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26edt: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-03 23:32:41 UTC</tt>.<br>
 
: The original revision id was <tt>250562236</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
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]].
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
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).
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 26 equal division of 3 (the tritave), divides it into 26 equal parts of 73.152 cents each. 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|tritave zeta peak tuning]].</pre></div>
 
<h4>Original HTML content:</h4>
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.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;26edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 26 equal division of 3 (the tritave), divides it into 26 equal parts of 73.152 cents each. It is contorted in the 7-limit, tempering out the same commas, 245/243 and 3125/3087, as &lt;a class="wiki_link" href="/13edt"&gt;13edt&lt;/a&gt;. 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 &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing prime"&gt;tritave zeta peak tuning&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
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 ==
{| class="wikitable center-all right-2 right-3"
|-
! Steps
! [[Cent]]s
! [[Hekt]]s
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree
! Corresponding<br>3.5.7.17 subgroup intervals
! Dubhe<br>(LLLLLLLLs,<br />J = 1/1)
! [[Lambda ups and downs notation|Lambda]]<br>(sLsLsLsLs,<br />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}} 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"
|-
! 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'' – [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]
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