130edo
| ← 129edo | 130edo | 131edo → |
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 21/130, 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. 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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.42 | +1.38 | +0.40 | +2.53 | -0.53 | -3.42 | -2.13 | -0.58 | +4.27 | -0.42 |
| Relative (%) | +0.0 | -4.5 | +14.9 | +4.4 | +27.4 | -5.7 | -37.0 | -23.1 | -6.3 | +46.2 | -4.6 | |
| Steps (reduced) |
130 (0) |
206 (76) |
302 (42) |
365 (105) |
450 (60) |
481 (91) |
531 (11) |
552 (32) |
588 (68) |
632 (112) |
644 (124) | |
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 | 0.000 | 1/1 |
| 1 | 9.231 | 126/125, 144/143, 169/168, 176/175, 196/195, 225/224 |
| 2 | 18.462 | 78/77, 81/80, 91/90, 99/98, 100/99, 105/104, 121/120 |
| 3 | 27.692 | 56/55, 64/63, 65/64, 66/65 |
| 4 | 36.923 | 45/44, 49/48, 50/49, 55/54 |
| 5 | 46.154 | 36/35, 40/39 |
| 6 | 55.385 | 33/32 |
| 7 | 64.615 | 27/26, 28/27 |
| 8 | 73.846 | 25/24, 26/25 |
| 9 | 83.077 | 21/20, 22/21 |
| 10 | 92.308 | 135/128 |
| 11 | 101.538 | 35/33 |
| 12 | 110.769 | 16/15 |
| 13 | 120.000 | 15/14 |
| 14 | 129.231 | 14/13 |
| 15 | 138.462 | 13/12 |
| 16 | 147.692 | 12/11 |
| 17 | 156.923 | 35/32 |
| 18 | 166.154 | 11/10 |
| 19 | 175.385 | 72/65 |
| 20 | 184.615 | 10/9 |
| 21 | 193.846 | 28/25 |
| 22 | 203.077 | 9/8 |
| 23 | 212.308 | 44/39 |
| 24 | 221.538 | 25/22 |
| 25 | 230.769 | 8/7 |
| 26 | 240.000 | 55/48 |
| 27 | 249.231 | 15/13 |
| 28 | 258.462 | 64/55 |
| 29 | 267.692 | 7/6 |
| 30 | 276.923 | 75/64 |
| 31 | 286.154 | 13/11 |
| 32 | 295.385 | 32/27 |
| 33 | 304.615 | 25/21 |
| 34 | 313.846 | 6/5 |
| 35 | 323.077 | 65/54 |
| 36 | 332.308 | 40/33 |
| 37 | 341.538 | 39/32 |
| 38 | 350.769 | 11/9, 27/22 |
| 39 | 360.000 | 16/13 |
| 40 | 369.231 | 26/21 |
| 41 | 378.462 | 56/45 |
| 42 | 387.692 | 5/4 |
| 43 | 396.923 | 44/35 |
| 44 | 406.154 | 81/64 |
| 45 | 415.385 | 14/11 |
| 46 | 424.615 | 32/25 |
| 47 | 433.846 | 9/7 |
| 48 | 443.077 | 84/65, 128/99 |
| 49 | 452.308 | 13/10 |
| 50 | 461.538 | 64/49, 72/55 |
| 51 | 470.769 | 21/16 |
| 52 | 480.000 | 33/25 |
| 53 | 489.231 | 65/49 |
| 54 | 498.462 | 4/3 |
| 55 | 507.692 | 75/56 |
| 56 | 516.923 | 27/20 |
| 57 | 526.154 | 65/48 |
| 58 | 535.385 | 15/11 |
| 59 | 544.615 | 48/35 |
| 60 | 553.846 | 11/8 |
| 61 | 563.077 | 18/13 |
| 62 | 572.308 | 25/18 |
| 63 | 581.538 | 7/5 |
| 64 | 590.769 | 45/32 |
| 65 | 600.000 | 99/70, 140/99 |
| … | … | … |
Notation
Sagittal
| Steps | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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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.
| 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 | Sextilififths |
| 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) |
Hemipental |
| 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) |
Decal |
| 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 it is distinct
Scales
| 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 ¢) |
Music
- See also: Category:130edo tracks
- wazzock (2024)
- The Paradise of Cantor play (2006)