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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 {{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.
 
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-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 26: 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 35: 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 44: 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 53: Line 64:
| 200
| 200
| M2/d3
| M2/d3
| M2
| [[25/21]] (−9.2¢)
| 25/21~13/11
| K#
|  
| F#, Gb
|  
|-
|-
| 5
| 5
Line 62: 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 71: Line 80:
| 300
| 300
| A2/P3/d4
| A2/P3/d4
| AA2/sm3
| [[9/7]] (+3.8¢)
| 9/7
| L
|  
| G
|  
|-
|-
| 7
| 7
Line 80: Line 88:
| 350
| 350
| Sa3/sd4
| Sa3/sd4
| m3
| [[85/63]] (−6.5¢)
| 27/20
| L#
| False 21/16
| ^G, vH
|  
|-
|-
| 8
| 8
Line 89: Line 96:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| M3
| [[7/5]] (+2.7¢)
| 7/5
| Mb
|  
| H
|  
|-
|-
| 9
| 9
Line 98: 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 107: 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 116: 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 134: 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 143: 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 152: 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 161: 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 170: 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 179: Line 176:
| 900
| 900
| A6/M7/d8
| A6/M7/d8
| m6
| [[15/7]] (−2.7¢)
| 15/7
| P
|  
| B
|  
|-
|-
| 19
| 19
Line 188: 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 206: 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 215: Line 208:
| 1100
| 1100
| A8/m9
| A8/m9
| m7
| [[63/25]] (+9.2¢)
| 63/25~33/13
| Q#
|  
| C#, Db
|  
|-
|-
| 23
| 23
Line 224: 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 233: 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 242: 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 251: 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 267: Line 256:
| 365.761
| 365.761
| 369.231
| 369.231
| -3.470
| −3.470
|-
|-
| 512.065
| 512.065
Line 279: Line 268:
| 1243.586
| 1243.586
| 1246.154
| 1246.154
| -2.168
| −2.168
|-
|-
| 1389.890
| 1389.890
Line 291: Line 280:
| 2121.411
| 2121.411
| 2123.077
| 2123.077
| -1.666
| −1.666
|-
|-
| 2633.476
| 2633.476
Line 300: 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"