39edt
| ← 38edt | 39edt | 40edt → |
39 equal divisions of the tritave (39edt) is the nonoctave tuning system derived by dividing the tritave (3/1) into 39 equal steps of approximately 48.7 cents each, 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.
It is a strong no-twos 13-limit system, a fact first noted by Paul Erlich, and like 26edt and 52edt, it is a multiple of 13edt and so contains the Bohlen-Pierce scale. It is contorted in the 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, 847/845 and 1575/1573. It is related to the 49f&172f temperament 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.
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +19.2 | +0.0 | -6.5 | -3.8 | -6.0 | -2.6 | +20.6 | +23.1 | -15.0 | +22.6 | +4.7 |
| relative (%) | +39 | +0 | -13 | -8 | -12 | -5 | +42 | +47 | -31 | +46 | +10 | |
| Steps (reduced) |
25 (25) |
39 (0) |
57 (18) |
69 (30) |
85 (7) |
91 (13) |
101 (23) |
105 (27) |
111 (33) |
120 (3) |
122 (5) | |
Intervals
| Degree | Cents | Cents (octave-reduced) |
Hekts | Degree of BP | Comments |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1/1 | |
| 1 | 48.768 | 33.333 | 39th root of 3 | ||
| 2 | 97.536 | 66.667 | |||
| 3 | 146.304 | 100 | 1 | 13th root of 3 | |
| 4 | 195.072 | 133.333 | |||
| 5 | 243.840 | 166.667 | |||
| 6 | 292.608 | 200 | 2 | ||
| 7 | 341.377 | 233.333 | |||
| 8 | 390.145 | 266.667 | |||
| 9 | 438.913 | 300 | 3 | ||
| 10 | 487.681 | 333.333 | |||
| 11 | 536.449 | 366.667 | |||
| 12 | 585.217 | 400 | 4 | ||
| 13 | 633.985 | 433.333 | cube root of 3 | ||
| 14 | 682.753 | 466.667 | |||
| 15 | 731.521 | 500 | 5 | ||
| 16 | 780.289 | 533.333 | |||
| 17 | 829.057 | 566.667 | |||
| 18 | 877.825 | 600 | 6 | ||
| 19 | 926.593 | 633.333 | |||
| 20 | 975.362 | 666.667 | |||
| 21 | 1024.13 | 700 | 7 | ||
| 22 | 1072.898 | 733.333 | |||
| 23 | 1121.666 | 766.667 | |||
| 24 | 1170.434 | 800 | 8 | ||
| 25 | 1219.202 | 19.202 | 833.333 | ||
| 26 | 1267.97 | 67.97 | 866.667 | ||
| 27 | 1316.738 | 116.738 | 900 | 9 | |
| 28 | 1365.506 | 165.506 | 933.333 | ||
| 29 | 1414.274 | 214.274 | 966.667 | ||
| 30 | 1463.042 | 263.042 | 1000 | 10 | |
| 31 | 1511.81 | 311.81 | 1033.333 | ||
| 32 | 1560.578 | 360.578 | 1066.667 | ||
| 33 | 1609.347 | 409.347 | 1100 | 11 | |
| 34 | 1658.115 | 458.115 | 1133.333 | ||
| 35 | 1706.883 | 506.883 | 1166.667 | ||
| 36 | 1755.651 | 555.651 | 1200 | 12 | |
| 37 | 1804.419 | 604.419 | 1233.333 | ||
| 38 | 1853.187 | 653.187 | 1266.667 | ||
| 39 | 1901.955 | 701.955 | 1300 | 13 | 3/1 (tritave) |
| 40 | 1950.723 | 750.723 | 1333.333 | ||
| 41 | 1999.491 | 799.491 | 1366.667 | ||
| 42 | 2048.259 | 848.259 | 1400 | 14 | |
| 43 | 2097.027 | 897.027 | 1433.333 | ||
| 44 | 2145.795 | 945.795 | 1466.667 | ||
| 45 | 2194.563 | 994.563 | 1500 | 15 | |
| 46 | 2243.332 | 1043.332 | 1533.333 | ||
| 47 | 2292.100 | 1092.100 | 1566.667 | ||
| 48 | 2340.868 | 1140.868 | 1600 | 16 | |
| 49 | 2389.636 | 1189.636 | 1633.333 | ||
| 50 | 2438.404 | 38.404 | 1666.667 | ||
| 51 | 2487.172 | 87.172 | 1700 | 17 | |
| 52 | 2535.940 | 135.940 | 1733.333 | ||
| 53 | 2584.708 | 184.708 | 1766.667 | ||
| 54 | 2633.476 | 233.476 | 1800 | 18 | |
| 55 | 2682.244 | 282.244 | 1833.333 | ||
| 56 | 2731.012 | 331.012 | 1866.667 | ||
| 57 | 2779.78 | 379.78 | 1900 | 19 | |
| 58 | 2828.548 | 428.548 | 1933.333 | ||
| 59 | 2877.317 | 477.317 | 1966.667 | ||
| 60 | 2926.085 | 526.085 | 2000 | 20 | |
| 61 | 2974.853 | 574.853 | 2033.333 | ||
| 62 | 3023.621 | 623.621 | 2066.667 | ||
| 63 | 3072.389 | 672.389 | 2100 | 21 | |
| 64 | 3121.157 | 721.157 | 2133.333 | ||
| 65 | 3169.925 | 769.925 | 2166.667 | ||
| 66 | 3218.693 | 818.693 | 2200 | 22 | |
| 67 | 3267.461 | 867.461 | 2233.333 | ||
| 68 | 3316.229 | 916.229 | 2266.667 | ||
| 69 | 3364.997 | 964.997 | 2300 | 23 | |
| 70 | 3413.765 | 1013.765 | 2333.333 | ||
| 71 | 3462.533 | 1062.533 | 2366.667 | ||
| 72 | 3511.302 | 1111.302 | 2400 | 24 | |
| 73 | 3560.07 | 1160.07 | 2433.333 | ||
| 74 | 3608.838 | 8.838 | 2466.667 | ||
| 75 | 3657.606 | 57.606 | 2500 | 25 | |
| 76 | 3706.374 | 106.374 | 2533.333 | ||
| 77 | 3755.142 | 155.142 | 2566.667 | ||
| 78 | 3803.91 | 203.91 | 2600 | 26 | 9/1 |