15ed4: 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}} It is every second step of [[15edo]].
: This revision was by author [[User:k9assassin|k9assassin]] and made on <tt>2014-12-29 13:49:34 UTC</tt>.<br>
: The original revision id was <tt>535991244</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">=Basics=
15ED4 is a macrotonal tuning system that divides the double-octave (4/1) into 15 equally spaced pitches each about 160 cents apart. It can be viewed as a subset of 15 EDO that repeats at two octaves rather than one. It has a big xen appeal as it is non octave but at the same time contains the double octave allowing for a sort of best of both worlds approach. 15ED4 doesn't do Just Intonation well for the most part but it does represent 7/4 and 11/8 rather well.  


=MOS Scales=
Lookalikes: [[12edt]]
There are a variety of Mos scales available. William Lynch considers the symmetrical MOS: LsLsLsLs to be the most consonant scale in 15ED4. There is also a LsLLsLLLs scale which is more spacey sounding than the symmetrical MOS. The symmetrical MOS can be harmonized with c


= =  
== Basics ==
15ED4 is a macrotonal tuning system that divides the double octave (4/1) into 15 equally spaced pitches each 160 cents apart. It can be viewed as a subset of 15 EDO that repeats at two octaves rather than one. It has [[The Xen|a big xen]] appeal as it is nonoctave but at the same time contains the double octave allowing for a sort of best of both worlds approach. 15ED4 doesn't do Just Intonation well for the most part but it does represent 7/4 and 11/10 rather well, so it can be viewed as a 3.4.7.10.11 subgroup temperament tempering out 28/27, 49/48, 55/54, and 77/75 (as well as [[12edt|12EDT]]).
{{Harmonics in equal
| steps = 15
| num = 4
| denom = 1
}}
{{Harmonics in equal
| steps = 15
| num = 4
| denom = 1
| start = 12
| collapsed = 1
}}


== MOS Scales ==
There are a variety of Mos scales available. William Lynch considers the symmetrical MOS: LsLsLsLs to be the most consonant scale in 15ED4. There is also a LsLLsLLLs scale which is more spacey sounding than the symmetrical MOS. The symmetrical MOS can be harmonized with c.


== Intervals ==
{{Interval table}}


= = </pre></div>
[[Category:macrotonal]]
<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;15ED4&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="Basics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Basics&lt;/h1&gt;
15ED4 is a macrotonal tuning system that divides the double-octave (4/1) into 15 equally spaced pitches each about 160 cents apart. It can be viewed as a subset of 15 EDO that repeats at two octaves rather than one. It has a big xen appeal as it is non octave but at the same time contains the double octave allowing for a sort of best of both worlds approach. 15ED4 doesn't do Just Intonation well for the most part but it does represent 7/4 and 11/8 rather well. &lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="MOS Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;MOS Scales&lt;/h1&gt;
There are a variety of Mos scales available. William Lynch considers the symmetrical MOS: LsLsLsLs to be the most consonant scale in 15ED4. There is also a LsLLsLLLs scale which is more spacey sounding than the symmetrical MOS. The symmetrical MOS can be harmonized with c&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt; &lt;/h1&gt;
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&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt; &lt;/h1&gt;
&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 21:18, 31 July 2025

← 13ed4 15ed4 17ed4 →
Prime factorization 3 × 5
Step size 160 ¢ 
Octave 8\15ed4 (1280 ¢)
Twelfth 12\15ed4 (1920 ¢) (→ 4\5ed4)
Consistency limit 1
Distinct consistency limit 1

15 equal divisions of the 4th harmonic (abbreviated 15ed4) is a nonoctave tuning system that divides the interval of 4/1 into 15 equal parts of exactly 160 ¢ each. Each step represents a frequency ratio of 41/15, or the 15th root of 4. It is every second step of 15edo.

Lookalikes: 12edt

Basics

15ED4 is a macrotonal tuning system that divides the double octave (4/1) into 15 equally spaced pitches each 160 cents apart. It can be viewed as a subset of 15 EDO that repeats at two octaves rather than one. It has a big xen appeal as it is nonoctave but at the same time contains the double octave allowing for a sort of best of both worlds approach. 15ED4 doesn't do Just Intonation well for the most part but it does represent 7/4 and 11/10 rather well, so it can be viewed as a 3.4.7.10.11 subgroup temperament tempering out 28/27, 49/48, 55/54, and 77/75 (as well as 12EDT).

Approximation of harmonics in 15ed4
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +80.0 +18.0 +0.0 -66.3 -62.0 -8.8 +80.0 +36.1 +13.7 +8.7 +18.0
Relative (%) +50.0 +11.3 +0.0 -41.4 -38.7 -5.5 +50.0 +22.6 +8.6 +5.4 +11.3
Steps
(reduced) 		8
(8) 		12
(12) 		15
(0) 		17
(2) 		19
(4) 		21
(6) 		23
(8) 		24
(9) 		25
(10) 		26
(11) 		27
(12)
Approximation of harmonics in 15ed4
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +39.5 +71.2 -48.3 +0.0 +55.0 -43.9 +22.5 -66.3 +9.2 -71.3 +11.7
Relative (%) +24.7 +44.5 -30.2 +0.0 +34.4 -27.4 +14.1 -41.4 +5.8 -44.6 +7.3
Steps
(reduced) 		28
(13) 		29
(14) 		29
(14) 		30
(0) 		31
(1) 		31
(1) 		32
(2) 		32
(2) 		33
(3) 		33
(3) 		34
(4)

MOS Scales

There are a variety of Mos scales available. William Lynch considers the symmetrical MOS: LsLsLsLs to be the most consonant scale in 15ED4. There is also a LsLLsLLLs scale which is more spacey sounding than the symmetrical MOS. The symmetrical MOS can be harmonized with c.

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 160 11/10, 21/19, 23/21
2 320 17/14, 23/19
3 480 17/13
4 640 13/9, 19/13
5 800 11/7
6 960 19/11
7 1120 17/9, 19/10, 21/11
8 1280 19/9, 21/10, 23/11
9 1440 23/10
10 1600
11 1760
12 1920 3/1
13 2080 10/3, 23/7
14 2240 11/3
15 2400
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