95ed5
| ← 94ed5 | 95ed5 | 96ed5 → |
95 equal divisions of the 5th harmonic (abbreviated 95ed5) is a nonoctave tuning system that divides the interval of 5/1 into 95 equal parts of about 29.3 ¢ each. Each step represents a frequency ratio of 51/95, or the 95th root of 5.
Theory
95ed5 is related to 41edo, but with the 5/1 rather than the 2/1 being just. The octave is about 2.5143 cents stretched. This tuning has a generally sharp tendency for harmonics up to 12. Unlike 41edo, it is only consistent up to the 12-integer-limit, with discrepancy for the 13th harmonic.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.5 | +4.5 | +5.0 | +0.0 | +7.0 | +4.1 | +7.5 | +8.9 | +2.5 | +13.5 | +9.5 |
| Relative (%) | +8.6 | +15.2 | +17.1 | +0.0 | +23.8 | +13.9 | +25.7 | +30.5 | +8.6 | +46.0 | +32.4 | |
| Steps (reduced) |
41 (41) |
65 (65) |
82 (82) |
95 (0) |
106 (11) |
115 (20) |
123 (28) |
130 (35) |
136 (41) |
142 (47) |
147 (52) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -11.8 | +6.6 | +4.5 | +10.1 | -6.9 | +11.5 | +5.8 | +5.0 | +8.6 | -13.3 | -2.3 | +12.0 |
| Relative (%) | -40.1 | +22.5 | +15.2 | +34.3 | -23.6 | +39.1 | +19.9 | +17.1 | +29.2 | -45.4 | -7.8 | +41.0 | |
| Steps (reduced) |
151 (56) |
156 (61) |
160 (65) |
164 (69) |
167 (72) |
171 (76) |
174 (79) |
177 (82) |
180 (85) |
182 (87) |
185 (90) |
188 (93) | |
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 29.3 | 49/48, 50/49, 64/63, 81/80 |
| 2 | 58.7 | 25/24, 28/27, 33/32, 36/35 |
| 3 | 88.0 | 19/18, 20/19, 21/20, 22/21 |
| 4 | 117.3 | 14/13, 15/14, 16/15 |
| 5 | 146.6 | [[12/11], 13/12 |
| 6 | 176.0 | 10/9, 11/10, 21/19 |
| 7 | 205.3 | 9/8 |
| 8 | 234.6 | 8/7, 15/13 |
| 9 | 264.0 | 7/6, 22/19 |
| 10 | 293.3 | 13/11, 19/16, 32/27 |
| 11 | 322.6 | 6/5 |
| 12 | 352.0 | 11/9, 16/13 |
| 13 | 381.3 | 5/4, 26/21 |
| 14 | 410.6 | 19/15 |
| 15 | 439.9 | 9/7, 32/25 |
| 16 | 469.3 | 21/16, 13/10 |
| 17 | 498.6 | 4/3 |
| 18 | 527.9 | 15/11, 19/14, 27/20 |
| 19 | 557.3 | 11/8, 18/13, 26/19 |
| 20 | 586.6 | 7/5, 45/32 |
| 21 | 615.9 | 10/7, 64/45 |
| 22 | 645.3 | 13/9, 16/11, 19/13 |
| 23 | 674.6 | 22/15, 28/19, 40/27 |
| 24 | 703.9 | 3/2 |
| 25 | 733.2 | 20/13, 32/21 |
| 26 | 762.6 | 14/9, 25/16 |
| 27 | 791.9 | 11/7, 19/12, 30/19 |
| 28 | 821.2 | 8/5, 21/13 |
| 29 | 850.6 | 13/8, 18/11 |
| 30 | 879.9 | 5/3 |
| 31 | 909.2 | 22/13, 27/16, 32/19 |
| 32 | 938.5 | 12/7, 19/11 |
| 33 | 967.9 | 7/4, 26/15 |
| 34 | 997.2 | 16/9 |
| 35 | 1026.5 | 9/5 |
| 36 | 1055.9 | 11/6 |
| 37 | 1085.2 | 13/7, 15/8 |
| 38 | 1114.5 | 19/10, 21/11 |
| 39 | 1143.9 | 27/14, 35/18 |
| 40 | 1173.2 | 49/25, 55/28, 63/32 |
| 41 | 1202.5 | 2/1 |
| 42 | 1231.8 | 45/22, 49/24, 55/27, 81/40 |
| 43 | 1261.2 | 25/12, 33/16 |
| 44 | 1290.5 | 19/9, 21/10 |
| 45 | 1319.8 | 15/7 |
| 46 | 1349.2 | 13/6 |
| 47 | 1378.5 | 11/5 |
| 48 | 1407.8 | 9/4 |
| 49 | 1437.2 | 16/7 |
| 50 | 1466.5 | 7/3 |
| 51 | 1495.8 | 19/8 |
| 52 | 1525.1 | 12/5 |
| 53 | 1554.5 | 22/9, 27/11 |
| 54 | 1583.8 | 5/2 |
| 55 | 1613.1 | 28/11, 33/13 |
| 56 | 1642.5 | 18/7 |
| 57 | 1671.8 | 21/8 |
| 58 | 1701.1 | 8/3 |
| 59 | 1730.4 | 19/7 |
| 60 | 1759.8 | 11/4 |
| 61 | 1789.1 | 14/5 |
| 62 | 1818.4 | 20/7 |
| 63 | 1847.8 | 26/9 |
| 64 | 1877.1 | 44/15 |
| 65 | 1906.4 | 3/1 |
| 66 | 1935.8 | 40/13 |
| 67 | 1965.1 | 25/8, 28/9 |
| 68 | 1994.4 | 19/6, 22/7 |
| 69 | 2023.7 | 16/5 |
| 70 | 2053.1 | 13/4 |
| 71 | 2082.4 | 10/3 |
| 72 | 2111.7 | 27/8 |
| 73 | 2141.1 | 24/7 |
| 74 | 2170.4 | 7/2 |
| 75 | 2199.7 | 25/7 |
| 76 | 2229.1 | 18/5 |
| 77 | 2258.4 | 11/3 |
| 78 | 2287.7 | 15/4 |
| 79 | 2317.0 | 19/5 |
| 80 | 2346.4 | 27/7, 35/9 |
| 81 | 2375.7 | 55/14, 63/16 |
| 82 | 2405.0 | 4/1 |
| 83 | 2434.4 | 49/12, 81/20 |
| 84 | 2463.7 | 25/6, 33/8 |
| 85 | 2493.0 | 21/5 |
| 86 | 2522.3 | 30/7 |
| 87 | 2551.7 | 13/3 |
| 88 | 2581.0 | 22/5 |
| 89 | 2610.3 | 9/2 |
| 90 | 2639.7 | 16/7 |
| 91 | 2669.0 | 14/3 |
| 92 | 2698.3 | 19/4 |
| 93 | 2727.7 | 24/5 |
| 94 | 2757.0 | 39/8 |
| 95 | 2786.3 | 5/1 |
As a generator
95ed5 can also be thought of as a generator of the 2.3.5.7.11.19-subgroup temperament which tempers out 1540/1539, 3025/3024, 6875/6859, and 184877/184320, which is a cluster temperament with 41 clusters of notes in an octave. While the small chroma interval between adjacent notes in each cluster represents 385/384 ~ 441/440 ~ 1479016/1476225 ~ 194579/194400 ~ 204800/204687 ~ 176000/175959 tempered together, the step interval is very versatile, representing 16807/16500 ~ 19551/19200 ~ 18000/17689 ~ 72900/71687 ~ 273375/268912 ~ 295245/290521 ~ 12100/11907 ~ 64/63 all tempered together. This temperament is supported by 41edo, 491edo (491e val), and 532edo (532d val) among others.