83ed11
83 equal divisions of the 11th harmonic (abbreviated 83ed11) is a nonoctave tuning system that divides the interval of 11/1 into 83 equal parts of about 50 ¢ each. Each step represents a frequency ratio of 111/83, or the 83rd root of 11.
Theory
83ed11 is closely related to 24edo (quarter-tone tuning), but with the 11th harmonic rather than the octave being just, which stretches the octave by about 0.381 cents. Like 24edo, 83ed11 is consistent to the 6-integer-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.4 | -1.4 | +0.8 | +14.6 | -1.0 | -17.8 | +1.1 | -2.7 | +15.0 | +0.0 | -0.6 |
| Relative (%) | +0.8 | -2.7 | +1.5 | +29.1 | -1.9 | -35.5 | +2.3 | -5.4 | +29.9 | +0.0 | -1.2 | |
| Steps (reduced) |
24 (24) |
38 (38) |
48 (48) |
56 (56) |
62 (62) |
67 (67) |
72 (72) |
76 (76) |
80 (80) |
83 (0) |
86 (3) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.9 | -17.4 | +13.2 | +1.5 | -3.4 | -2.3 | +4.1 | +15.3 | -19.1 | +0.4 | +23.5 | -0.2 |
| Relative (%) | +21.8 | -34.8 | +26.4 | +3.0 | -6.8 | -4.6 | +8.2 | +30.7 | -38.2 | +0.8 | +46.9 | -0.4 | |
| Steps (reduced) |
89 (6) |
91 (8) |
94 (11) |
96 (13) |
98 (15) |
100 (17) |
102 (19) |
104 (21) |
105 (22) |
107 (24) |
109 (26) |
110 (27) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 50 | 34/33, 35/34, 36/35, 37/36 |
| 2 | 100 | 18/17, 35/33 |
| 3 | 150 | 12/11 |
| 4 | 200.1 | 37/33 |
| 5 | 250.1 | 15/13, 37/32 |
| 6 | 300.1 | 19/16 |
| 7 | 350.1 | 11/9, 38/31 |
| 8 | 400.1 | 29/23, 34/27, 39/31 |
| 9 | 450.1 | 35/27 |
| 10 | 500.2 | 4/3 |
| 11 | 550.2 | 11/8 |
| 12 | 600.2 | 17/12 |
| 13 | 650.2 | 16/11, 35/24 |
| 14 | 700.2 | 3/2 |
| 15 | 750.2 | 37/24 |
| 16 | 800.3 | 27/17 |
| 17 | 850.3 | 18/11, 31/19 |
| 18 | 900.3 | 32/19, 37/22 |
| 19 | 950.3 | 26/15 |
| 20 | 1000.3 | |
| 21 | 1050.3 | 11/6 |
| 22 | 1100.3 | 17/9 |
| 23 | 1150.4 | 33/17, 35/18 |
| 24 | 1200.4 | 2/1 |
| 25 | 1250.4 | 33/16, 35/17, 37/18 |
| 26 | 1300.4 | 36/17 |
| 27 | 1350.4 | 24/11 |
| 28 | 1400.4 | |
| 29 | 1450.5 | 30/13, 37/16 |
| 30 | 1500.5 | 19/8 |
| 31 | 1550.5 | 22/9 |
| 32 | 1600.5 | |
| 33 | 1650.5 | |
| 34 | 1700.5 | 8/3 |
| 35 | 1750.6 | 11/4 |
| 36 | 1800.6 | 17/6 |
| 37 | 1850.6 | 32/11, 35/12 |
| 38 | 1900.6 | 3/1 |
| 39 | 1950.6 | 34/11, 37/12 |
| 40 | 2000.6 | |
| 41 | 2050.7 | 36/11 |
| 42 | 2100.7 | 37/11 |
| 43 | 2150.7 | |
| 44 | 2200.7 | |
| 45 | 2250.7 | 11/3 |
| 46 | 2300.7 | 34/9 |
| 47 | 2350.7 | 35/9 |
| 48 | 2400.8 | 4/1 |
| 49 | 2450.8 | 33/8 |
| 50 | 2500.8 | |
| 51 | 2550.8 | |
| 52 | 2600.8 | 9/2 |
| 53 | 2650.8 | 37/8 |
| 54 | 2700.9 | |
| 55 | 2750.9 | |
| 56 | 2800.9 | |
| 57 | 2850.9 | |
| 58 | 2900.9 | 16/3 |
| 59 | 2950.9 | 11/2 |
| 60 | 3001 | 17/3 |
| 61 | 3051 | 35/6 |
| 62 | 3101 | 6/1 |
| 63 | 3151 | 37/6 |
| 64 | 3201 | |
| 65 | 3251 | |
| 66 | 3301 | |
| 67 | 3351.1 | |
| 68 | 3401.1 | |
| 69 | 3451.1 | 22/3 |
| 70 | 3501.1 | |
| 71 | 3551.1 | |
| 72 | 3601.1 | 8/1 |
| 73 | 3651.2 | 33/4 |
| 74 | 3701.2 | |
| 75 | 3751.2 | |
| 76 | 3801.2 | 9/1 |
| 77 | 3851.2 | 37/4 |
| 78 | 3901.2 | |
| 79 | 3951.3 | |
| 80 | 4001.3 | |
| 81 | 4051.3 | |
| 82 | 4101.3 | |
| 83 | 4151.3 | 11/1 |