130edo

← 129edo

130edo

131edo →

Prime factorization	2 × 5 × 13
Step size	9.23077 ¢
Fifth	76\130 (701.538 ¢) (→ 38\65)
Semitones (A1:m2)	12:10 (110.8 ¢ : 92.31 ¢)
Consistency limit	15
Distinct consistency limit	15

130 equal divisions of the octave (abbreviated 130edo or 130ed2), also called 130-tone equal temperament (130tet) or 130 equal temperament (130et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 130 equal parts of about 9.23 ¢ each. Each step represents a frequency ratio of 2^1/130, or the 130th root of 2.

Theory

130edo is a zeta peak edo, a zeta peak integer edo, and a zeta integral edo but not a gap edo. It is distinctly consistent to the 15-odd-limit and is the first nontrivial edo to be consistent in the 14-odd-prime-sum-limit. It is also almost consistent in the no-29 31-odd-limit, missing 19/11 (50.5%), 25/19 (52.9%), 17/11 (64,4%), 25/17 (66.8%), and octave complements.

As an equal temperament, it tempers out 2401/2400, 3136/3125, 6144/6125, and 19683/19600 in the 7-limit; 243/242, 441/440, 540/539, and 4000/3993 in the 11-limit; and 351/350, 364/363, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit. It can be used to tune a variety of temperaments, including hemiwürschmidt, sesquiquartififths, harry and hemischis. It also can be used to tune the rank-3 temperament jove, tempering out 243/242 and 441/440, plus 364/363 for the 13-limit and 595/594 for the 17-limit. It gives the optimal patent val for 11-limit hemiwürschmidt and sesquart and 13-limit harry.

Prime harmonics

Approximation of prime harmonics in 130edo

Harmonic		2	3	5	7	11	13	17	19	23
Error	Absolute (¢)	+0.00	-0.42	+1.38	+0.40	+2.53	-0.53	-3.42	-2.13	-0.58
	Relative (%)	+0.0	-4.5	+14.9	+4.4	+27.4	-5.7	-37.0	-23.1	-6.3
Steps (reduced)		130 (0)	206 (76)	302 (42)	365 (105)	450 (60)	481 (91)	531 (11)	552 (32)	588 (68)

Approximation of prime harmonics in 130edo (continued)

Harmonic		29	31	37	41	43	47	53	59	61
Error	Absolute (¢)	+4.27	-0.42	-2.11	-4.45	-3.83	-0.89	+3.42	+2.37	+0.04
	Relative (%)	+46.2	-4.6	-22.9	-48.2	-41.4	-9.7	+37.0	+25.6	+0.4
Steps (reduced)		632 (112)	644 (124)	677 (27)	696 (46)	705 (55)	722 (72)	745 (95)	765 (115)	771 (121)

Subsets and supersets

Since 130 factors into 2 × 5 × 13, 130edo has subset edos 2, 5, 10, 13, 26, and 65.

260edo, which divides the edostep in two, provides a strong correction for the 29th harmonic.

Intervals

Degree	Cents	Approximate ratios
0	0.00	1/1
1	9.23	126/125, 144/143, 169/168, 176/175, 196/195, 225/224
2	18.46	78/77, 81/80, 91/90, 99/98, 100/99, 105/104, 121/120
3	27.69	56/55, 64/63, 65/64, 66/65
4	36.92	45/44, 49/48, 50/49, 55/54
5	46.15	36/35, 40/39
6	55.38	33/32
7	64.62	27/26, 28/27
8	73.85	25/24, 26/25
9	83.08	21/20, 22/21
10	92.31	135/128
11	101.54	35/33
12	110.77	16/15
13	120.00	15/14
14	129.23	14/13
15	138.46	13/12
16	147.69	12/11
17	156.92	35/32
18	166.15	11/10
19	175.38	72/65
20	184.62	10/9
21	193.85	28/25
22	203.08	9/8
23	212.31	44/39
24	221.54	25/22
25	230.77	8/7
26	240.00	55/48
27	249.23	15/13
28	258.46	64/55
29	267.69	7/6
30	276.92	75/64
31	286.15	13/11
32	295.38	32/27
33	304.62	25/21
34	313.85	6/5
35	323.08	65/54
36	332.31	40/33
37	341.54	39/32
38	350.77	11/9, 27/22
39	360.00	16/13
40	369.23	26/21
41	378.46	56/45
42	387.69	5/4
43	396.92	44/35
44	406.15	81/64
45	415.38	14/11
46	424.62	32/25
47	433.85	9/7
48	443.08	84/65, 128/99
49	452.31	13/10
50	461.54	64/49, 72/55
51	470.77	21/16
52	480.00	33/25
53	489.23	65/49
54	498.46	4/3
55	507.69	75/56
56	516.92	27/20
57	526.15	65/48
58	535.38	15/11
59	544.62	48/35
60	553.85	11/8
61	563.08	18/13
62	572.31	25/18
63	581.54	7/5
64	590.77	45/32
65	600.00	99/70, 140/99
…	…	…

Notation

Ups and downs notation

130edo can be notated using ups and downs and quarter-tone accidentals:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
Sharp symbol
Flat symbol

Sagittal notation

130edo can be notated in Sagittal using the Spartan extension, with the apotome equal to 12 edosteps and the limma to 10 edosteps. Since the the rastma is tempered out, a SZ half-sharp and a half-flat may be used instead of pakai/pakao. Here is a simplified table:

Steps		0	1	2	3	4	5	6	7	8	9	10	11	12
Symbol	Evo+SZ													
	Evo												
	Revo													

Because it uses the entire Spartan extension, it allows no accidental enharmonic respellings.

Approximation to JI

Zeta peak index

Tuning			Strength				Octave (cents)		Integer limit
ZPI	Steps per 8ve	Step size (cents)	Height		Integral	Gap	Size	Stretch	Consistent	Distinct
			Tempered	Pure
796zpi	130.00391	9.230492	10.355108	10.339572	1.634018	19.594551	1199.963905	−0.036095	16	16

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5.7	2401/2400, 3136/3125, 19683/19600	[⟨130 206 302 365]]	−0.119	0.311	3.37
2.3.5.7.11	243/242, 441/440, 3136/3125, 4000/3993	[⟨130 206 302 365 450]]	−0.241	0.370	4.02
2.3.5.7.11.13	243/242, 351/350, 364/363, 441/440, 3136/3125	[⟨130 206 302 365 450 481]]	−0.177	0.367	3.98

Rank-2 temperaments

Note: temperaments supported by 65et are not included.

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	3\130	27.69	64/63	Arch
1	7\130	64.62	26/25	Rectified hebrew
1	9\130	83.08	21/20	Sextilifourths
1	19\130	175.38	72/65	Sesquiquartififths / sesquart
1	21\130	193.85	28/25	Hemiwürschmidt
1	27\130	249.23	15/13	Hemischis
1	41\130	378.46	56/45	Subpental
2	6\130	55.38	33/32	Septisuperfourth
2	9\130	83.08	21/20	Harry
2	17\130	156.92	35/32	Bison
2	19\130	175.38	448/405	Bisesqui
2	54\130 (11\130)	498.46 (101.54)	4/3 (35/33)	Bischismic
5	27\130 (1\130)	249.23 (9.23)	81/70 (176/175)	Hemiquintile
10	27\130 (1\130)	249.23 (9.23)	15/13 (176/175)	Decoid
10	54\130 (2\130)	498.46 (18.46)	4/3 (81/80)	Decile
26	54\130 (1\130)	498.46 (9.23)	4/3 (225/224)	Bosonic

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

Scales

14-tone temperament of "Narrative Wars"
as an example of using 130edo:

Step	Cents	Distance to the nearest JI interval (selected ratios)
13 (13/130)	120.000	15/14 (+0.557 ¢)
7 (20/130)	184.615	10/9 (+2.211 ¢)
9 (29/130)	267.692	7/6 (+0,821 ¢)
9 (38/130)	350.769	11/9 (+3.361 ¢)
9 (47/130)	433.846	9/7 (−1.238 ¢)
7 (54/130)	498.462	4/3 (+0.417 ¢)
13 (67/130)	618.462	10/7 (+0.974 ¢)
9 (76/130)	701.538	3/2 (−0.417 ¢)
7 (83/130)	766.154	14/9 (+1.238 ¢)
13 (96/130)	886.154	5/3 (+1.795 ¢)
5 (101/130)	932.308	12/7 (−0.821 ¢)
13 (114/130)	1052.308	11/6 (+2.945 ¢)
7 (121/130)	1116.923	21/11 (−2.540 ¢)
9 (130/130)	1200.000	Octave (2/1, 0 ¢)

Instruments

Lumatone mapping for 130edo

Music

See also: Category:130edo tracks

birdshite stalactite

• wazzock (2024)

Sevish

• Narrative Wars

Gene Ward Smith

• The Paradise of Cantor play (2006)