39edt
| ← 38edt | 39edt | 40edt → |
39 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 39edt or 39ed3), is a nonoctave tuning system that divides the interval of 3/1 into 39 equal parts of about 48.8 ¢ each. Each step represents a frequency ratio of 31/39, or the 39th root of 3. It is also known as the Triple Bohlen–Pierce scale (Triple BP), since it divides each step of the equal-tempered Bohlen–Pierce scale (13edt) into three equal parts.
39edt can be described as approximately 24.606edo. This implies that each step of 39edt can be approximated by 5 steps of 123edo. 39edt contains within it a close approximation of 4ed11/5: every seventh step of 39edt equates to a step of 4ed11/5.
Theory
It is a strong no-twos 13-limit system, a fact first noted by Paul Erlich; in fact it has a better no-twos 13-odd limit relative error than any other edt up to 914edt. Like 26edt and 52edt, it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being contorted in the no-twos 7-limit, tempering out the same BP commas, 245/243 and 3125/3087, as 13edt. In the 11-limit it tempers out 1331/1323 and in the 13-limit 275/273, 1575/1573, and 847/845. An efficient traversal is therefore given by Mintra temperament, which in the 13-limit tempers out 275/273 and 1575/1573 alongside 245/243, and is generated by the interval of 11/7, which serves as a macrodiatonic "superpyth" fourth and splits the BPS generator of 9/7, up a tritave, in three.
If octaves are inserted, 39edt is related to the 49f & 172f temperament in the full 13-limit, known as triboh, tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [⟨1 0 0 0 0 0], ⟨0 39 57 69 85 91]]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth no-twos zeta peak edt.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +19.2 | +0.0 | -6.5 | -3.8 | -6.0 | -2.6 | +20.6 | +23.1 | -15.0 | +22.6 | +4.7 | -9.0 |
| Relative (%) | +39.4 | +0.0 | -13.4 | -7.9 | -12.4 | -5.4 | +42.3 | +47.4 | -30.8 | +46.3 | +9.6 | -18.5 | |
| Steps (reduced) |
25 (25) |
39 (0) |
57 (18) |
69 (30) |
85 (7) |
91 (13) |
101 (23) |
105 (27) |
111 (33) |
120 (3) |
122 (5) |
128 (11) | |
Intervals
All intervals shown are within the 91-throdd limit and are consistently represented.
| Steps | Cents | Hekts | Enneatonic degree | Corresponding 3.5.7.11.13 subgroup intervals |
Lambda (sLsLsLsLs, J = 1/1) |
Mintaka[7] (E macro-Phrygian) |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | P1 | 1/1 | J | E |
| 1 | 48.8 | 33.3 | SP1 | 77/75 (+3.2¢); 65/63 (−5.3¢) | ^J | ^E, vF |
| 2 | 97.5 | 66.7 | sA1/sm2 | 35/33 (−4.3¢); 81/77 (+9.9¢) | vK | F |
| 3 | 146.3 | 100 | A1/m2 | 99/91 (+0.4¢); 49/45 (−1.1¢); 27/25 (+13.1¢) | K | ^F, vGb, Dx |
| 4 | 195.1 | 133.3 | SA1/Sm2 | 55/49 (−4.9¢); 91/81 (−6.5¢); 39/35 (+7.7¢) | ^K | Gb, vE# |
| 5 | 243.8 | 166.7 | sM2/sd3 | 15/13 (−3.9¢); 63/55 (+8.7¢) | vK#, vLb | ^Gb, E# |
| 6 | 292.6 | 200 | M2/d3 | 77/65 (−0.7¢); 13/11 (+3.4¢); 25/21 (−9.2¢) | K#, Lb | vF#, ^E# |
| 7 | 341.4 | 233.3 | SM2/Sd3 | 11/9 (−6.0¢); 91/75 (+6.6¢) | ^K#, ^Lb | F# |
| 8 | 390.1 | 266.7 | sA2/sP3/sd4 | 49/39 (−5.0¢); 81/65 (+9.2¢) | vL | vG, ^F# |
| 9 | 438.9 | 300 | A2/P3/d4 | 9/7 (+3.8¢); 35/27 (−10.3¢) | L | G |
| 10 | 487.7 | 333.3 | SA2/SP3/Sd4 | 65/49 (−1.5¢); 33/25 (+7.0¢) | ^L | ^G, vAb |
| 11 | 536.4 | 366.7 | sA3/sm4/sd5 | 15/11 (−0.5¢) | vM | Ab |
| 12 | 585.2 | 400 | A3/m4/d5 | 7/5 (+2.7¢) | M | ^Ab, Fx |
| 13 | 634.0 | 433.3 | SA3/Sm4/Sd5 | 13/9 (−2.6¢) | ^M | vG# |
| 14 | 682.7 | 466.7 | sM4/sm5 | 135/91 (+0.07¢); 49/33 (−1.6¢); 81/55 (+12.6¢) | vM#, vNb | G# |
| 15 | 731.5 | 500 | M4/m5 | 75/49 (−5.4¢); 117/77 (+7.2¢) | M#, Nb | vA, ^G# |
| 16 | 780.3 | 533.3 | SM4/Sm5 | 11/7 (−2.2¢); 39/25 (+10.4¢) | ^M#, ^Nb | A |
| 17 | 829.0 | 566.7 | sA4/sM5 | 21/13 (−1.2¢) | vN | ^A, vBb |
| 18 | 877.8 | 600 | A4/M5 | 91/55 (+6.1¢); 5/3 (−6.5¢); 81/49 (+7.7¢) | N | Bb |
| 19 | 926.6 | 633.3 | SA4/SM5 | 77/45 (−3.3¢) | ^N | ^Bb, vCb, Gx |
| 20 | 975.3 | 666.7 | sA5/sm6/sd7 | 135/77 (+3.3¢) | vO | vA#, Cb |
| 21 | 1024.1 | 700 | A5/m6/d7 | 165/91 (−6.1¢); 9/5 (+6.5¢); 49/27 (−7.7¢) | O | A#, ^Cb |
| 22 | 1072.9 | 733.3 | SA5/Sm6/Sd7 | 13/7 (+1.2¢) | ^O | vB, ^A# |
| 23 | 1121.6 | 766.7 | sM6/sm7 | 21/11 (+2.2¢); 25/13 (−10.4¢) | vO#, vPb | B |
| 24 | 1170.4 | 800 | M6/m7 | 49/25 (+5.4¢); 77/39 (−7.2¢) | O#, Pb | ^B, vC |
| 25 | 1219.2 | 833.3 | SM6/Sm7 | 91/45 (+0.07¢); 99/49 (+1.6¢); 55/27 (−12.6¢) | ^O#, ^Pb | C |
| 26 | 1267.9 | 866.7 | sA6/sM7/sd8 | 27/13 (+2.6¢) | vP | ^C, vDb |
| 27 | 1316.7 | 900 | A6/M7/d8 | 15/7 (−2.7¢) | P | Db, vB# |
| 28 | 1365.5 | 933.3 | SA6/SM7/Sd8 | 11/5 (+0.5¢) | ^P | ^Db, B# |
| 29 | 1414.2 | 966.7 | sP8/sd9 | 147/65 (+1.5¢); 25/11 (−7.0¢) | vQ | vC#, ^B# |
| 30 | 1463.0 | 1000 | P8/d9 | 7/3 (−3.8¢); 81/35 (+10.3¢) | Q | C# |
| 31 | 1511.8 | 1033.3 | SP8/Sd9 | 117/49 (+5.0¢); 65/27 (−9.2¢) | ^Q | vD, ^C# |
| 32 | 1560.5 | 1066.7 | sA8/sm9 | 27/11 (+6.0¢); 225/91 (+6.6¢) | vQ#, vRb | D |
| 33 | 1609.3 | 1100 | A8/m9 | 195/77 (−0.7¢); 33/13 (−3.4¢); 63/25 (+9.2¢) | Q#, Rb | ^D, vEb |
| 34 | 1658.1 | 1133.3 | SA8/Sm9 | 13/5 (+3.9¢); 55/21 (−8.7¢) | ^Q#, ^Rb | Eb |
| 35 | 1706.9 | 1166.7 | sM9/sd10 | 147/55 (+4.9¢); 243/91 (+6.5¢); 35/13 (−7.7¢) | vR | ^Eb, vFb, Cx |
| 36 | 1755.7 | 1200 | M9/d10 | 91/33 (+0.4¢); 135/49 (+1.1¢); 25/9 (−13.1¢) | R | vD#, Fb |
| 37 | 1804.5 | 1233.3 | SM9/Sd10 | 99/35 (+4.3¢); 77/27 (−9.9¢) | ^R | D#, ^Fb |
| 38 | 1853.2 | 1266.7 | sA9/sP10 | 225/77 (−3.2¢); 189/65 (+5.3¢) | vJ | vE, ^D# |
| 39 | 1902.0 | 1300 | A9/P10 | 3/1 | J | E |