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{{Infobox ET}} | {{Infobox ET}} | ||
26edt | {{ED intro}} | ||
26edt is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes|zeta peak tritave division]]. | |||
A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well{{Dubious}}. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63). | A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well{{Dubious}}. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63). | ||
== Theory == | == Theory == | ||
While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[Bohlen−Pierce−Stearns]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and MOS scales of {{sl|8L 1s | While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[Bohlen−Pierce−Stearns]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and MOS scales of {{sl|8L 1s}} and {{sl|9L 8s}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17 [[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt. | ||
Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos 21-[[odd limit]], and is in fact the first edt to achieve this. | Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos 21-[[odd limit]], and is in fact the first edt to achieve this. | ||
Revision as of 15:40, 10 February 2025
| ← 25edt | 26edt | 27edt → |
26 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 26edt or 26ed3), is a nonoctave tuning system that divides the interval of 3/1 into 26 equal parts of about 73.2 ¢ each. Each step represents a frequency ratio of 31/26, or the 26th root of 3.
26edt is contorted in the 7-limit, tempering out the same commas, 245/243 and 3125/3087, as 13edt. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh zeta peak tritave division.
A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well[dubious – discuss]. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).
Theory
While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out 2025/2023 to split the Bohlen−Pierce−Stearns generator of 9/7 into two intervals of 17/15. This 17/15 generates Dubhe temperament and MOS scales of Template:Sl and Template:Sl that can be used as a simple traversal of 26edt. Among the 3.5.7.17 subgroup intervals, the accuracy of 21/17 should be highlighted, forming a 21-strong consistent circle that traverses the edt.
Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be consistent to the no-twos 21-odd limit, and is in fact the first edt to achieve this.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -29.6 | +0.0 | -6.5 | -3.8 | +18.4 | +21.8 | -3.8 | +23.1 | -15.0 | +22.6 | -19.7 |
| Relative (%) | -40.4 | +0.0 | -8.9 | -5.2 | +25.1 | +29.7 | -5.1 | +31.6 | -20.5 | +30.9 | -26.9 | |
| Steps (reduced) |
16 (16) |
26 (0) |
38 (12) |
46 (20) |
57 (5) |
61 (9) |
67 (15) |
70 (18) |
74 (22) |
80 (2) |
81 (3) | |
Intervals
| Steps | Cents | Hekts | Enneatonic degree | Corresponding 3.5.7.17 subgroup intervals |
Dubhe (LLLLLLLLs, J = 1/1) |
Lambda (sLsLsLsLs, E = 1/1) |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | P1 | 1/1 | J | E |
| 1 | 73.2 | 50 | Sa1/sd2 | 51/49 (+3.9¢); 85/81 (−10.3¢) | J# | ^E, vF |
| 2 | 146.3 | 100 | A1/m2 | 49/45 (−1.1¢); 27/25 (+13.1¢) | Kb | F |
| 3 | 219.5 | 150 | N2 | 135/119 (+1.1¢); 17/15 (+2.8¢) | K | ^F, vF#, vGb |
| 4 | 292.6 | 200 | M2/d3 | 25/21 (−9.2¢) | K# | F#, Gb |
| 5 | 365.8 | 250 | Sa2/sd3 | 21/17 (−0.06¢) | Lb | vG, ^F#, ^Gb |
| 6 | 438.9 | 300 | A2/P3/d4 | 9/7 (+3.8¢) | L | G |
| 7 | 512.1 | 350 | Sa3/sd4 | 85/63 (−6.5¢) | L# | ^G, vH |
| 8 | 585.2 | 400 | A3/m4/d5 | 7/5 (+2.7¢) | Mb | H |
| 9 | 658.4 | 450 | N4/sd5 | 51/35 (+6.6¢); 119/81 (−7.6¢); 25/17 (−9.3¢) | M | ^H, vH#, vJb |
| 10 | 731.5 | 500 | M4/m5 | 75/49 (−5.4¢) | M# | H#, Jb |
| 11 | 804.7 | 550 | Sa4/N5 | 119/75 (+5.5¢); 27/17 (+3.8¢) | Nb | vJ, ^H#, ^Jb |
| 12 | 877.8 | 600 | A4/M5 | 5/3 (−6.5¢) | N | J |
| 13 | 951.0 | 650 | Sa5/sd6 | 85/49 (−2.6¢), 147/85 (+2.6¢) | N# | ^J, vA |
| 14 | 1024.1 | 700 | A5/m6/d7 | 9/5 (+6.5¢) | Ob | A |
| 15 | 1097.3 | 750 | N6/sd7 | 225/119 (−5.5¢); 17/9 (−3.8¢) | O | ^A, vA#, vBb |
| 16 | 1170.4 | 800 | M6/m7 | 49/25 (+5.4¢) | O# | A#, Bb |
| 17 | 1243.6 | 850 | Sa6/N7 | 35/17 (−6.6¢); 243/119 (+7.6¢); 51/25 (+9.3¢) | Pb | vB, ^A#, ^Bb |
| 18 | 1316.7 | 900 | A6/M7/d8 | 15/7 (−2.7¢) | P | B |
| 19 | 1389.9 | 950 | Sa7/sd8 | 189/85 (+6.5¢) | P# | ^B, vC |
| 20 | 1463.0 | 1000 | P8/d9 | 7/3 (−3.8¢) | Qb | C |
| 21 | 1536.2 | 1050 | Sa8/sd9 | 17/7 (+0.06¢) | Q | ^C, vC#, vDb |
| 22 | 1609.3 | 1100 | A8/m9 | 63/25 (+9.2¢) | Q# | C#, Db |
| 23 | 1682.5 | 1150 | N9 | 119/45 (−1.1¢); 45/17 (−2.8¢) | Rb | vD, ^C#, ^Db |
| 24 | 1755.7 | 1200 | M9/d10 | 135/49 (+1.1¢); 25/9 (−13.1¢) | R | D |
| 25 | 1828.8 | 1250 | Sa9/sd10 | 49/17 (−3.9¢); 243/85 (+10.3¢) | R#, Jb | ^D, vE |
| 26 | 1902.0 | 1300 | A9/P10 | 3/1 | J | E |
Connection to 26edo
It is a weird coincidence[dubious – discuss] how 26edt intones many 26edo intervals within ±6.5 ¢ when it is supposed to have nothing to do with this other tuning:
| 26edt | 26edo | Delta |
|---|---|---|
| 365.761 | 369.231 | −3.470 |
| 512.065 | 507.692 | +4.373 |
| 877.825 | 876.923 | +0.902 |
| 1243.586 | 1246.154 | −2.168 |
| 1389.890 | 1384.615 | +5.275 |
| 1755.651 | 1753.846 | +1.805 |
| 2121.411 | 2123.077 | −1.666 |
| 2633.476 | 2630.769 | +2.647 |
etc.