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Neutral third scales: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
'''Neutral third scales''' are characterized by the division of the 3/2 "perfect fifth" into two equal "neutral thirds". It usually refers to a [[MOS]] or [[MODMOS]] generated by a neutral third (usually from 342.9c in [[7edo]] to 360c in [[10edo]]).
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-18 13:57:29 UTC</tt>.<br>
: The original revision id was <tt>246753671</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">Mohajira is characterized by the division of the 3/2 "perfect fifth" into two equal "neutral thirds", which are taken to represent either 11/9 or 27/22. This means the comma 243/242 is tempered out.


If the fifth harmonic is used at all, it makes sense to map it to 8 generators by tempering out 81/80, making it [[Meantone family#Mohajira|a meantone system]], sometimes called [[Chromatic pairs#Mohaha|"mohaha"]]. Alternatively, 64/63 could be tempered out, leading to a 2.3.7.11 version of [[Archytas clan#Beatles|beatles]]. But neither of these adjustments to the farther-out intervals affects its basic nature as a "neutral thirds" scale.
One possible temperament interpretation is that the neutral third generator can be taken to represent either 11/9 or 27/22. This means the comma [[243/242]] is tempered out, resulting in the [[Rastmic clan#Neutral|neutral]] temperament and its ''extensions'' (described below).


==Interval chains==
If the fifth harmonic is used at all, it makes sense to map it to 8 generators by tempering out 81/80, making it [[Meantone family #Mohaha|a meantone system]], sometimes called [[Rastmic clan #Mohaha|mohaha]]. If 7 is also mapped a particular way it is called [[mohajira]]. Alternatively, 64/63 could be tempered out, leading to a 2.3.7.11 version of [[neutrominant]] temperament. But neither of these adjustments to the farther-out intervals affects its basic nature as a "neutral thirds" scale, which equally subdivides the 3/2 into two equal parts.
===Basic 2.3.11 mohajira===
|| 296.85 || 647.37 || 997.90 || 148.42 || 498.95 || 849.47 || 0. || 350.53 || 701.05 || 1051.58 || 202.10 || 552.63 || 903.15 ||
|| 32/27 || 16/11 || 16/9 || 12/11 || 4/3 || 18/11 || 1/1 || 11/9 || 3/2 || 11/6 || 9/8 || 11/8 || 27/16 ||
===2.3.5.11 meantone-like mohajira ("mohaha")===
|| 815.25 || 1163.34 || 311.44 || 659.53 || 1007.63 || 155.72 || 503.81 || 851.91 || 0. || 348.09 || 696.19 || 1044.28 || 192.37 || 540.47 || 888.56 || 36.66 || 384.75 ||
|| 8/5 ||  || 6/5 || 16/11 || 16/9~9/5 || 12/11~11/10 || 4/3 || 18/11 || 1/1 || 11/9 || 3/2 || 11/6 || 10/9~9/8 || 11/8 || 5/3 ||  || 5/4 ||
===2.3.7.11 beatles===
|| 414.38 || 768.34 || 1122.29 || 276.25 || 630.21 || 984.17 || 138.13 || 492.08 || 846.04 || 0. || 353.96 || 707.92 || 1061.87 || 215.83 || 569.79 || 923.75 || 77.71 || 431.66 || 785.62 ||
|| 14/11 || 14/9 ||  || 7/6 || 16/11 || 7/4~16/9 || 12/11 || 4/3 || 18/11 || 1/1 || 11/9 || 3/2 || 11/6 || 9/8~8/7 || 11/8 || 12/7 ||  || 9/7 || 11/7 ||</pre></div>
<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;Neutral third scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Mohajira is characterized by the division of the 3/2 &amp;quot;perfect fifth&amp;quot; into two equal &amp;quot;neutral thirds&amp;quot;, which are taken to represent either 11/9 or 27/22. This means the comma 243/242 is tempered out.&lt;br /&gt;
&lt;br /&gt;
If the fifth harmonic is used at all, it makes sense to map it to 8 generators by tempering out 81/80, making it &lt;a class="wiki_link" href="/Meantone%20family#Mohajira"&gt;a meantone system&lt;/a&gt;, sometimes called &lt;a class="wiki_link" href="/Chromatic%20pairs#Mohaha"&gt;&amp;quot;mohaha&amp;quot;&lt;/a&gt;. Alternatively, 64/63 could be tempered out, leading to a 2.3.7.11 version of &lt;a class="wiki_link" href="/Archytas%20clan#Beatles"&gt;beatles&lt;/a&gt;. But neither of these adjustments to the farther-out intervals affects its basic nature as a &amp;quot;neutral thirds&amp;quot; scale.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Interval chains"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Interval chains&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Interval chains-Basic 2.3.11 mohajira"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Basic 2.3.11 mohajira&lt;/h3&gt;


&lt;table class="wiki_table"&gt;
Any temperament tempering out 243/242 lends itself to neutral thirds; this becomes more significant when fifths are of low complexity, such as hemififths or the rank three temperament jove, but also includes miracle, harry and so forth. Nor does a scale need to be a MOS to qualify as a neutral thirds scale; that name could reasonably be given to [[Graph-theoretic properties of scales #Oktone|Oktone]], for example.
    &lt;tr&gt;
        &lt;td&gt;296.85&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;647.37&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;997.90&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;148.42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;498.95&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;849.47&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;350.53&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;701.05&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1051.58&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;202.10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;552.63&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;903.15&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;32/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/16&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Interval chains-2.3.5.11 meantone-like mohajira (&amp;quot;mohaha&amp;quot;)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;2.3.5.11 meantone-like mohajira (&amp;quot;mohaha&amp;quot;)&lt;/h3&gt;
== Interval chains ==
=== Basic 2.3.11 neutral thirds scale ===


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;td&gt;815.25&lt;br /&gt;
| | 296.85
&lt;/td&gt;
| | 647.37
        &lt;td&gt;1163.34&lt;br /&gt;
| | 997.90
&lt;/td&gt;
| | 148.42
        &lt;td&gt;311.44&lt;br /&gt;
| | 498.95
&lt;/td&gt;
| | 849.47
        &lt;td&gt;659.53&lt;br /&gt;
| | 0.
&lt;/td&gt;
| | 350.53
        &lt;td&gt;1007.63&lt;br /&gt;
| | 701.05
&lt;/td&gt;
| | 1051.58
        &lt;td&gt;155.72&lt;br /&gt;
| | 202.10
&lt;/td&gt;
| | 552.63
        &lt;td&gt;503.81&lt;br /&gt;
| | 903.15
&lt;/td&gt;
|-
        &lt;td&gt;851.91&lt;br /&gt;
| | 32/27
&lt;/td&gt;
| | 16/11
        &lt;td&gt;0.&lt;br /&gt;
| | 16/9
&lt;/td&gt;
| | 12/11
        &lt;td&gt;348.09&lt;br /&gt;
| | 4/3
&lt;/td&gt;
| | 18/11
        &lt;td&gt;696.19&lt;br /&gt;
| | 1/1
&lt;/td&gt;
| | 11/9
        &lt;td&gt;1044.28&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 11/6
        &lt;td&gt;192.37&lt;br /&gt;
| | 9/8
&lt;/td&gt;
| | 11/8
        &lt;td&gt;540.47&lt;br /&gt;
| | 27/16
&lt;/td&gt;
|}
        &lt;td&gt;888.56&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;36.66&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;384.75&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9~9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11~11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9~9/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Interval chains-2.3.7.11 beatles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;2.3.7.11 beatles&lt;/h3&gt;
=== 2.3.5.11 mohaha ===


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;td&gt;414.38&lt;br /&gt;
| | 815.25
&lt;/td&gt;
| | 1163.34
        &lt;td&gt;768.34&lt;br /&gt;
| | 311.44
&lt;/td&gt;
| | 659.53
        &lt;td&gt;1122.29&lt;br /&gt;
| | 1007.63
&lt;/td&gt;
| | 155.72
        &lt;td&gt;276.25&lt;br /&gt;
| | 503.81
&lt;/td&gt;
| | 851.91
        &lt;td&gt;630.21&lt;br /&gt;
| | 0.
&lt;/td&gt;
| | 348.09
        &lt;td&gt;984.17&lt;br /&gt;
| | 696.19
&lt;/td&gt;
| | 1044.28
        &lt;td&gt;138.13&lt;br /&gt;
| | 192.37
&lt;/td&gt;
| | 540.47
        &lt;td&gt;492.08&lt;br /&gt;
| | 888.56
&lt;/td&gt;
| | 36.66
        &lt;td&gt;846.04&lt;br /&gt;
| | 384.75
&lt;/td&gt;
|-
        &lt;td&gt;0.&lt;br /&gt;
| | 8/5
&lt;/td&gt;
| |
        &lt;td&gt;353.96&lt;br /&gt;
| | 6/5
&lt;/td&gt;
| | 16/11
        &lt;td&gt;707.92&lt;br /&gt;
| | 16/9~9/5
&lt;/td&gt;
| | 12/11~11/10
        &lt;td&gt;1061.87&lt;br /&gt;
| | 4/3
&lt;/td&gt;
| | 18/11
        &lt;td&gt;215.83&lt;br /&gt;
| | 1/1
&lt;/td&gt;
| | 11/9
        &lt;td&gt;569.79&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 11/6
        &lt;td&gt;923.75&lt;br /&gt;
| | 10/9~9/8
&lt;/td&gt;
| | 11/8
        &lt;td&gt;77.71&lt;br /&gt;
| | 5/3
&lt;/td&gt;
| |
        &lt;td&gt;431.66&lt;br /&gt;
| | 5/4
&lt;/td&gt;
|}
        &lt;td&gt;785.62&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4~16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8~8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
=== 2.3.7.11 suhajira ===
 
{| class="wikitable"
|-
| | 414.38
| | 768.34
| | 1122.29
| | 276.25
| | 630.21
| | 984.17
| | 138.13
| | 492.08
| | 846.04
| | 0.
| | 353.96
| | 707.92
| | 1061.87
| | 215.83
| | 569.79
| | 923.75
| | 77.71
| | 431.66
| | 785.62
|-
| | 14/11
| | 14/9
| |
| | 7/6
| | 16/11
| | 7/4~16/9
| | 12/11
| | 4/3
| | 18/11
| | 1/1
| | 11/9
| | 3/2
| | 11/6
| | 9/8~8/7
| | 11/8
| | 12/7
| |
| | 9/7
| | 11/7
|}
 
== MOSes ==
Neutral third scales are associated with the [[MOS scales|MOS patterns]]: [[3L 4s]] and [[7L 3s]].
 
In the 3L4s MOS, the large step is a 9/8 and the small step is a 12/11 neutral second. In the 7L3s MOS, the 12/11 is now the large step and the small step is a 33/32 quarter tone.
 
== Munits ==
Though they differ in the specifics of their temperaments, all neutral third scales have certain [[munit]]s in common.
 
* 4/3 Lss, 4/3 sLs: Rather than being divided into two large steps and one small step as in the diatonic scale, 4/3 can always be divided into one large step and two small steps. The large step is 9/8, and the two small steps are "neutral seconds" representing the sub-munit "32/27 ss".
* 11/8 LsL: In contrast, two large steps and a small step make a much less consonant and more colorful interval, 11/8.
 
== Lumatone mapping ==
''See: [[Lumatone mapping for neutral thirds scales]]''.
 
== Neutral thirds scales by EDO ==
Not an exhaustive list.
 
 
=== 7edo ===
 
 
1 1 1 1 1 1 1
 
7edo
 
 
=== 10edo ===
 
 
2 1 2 1 2 1 1
 
Neutral Hypolydian
 
 
2 1 2 1 1 2 1
 
Neutral Hypodorian
 
 
2 1 1 2 1 2 1
 
Neutral Lydian
 
 
1 2 1 2 1 2 1
 
Neutral Dorian
 
 
1 2 1 2 1 1 2
 
Neutral Hypophrygian
 
 
''Any other scales which contain the 3rd (360c) and 6th (720c) degrees of 10edo are also considered neutral thirds scales.''
 
 
=== 17edo ===
''See also: [[17edo neutral scale]]''
 
 
3 2 3 2 3 2 2
 
Neutral Hypolydian
 
 
3 2 3 2 2 3 2
 
Neutral Hypodorian
 
 
3 2 2 3 2 3 2
 
Neutral Lydian
 
 
2 3 2 3 2 3 2
 
Neutral Dorian, Misaelides 2nd Byzantine mode, Maqam Sikah Baladi
 
 
2 3 2 3 2 2 3
 
Neutral Hypophrygian
 
 
''Any other scales which contain the 5th (~353c) and 10th (~706c) degrees of 10edo are also considered neutral thirds scales.''
 
 
=== 24edo ===
''See also: [[24edo scales]]''
 
 
4 3 4 3 4 3 3
 
Neutral Hypolydian
 
 
4 3 4 3 3 4 3
 
Neutral Hypodorian
 
 
4 3 3 4 3 4 3
 
Neutral Lydian
 
 
3 4 3 4 3 4 3
 
Neutral Dorian, Misaelides 2nd Byzantine mode, Maqam Sikah Baladi
 
 
3 4 3 4 3 3 4
 
Neutral Hypophrygian
 
 
''Any other scales which contain the 7th (350c) and 14th (700c) degrees of 24edo are also considered neutral thirds scales.''
 
== See also ==
* [[Songs with neutral intervals]] (not a complete list, only a small sampling)
 
 
{{Navbox scale gallery}}
[[Category:Lists of scales]]
[[Category:Neutral| ]] <!-- main article -->
[[Category:Rastmic]]
Retrieved from "https://en.xen.wiki/w/Neutral_third_scales"