12edf: 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>
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]]. This is similar to every second step of [[41edo]], and thus to the fretting of one string on the [[Kite Guitar]]. It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-12-26 16:14:28 UTC</tt>.<br>
: The original revision id was <tt>288516036</tt>.<br>
: The revision comment was: <tt></tt><br>
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>
<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">=12 ED 3/2=


It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|microdiatonic]] scale that corresponds to 12edo's [[5L 2s|diatonic scale]] with [[2/1]] compressed to [[3/2]].
==Harmonics==
{{Harmonics in equal|12|3|2}}
{{Harmonics in equal|12|3|2|start=12|collapsed=1}}
==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding <br>JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| |
|-
| | 1
| | 58.49625
| | [[28/27]], 91/88, 88/85
| |
|-
| | 2
| | 116.9925
| | [[15/14]]
| |
|-
| | 3
| | 175.48875
| | [[10/9]], [[21/19]]
| |
|-
| | 4
| | 233.9850
| | [[8/7]]
| |
|-
| | 5
| | 292.48125
| | 45/38
| |
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| |
|-
| | 7
| | 409.47375
| | [[19/15]], [[63/50]]
| |
|-
| | 8
| | 467.9700
| | [[21/16]]
| |
|-
| | 9
| | 526.46625
| | [[19/14]]
| |
|-
| | 10
| | 584.9625
| | [[7/5]]
| |
|-
| | 11
| | 643.4588
| | [[13/9]]
| |
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|
0: 1/1 0.000 unison, perfect prime
|}
1: 58.496 cents 58.496
2: 116.993 cents 116.993
3: 175.489 cents 175.489
4: 233.985 cents 233.985
5: 292.481 cents 292.481
6: 350.978 cents 350.978
7: 409.474 cents 409.474
8: 467.970 cents 467.970
9: 526.466 cents 526.466
10: 584.963 cents 584.963
11: 643.459 cents 643.459
12: 3/2 701.955 perfect fifth


Lookalikes:</pre></div>
{{todo|expand}}
<h4>Original HTML content:</h4>
<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;12edf&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x12 ED 3/2"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;12 ED 3/2&lt;/h1&gt;
&lt;br /&gt;
|&lt;br /&gt;
0: 1/1 0.000 unison, perfect prime&lt;br /&gt;
1: 58.496 cents 58.496&lt;br /&gt;
2: 116.993 cents 116.993&lt;br /&gt;
3: 175.489 cents 175.489&lt;br /&gt;
4: 233.985 cents 233.985&lt;br /&gt;
5: 292.481 cents 292.481&lt;br /&gt;
6: 350.978 cents 350.978&lt;br /&gt;
7: 409.474 cents 409.474&lt;br /&gt;
8: 467.970 cents 467.970&lt;br /&gt;
9: 526.466 cents 526.466&lt;br /&gt;
10: 584.963 cents 584.963&lt;br /&gt;
11: 643.459 cents 643.459&lt;br /&gt;
12: 3/2 701.955 perfect fifth&lt;br /&gt;
&lt;br /&gt;
Lookalikes:&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 06:59, 30 August 2026

← 11edf 12edf 13edf →
Prime factorization 22 × 3 (highly composite)
Step size 58.4963 ¢ 
Octave 21\12edf (1228.42 ¢) (→ 7\4edf)
Twelfth 33\12edf (1930.38 ¢) (→ 11\4edf)
Consistency limit 3
Distinct consistency limit 3

12EDF is the equal division of the just perfect fifth into 12 parts of 58.49625 cents each, corresponding to 20.5141 edo. This is similar to every second step of 41edo, and thus to the fretting of one string on the Kite Guitar. It is an intersection of 3edf~5edo and 4edf~7edo relations, and could pass as both 20edo and 21edo, with both relations nearly breaking down by this point. It is related to the dodecacot temperament, which tempers out 3087/3125 and 10976/10935 in the 7-limit.

It is a strong 3/2.5/2.7/2 subgroup system, a fact first noted by CompactStar, tempering out the commas 10976/10935 and 3125/3087, although the representation of 11/2 is more questionable. 24edf (effectively 41edo) provides a correction for 11/2. It contains the microdiatonic scale that corresponds to 12edo's diatonic scale with 2/1 compressed to 3/2.

Harmonics

Approximation of harmonics in 12edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +28.4 +28.4 -1.7 +21.5 -1.7 +24.0 +26.8 -1.7 -8.6 +1.9 +26.8
Relative (%) +48.6 +48.6 -2.8 +36.8 -2.8 +41.0 +45.8 -2.8 -14.6 +3.3 +45.8
Steps
(reduced) 		21
(9) 		33
(9) 		41
(5) 		48
(0) 		53
(5) 		58
(10) 		62
(2) 		65
(5) 		68
(8) 		71
(11) 		74
(2)
Approximation of harmonics in 12edf
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +5.2 -6.1 -8.6 -3.3 +8.7 +26.8 -8.3 +19.9 -6.1 -28.2 +11.9
Relative (%) +8.9 -10.5 -14.6 -5.7 +14.9 +45.8 -14.3 +33.9 -10.5 -48.1 +20.3
Steps
(reduced) 		76
(4) 		78
(6) 		80
(8) 		82
(10) 		84
(0) 		86
(2) 		87
(3) 		89
(5) 		90
(6) 		91
(7) 		93
(9)

Intervals

degree cents value corresponding
JI intervals 		comments
0 exact 1/1
1 58.49625 28/27, 91/88, 88/85
2 116.9925 15/14
3 175.48875 10/9, 21/19
4 233.9850 8/7
5 292.48125 45/38
6 350.9775 11/9, 27/22
7 409.47375 19/15, 63/50
8 467.9700 21/16
9 526.46625 19/14
10 584.9625 7/5
11 643.4588 13/9
12 701.9550 exact 3/2 just perfect fifth
13 760.45125 273/176, 132/85
14 818.9475 8/5
15 877.44375 63/38
16 935.94 12/7
17 994.43625 135/76
18 1052.9325 11/6, 81/44
19 1111.42875 19/10
20 1169.925 63/32
21 1228.42125 57/28
22 1286.9175 21/10
23 1345.41375 13/6
24 1403.91 exact 9/4
Retrieved from "https://en.xen.wiki/index.php?title=12edf&oldid=236433"