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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 {{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 ==
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.


== Theory ==
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.
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.


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|intervals=prime}}
{{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 18: 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 <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 40: Line 38:
| 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 48: Line 46:
| 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 56: Line 54:
| 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 64: Line 62:
| 200
| 200
| M2/d3
| M2/d3
| [[25/21]] (-9.2c)
| [[25/21]] (−9.2¢)
| K#
| K#
| F#, Gb
| F#, Gb
Line 72: Line 70:
| 250
| 250
| Sa2/sd3
| Sa2/sd3
| [[21/17]] (-0.06c)
| [[21/17]] (−0.06¢)
| Lb
| Lb
| vG, ^F#, ^Gb
| vG, ^F#, ^Gb
Line 80: Line 78:
| 300
| 300
| A2/P3/d4
| A2/P3/d4
| [[9/7]] (+3.8c)
| [[9/7]] (+3.8¢)
| L
| L
| G
| G
Line 88: Line 86:
| 350
| 350
| Sa3/sd4
| Sa3/sd4
| [[85/63]] (-6.5c)
| [[85/63]] (−6.5¢)
| L#
| L#
| ^G, vH
| ^G, vH
Line 96: Line 94:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| [[7/5]] (+2.7c)
| [[7/5]] (+2.7¢)
| Mb
| Mb
| H
| H
Line 104: Line 102:
| 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 112: Line 110:
| 500
| 500
| M4/m5
| M4/m5
| [[75/49]] (-5.4c)
| [[75/49]] (−5.4¢)
| M#
| M#
| H#, Jb
| H#, Jb
Line 120: Line 118:
| 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 128: Line 126:
| 600
| 600
| A4/M5
| A4/M5
| [[5/3]] (-6.5c)
| [[5/3]] (−6.5¢)
| N
| N
| J
| J
Line 136: Line 134:
| 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 144: Line 142:
| 700
| 700
| A5/m6/d7
| A5/m6/d7
| [[9/5]] (+6.5c)
| [[9/5]] (+6.5¢)
| Ob
| Ob
| A
| A
Line 152: Line 150:
| 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 160: Line 158:
| 800
| 800
| M6/m7
| M6/m7
| [[49/25]] (+5.4c)
| [[49/25]] (+5.4¢)
| O#
| O#
| A#, Bb
| A#, Bb
Line 168: Line 166:
| 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 176: Line 174:
| 900
| 900
| A6/M7/d8
| A6/M7/d8
| [[15/7]] (-2.7c)
| [[15/7]] (−2.7¢)
| P
| P
| B
| B
Line 184: Line 182:
| 950
| 950
| Sa7/sd8
| Sa7/sd8
| [[189/85]] (+6.5c)
| [[189/85]] (+6.5¢)
| P#
| P#
| ^B, vC
| ^B, vC
Line 192: Line 190:
| 1000
| 1000
| P8/d9
| P8/d9
| [[7/3]] (-3.8c)
| [[7/3]] (−3.8¢)
| Qb
| Qb
| C
| C
Line 200: Line 198:
| 1050
| 1050
| Sa8/sd9
| Sa8/sd9
| [[17/7]] (+0.06c)
| [[17/7]] (+0.06¢)
| Q
| Q
| ^C, vC#, vDb
| ^C, vC#, vDb
Line 208: Line 206:
| 1100
| 1100
| A8/m9
| A8/m9
| [[63/25]] (+9.2c)
| [[63/25]] (+9.2¢)
| Q#
| Q#
| C#, Db
| C#, Db
Line 216: Line 214:
| 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 224: Line 222:
| 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 232: Line 230:
| 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 246: 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 {{Dubious}} 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 257: Line 254:
| 365.761
| 365.761
| 369.231
| 369.231
| -3.470
| −3.470
|-
|-
| 512.065
| 512.065
Line 269: Line 266:
| 1243.586
| 1243.586
| 1246.154
| 1246.154
| -2.168
| −2.168
|-
|-
| 1389.890
| 1389.890
Line 281: Line 278:
| 2121.411
| 2121.411
| 2123.077
| 2123.077
| -1.666
| −1.666
|-
|-
| 2633.476
| 2633.476
Line 290: Line 287:


== 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"