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{{Infobox ET}}
{{Infobox ET}}
26edt divides the tritave ([[3/1]]) into 26 equal parts of 73.152 cents each, corresponding 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 prime|zeta peak tritave division]].
26edt divides the tritave ([[3/1]]) into 26 equal parts of 73.152 cents each, corresponding 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 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 BPS generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and a [[8L 1s (3/1-equivalent)|8L 1s]] MOS scale 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 [[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|3/1}} and {{sl|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.
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.
Line 11: Line 11:


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 right-3"
{| class="wikitable center-all right-2 right-3"
|-
|-
Line 18: Line 17:
! [[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 intervals
! Dubhe<br />(LLLLLLLLs,<br />J = 1/1)
! Dubhe  
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs,<br />E = 1/1)
(LLLLLLLLs, <br>
J = 1/1)
! [[Lambda ups and downs notation|Lambda]]  
(sLsLsLsLs, <br>  
E = 1/1)
|-
|-
| 0
| 0
Line 39: Line 33:
| 50
| 50
| Sa1/sd2
| Sa1/sd2
| [[51/49]] (+3.9c); [[85/81]] (-10.3c)
| [[51/49]] (+3.9¢); [[85/81]] (−10.3¢)
| J#
| J#
| ^E, vF
| ^E, vF
Line 47: Line 41:
| 100
| 100
| A1/m2
| A1/m2
| [[49/45]] (-1.1c); [[27/25]] (+13.1c)
| [[49/45]] (−1.1¢); [[27/25]] (+13.1¢)
| Kb
| Kb
| F
| F
Line 55: Line 49:
| 150
| 150
| N2
| N2
| [[135/119]] (+1.1c); [[17/15]] (+2.8c)
| [[135/119]] (+1.1¢); [[17/15]] (+2.8¢)
| K
| K
| ^F, vF#, vGb
| ^F, vF#, vGb
Line 63: Line 57:
| 200
| 200
| M2/d3
| M2/d3
| [[25/21]] (-9.2c)
| [[25/21]] (−9.2¢)
| K#
| K#
| F#, Gb
| F#, Gb
Line 71: Line 65:
| 250
| 250
| Sa2/sd3
| Sa2/sd3
| [[21/17]] (-0.06c)
| [[21/17]] (−0.06¢)
| Lb
| Lb
| vG, ^F#, ^Gb
| vG, ^F#, ^Gb
Line 79: Line 73:
| 300
| 300
| A2/P3/d4
| A2/P3/d4
| [[9/7]] (+3.8c)
| [[9/7]] (+3.8¢)
| L
| L
| G
| G
Line 87: Line 81:
| 350
| 350
| Sa3/sd4
| Sa3/sd4
| [[85/63]] (-6.5c)
| [[85/63]] (−6.5¢)
| L#
| L#
| ^G, vH
| ^G, vH
Line 95: Line 89:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| [[7/5]] (+2.7c)
| [[7/5]] (+2.7¢)
| Mb
| Mb
| H
| H
Line 103: Line 97:
| 450
| 450
| N4/sd5
| N4/sd5
| [[51/35]] (+6.6c); [[119/81]] (-7.6c); [[25/17]] (-9.3c)
| [[51/35]] (+6.6¢); [[119/81]] (−7.6¢); [[25/17]] (−9.3¢)
| M
| M
| ^H, vH#, vJb
| ^H, vH#, vJb
Line 111: Line 105:
| 500
| 500
| M4/m5
| M4/m5
| [[75/49]] (-5.4c)
| [[75/49]] (−5.4¢)
| M#
| M#
| H#, Jb
| H#, Jb
Line 119: Line 113:
| 550
| 550
| Sa4/N5
| Sa4/N5
| [[119/75]] (+5.5c); [[27/17]] (+3.8c)
| [[119/75]] (+5.5¢); [[27/17]] (+3.8¢)
| Nb
| Nb
| vJ, ^H#, ^Jb
| vJ, ^H#, ^Jb
Line 127: Line 121:
| 600
| 600
| A4/M5
| A4/M5
| [[5/3]] (-6.5c)
| [[5/3]] (−6.5¢)
| N
| N
| J
| J
Line 135: Line 129:
| 650
| 650
| Sa5/sd6
| Sa5/sd6
| [[85/49]] (-2.6c), [[147/85]] (+2.6c)
| [[85/49]] (−2.6¢), [[147/85]] (+2.6¢)
| N#
| N#
| ^J, vA
| ^J, vA
Line 143: Line 137:
| 700
| 700
| A5/m6/d7
| A5/m6/d7
| [[9/5]] (+6.5c)
| [[9/5]] (+6.5¢)
| Ob
| Ob
| A
| A
Line 151: Line 145:
| 750
| 750
| N6/sd7
| N6/sd7
| [[225/119]] (-5.5c); [[17/9]] (-3.8c)
| [[225/119]] (−5.5¢); [[17/9]] (−3.8¢)
| O
| O
| ^A, vA#, vBb
| ^A, vA#, vBb
Line 159: Line 153:
| 800
| 800
| M6/m7
| M6/m7
| [[49/25]] (+5.4c)
| [[49/25]] (+5.4¢)
| O#
| O#
| A#, Bb
| A#, Bb
Line 167: Line 161:
| 850
| 850
| Sa6/N7
| Sa6/N7
| [[35/17]] (-6.6c); [[243/119]] (+7.6c); [[51/25]] (+9.3c)
| [[35/17]] (−6.6¢); [[243/119]] (+7.6¢); [[51/25]] (+9.3¢)
| Pb
| Pb
| vB, ^A#, ^Bb
| vB, ^A#, ^Bb
Line 175: Line 169:
| 900
| 900
| A6/M7/d8
| A6/M7/d8
| [[15/7]] (-2.7c)
| [[15/7]] (−2.7¢)
| P
| P
| B
| B
Line 183: Line 177:
| 950
| 950
| Sa7/sd8
| Sa7/sd8
| [[189/85]] (+6.5c)
| [[189/85]] (+6.5¢)
| P#
| P#
| ^B, vC
| ^B, vC
Line 191: Line 185:
| 1000
| 1000
| P8/d9
| P8/d9
| [[7/3]] (-3.8c)
| [[7/3]] (−3.8¢)
| Qb
| Qb
| C
| C
Line 199: Line 193:
| 1050
| 1050
| Sa8/sd9
| Sa8/sd9
| [[17/7]] (+0.06c)
| [[17/7]] (+0.06¢)
| Q
| Q
| ^C, vC#, vDb
| ^C, vC#, vDb
Line 207: Line 201:
| 1100
| 1100
| A8/m9
| A8/m9
| [[63/25]] (+9.2c)
| [[63/25]] (+9.2¢)
| Q#
| Q#
| C#, Db
| C#, Db
Line 215: Line 209:
| 1150
| 1150
| N9
| N9
| [[119/45]] (-1.1c); [[45/17]] (-2.8c)
| [[119/45]] (−1.1¢); [[45/17]] (−2.8¢)
| Rb
| Rb
| vD, ^C#, ^Db
| vD, ^C#, ^Db
Line 223: Line 217:
| 1200
| 1200
| M9/d10
| M9/d10
| [[135/49]] (+1.1c); [[25/9]] (-13.1c)
| [[135/49]] (+1.1¢); [[25/9]] (−13.1¢)
| R
| R
| D
| D
Line 231: Line 225:
| 1250
| 1250
| Sa9/sd10
| Sa9/sd10
| [[49/17]] (-3.9c); [[243/85]] (+10.3c)
| [[49/17]] (−3.9¢); [[243/85]] (+10.3¢)
| R#, Jb
| R#, Jb
| ^D, vE
| ^D, vE
Line 256: Line 250:
| 365.761
| 365.761
| 369.231
| 369.231
| -3.470
| −3.470
|-
|-
| 512.065
| 512.065
Line 268: Line 262:
| 1243.586
| 1243.586
| 1246.154
| 1246.154
| -2.168
| −2.168
|-
|-
| 1389.890
| 1389.890
Line 280: Line 274:
| 2121.411
| 2121.411
| 2123.077
| 2123.077
| -1.666
| −1.666
|-
|-
| 2633.476
| 2633.476
Line 289: Line 283:


== Music ==
== Music ==
 
* ''The Eel And Loach To Attack In Lasciviousness Are Insane'': [https://www.youtube.com/watch?v=AhWJ2yJsODs video] | [http://micro.soonlabel.com/gene_ward_smith/Others/Omega9/Omega9%20-%20The%20Eel%20And%20Loach%20To%20Attack%20In%20Lasciviousness%20Are%20Insane.mp3 MP3] by Omega9
*''The Eel And Loach To Attack In Lasciviousness Are Insane'': [https://www.youtube.com/watch?v=AhWJ2yJsODs video] | [http://micro.soonlabel.com/gene_ward_smith/Others/Omega9/Omega9%20-%20The%20Eel%20And%20Loach%20To%20Attack%20In%20Lasciviousness%20Are%20Insane.mp3 MP3] by Omega9


[[Category:edt]]
[[Category:edt]]
[[Category:tritave]]
[[Category:tritave]]
[[category:nonoctave]]
[[category:nonoctave]]

Revision as of 14:02, 9 January 2025

← 25edt 26edt 27edt →
Prime factorization 2 × 13
Step size 73.1521 ¢ 
Octave 16\26edt (1170.43 ¢) (→ 8\13edt)
Consistency limit 3
Distinct consistency limit 3

26edt divides the tritave (3/1) into 26 equal parts of 73.152 cents each, corresponding to 16.404edo. 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 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.

Approximation of prime harmonics in 26edt
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 plus or minus 6.5 cents 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.

Music

  • The Eel And Loach To Attack In Lasciviousness Are Insane: video | MP3 by Omega9
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