43edt
| ← 42edt | 43edt | 44edt → |
43 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 43edt or 43ed3), is a nonoctave tuning system that divides the interval of 3/1 into 43 equal parts of about 44.2 ¢ each. Each step represents a frequency ratio of 31/43, or the 43rd root of 3.
Theory
43edt is related to 27edo, but with the 3/1 rather than the 2/1 being just. It has octaves compressed by about 5.7492 ¢ and is consistent to the 10-integer-limit. The octave compression is a small but significant deviation. This is particularly relevant because the harmonics 27edo approximates well—3, 5, 7, and 13—are all tuned sharp, so 43edt improves those approximations.
However, in addition to its rich octave-based harmony, the 43edt is also a fine tritave-based tuning: with a 7/3 of 1460 cents and such a near perfect 5/3, Bohlen–Pierce harmony is very clear and hearty, as well as capable of extended enharmonic distinctions that 13edt is not. The 4L 5s mos has L = 7, s = 3.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.7 | +0.0 | -11.5 | +0.3 | -5.7 | -7.2 | -17.2 | +0.0 | -5.5 | +6.4 | -11.5 |
| Relative (%) | -13.0 | +0.0 | -26.0 | +0.6 | -13.0 | -16.3 | -39.0 | +0.0 | -12.4 | +14.6 | -26.0 | |
| Steps (reduced) |
27 (27) |
43 (0) |
54 (11) |
63 (20) |
70 (27) |
76 (33) |
81 (38) |
86 (0) |
90 (4) |
94 (8) |
97 (11) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -17.4 | -13.0 | +0.3 | +21.2 | +4.7 | -5.7 | -10.9 | -11.2 | -7.2 | +0.7 | +12.2 | -17.2 |
| Relative (%) | -39.3 | -29.3 | +0.6 | +48.0 | +10.7 | -13.0 | -24.6 | -25.4 | -16.3 | +1.6 | +27.6 | -39.0 | |
| Steps (reduced) |
100 (14) |
103 (17) |
106 (20) |
109 (23) |
111 (25) |
113 (27) |
115 (29) |
117 (31) |
119 (33) |
121 (35) |
123 (37) |
124 (38) | |
Subsets and supersets
43edt is the 14th prime edt, following 41edt and coming before 47edt.
Intervals
| # | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 1 | 44.2 | 30.2 | 39/38, 40/39 |
| 2 | 88.5 | 60.5 | 20/19 |
| 3 | 132.7 | 90.7 | 27/25 |
| 4 | 176.9 | 120.9 | 10/9 |
| 5 | 221.2 | 151.2 | 25/22 |
| 6 | 265.4 | 181.4 | 7/6 |
| 7 | 309.6 | 211.6 | 6/5 |
| 8 | 353.9 | 241.9 | 27/22 |
| 9 | 398.1 | 272.1 | 24/19 |
| 10 | 442.3 | 302.3 | 9/7 |
| 11 | 486.5 | 332.6 | 45/34 |
| 12 | 530.8 | 362.8 | 34/25 |
| 13 | 575.0 | 393.0 | 39/28 |
| 14 | 619.2 | 423.3 | 10/7 |
| 15 | 663.5 | 453.5 | 22/15 |
| 16 | 707.7 | 483.7 | 3/2 |
| 17 | 751.9 | 514.0 | 20/13, 105/68 |
| 18 | 796.2 | 544.2 | 19/12 |
| 19 | 840.4 | 574.4 | 13/8 |
| 20 | 884.6 | 604.7 | 5/3 |
| 21 | 928.9 | 634.9 | 12/7 |
| 22 | 973.1 | 665.1 | 7/4 |
| 23 | 1017.3 | 695.3 | 9/5 |
| 24 | 1061.6 | 725.6 | 24/13 |
| 25 | 1105.8 | 755.8 | 36/19 |
| 26 | 1150.0 | 786.0 | 39/20, 68/35 |
| 27 | 1194.3 | 816.3 | 2/1 |
| 28 | 1238.5 | 846.5 | 45/22 |
| 29 | 1282.7 | 876.7 | 21/10 |
| 30 | 1326.9 | 907.0 | 28/13 |
| 31 | 1371.2 | 937.2 | 75/34 |
| 32 | 1415.4 | 967.4 | 34/15 |
| 33 | 1459.6 | 997.7 | 7/3 |
| 34 | 1503.9 | 1027.9 | 19/8 |
| 35 | 1548.1 | 1058.1 | 22/9 |
| 36 | 1592.3 | 1088.3 | 5/2 |
| 37 | 1636.6 | 1118.6 | 18/7 |
| 38 | 1680.8 | 1148.8 | 66/25 |
| 39 | 1725.0 | 1179.1 | 27/10 |
| 40 | 1769.3 | 1209.3 | 25/9 |
| 41 | 1813.5 | 1239.5 | 57/20 |
| 42 | 1857.7 | 1269.8 | 38/13, 117/40 |
| 43 | 1902.0 | 1300.0 | 3/1 |
Related regular temperaments
43edt tempers out the no-twos comma of [0 63 -43⟩, leading to the regular temperament supported by 27-, 190-, and 217edo.
27 & 190 temperament
5-limit
Subgroup: 2.3.5
Comma list: [0 63 -43⟩
Mapping: [⟨1 0 0], ⟨0 43 63]]
Optimal tuning (POTE): ~[0 -41 28⟩ = 44.2294
Optimal ET sequence: 27, 190, 217, 407, 597, 624, 841
7-limit
Subgroup: 2.3.5.7
Comma list: 4375/4374, 40353607/40000000
Mapping: [⟨1 0 0 1], ⟨0 43 63 49]]
Optimal tuning (POTE): ~1029/1000 = 44.2288
Optimal ET sequence: 27, 190, 217
Badness: 0.1659
217 & 407 temperament
7-limit
Subgroup: 2.3.5.7
Comma list: 134217728/133984375, 512557306947/512000000000
Mapping: [⟨1 0 0 9], ⟨0 43 63 -168]]
Optimal tuning (POTE): ~525/512 = 44.2320
Optimal ET sequence: 217, 407, 624, 841, 1058, 1465
Badness: 0.3544
11-limit
Subgroup: 2.3.5.7.11
Comma list: 46656/46585, 131072/130977, 234375/234256
Mapping: [⟨1 0 0 9 -1], ⟨0 43 63 -168 121]]
Optimal tuning (POTE): ~525/512 = 44.2312
Optimal ET sequence: 217, 407, 624
Badness: 0.1129
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 4096/4095, 39366/39325, 109512/109375
Mapping: [⟨1 0 0 9 -1 3], ⟨0 43 63 -168 121 19]]
Optimal tuning (POTE): ~40/39 = 44.2312
Optimal ET sequence: 217, 407, 624
Badness: 0.0503