494edo

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494edo

495edo →

Prime factorization	2 × 13 × 19
Step size	2.42915 ¢
Fifth	289\494 (702.024 ¢)
Semitones (A1:m2)	47:37 (114.2 ¢ : 89.88 ¢)
Consistency limit	17
Distinct consistency limit	17

494 equal divisions of the octave (abbreviated 494edo or 494ed2), also called 494-tone equal temperament (494tet) or 494 equal temperament (494et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 494 equal parts of about 2.43 ¢ each. Each step represents a frequency ratio of 2^1/494, or the 494th root of 2.

Theory

494 is a very strong 13- and 17-limit equal temperament. 494edo is a zeta peak and zeta peak integer edo and distinctly consistent through the 17-odd-limit. It tempers out the enneadeca, [-14 -19 19⟩, the tricot comma, [39 -29 3⟩, and the kwazy comma, [-53 10 16⟩ in the 5-limit. In the 7-limit, it tempers out 4375/4374 and 703125/702464; in the 11-limit 3025/3024 and 9801/9800; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224 and 6656/6655; and in the 17-limit, 1156/1155, 1275/1274, 2431/2430, and 2500/2499.

Since the step size is close to 729/728, the squbema, the accepted name for 494edo's step is squb.

Prime harmonics

Approximation of prime harmonics in 494edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.069	-0.079	+0.405	+0.099	-0.042	-0.502	-1.157	+0.875	+0.382	-0.906
	Relative (%)	+0.0	+2.9	-3.2	+16.7	+4.1	-1.7	-20.7	-47.6	+36.0	+15.7	-37.3
Steps (reduced)		494 (0)	783 (289)	1147 (159)	1387 (399)	1709 (227)	1828 (346)	2019 (43)	2098 (122)	2235 (259)	2400 (424)	2447 (471)

Subsets and supersets

Since 494 factors into 2 × 13 × 19, 494edo has subset edos 2, 13, 19, 26, 38, and 247.

988edo, which slices the edostep in two, provides a good correction of the 19th harmonic. 2964edo, which slices the edostep in six, provides an extremely precise correction of the 7th harmonic.

Intervals

Main article: Table of 494edo intervals

The following table shows how 15-odd-limit intervals are represented in 494edo. Prime harmonics are in bold.

As 494edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 494edo

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
15/8, 16/15	0.010	0.4
11/6, 12/11	0.030	1.2
15/13, 26/15	0.032	1.3
13/10, 20/13	0.037	1.5
11/9, 18/11	0.040	1.6
13/8, 16/13	0.042	1.7
3/2, 4/3	0.069	2.9
5/4, 8/5	0.079	3.2
11/8, 16/11	0.099	4.1
15/11, 22/15	0.109	4.5
13/12, 24/13	0.111	4.6
9/8, 16/9	0.139	5.7
13/11, 22/13	0.141	5.8
5/3, 6/5	0.148	6.1
11/10, 20/11	0.178	7.3
13/9, 18/13	0.180	7.4
9/5, 10/9	0.217	9.0
9/7, 14/9	0.266	11.0
11/7, 14/11	0.306	12.6
7/6, 12/7	0.336	13.8
7/4, 8/7	0.405	16.7
15/14, 28/15	0.414	17.1
13/7, 14/13	0.447	18.4
7/5, 10/7	0.484	19.9

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[783 -494⟩	[⟨494 783]]	−0.0219	0.0219	0.90
2.3.5	[-14 -19 19⟩, [39 -23 3⟩	[⟨494 783 1147]]	−0.0032	0.0318	1.31
2.3.5.7	4375/4374, 703125/702464, [21 3 1 -10⟩	[⟨494 783 1147 1387]]	−0.0385	0.0670	2.76
2.3.5.7.11	3025/3024, 4375/4374, 131072/130977, 234375/234256	[⟨494 783 1147 1387 1709]]	−0.0365	0.0600	2.47
2.3.5.7.11.13	1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213	[⟨494 783 1147 1387 1709 1828]]	−0.0286	0.0576	2.37
2.3.5.7.11.13.17	1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095	[⟨494 783 1147 1387 1709 1828 2019]]	−0.0069	0.0752	3.09

• 494et has lower relative errors than any previous equal temperaments in the 13- and 17-limit. It is the first past 270 with a lower 13-limit relative error, and the first past 72 with a lower 17-limit relative error. It is narrowly beaten by 684 in terms of 13-limit absolute error and by 581 in terms of 17-limit absolute error. Not until 1506 do we reach an equal temperament with a lower relative error in either subgroup.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	27\494	65.59	27/26	Luminal
1	119\494	289.07	13/11	Moulin
1	233\494	565.99	104/75	Trillium
2	67\494	162.75	1125/1024	Crazy
2	86\494	208.91	44/39	Abigail
13	205\494 (15\494)	497.98 (36.43)	4/3 (?)	Aluminium
19	205\494 (3\494)	497.98 (7.29)	4/3 (225/224)	Enneadecal
38	205\494 (3\494)	497.98 (7.29)	4/3 (225/224)	Hemienneadecal
38	109\494 (5\494)	264.78 (12.15)	500/429 (144/143)	Semihemienneadecal

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Eliora

• Unknown piece in Abigail (2023)