26edt: Difference between revisions

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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]].
{{ED intro}}


A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well. 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 ==
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.
 
26 also supports the temperaments: [[mizar]] (generators ~1097.8c, ~49.7c) and [[bohlenic]] (1\13edt, ~11/1).
 
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 ===
{{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-1 right-2 right-3"
|-
|-
! Steps
! Steps
! [[Cent]]s
! [[Cent]]s
! [[Hekt]]s
! [[Hekt]]s
! BP nonatonic degree
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree
! Diatonic degree
! Corresponding<br>3.5.7.17 subgroup intervals
! Corresponding JI intervals
! Dubhe<br>(LLLLLLLLs,<br />J = 1/1)
! Comments
! [[Lambda ups and downs notation|Lambda]]<br>(sLsLsLsLs,<br />E = 1/1)
! Generator for...
|-
| 0
| 0
| 0
| P1
| 1/1
| J
| E
|-
|-
| 1
| 1
Line 21: Line 40:
| 50
| 50
| Sa1/sd2
| Sa1/sd2
| A1/dd2
| [[51/49]] (+3.9¢); [[85/81]] (−10.3¢)
| 25/24
| J#
|  
| ^E, vF
|  
|-
|-
| 2
| 2
Line 30: Line 48:
| 100
| 100
| A1/m2
| A1/m2
| AA1/sm2
| [[49/45]] (−1.1¢); [[27/25]] (+13.1¢)
| 27/25~49/45
| Kb
|  
| F
|  
|-
|-
| 3
| 3
Line 39: Line 56:
| 150
| 150
| N2
| N2
| m2
| [[135/119]] (+1.1¢); [[17/15]] (+2.8¢)
| 9/8~312/275
| K
|  
| ^F, vF#, vGb
|  
|-
|-
| 4
| 4
Line 48: Line 64:
| 200
| 200
| M2/d3
| M2/d3
| M2
| [[25/21]] (−9.2¢)
| 25/21~13/11
| K#
|  
| F#, Gb
|  
|-
|-
| 5
| 5
Line 57: Line 72:
| 250
| 250
| Sa2/sd3
| Sa2/sd3
| SM2/dd3
| [[21/17]] (−0.06¢)
| 5/4~243/196
| Lb
| False 11/9
| vG, ^F#, ^Gb
|  
|-
|-
| 6
| 6
Line 66: Line 80:
| 300
| 300
| A2/P3/d4
| A2/P3/d4
| AA2/sm3
| [[9/7]] (+3.8¢)
| 9/7
| L
|  
| G
|  
|-
|-
| 7
| 7
Line 75: Line 88:
| 350
| 350
| Sa3/sd4
| Sa3/sd4
| m3
| [[85/63]] (−6.5¢)
| 27/20
| L#
| False 21/16
| ^G, vH
|  
|-
|-
| 8
| 8
Line 84: Line 96:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| M3
| [[7/5]] (+2.7¢)
| 7/5
| Mb
|  
| H
|  
|-
|-
| 9
| 9
Line 93: Line 104:
| 450
| 450
| N4/sd5
| N4/sd5
| SM3/dd4
| [[51/35]] (+6.6¢); [[119/81]] (−7.6¢); [[25/17]] (−9.3¢)
| 16/11
| M
| False 13/9
| ^H, vH#, vJb
|  
|-
|-
| 10
| 10
Line 102: Line 112:
| 500
| 500
| M4/m5
| M4/m5
| AA3/d4
| [[75/49]] (−5.4¢)
| 75/49
| M#
| False 3/2
| H#, Jb
|  
|-
|-
| 11
| 11
Line 111: Line 120:
| 550
| 550
| Sa4/N5
| Sa4/N5
| P4
| [[119/75]] (+5.5¢); [[27/17]] (+3.8¢)
| 8/5
| Nb
| False 11/7
| vJ, ^H#, ^Jb
|  
|-
|-
| 12
| 12
| 877.8
| 877.8
| 600
| 600
| A4/M5/d6
| A4/M5
| A4
| [[5/3]] (−6.5¢)
| 5/3
| N
| False 27/16
| J
|  
|-
|-
| 13
| 13
Line 129: Line 136:
| 650
| 650
| Sa5/sd6
| Sa5/sd6
| AA4/dd5
| [[85/49]] (−2.6¢), [[147/85]] (+2.6¢)
| 125/72
| N#
|  
| ^J, vA
|  
|-
|-
| 14
| 14
Line 138: Line 144:
| 700
| 700
| A5/m6/d7
| A5/m6/d7
| d5
| [[9/5]] (+6.5¢)
| 9/5
| Ob
| False 16/9
| A
|  
|-
|-
| 15
| 15
Line 147: Line 152:
| 750
| 750
| N6/sd7
| N6/sd7
| P5
| [[225/119]] (−5.5¢); [[17/9]] (−3.8¢)
| 15/8
| O
| False 21/11
| ^A, vA#, vBb
|  
|-
|-
| 16
| 16
Line 156: Line 160:
| 800
| 800
| M6/m7
| M6/m7
| A5/dd6
| [[49/25]] (+5.4¢)
| 49/25
| O#
| False 2/1
| A#, Bb
|  
|-
|-
| 17
| 17
Line 165: Line 168:
| 850
| 850
| Sa6/N7
| Sa6/N7
| AA5/sm6
| [[35/17]] (−6.6¢); [[243/119]] (+7.6¢); [[51/25]] (+9.3¢)
| 33/16
| Pb
| False 27/13
| vB, ^A#, ^Bb
|  
|-
|-
| 18
| 18
Line 174: Line 176:
| 900
| 900
| A6/M7/d8
| A6/M7/d8
| m6
| [[15/7]] (−2.7¢)
| 15/7
| P
|  
| B
|  
|-
|-
| 19
| 19
Line 183: Line 184:
| 950
| 950
| Sa7/sd8
| Sa7/sd8
| M6
| [[189/85]] (+6.5¢)
| 20/9
| P#
| False 16/7
| ^B, vC
|  
|-
|-
| 20
| 20
| 1463.0
| 1463.0
| 1000
| 1000
| A7/P8/d9
| P8/d9
| SM6/dd7
| [[7/3]] (−3.8¢)
| 7/3
| Qb
|  
| C
|  
|-
|-
| 21
| 21
Line 201: Line 200:
| 1050
| 1050
| Sa8/sd9
| Sa8/sd9
| AA6/sm7
| [[17/7]] (+0.06¢)
| 12/5~196/81
| Q
| False 27/11
| ^C, vC#, vDb
|  
|-
|-
| 22
| 22
Line 210: Line 208:
| 1100
| 1100
| A8/m9
| A8/m9
| m7
| [[63/25]] (+9.2¢)
| 63/25~33/13
| Q#
|  
| C#, Db
|  
|-
|-
| 23
| 23
Line 219: Line 216:
| 1150
| 1150
| N9
| N9
| M7
| [[119/45]] (−1.1¢); [[45/17]] (−2.8¢)
| 8/3~275/104
| Rb
|  
| vD, ^C#, ^Db
|  
|-
|-
| 24
| 24
Line 228: Line 224:
| 1200
| 1200
| M9/d10
| M9/d10
| SM7/dd8
| [[135/49]] (+1.1¢); [[25/9]] (−13.1¢)
| 25/9~135/49
| R
|  
| D
|  
|-
|-
| 25
| 25
Line 237: Line 232:
| 1250
| 1250
| Sa9/sd10
| Sa9/sd10
| A7/d8
| [[49/17]] (−3.9¢); [[243/85]] (+10.3¢)
| 72/25
| R#, Jb
|  
| ^D, vE
|  
|-
|-
| 26
| 26
Line 246: Line 240:
| 1300
| 1300
| A9/P10
| A9/P10
| P8
| [[3/1]]
| 3/1
| J
| Tritave
| E
|  
|}
|}


It is a weird coincidence how 26edt intones any [[26edo]] intervals within plus or minus 6.5 cents when it is supposed to have nothing to do with this other tuning:
=== 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:


{| class="wikitable right-all"
{| class="wikitable right-all"
Line 262: Line 256:
| 365.761
| 365.761
| 369.231
| 369.231
| -3.470
| −3.470
|-
|-
| 512.065
| 512.065
Line 274: Line 268:
| 1243.586
| 1243.586
| 1246.154
| 1246.154
| -2.168
| −2.168
|-
|-
| 1389.890
| 1389.890
Line 286: Line 280:
| 2121.411
| 2121.411
| 2123.077
| 2123.077
| -1.666
| −1.666
|-
|-
| 2633.476
| 2633.476
Line 295: Line 289:


== Music ==
== Music ==
 
; [[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
* ''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]
 
[[Category:edt]]
[[Category:tritave]]
[[category:nonoctave]]

Latest revision as of 04:31, 23 March 2026

← 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

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.

26 also supports the temperaments: mizar (generators ~1097.8c, ~49.7c) and bohlenic (1\13edt, ~11/1).

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

Approximation of harmonics in 26edt
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)
Approximation of harmonics in 26edt (continued)
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.

Music

Omega9
  • The Eel And Loach To Attack In Lasciviousness Are Insane – video | play
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