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 5th harmonic rather than the octave being just. The octave is about 2.51 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) | |
Subsets and supersets
Since 95 factors into primes as 5 × 19, 95ed5 contains 5ed5 and 19ed5 as subset ed5's.
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.