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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd | == Theory == | ||
26edt corresponds to 16.404…[[edo]]. It 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 [[8/1|8th]], [[13/1|13th]], [[17/1|17th]], [[20/1|20th]], and [[22/1|22nd]] [[harmonic]]s particularly well{{dubious}}. Moreover, it has an exaggerated [[5L 2s (3/1-equivalent)|triatonic]] 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). | |||
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 [[BPS]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and mos scales of {{mos scalesig|8L 1s<3/1>|link=1}} and {{mos scalesig|9L 8s<3/1>|link=1}} 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. | |||
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 BPS generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and | |||
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- | 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. | ||
{{Harmonics in equal|26|3|1| | === Harmonics === | ||
{{Harmonics in equal|26|3|1}} | |||
{{Harmonics in equal|26|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 26edt (continued)}} | |||
== Intervals == | == Intervals == | ||
{| class="wikitable center-all right-2 right-3" | {| class="wikitable center-all right-2 right-3" | ||
|- | |- | ||
Line 19: | Line 22: | ||
! [[Hekt]]s | ! [[Hekt]]s | ||
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree | ! [[4L 5s (3/1-equivalent)|Enneatonic]] degree | ||
! Corresponding | ! Corresponding<br>3.5.7.17 subgroup intervals | ||
3.5.7.17 subgroup | ! Dubhe<br>(LLLLLLLLs,<br />J = 1/1) | ||
intervals | ! [[Lambda ups and downs notation|Lambda]]<br>(sLsLsLsLs,<br />E = 1/1) | ||
! Dubhe | |||
(LLLLLLLLs, <br> | |||
J = 1/1) | |||
! [[Lambda ups and downs notation|Lambda]] | |||
(sLsLsLsLs, <br> | |||
E = 1/1) | |||
|- | |- | ||
| 0 | | 0 | ||
Line 41: | Line 38: | ||
| 50 | | 50 | ||
| Sa1/sd2 | | Sa1/sd2 | ||
| [[51/49]] (+3. | | [[51/49]] (+3.9¢); [[85/81]] (−10.3¢) | ||
| J# | | J# | ||
| ^E, vF | | ^E, vF | ||
Line 49: | Line 46: | ||
| 100 | | 100 | ||
| A1/m2 | | A1/m2 | ||
| [[49/45]] ( | | [[49/45]] (−1.1¢); [[27/25]] (+13.1¢) | ||
| Kb | | Kb | ||
| F | | F | ||
Line 57: | Line 54: | ||
| 150 | | 150 | ||
| N2 | | N2 | ||
| [[17/15]] (+2. | | [[135/119]] (+1.1¢); [[17/15]] (+2.8¢) | ||
| K | | K | ||
| ^F, vF#, vGb | | ^F, vF#, vGb | ||
Line 65: | Line 62: | ||
| 200 | | 200 | ||
| M2/d3 | | M2/d3 | ||
| [[25/21]] ( | | [[25/21]] (−9.2¢) | ||
| K# | | K# | ||
| F#, Gb | | F#, Gb | ||
Line 73: | Line 70: | ||
| 250 | | 250 | ||
| Sa2/sd3 | | Sa2/sd3 | ||
| [[21/17]] ( | | [[21/17]] (−0.06¢) | ||
| Lb | | Lb | ||
| vG, ^F#, ^Gb | | vG, ^F#, ^Gb | ||
Line 81: | Line 78: | ||
| 300 | | 300 | ||
| A2/P3/d4 | | A2/P3/d4 | ||
| [[9/7]] (+3. | | [[9/7]] (+3.8¢) | ||
| L | | L | ||
| G | | G | ||
Line 89: | Line 86: | ||
| 350 | | 350 | ||
| Sa3/sd4 | | Sa3/sd4 | ||
| [[85/63]] ( | | [[85/63]] (−6.5¢) | ||
| L# | | L# | ||
| ^G, vH | | ^G, vH | ||
Line 97: | Line 94: | ||
| 400 | | 400 | ||
| A3/m4/d5 | | A3/m4/d5 | ||
| [[7/5]] (+2. | | [[7/5]] (+2.7¢) | ||
| Mb | | Mb | ||
| H | | H | ||
Line 105: | Line 102: | ||
| 450 | | 450 | ||
| N4/sd5 | | N4/sd5 | ||
| [[51/35]] (+6. | | [[51/35]] (+6.6¢); [[119/81]] (−7.6¢); [[25/17]] (−9.3¢) | ||
| M | | M | ||
| ^H, vH#, vJb | | ^H, vH#, vJb | ||
Line 113: | Line 110: | ||
| 500 | | 500 | ||
| M4/m5 | | M4/m5 | ||
| [[75/49]] ( | | [[75/49]] (−5.4¢) | ||
| M# | | M# | ||
| H#, Jb | | H#, Jb | ||
Line 121: | Line 118: | ||
| 550 | | 550 | ||
| Sa4/N5 | | Sa4/N5 | ||
| [[27/17]] (+3. | | [[119/75]] (+5.5¢); [[27/17]] (+3.8¢) | ||
| Nb | | Nb | ||
| vJ, ^H#, ^Jb | | vJ, ^H#, ^Jb | ||
Line 129: | Line 126: | ||
| 600 | | 600 | ||
| A4/M5 | | A4/M5 | ||
| [[5/3]] ( | | [[5/3]] (−6.5¢) | ||
| N | | N | ||
| J | | J | ||
Line 137: | Line 134: | ||
| 650 | | 650 | ||
| Sa5/sd6 | | Sa5/sd6 | ||
| [[85/49]] ( | | [[85/49]] (−2.6¢), [[147/85]] (+2.6¢) | ||
| N# | | N# | ||
| ^J, vA | | ^J, vA | ||
Line 145: | Line 142: | ||
| 700 | | 700 | ||
| A5/m6/d7 | | A5/m6/d7 | ||
| [[9/5]] (+6. | | [[9/5]] (+6.5¢) | ||
| Ob | | Ob | ||
| A | | A | ||
Line 153: | Line 150: | ||
| 750 | | 750 | ||
| N6/sd7 | | N6/sd7 | ||
| [[17/9]] ( | | [[225/119]] (−5.5¢); [[17/9]] (−3.8¢) | ||
| O | | O | ||
| ^A, vA#, vBb | | ^A, vA#, vBb | ||
Line 161: | Line 158: | ||
| 800 | | 800 | ||
| M6/m7 | | M6/m7 | ||
| [[49/25]] (+5. | | [[49/25]] (+5.4¢) | ||
| O# | | O# | ||
| A#, Bb | | A#, Bb | ||
Line 169: | Line 166: | ||
| 850 | | 850 | ||
| Sa6/N7 | | Sa6/N7 | ||
| [[35/17]] ( | | [[35/17]] (−6.6¢); [[243/119]] (+7.6¢); [[51/25]] (+9.3¢) | ||
| Pb | | Pb | ||
| vB, ^A#, ^Bb | | vB, ^A#, ^Bb | ||
Line 177: | Line 174: | ||
| 900 | | 900 | ||
| A6/M7/d8 | | A6/M7/d8 | ||
| [[15/7]] ( | | [[15/7]] (−2.7¢) | ||
| P | | P | ||
| B | | B | ||
Line 185: | Line 182: | ||
| 950 | | 950 | ||
| Sa7/sd8 | | Sa7/sd8 | ||
| [[189/85]] (+6. | | [[189/85]] (+6.5¢) | ||
| P# | | P# | ||
| ^B, vC | | ^B, vC | ||
Line 193: | Line 190: | ||
| 1000 | | 1000 | ||
| P8/d9 | | P8/d9 | ||
| [[7/3]] ( | | [[7/3]] (−3.8¢) | ||
| Qb | | Qb | ||
| C | | C | ||
Line 201: | Line 198: | ||
| 1050 | | 1050 | ||
| Sa8/sd9 | | Sa8/sd9 | ||
| [[17/7]] (+0. | | [[17/7]] (+0.06¢) | ||
| Q | | Q | ||
| ^C, vC#, vDb | | ^C, vC#, vDb | ||
Line 209: | Line 206: | ||
| 1100 | | 1100 | ||
| A8/m9 | | A8/m9 | ||
| [[63/25]] (+9. | | [[63/25]] (+9.2¢) | ||
| Q# | | Q# | ||
| C#, Db | | C#, Db | ||
Line 217: | Line 214: | ||
| 1150 | | 1150 | ||
| N9 | | N9 | ||
| [[45/17]] ( | | [[119/45]] (−1.1¢); [[45/17]] (−2.8¢) | ||
| Rb | | Rb | ||
| vD, ^C#, ^Db | | vD, ^C#, ^Db | ||
Line 225: | Line 222: | ||
| 1200 | | 1200 | ||
| M9/d10 | | M9/d10 | ||
| [[135/49]] (+1. | | [[135/49]] (+1.1¢); [[25/9]] (−13.1¢) | ||
| R | | R | ||
| D | | D | ||
Line 233: | Line 230: | ||
| 1250 | | 1250 | ||
| Sa9/sd10 | | Sa9/sd10 | ||
| [[49/17]] ( | | [[49/17]] (−3.9¢); [[243/85]] (+10.3¢) | ||
| R#, Jb | | R#, Jb | ||
| ^D, vE | | ^D, vE | ||
Line 247: | Line 244: | ||
=== Connection to 26edo === | === Connection to 26edo === | ||
It is a weird coincidence{{dubious}} how 26edt intones many [[26edo]] intervals within ±6.5{{c}} when it is supposed to have nothing to do with this other tuning: | |||
It is a weird coincidence {{ | |||
{| class="wikitable right-all" | {| class="wikitable right-all" | ||
Line 258: | Line 254: | ||
| 365.761 | | 365.761 | ||
| 369.231 | | 369.231 | ||
| | | −3.470 | ||
|- | |- | ||
| 512.065 | | 512.065 | ||
Line 270: | Line 266: | ||
| 1243.586 | | 1243.586 | ||
| 1246.154 | | 1246.154 | ||
| | | −2.168 | ||
|- | |- | ||
| 1389.890 | | 1389.890 | ||
Line 282: | Line 278: | ||
| 2121.411 | | 2121.411 | ||
| 2123.077 | | 2123.077 | ||
| | | −1.666 | ||
|- | |- | ||
| 2633.476 | | 2633.476 | ||
Line 291: | Line 287: | ||
== Music == | == Music == | ||
; [[Omega9]] | |||
*''The Eel And Loach To Attack In Lasciviousness Are Insane'' | * ''The Eel And Loach To Attack In Lasciviousness Are Insane'' – [https://www.youtube.com/watch?v=AhWJ2yJsODs video] | [https://web.archive.org/web/20201127012842/http://micro.soonlabel.com/gene_ward_smith/Others/Omega9/Omega9%20-%20The%20Eel%20And%20Loach%20To%20Attack%20In%20Lasciviousness%20Are%20Insane.mp3 play] | ||
Latest revision as of 00:27, 8 March 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.
Theory
26edt corresponds to 16.404…edo. It 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 triatonic 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).
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 BPS generator of 9/7 into two intervals of 17/15. This 17/15 generates Dubhe temperament and mos scales of 8L 1s⟨3/1⟩ and 9L 8s⟨3/1⟩ 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.
Harmonics
Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | -29.6 | +0.0 | +14.0 | -6.5 | -29.6 | -3.8 | -15.5 | +0.0 | -36.1 | +18.4 | +14.0 |
Relative (%) | -40.4 | +0.0 | +19.2 | -8.9 | -40.4 | -5.2 | -21.3 | +0.0 | -49.3 | +25.1 | +19.2 | |
Steps (reduced) |
16 (16) |
26 (0) |
33 (7) |
38 (12) |
42 (16) |
46 (20) |
49 (23) |
52 (0) |
54 (2) |
57 (5) |
59 (7) |
Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +21.8 | -33.4 | -6.5 | +28.0 | -3.8 | -29.6 | +23.1 | +7.5 | -3.8 | -11.2 | -15.0 | -15.5 |
Relative (%) | +29.7 | -45.7 | -8.9 | +38.3 | -5.1 | -40.4 | +31.6 | +10.2 | -5.2 | -15.3 | -20.5 | -21.3 | |
Steps (reduced) |
61 (9) |
62 (10) |
64 (12) |
66 (14) |
67 (15) |
68 (16) |
70 (18) |
71 (19) |
72 (20) |
73 (21) |
74 (22) |
75 (23) |
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.