18edf: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
VisualWikitext
Wikispaces>Kosmorsky
**Imported revision 288516678 - Original comment: **
 
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,134 edits
Add comparison with 20edf
 
(28 intermediate revisions by 10 users not shown)
Line 1: Line 1:
<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>2011-12-26 16:22:16 UTC</tt>.<br>
: The original revision id was <tt>288516678</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">=18edf=


0: 1/1 0.000 unison, perfect prime
== Theory ==
1: 38.998 cents 38.998
18edf corresponds to [[31edo]] with an [[octave stretching]] of about 9 [[cent]]s. Consequently, it does not provide 31edo's good approximations of most low harmonics, but it provides good approximations to many simple ratios in the thirds region: subminor [[7/6]] (+6{{cent}}), minor [[6/5]] (-3{{cent}}), neutral [[11/9]] (+4{{cent}}), major [[5/4]] (+4{{cent}}), and supermajor [[9/7]] (-6{{cent}}). These intervals may be used to form a variety of [[triad]]s and [[tetrad]]s in close harmony along with the tuning's pure fifth.
2: 77.995 cents 77.995
3: 116.993 cents 116.993
4: 155.990 cents 155.990
5: 194.988 cents 194.988
6: 233.985 cents 233.985
7: 272.983 cents 272.983
8: 311.980 cents 311.980
9: 350.978 cents 350.978
10: 389.975 cents 389.975
11: 428.973 cents 428.973
12: 467.970 cents 467.970
13: 506.968 cents 506.968
14: 545.965 cents 545.965
15: 584.963 cents 584.963
16: 623.960 cents 623.960
17: 662.958 cents 662.958
18: 3/2 701.955 perfect fifth


Lookalikes: [[31edo]]</pre></div>
In comparison, [[20edf]] (and [[Carlos Gamma]]) offers more accurate pental (minor and major) and undecimal (neutral) thirds, but less accurate septimal (subminor and supermajor) thirds.
<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;18edf&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="x18edf"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;18edf&lt;/h1&gt;
=== Regular temperaments ===
&lt;br /&gt;
18edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]].
0: 1/1 0.000 unison, perfect prime&lt;br /&gt;
 
1: 38.998 cents 38.998&lt;br /&gt;
=== Harmonics ===
2: 77.995 cents 77.995&lt;br /&gt;
{{Harmonics in equal|18|3|2|intervals=integer|columns=11}}
3: 116.993 cents 116.993&lt;br /&gt;
{{Harmonics in equal|18|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edf (continued)}}
4: 155.990 cents 155.990&lt;br /&gt;
 
5: 194.988 cents 194.988&lt;br /&gt;
=== Subsets and supersets ===
6: 233.985 cents 233.985&lt;br /&gt;
Since 18 factors into primes as {{nowrap| 2 × 3<sup>2</sup> }}, 18edf has subset edfs {{EDs|equave=f| 2, 3, 6, and 9 }}.
7: 272.983 cents 272.983&lt;br /&gt;
 
8: 311.980 cents 311.980&lt;br /&gt;
== Intervals ==
9: 350.978 cents 350.978&lt;br /&gt;
{| class="wikitable center-1 right-2 mw-collapsible"
10: 389.975 cents 389.975&lt;br /&gt;
|-
11: 428.973 cents 428.973&lt;br /&gt;
! #
12: 467.970 cents 467.970&lt;br /&gt;
! Cents
13: 506.968 cents 506.968&lt;br /&gt;
! Approximate ratios
14: 545.965 cents 545.965&lt;br /&gt;
|-
15: 584.963 cents 584.963&lt;br /&gt;
| 0
16: 623.960 cents 623.960&lt;br /&gt;
| 0.0
17: 662.958 cents 662.958&lt;br /&gt;
| [[1/1]]
18: 3/2 701.955 perfect fifth&lt;br /&gt;
|-
&lt;br /&gt;
| 1
Lookalikes: &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
| 39.0
| [[33/32]], [[36/35]], [[49/48]], [[50/49]], [[64/63]]
|-
| 2
| 78.0
| [[21/20]], [[22/21]], [[25/24]], [[28/27]]
|-
| 3
| 117.0
| [[15/14]], [[16/15]]
|-
| 4
| 156.0
| [[11/10]], [[12/11]]
|-
| 5
| 195.0
| [[9/8]], [[10/9]]
|-
| 6
| 234.0
| [[8/7]]
|-
| 7
| 273.0
| [[7/6]]
|-
| 8
| 312.0
| [[6/5]]
|-
| 9
| 351.0
| [[11/9]], [[16/13]]
|-
| 10
| 390.0
| [[5/4]]
|-
| 11
| 429.0
| [[9/7]], [[14/11]]
|-
| 12
| 468.0
| [[13/10]], [[21/16]]
|-
| 13
| 507.0
| [[4/3]]
|-
| 14
| 546.0
| [[11/8]], [[15/11]]
|-
| 15
| 585.0
| [[7/5]]
|-
| 16
| 624.0
| [[10/7]]
|-
| 17
| 663.0
| [[16/11]], [[22/15]]
|-
| 18
| 702.0
| [[3/2]]
|-
| 19
| 741.0
| [[20/13]], [[32/21]]
|-
| 20
| 780.0
| [[11/7]], [[14/9]]
|-
| 21
| 818.9
| [[8/5]]
|-
| 22
| 857.9
| [[18/11]]
|-
| 23
| 896.9
| [[5/3]]
|-
| 24
| 935.9
| [[12/7]]
|-
| 25
| 974.9
| [[7/4]]
|-
| 26
| 1013.9
| [[9/5]]
|-
| 27
| 1052.9
| [[11/6]]
|-
| 28
| 1091.9
| [[15/8]]
|-
| 29
| 1130.9
| [[27/14]]
|-
| 30
| 1169.9
| [[35/18]], [[49/25]], [[63/32]]
|-
| 31
| 1208.9
| [[2/1]]
|-
| 32
| 1247.9
| [[33/16]], [[45/22]], [[49/24]], [[55/27]]
|-
| 33
| 1286.9
| [[21/10]], [[25/12]]
|-
| 34
| 1325.9
| [[15/7]]
|-
| 35
| 1364.9
| [[11/5]]
|-
| 36
| 1403.9
| [[9/4]]
|}
 
== Related regular temperaments ==
The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.
 
=== 7-limit 31 &amp; 369 ===
Commas: 2401/2400, 8589934592/8544921875
 
POTE generator: ~5/4 = 386.997
 
Mapping: [{{map| 1 19 2 7 }}, {{map| 0 -54 1 -13 }}]
 
EDOs: {{EDOs|31, 369, 400, 431, 462}}
 
=== 11-limit 31 &amp; 369 ===
Commas: 2401/2400, 5632/5625, 46656/46585
 
POTE generator: ~5/4 = 386.999
 
Mapping: [{{map| 1 19 2 7 37 }}, {{map| 0 -54 1 -13 -104 }}]
 
EDOs: 31, 369, 400, 431, 462
 
=== 13-limit 31 &amp; 369 ===
Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585
 
POTE generator: ~5/4 = 387.003
 
Mapping: [{{map| 1 19 2 7 37 -35 }}, {{map| 0 -54 1 -13 -104 120 }}]
 
EDOs: 31, 369, 400, 431, 462
 
{{Todo|cleanup|expand|inline=1|comment=say what the temperaments are like and why one would want to use them, and for what}}
 
== See also ==
* [[31edo]] – relative edo
* [[49edt]] – relative edt
* [[72ed5]] – relative ed5
* [[80ed6]] – relative ed6
* [[87ed7]] – relative ed7
* [[107ed11]] – relative ed11
* [[111ed12]] – relative ed12
* [[138ed22]] – relative ed22
* [[204ed96]] – close to the zeta-optimized tuning for 31edo
* [[39cET]]
 
[[Category:31edo]]

Latest revision as of 05:43, 1 August 2025

← 17edf 18edf 19edf →
Prime factorization 2 × 32
Step size 38.9975 ¢ 
Octave 31\18edf (1208.92 ¢)
Twelfth 49\18edf (1910.88 ¢)
Consistency limit 4
Distinct consistency limit 4

18 equal divisions of the perfect fifth (abbreviated 18edf or 18ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 18 equal parts of about 39 ¢ each. Each step represents a frequency ratio of (3/2)1/18, or the 18th root of 3/2.

Theory

18edf corresponds to 31edo with an octave stretching of about 9 cents. Consequently, it does not provide 31edo's good approximations of most low harmonics, but it provides good approximations to many simple ratios in the thirds region: subminor 7/6 (+6 ¢), minor 6/5 (-3 ¢), neutral 11/9 (+4 ¢), major 5/4 (+4 ¢), and supermajor 9/7 (-6 ¢). These intervals may be used to form a variety of triads and tetrads in close harmony along with the tuning's pure fifth.

In comparison, 20edf (and Carlos Gamma) offers more accurate pental (minor and major) and undecimal (neutral) thirds, but less accurate septimal (subminor and supermajor) thirds.

Regular temperaments

18edf is related to the regular temperament which tempers out 2401/2400 and 8589934592/8544921875 in the 7-limit; with 5632/5625, 46656/46585, and 166698/166375 in the 11-limit, which is supported by 31edo, 369edo, 400edo, 431edo, and 462edo.

Harmonics

Approximation of harmonics in 18edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +8.9 +8.9 +17.8 -17.5 +17.8 -15.0 -12.2 +17.8 -8.6 -17.6 -12.2
Relative (%) +22.9 +22.9 +45.8 -44.9 +45.8 -38.6 -31.4 +45.8 -22.0 -45.1 -31.4
Steps
(reduced) 		31
(13) 		49
(13) 		62
(8) 		71
(17) 		80
(8) 		86
(14) 		92
(2) 		98
(8) 		102
(12) 		106
(16) 		110
(2)
Approximation of harmonics in 18edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.2 -6.1 -8.6 -3.3 +8.7 -12.2 +11.2 +0.4 -6.1 -8.7 -7.6 -3.3
Relative (%) +13.3 -15.7 -22.0 -8.5 +22.4 -31.4 +28.6 +0.9 -15.7 -22.2 -19.5 -8.5
Steps
(reduced) 		114
(6) 		117
(9) 		120
(12) 		123
(15) 		126
(0) 		128
(2) 		131
(5) 		133
(7) 		135
(9) 		137
(11) 		139
(13) 		141
(15)

Subsets and supersets

Since 18 factors into primes as 2 × 32, 18edf has subset edfs 2, 3, 6, and 9.

Intervals

# Cents Approximate ratios
0 0.0 1/1
1 39.0 33/32, 36/35, 49/48, 50/49, 64/63
2 78.0 21/20, 22/21, 25/24, 28/27
3 117.0 15/14, 16/15
4 156.0 11/10, 12/11
5 195.0 9/8, 10/9
6 234.0 8/7
7 273.0 7/6
8 312.0 6/5
9 351.0 11/9, 16/13
10 390.0 5/4
11 429.0 9/7, 14/11
12 468.0 13/10, 21/16
13 507.0 4/3
14 546.0 11/8, 15/11
15 585.0 7/5
16 624.0 10/7
17 663.0 16/11, 22/15
18 702.0 3/2
19 741.0 20/13, 32/21
20 780.0 11/7, 14/9
21 818.9 8/5
22 857.9 18/11
23 896.9 5/3
24 935.9 12/7
25 974.9 7/4
26 1013.9 9/5
27 1052.9 11/6
28 1091.9 15/8
29 1130.9 27/14
30 1169.9 35/18, 49/25, 63/32
31 1208.9 2/1
32 1247.9 33/16, 45/22, 49/24, 55/27
33 1286.9 21/10, 25/12
34 1325.9 15/7
35 1364.9 11/5
36 1403.9 9/4

Related regular temperaments

The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.

7-limit 31 & 369

Commas: 2401/2400, 8589934592/8544921875

POTE generator: ~5/4 = 386.997

Mapping: [⟨1 19 2 7], ⟨0 -54 1 -13]]

EDOs: 31, 369, 400, 431, 462

11-limit 31 & 369

Commas: 2401/2400, 5632/5625, 46656/46585

POTE generator: ~5/4 = 386.999

Mapping: [⟨1 19 2 7 37], ⟨0 -54 1 -13 -104]]

EDOs: 31, 369, 400, 431, 462

13-limit 31 & 369

Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585

POTE generator: ~5/4 = 387.003

Mapping: [⟨1 19 2 7 37 -35], ⟨0 -54 1 -13 -104 120]]

EDOs: 31, 369, 400, 431, 462

Todo: cleanup , expand

say what the temperaments are like and why one would want to use them, and for what

See also

Retrieved from "https://en.xen.wiki/index.php?title=18edf&oldid=205674"