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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 {{Citation needed}}. 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]].


== Theory ==
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 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 [[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 in equal|26|3|1|intervals=prime}}
=== 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 17: Line 24:
! [[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 <br>
! 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 39: Line 40:
| 50
| 50
| Sa1/sd2
| Sa1/sd2
| [[51/49]], [[85/81]]
| [[51/49]] (+3.9¢); [[85/81]] (−10.3¢)
| J#
| J#
| ^E, vF
| ^E, vF
Line 47: Line 48:
| 100
| 100
| A1/m2
| A1/m2
| [[27/25]], [[49/45]]
| [[49/45]] (−1.1¢); [[27/25]] (+13.1¢)
| Kb
| Kb
| F
| F
Line 55: Line 56:
| 150
| 150
| N2
| N2
| [[17/15]]
| [[135/119]] (+1.1¢); [[17/15]] (+2.8¢)
| K
| K
| ^F, vF#, vGb
| ^F, vF#, vGb
Line 63: Line 64:
| 200
| 200
| M2/d3
| M2/d3
| [[25/21]]
| [[25/21]] (−9.2¢)
| K#
| K#
| F#, Gb
| F#, Gb
Line 71: Line 72:
| 250
| 250
| Sa2/sd3
| Sa2/sd3
| [[21/17]]
| [[21/17]] (−0.06¢)
| Lb
| Lb
| vG, ^F#, ^Gb
| vG, ^F#, ^Gb
Line 79: Line 80:
| 300
| 300
| A2/P3/d4
| A2/P3/d4
| [[9/7]]
| [[9/7]] (+3.8¢)
| L
| L
| G
| G
Line 87: Line 88:
| 350
| 350
| Sa3/sd4
| Sa3/sd4
| [[85/63]]
| [[85/63]] (−6.5¢)
| L#
| L#
| ^G, vH
| ^G, vH
Line 95: Line 96:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| [[7/5]]
| [[7/5]] (+2.7¢)
| Mb
| Mb
| H
| H
Line 103: Line 104:
| 450
| 450
| N4/sd5
| N4/sd5
| [[25/17]], [[51/35]]
| [[51/35]] (+6.6¢); [[119/81]] (−7.6¢); [[25/17]] (−9.3¢)
| M
| M
| ^H, vH#, vJb
| ^H, vH#, vJb
Line 111: Line 112:
| 500
| 500
| M4/m5
| M4/m5
| [[75/49]]
| [[75/49]] (−5.4¢)
| M#
| M#
| H#, Jb
| H#, Jb
Line 119: Line 120:
| 550
| 550
| Sa4/N5
| Sa4/N5
| [[27/17]]
| [[119/75]] (+5.5¢); [[27/17]] (+3.8¢)
| Nb
| Nb
| vJ, ^H#, ^Jb
| vJ, ^H#, ^Jb
Line 127: Line 128:
| 600
| 600
| A4/M5
| A4/M5
| [[5/3]]
| [[5/3]] (−6.5¢)
| N
| N
| J
| J
Line 135: Line 136:
| 650
| 650
| Sa5/sd6
| Sa5/sd6
| [[85/49]], [[147/85]]
| [[85/49]] (−2.6¢), [[147/85]] (+2.6¢)
| N#
| N#
| ^J, vA
| ^J, vA
Line 143: Line 144:
| 700
| 700
| A5/m6/d7
| A5/m6/d7
| [[9/5]]
| [[9/5]] (+6.5¢)
| Ob
| Ob
| A
| A
Line 151: Line 152:
| 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 159: Line 160:
| 800
| 800
| M6/m7
| M6/m7
| [[49/25]]
| [[49/25]] (+5.4¢)
| O#
| O#
| A#, Bb
| A#, Bb
Line 167: Line 168:
| 850
| 850
| Sa6/N7
| Sa6/N7
| [[35/17]], [[51/25]]
| [[35/17]] (−6.6¢); [[243/119]] (+7.6¢); [[51/25]] (+9.3¢)
| Pb
| Pb
| vB, ^A#, ^Bb
| vB, ^A#, ^Bb
Line 175: Line 176:
| 900
| 900
| A6/M7/d8
| A6/M7/d8
| [[15/7]]
| [[15/7]] (−2.7¢)
| P
| P
| B
| B
Line 183: Line 184:
| 950
| 950
| Sa7/sd8
| Sa7/sd8
| [[189/85]]
| [[189/85]] (+6.5¢)
| P#
| P#
| ^B, vC
| ^B, vC
Line 191: Line 192:
| 1000
| 1000
| P8/d9
| P8/d9
| [[7/3]]
| [[7/3]] (−3.8¢)
| Qb
| Qb
| C
| C
Line 199: Line 200:
| 1050
| 1050
| Sa8/sd9
| Sa8/sd9
| [[17/7]]
| [[17/7]] (+0.06¢)
| Q
| Q
| ^C, vC#, vDb
| ^C, vC#, vDb
Line 207: Line 208:
| 1100
| 1100
| A8/m9
| A8/m9
| [[63/25]]
| [[63/25]] (+9.2¢)
| Q#
| Q#
| C#, Db
| C#, Db
Line 215: Line 216:
| 1150
| 1150
| N9
| N9
| [[45/17]]
| [[119/45]] (−1.1¢); [[45/17]] (−2.8¢)
| Rb
| Rb
| vD, ^C#, ^Db
| vD, ^C#, ^Db
Line 223: Line 224:
| 1200
| 1200
| M9/d10
| M9/d10
| [[25/9]], [[135/49]]
| [[135/49]] (+1.1¢); [[25/9]] (−13.1¢)
| R
| R
| D
| D
Line 231: Line 232:
| 1250
| 1250
| Sa9/sd10
| Sa9/sd10
| [[49/17]], [[243/85]]
| [[49/17]] (−3.9¢); [[243/85]] (+10.3¢)
| R#, Jb
| R#, Jb
| ^D, vE
| ^D, vE
Line 245: Line 246:


=== 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 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:


{| class="wikitable right-all"
{| class="wikitable right-all"
Line 256: Line 256:
| 365.761
| 365.761
| 369.231
| 369.231
| -3.470
| −3.470
|-
|-
| 512.065
| 512.065
Line 268: Line 268:
| 1243.586
| 1243.586
| 1246.154
| 1246.154
| -2.168
| −2.168
|-
|-
| 1389.890
| 1389.890
Line 280: Line 280:
| 2121.411
| 2121.411
| 2123.077
| 2123.077
| -1.666
| −1.666
|-
|-
| 2633.476
| 2633.476
Line 289: 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]]
Retrieved from "https://en.xen.wiki/w/26edt"