, 7/4
minor	fourthward wa	(a, b) with b < −1	32/27, 16/9
upminor	gu	(a, b, −1)	6/5, 9/5
dupminor	ilo	(a, b, 0, 0, 1)	11/9, 11/6
dudmajor	lu	(a, b, 0, 0, −1)	12/11, 18/11
downmajor	yo	(a, b, 1)	5/4, 5/3
major	fifthward wa	(a, b) with b > 1	9/8, 27/16
upmajor	ru	(a, b, 0, −1)	9/7, 12/7

All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked−3rds chord 6 – 1 – 3 – 5 – 7 – 9 – 11 – 13). Alterations are always enclosed in parentheses, additions never are.

Here are the zo, gu, ilo, lu, yo and ru triads:

Color of the 3rd	JI chord	Notes as edosteps	Notes of C chord	Written name	Spoken name
zo	6:7:9	0–12–31	C vEb G	Cvm	C downminor
gu	10:12:15	0–14–31	C ^Eb G	C^m	C upminor
ilo	18:22:27	0–15–31	C ^^Eb G	C^^m	C dupminor
lu	22:27:33	0–16–31	C vvE G	Cvv	C dudmajor or C dud
yo	4:5:6	0–17–31	C vE G	Cv	C downmajor or C down
ru	14:18:21	0–19–31	C ^E G	C^	C upmajor or C up

For a more complete list, see Ups and downs notation #Chords and chord progressions.

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats with arrows:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12
Sharp symbol													
Flat symbol													

Kite's ups and downs notation

53edo can also be notated with Kite's ups and downs, spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

Step offset	0	1	2	3	4	5	6	7	8	9	10	11
Sharp symbol
Flat symbol

Sagittal notation

53edo can be notated in Sagittal using the Spartan set, with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:

Steps		0	1	2	3	4	5
Symbol	Evo						
Symbol	Revo						

The following enharmonics from the Spartan set are present (comma tempered out):

 =  =  (325/324, 352/351)
 =  (225/224)
 =  (5120/5103)

See apotome complements for equivalent accidental pairs.

Featured below is the 53edo gamut notated using the best accidental approximants; in this case, pai/pao and phai/phao.

Evo flavor

Revo flavor

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

Relationship to 12edo

53edo's circle of 53 fifths can be bent into a 12-spoked "spiral of fifths". This makes sense to do because going up by 12 fifths results in the Pythagorean comma (by definition), which is mapped to one edostep and is thus also the syntonic and septimal comma, introducing a simple second accidental in the form of the arrow to reach useful intervals from the basic 12-chromatic scale. The one-edostep comma is a requirement in Kite's theory, and implies that 31\53 is on the 7\12 kite in the scale tree.

This "spiral of fifths" can be a useful construct for introducing 53edo to musicians unfamiliar with microtonal music. It may help composers and musicians to make visual sense of the notation, and to understand what size of a jump is likely to land them where compared to 12edo.

The two innermost and two outermost intervals on the spiral are duplicates, reflecting the fact that it is a repeating circle at heart and the spiral shape is only a helpful illusion.

Approximation to JI

53edo provides excellent approximations for the classic 5-limit just chords and scales, such as the Ptolemy–Zarlino "just major" scale.

Interval	Ratio	Size	Difference
Perfect fifth	3/2	31	−0.07 cents
Major third	5/4	17	−1.40 cents
Minor third	6/5	14	+1.34 cents
Major second	9/8	9	−0.14 cents
Major second	10/9	8	−1.27 cents
Minor second	16/15	5	+1.48 cents

Because the 5th is so incredibly accurate (next edo with a more accurate fifth is 200edo), 53edo also offers a great approximation to Pythagorean tuning. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the septimal kleisma, 225/224.

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 53edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 53edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
3/2, 4/3	0.068	0.3
9/8, 16/9	0.136	0.6
9/5, 10/9	1.272	5.6
15/13, 26/15	1.316	5.8
5/3, 6/5	1.340	5.9
13/10, 20/13	1.384	6.1
5/4, 8/5	1.408	6.2
15/8, 16/15	1.476	6.5
13/9, 18/13	2.655	11.7
13/12, 24/13	2.724	12.0
13/8, 16/13	2.792	12.3
7/4, 8/7	4.759	21.0
7/6, 12/7	4.827	21.3
9/7, 14/9	4.895	21.6
13/11, 22/13	5.130	22.7
7/5, 10/7	6.167	27.2
15/14, 28/15	6.235	27.5
15/11, 22/15	6.445	28.5
11/10, 20/11	6.514	28.8
13/7, 14/13	7.551	33.3
11/9, 18/11	7.785	34.4
11/6, 12/11	7.854	34.7
11/8, 16/11	7.922	35.0
11/7, 14/11	9.961	44.0

15-odd-limit intervals in 53edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
3/2, 4/3	0.068	0.3
9/8, 16/9	0.136	0.6
9/5, 10/9	1.272	5.6
15/13, 26/15	1.316	5.8
5/3, 6/5	1.340	5.9
13/10, 20/13	1.384	6.1
5/4, 8/5	1.408	6.2
15/8, 16/15	1.476	6.5
13/9, 18/13	2.655	11.7
13/12, 24/13	2.724	12.0
13/8, 16/13	2.792	12.3
7/4, 8/7	4.759	21.0
7/6, 12/7	4.827	21.3
9/7, 14/9	4.895	21.6
13/11, 22/13	5.130	22.7
7/5, 10/7	6.167	27.2
15/14, 28/15	6.235	27.5
15/11, 22/15	6.445	28.5
11/10, 20/11	6.514	28.8
13/7, 14/13	7.551	33.3
11/9, 18/11	7.785	34.4
11/6, 12/11	7.854	34.7
11/8, 16/11	7.922	35.0
11/7, 14/11	12.681	56.0

Higher-limit JI

53edo has only 5 pairs of inconsistent intervals in the full 27-odd-limit: {11/7, 14/11}, {17/11, 22/17}, {19/17, 34/19}, {21/11, 22/21}, and {23/22, 44/23}. This is perhaps remarkable compared to 9 pairs in 46edo and 11 pairs in 41edo, because the smallest edo after 53edo to get 5 or less inconsistencies in the 27-odd-limit is 99edo (using the 99ef val), followed by 111edo (patent val), 130edo (patent val) and 159edo (also patent); all of these get 5 inconsistencies as well except 159edo which gets 1 and which is itself a superset of 53edo. However, most interpret the approximation of prime 17 in 53edo as too off for all but the most opportunistic harmonies, and some question the 23 and possibly also 11, so the practical significance of this is debatable.

As shown below, there is also a cluster of usable higher primes starting at 71; even 89 (4.84 ¢ flat), 97 (4.63 ¢ sharp) and 101 (2.6 ¢ sharp) are usable if placed in just the right context. (Note that prime 67 is almost perfectly off.)

Approximation of large prime harmonics in 53edo
Harmonic		71	73	79	83
Error	Absolute (¢)	+1.44	-1.37	-2.27	+2.78
Error	Relative (%)	+6.3	-6.1	-10.0	+12.3
Steps (reduced)		326 (8)	328 (10)	334 (16)	338 (20)

This makes 53edo excellent (for its size) in the 2.3.5.7.11.13.19.23.37.41.71.73(.79.83.101) subgroup, although some higher error primes like 11 and 23 require the right context to be convincing.

Note that the high primes, in rooted (/2ⁿ) position, essentially act as alternate interpretations of LCJI intervals, if you want to force a rooted interpretation; namely: 71/64 as ~10/9, 73/64 as ~8/7, 79/64 as ~16/13, and perhaps most questionably in the context of 53edo, 83/64 as ~13/10.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-84 53⟩	[⟨53 84]]	+0.022	0.022	0.10
2.3.5	15625/15552, 32805/32768	[⟨53 84 123]]	+0.216	0.276	1.22
2.3.5.7	225/224, 1728/1715, 3125/3087	[⟨53 84 123 149]]	−0.262	0.861	3.81
2.3.5.7.11	99/98, 121/120, 176/175, 2200/2187	[⟨53 84 123 149 183]]	+0.248	1.279	5.64
2.3.5.7.11.13	99/98, 121/120, 169/168, 176/175, 275/273	[⟨53 84 123 149 183 196]]	+0.332	1.183	5.22
2.3.5.7.11.13.19	99/98, 121/120, 169/168, 176/175, 209/208, 275/273	[⟨53 84 123 149 183 196 225]]	+0.391	1.105	4.88

53et is lower in relative error than any previous equal temperaments in the 3-, 5-, and 13-limit. The next equal temperaments doing better in these subgroups are 306, 118, and 58, respectively. It is even more prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups, and the next equal temperament doing better in either subgroup is 130.

Commas

Commas that 53edo tempers out using its patent val, ⟨53 84 123 149 183 196 225], include:

Prime limit	Ratio^{[note 2]}	Monzo	Cents	Color name	Name(s)
3	(52 digits)	[-84 53⟩	3.62	Tribilawa	53-comma, Mercator's comma
5	(14 digits)	[-21 3 7⟩	10.06	Lasepyo	Semicomma
5	15625/15552	[-6 -5 6⟩	8.11	Tribiyo	Kleisma
5	(14 digits)	[9 -13 5⟩	6.15	Saquinyo	Amity comma
5	(22 digits)	[24 -21 4⟩	4.20	Sasaquadyo	Vulture comma
5	32805/32768	[-15 8 1⟩	1.95	Layo	Schisma
7	3125/3087	[0 -2 5 -3⟩	21.18	Triru-aquinyo	Gariboh comma
7	1728/1715	[6 3 -1 -3⟩	13.07	Triru-agu	Orwellisma
7	225/224	[-5 2 2 -1⟩	7.71	Ruyoyo	Marvel comma, septimal kleisma
7	4375/4374	[-1 -7 4 1⟩	0.40	Zoquadyo	Ragisma
11	99/98	[-1 2 0 -2 1⟩	17.58	Loruru	Mothwellsma
11	121/120	[-3 -1 -1 0 2⟩	14.37	Lologu	Biyatisma
11	176/175	[4 0 -2 -1 1⟩	9.86	Lorugugu	Valinorsma
11	(22 digits)	[33 -23 0 0 1⟩	6.35	Trisalo	Pythrabian comma
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
11	540/539	[2 3 1 -2 -1⟩	3.21	Lururuyo	Swetisma
13	275/273	[0 -1 2 -1 1 -1⟩	12.64	Thuloruyoyo	Gassorma
13	169/168	[-3 -1 0 -1 0 2⟩	10.27	Thothoru	Buzurgisma, dhanvantarisma
13	625/624	[-4 -1 4 0 0 -1⟩	2.77	Thuquadyo	Tunbarsma
13	676/675	[2 -3 -2 0 0 2⟩	2.56	Bithogu	Island comma, parizeksma
13	1001/1000	[-3 0 -3 1 1 1⟩	1.73	Tholozotrigu	Fairytale comma, sinbadma
13	2080/2079	[5 -3 1 -1 -1 1⟩	0.83	Tholuruyo	Ibnsinma, sinaisma
13	4096/4095	[12 -2 -1 -1 0 -1⟩	0.42	Sathurugu	Minisma

Linear temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	2\53	45.3	36/35	Quartonic
1	5\53	113.2	16/15	Misneb
1	6\53	135.8	13/12~14/13	Doublethink
1	7\53	158.5	11/10	Hemikleismic
1	9\53	203.8	9/8	Baldy
1	10\53	226.4	8/7	Semaja
1	11\53	249.1	15/13	Hemischis / hemigari
1	12\53	271.7	7/6	Orwell
1	13\53	294.3	25/21	Kleiboh
1	14\53	317.0	6/5	Hanson / catakleismic / countercata
1	15\53	339.6	11/9	Amity / houborizic
1	16\53	362.3	16/13	Demibuzzard / submajor / interpental
1	18\53	407.5	1225/972	Ditonic / coditone
1	19\53	430.2	9/7	Hamity
1	20\53	452.8	13/10	Maja
1	21\53	475.5	21/16	Vulture / buzzard
1	22\53	498.1	4/3	Garibaldi / pontiac
1	23\53	520.8	4/3	Mavila (53bbcc)
1	25\53	566.0	18/13	Alphatrimot
1	26\53	588.7	45/32	Untriton / aufo

* Octave-reduced form, reduced to the first half-octave

Scales

Mos scales

While there is only one possible generator for the diatonic mos scale supported by this edo, there are a greater number of generators for other mosses such as the antidiatonic scale.

Scales approximated from JI

The eagle 53 scale described by John O'Sullivan
Ptolmey–Zarlino justly-intonated major: 9 8 5 9 8 9 5
Ptolmey–Zarlino justly-intonated minor: 9 5 8 9 5 9 8

From AFDOs

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.

Composite Cliffedge (approximated from 60afdo): 12 10 9 19 3
Composite Deja Vu (approximated from 101afdo): 14 17 5 9 8
Composite Dungeon (approximated from 30afdo): 17 5 9 4 18
Composite Freeway (approximated from 6afdo): 12 10 9 8 7 7
Composite Geode (approximated from 6afdo): 12 10 9 15 7
Composite Labyrinth (approximated from 30afdo): 7 7 17 5 17
Composite Mushroom (approximated from 30afdo): 12 10 9 3 19
Composite Underpass (approximated from 10afdo): 14 17 10 4 8
Spectral Arcade (approximated from 32afdo): 17 4 10 12 10
Spectral Mechanical (approximated from 16afdo): 13 4 14 12 10
Spectral Moonbeam (approximated from 16afdo): 9 4 18 17 5
Spectral Springwater (approximated from 8afdo): 9 8 14 12 10
Spectral Starship (approximated from 68ifdo): 4 13 4 10 12 10
Spectral Volcanic (approximated from 16afdo): 5 12 14 12 10

Other scales

cthon5m: 6 3 6 2 3 6 2 3 6 3 2 6 3 2
Palace^{[idiosyncratic term]} (approximated from Porky in 29edo): 7 7 8 9 7 7 8

Instruments

Music

See also: Category:53edo tracks

Modern renderings

Johann Sebastian Bach

"Jesus bleibet meine Freude" from Herz und Mund und Tat und Leben, BWV 147 (1723) – tuned in 53edo, rendered by Claudi Meneghin (2021)
Prelude and Fugue in C Major, No. 1, BWV 846, from The Well-Tempered Clavier, Book I (1722) – rendered by Mykhaylo Khramov
- Prelude · Fugue
"Ricercar a 3" from The Musical Offering, BWV 1079 (1747) – rendered by Claudi Meneghin (2024)
"Ricercar a 6" from The Musical Offering, BWV 1079 (1747) – rendered by Claudi Meneghin (2025)
"Ricercar a 6" from The Musical Offering, BWV 1079 (1747) – with syntonic-comma adjustment, rendered by Claudi Meneghin (2025)
"Contrapunctus 4" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)
"Contrapunctus 11" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)

Nicolaus Bruhns

Prelude in E Minor "The Great" – rendered by Claudi Meneghin (2023)
Prelude in E Minor "The Little" – rendered by Claudi Meneghin (2024)

Frédéric Chopin

Prelude Op. 28, No. 4 in E minor « Suffocation » (1839), arranged for harpsichord, tuned into 53-edo — rendered by Claudi Meneghin (2025)
- Near-Pythagorean version
- Schismatic version
"Waterfall" Étude from 12 Études, op. 10 (1829–1832) — rendered by Claudi Meneghin (2025)

George Frideric Handel

Suite in D minor HWV 428 for Harpsichord - Allemande (1720) – rendered by Claudi Meneghin (2024)

Scott Joplin

Maple Leaf Rag (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024 version; 2025 version))
Maple Leaf Rag (1899) – with syntonic comma adjustment, arranged for harpsichord and rendered by Claudi Meneghin (2024 version; (2025 version)

Shirō Sagisu

Qui veut faire l'ange fait la bete – rendered by MortisTheneRd (2024)
Bande-announce – rendered by MortisTheneRd (2024)

21st century

ALLY195

My Soul adaptation (2020)

Alxeusxiao

53edo exploration (2024)

Bryan Deister

microtonal improvisation in 53edo (2025)
Color & Electricity - muship (microtonal cover in 53edo) (2025)
53edo improv (2025)
Fantasy in 53edo (abstract version; visualizer version (2025)
Finale - Undertale (microtonal cover in 53edo) (2026)
Waltz in 53edo (2026)
- [Short] (with Lumatone view))
- [Full version]

Francium

Space Race (2022)
"strange worlds" from hope in dark times (2024) – Spotify | Bandcamp | YouTube – in Hanson[11], 53edo tuning
"Blasphemous Rumors" from TOTMC September to December 2024 (2024) – Spotify | Bandcamp | YouTube – in Blackdye, 53edo tuning
"It's a Good Idea to Have a Good Time." from Random Sentences (2025) – Spotify | Bandcamp | YouTube
"Decearing Egg" from Eggs (2025) – Spotify | Bandcamp | YouTube
"Husband Head Void" from Void (2025) – Spotify | Bandcamp | YouTube
"Lasagna Cat" from Microtonal Six-Dimensional Cats (2025) – Spotify | Bandcamp | YouTube
Opunish Bathomet (2025)

Andrew Heathwaite

Elf Dine on Ho Ho (2012) play^{[dead link]}
Spun (2012) play^{[dead link]}

Hideya

Like Uminari (2021)

Nathan Ho

53edo microtonal algorithmic IDM in SuperCollider (2023)

Aaron Krister Johnson (site^{[dead link]})

Desert Prayer^{[dead link]}

Logan02A4

53edo try (2024)

Claudi Meneghin

Orwell Micro Trio, for Organ (Just: 7 Orwells = 1 Twelfth) (2025) — actually in open-ended Orwell tuning, but with the generator adjusted to be extremely close to 12\53, at 271.71 ¢

Merct

drifting light (2025)
i don't want to die (2025)

MortisTheneRd

Mundoworld

from No Fun House (2025)
- "No Explanations" – Spotify | Bandcamp | YouTube – in Gorgo[11], 53edo tuning
- "Liminal" – Spotify | Bandcamp | YouTube – in Gorgo[11], 53edo tuning

Prent Rodgers

Whisper Song (2007) – blog | play | SoundCloud

Sevish

"Droplet", from Rhythm and Xen (2015) – Bandcamp | SoundCloud | YouTube – drum and bass in Orwell[9], 53edo tuning

Subhraag Singh

"Stranges" (2021)

Gene Ward Smith

Trio in Orwell (archived 2010) – detail | play – in Orwell[9], 53edo tuning

Nick Stephens

"Initialising", from Microwave (2019) – Bandcamp | SoundCloud

Cam Taylor

mothers (2014)
Schumann: The Poet Speaks in 53-equal (5-limit) on the Lumatone (2022)
53-equal: lydian/aeolian pentatonic (2023)
53-equal Luma MKI: around a drone on middle C (2023)
22 shrutis as Schismatic[22 in A446Hz (53-equal)] (2024)
A meander around 53-equal on the Lumatone (2025) (this is actually a keyboard mapping guide)

The Evil Doings Of An Intergalactic Skeleton

Metal Reindeer (2025)

Chris Vaisvil

The Fallen of Kleismic15 (2013) – blog | play – in Kleismic[15], 53edo tuning

Valeriana of the Night

Hero (2023)

VitaminCD

Orwellian in Nature (Orwell [9] Microtonal Lament) (2025)

Randy Wells

Ficta (2021)

Xotla

"Taking Flight" from Nano Particular (2019) – Spotify | Bandcamp | YouTube
"Detective Duckweed" from Jazzbeetle (2023) – Spotify | Bandcamp | YouTube – jazzy big band electronic hybrid

Notes

↑ Based on treating 53edo as a no-17's 19-limit temperament; other approaches are also possible. Italics represent inconsistent intervals which are mapped by the 19-limit patent val to their second-closest (as opposed to closest) approximation in 53edo.
↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

References

↑ Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". Empirical Musicology Review 15 (3–4):223–48. https://doi.org/10.18061/emr.v15i3-4.7647.

[2] Based on treating 53edo as a no-17's 19-limit temperament; other approaches are also possible. Italics represent inconsistent intervals which are mapped by the 19-limit patent val to their second-closest (as opposed to closest) approximation in 53edo.

[3] Ratios longer than 10 digits are presented by placeholders with informative hints.

[1] Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". Empirical Musicology Review 15 (3–4):223–48. https://doi.org/10.18061/emr.v15i3-4.7647.

[1]

[note 1]

[note 2]

53edo: Difference between revisions

Latest revision as of 03:10, 2 June 2026

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