13edf: Difference between revisions

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**Imported revision 248750263 - Original comment: **
m Text replacement - "{{infobox et}}↵{{ed intro}}" to "{{Infobox ET}} {{ED intro}}"
 
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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:Kosmorsky|Kosmorsky]] and made on <tt>2011-08-26 23:41:49 UTC</tt>.<br>
 
: The original revision id was <tt>248750263</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
13edf corresponds to 22.2236[[edo]]. It is nearly identical to every ninth step of [[200edo]], but not quite similar to [[22edo]]; the octave is compressed by 12.076{{c}}, a deviation that is small but significant enough to create a discrepancy for the [[7/1|7th]] and [[11/1|11th]] harmonics.
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>
== Harmonics ==
<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"> </pre></div>
{{Harmonics in equal|13|3|2|intervals=prime|columns=8}}
<h4>Original HTML content:</h4>
{{Harmonics in equal|13|3|2|start=9|intervals=prime|columns=8}}
<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;13edf&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== Intervals ==
{| class="wikitable mw-collapsible"
|+ style="font-size: 105%;" | Intervals of 13edf
|-
! Degree
! Cents
! Corresponding<br />JI intervals
! Comments
|-
! colspan="2" | 0
| '''exact [[1/1]]'''
|
|-
| 1
| 53.9965
| 33/32
| pseudo-[[25/24]]
|-
| 2
| 107.9931
| [[17/16]], 117/110, [[16/15]]
|
|-
| 3
| 161.9896
| [[11/10]]
|
|-
| 4
| 215.9862
| [[17/15]]
|
|-
| 5
| 269.9827
| [[7/6]]
|
|-
| 6
| 323.9792
| [[77/64]]
| pseudo-[[6/5]]
|-
| 7
| 377.9758
| 56/45
| pseudo-[[5/4]]
|-
| 8
| 431.9723
| [[9/7]]
|
|-
| 9
| 485.9688
| 45/34
| pseudo-[[4/3]]
|-
| 10
| 539.9654
| [[15/11]]
|
|-
| 11
| 593.9619
| 55/39, [[24/17]]
|
|-
| 12
| 647.9585
| [[16/11]]
|
|-
| 13
| 701.9550
| '''exact [[3/2]]'''
| just perfect fifth
|-
| 14
| 755.9515
| 99/64
|
|-
| 15
| 809.9481
| 51/32, [[8/5]]
|
|-
| 16
| 863.9446
| 33/20
|
|-
| 17
| 917.9412
| [[17/10]]
|
|-
| 18
| 971.9377
| [[7/4]]
|
|-
| 19
| 1025.9342
| [[29/16]]
| pseudo-[[9/5]]
|-
| 20
| 1079.9308
| [[28/15]]
| pseudo-[[15/8]]
|-
| 21
| 1133.9273
| 52/27, [[27/14]]
|
|-
| 22
| 1187.9238
| 135/68
| pseudo-[[octave]]
|-
| 23
| 1241.9204
| [[45/44|45/22]]
|
|-
| 24
| 1295.9169
| [[19/18|19/9]], [[18/17|36/17]]
|
|-
| 25
| 1349.9135
| [[12/11|24/11]]
|
|-
| 26
| 1403.9100
| '''exact [[9/4]]'''
| pythagorean major ninth
|}
 
{{stub}}
 
[[Category:22edo]]

Latest revision as of 11:56, 26 June 2025

← 12edf 13edf 14edf →
Prime factorization 13 (prime)
Step size 53.9965 ¢ 
Octave 22\13edf (1187.92 ¢)
Twelfth 35\13edf (1889.88 ¢)
Consistency limit 4
Distinct consistency limit 4

13 equal divisions of the perfect fifth (abbreviated 13edf or 13ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 13 equal parts of about 54 ¢ each. Each step represents a frequency ratio of (3/2)1/13, or the 13th root of 3/2.

Theory

13edf corresponds to 22.2236edo. It is nearly identical to every ninth step of 200edo, but not quite similar to 22edo; the octave is compressed by 12.076 ¢, a deviation that is small but significant enough to create a discrepancy for the 7th and 11th harmonics.

Harmonics

Approximation of prime harmonics in 13edf
Harmonic 2 3 5 7 11 13 17 19
Error Absolute (¢) -12.1 -12.1 +21.5 -21.0 +6.4 -12.8 +8.7 -21.8
Relative (%) -22.4 -22.4 +39.8 -39.0 +11.9 -23.7 +16.2 -40.4
Steps
(reduced) 		22
(9) 		35
(9) 		52
(0) 		62
(10) 		77
(12) 		82
(4) 		91
(0) 		94
(3)
Approximation of prime harmonics in 13edf
Harmonic 23 29 31 37 41 43 47 53
Error Absolute (¢) +25.4 +2.0 -5.4 +12.3 -3.5 +22.1 -23.9 -15.9
Relative (%) +47.0 +3.8 -10.0 +22.7 -6.4 +40.9 -44.3 -29.5
Steps
(reduced) 		101
(10) 		108
(4) 		110
(6) 		116
(12) 		119
(2) 		121
(4) 		123
(6) 		127
(10)

Intervals

Intervals of 13edf
Degree Cents Corresponding
JI intervals 		Comments
0 exact 1/1
1 53.9965 33/32 pseudo-25/24
2 107.9931 17/16, 117/110, 16/15
3 161.9896 11/10
4 215.9862 17/15
5 269.9827 7/6
6 323.9792 77/64 pseudo-6/5
7 377.9758 56/45 pseudo-5/4
8 431.9723 9/7
9 485.9688 45/34 pseudo-4/3
10 539.9654 15/11
11 593.9619 55/39, 24/17
12 647.9585 16/11
13 701.9550 exact 3/2 just perfect fifth
14 755.9515 99/64
15 809.9481 51/32, 8/5
16 863.9446 33/20
17 917.9412 17/10
18 971.9377 7/4
19 1025.9342 29/16 pseudo-9/5
20 1079.9308 28/15 pseudo-15/8
21 1133.9273 52/27, 27/14
22 1187.9238 135/68 pseudo-octave
23 1241.9204 45/22
24 1295.9169 19/9, 36/17
25 1349.9135 24/11
26 1403.9100 exact 9/4 pythagorean major ninth
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