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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| de = Mohajira
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<h4>Original Wikitext content:</h4>
{{Infobox regtemp
<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 a meantone temperament that splits 3/2 into two 11/9's and the 4/3 into two 11/10's. It can be thought of as meantone with quarter tones as it merely spits the generator 3/2 into two equally spaced neutral thirds. Among the most common is the seven note MOS mohajira[7] which consists of the steps LsLsLss, as well as the Modmos "Rast" which merely flats the fourth by one quarter tone.  
| Title = Mohajira
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[81/80]], [[6144/6125]] (7-limit); <br>[[81/80]], [[121/120]], [[176/175]] (11-limit)
| Edo join 1 = 24 | Edo join 2 = 31
| Mapping = 1; 2 8 -11 5
| Generators = 11/9
| Generators tuning = 348.5
| Optimization method = CWE
| MOS scales = [[3L 4s]], [[7L 3s]], [[7L 10s]], [[7L 17s]]
| Odd limit 1 = 9 | Mistuning 1 = 10.8 | Complexity 1 = 24
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 10.8 | Complexity 2 = 24
}}
'''Mohajira'''<ref group="note">(/mʊˈhɑːdʒɪɹə, -hæ-/ ''muu-HA(H)-jirr-ə'' or /məˈhɑːdʒɪɹə, -hæ-/ ''mə-HA(H)-jirr-ə'', named by Jacques Dudon from Arabic مهاجرة ''Muhājirah'')</ref> is a [[Meantone_family#Mohajira|meantone]] temperament that splits [[3/2]] into two [[11/9]] generators and the [[6/5]] into two [[11/10]]'s, thus equating 11/10 with [[12/11]] and 11/9 with [[27/22]]. As a result, eight 11/9's reach [[5/4]], and five reach [[11/8]]. It [[tempering out|tempers out]] the biyatisma ([[121/120]]) and the [[rastma]] (along with meantone's [[syntonic comma]]). It can be thought of as meantone with [[quartertone]]s as it merely splits the generator 3/2 into two equally spaced neutral thirds. Among the most common is the 7-note mos Mohajira[7] which consists of the steps LsLsLss, as well as the [[modmos]] "Rast" which merely flats the fourth by one quartertone.


=Modes and MOS=
The simplest form of this temperament, in 2.3.5.11, is sometimes called '''mohaha'''. To extend mohaha into full 11-limit mohajira, 7 is found at 11 neutral thirds down, tempering out the [[valinorsma]] such that 14/11 is equated with 32/25. Mohaha is tuned best sharp of 31edo's neutral third. An alternate extension, ''migration'', is optimized for flatter tunings and finds 7 at 20 generators up, the same mapping as in [[septimal meantone]] which tempers out [[225/224]].
There are seven diatonic modes of mohajira which are structured in an analogous way to the seven church modes of the major scale. They can be thought of as quarter tone altered modes in a 24 EDO setting, but also as microtonal altered scales in any equal temperament. While the Mohajira[7] MOS by itself is quite beautiful, the Modmos, Rast tends to be more expressive as it contain a few major and minor thirds allowing for more contrast.  


Because there are too many possible Mohajira Mos and Modmos, Kentaku has organized a way to think of the progression of accidentals in the modes of Mohajira.
As mohajira is a neutral third system, 13 also has a reasonable mapping at −1 generator, equating 11/9 and 16/13. Subsequently, 17 is mapped to −10 generators (the diatonic semitone) and 19 to −6 generators (the minor third).  
The scales are organized based on evoking slightly different colors and categorized by different levels of alteration from a meantone scale. A Mos of Mohajira[10] can be used in order to improvise on all the common Modmos of mohajira.  


Here are the scales with 1 2 3 4 5 6 7 being the major scale.
For technical data, see [[Meantone family #Mohajira]].
==Level 0 - Mohajira MOS - The Neutral Diatonic scale==
1 2 v3 ^4 5 6 v7 1 or C D Ed Ft G A Bd C


==Level 1 - Mohajira ModMOS by 1 alteration==  
== Interval chain ==
Rast - 1 2 v3 4 5 6 v7 1 or C D Ed F G A Bd C
In the following tables, odd harmonics 1–11 are shown in '''bold'''.


==Level 2 - Mohajira ModMOS by 2 alteration==
<div style="display: inline-grid; margin-right: 25px;">
Altered Melodic Minor - 1 2 b3 4 5 6 v7 1 or C D Eb F G A Bd C
{| class="wikitable center-1 right-2"
Turkish Minor - 1 2 b3 4 5 b6 7 1 or C D Ed F G A Bd C
|-
Bayati - 1 d2 b3 4 5 b6 b7 1
! #
==Level 3 - Meantone by 3 alterations==
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 348.5
| 11/9, 27/22
|-
| 2
| 697.0
| '''3/2'''
|-
| 3
| 1045.5
| 11/6, 20/11
|-
| 4
| 194.0
| '''9/8''', 10/9
|-
| 5
| 542.5
| '''11/8''', 15/11
|-
| 6
| 890.9
| 5/3
|-
| 7
| 39.4
| 33/32, 36/35, 45/44, <br>55/54, 64/63
|-
| 8
| 387.9
| '''5/4'''
|-
| 9
| 736.4
| 32/21
|-
| 10
| 1084.9
| 15/8
|-
| 11
| 233.4
| 8/7
|-
| 12
| 581.9
| 25/18, 45/32
|-
| 13
| 930.4
| 12/7
|-
| 14
| 78.9
| 22/21, 25/24
|-
| 15
| 427.4
| 9/7
|-
| 16
| 775.9
| 11/7
|-
| 17
| 1124.3
| 27/14, 40/21
|}


[[image:altered mohajira levels.PNG]]
</div>
<div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 1200.0
| '''2/1'''
|-
| -1
| 851.5
| 18/11, 44/27
|-
| -2
| 503.0
| 4/3
|-
| -3
| 154.5
| 11/10, 12/11
|-
| -4
| 1006.0
| 9/5, 16/9
|-
| -5
| 657.5
| 16/11, 22/15
|-
| -6
| 309.1
| 6/5
|-
| -7
| 1160.6
| 35/18, 63/32, 64/33, <br>88/45, 108/55
|-
| -8
| 812.1
| 8/5
|-
| -9
| 463.6
| 21/16
|-
| -10
| 115.1
| 16/15
|-
| -11
| 966.6
| '''7/4'''
|-
| -12
| 618.1
| 36/25, 64/45
|-
| -13
| 269.6
| 7/6
|-
| -14
| 1121.1
| 21/11, 48/25
|-
| -15
| 772.6
| 14/9
|-
| -16
| 424.1
| 14/11
|-
| -17
| 75.7
| 21/20, 28/27
|}
</div>
<nowiki/>* In 11-limit CWE tuning, octave reduced


=Harmonization of Mohajira=  
== Scales and harmony ==
While the structure of mohajra[7] is virtually the same as meantone[7] with neutral tones instead of semitones, the scale doesn't harmonize in the same manner. Because there is now only one middle sized third, it's best to imagine.
: {{Main| Mohajira/Scales }}


See also [[Meantone family#Mohajira]].</pre></div>
There are seven modes of mohajira's 7-note mos (named [[mosh]] due to being "mohajira-ish") which are structured in an analogous way to the seven modes of the major scale. They can be thought of as quarter tone altered modes in a 24edo setting, but also as microtonal altered scales in any equal temperament. While the Mohajira[7] mos by itself is quite beautiful, the modMOS Rast tends to be more expressive as it contains a few major and minor thirds allowing for more contrast.
<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;Mohajira&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Mohajira is a meantone temperament that splits 3/2 into two 11/9's and the 4/3 into two 11/10's. It can be thought of as meantone with quarter tones as it merely spits the generator 3/2 into two equally spaced neutral thirds. Among the most common is the seven note MOS mohajira[7] which consists of the steps LsLsLss, as well as the Modmos &amp;quot;Rast&amp;quot; which merely flats the fourth by one quarter tone. &lt;br /&gt;
== Tunings ==
&lt;br /&gt;
{| class="wikitable mw-collapsible mw-collapsed"
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Modes and MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Modes and MOS&lt;/h1&gt;
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
There are seven diatonic modes of mohajira which are structured in an analogous way to the seven church modes of the major scale. They can be thought of as quarter tone altered modes in a 24 EDO setting, but also as microtonal altered scales in any equal temperament. While the Mohajira[7] MOS by itself is quite beautiful, the Modmos, Rast tends to be more expressive as it contain a few major and minor thirds allowing for more contrast. &lt;br /&gt;
|-
&lt;br /&gt;
! rowspan="2" |
Because there are too many possible Mohajira Mos and Modmos, Kentaku has organized a way to think of the progression of accidentals in the modes of Mohajira.&lt;br /&gt;
! colspan="3" | Euclidean
The scales are organized based on evoking slightly different colors and categorized by different levels of alteration from a meantone scale. A Mos of Mohajira[10] can be used in order to improvise on all the common Modmos of mohajira. &lt;br /&gt;
|-
&lt;br /&gt;
! Constrained
Here are the scales with 1 2 3 4 5 6 7 being the major scale.&lt;br /&gt;
! Constrained & skewed
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Modes and MOS-Level 0 - Mohajira MOS - The Neutral Diatonic scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Level 0 - Mohajira MOS - The Neutral Diatonic scale&lt;/h2&gt;
! Destretched
1 2 v3 ^4 5 6 v7 1 or C D Ed Ft G A Bd C&lt;br /&gt;
|-
&lt;br /&gt;
! Tenney
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Modes and MOS-Level 1 - Mohajira ModMOS by 1 alteration"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Level 1 - Mohajira ModMOS by 1 alteration&lt;/h2&gt;
| CTE: ~128/105 = 348.4374{{c}}
Rast - 1 2 v3 4 5 6 v7 1 or C D Ed F G A Bd C&lt;br /&gt;
| CWE: ~128/105 = 348.4194{{c}}
&lt;br /&gt;
| POTE: ~128/105 = 348.4149{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Modes and MOS-Level 2 - Mohajira ModMOS by 2 alteration"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Level 2 - Mohajira ModMOS by 2 alteration&lt;/h2&gt;
|}
Altered Melodic Minor - 1 2 b3 4 5 6 v7 1 or C D Eb F G A Bd C&lt;br /&gt;
 
Turkish Minor - 1 2 b3 4 5 b6 7 1 or C D Ed F G A Bd C&lt;br /&gt;
{| class="wikitable mw-collapsible mw-collapsed"
Bayati - 1 d2 b3 4 5 b6 b7 1&lt;br /&gt;
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Modes and MOS-Level 3 - Meantone by 3 alterations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Level 3 - Meantone by 3 alterations&lt;/h2&gt;
|-
&lt;br /&gt;
! rowspan="2" |
&lt;!-- ws:start:WikiTextLocalImageRule:12:&amp;lt;img src=&amp;quot;/file/view/altered%20mohajira%20levels.PNG/519036492/altered%20mohajira%20levels.PNG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/altered%20mohajira%20levels.PNG/519036492/altered%20mohajira%20levels.PNG" alt="altered mohajira levels.PNG" title="altered mohajira levels.PNG" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:12 --&gt;&lt;br /&gt;
! colspan="3" | Euclidean
&lt;br /&gt;
|-
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Harmonization of Mohajira"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Harmonization of Mohajira&lt;/h1&gt;
! Constrained
While the structure of mohajra[7] is virtually the same as meantone[7] with neutral tones instead of semitones, the scale doesn't harmonize in the same manner. Because there is now only one middle sized third, it's best to imagine.&lt;br /&gt;
! Constrained & skewed
&lt;br /&gt;
! Destretched
See also &lt;a class="wiki_link" href="/Meantone%20family#Mohajira"&gt;Meantone family&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
|-
! Tenney
| CTE: ~11/9 = 348.5608{{c}}
| CWE: ~11/9 = 348.4910{{c}}
| POTE: ~11/9 = 348.4767{{c}}
|}
 
== Notes ==
<references group="note" />
 
[[Category:Mohajira| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Meantone family]]
[[Category:Porwell temperaments]]
[[Category:Rastmic clan]]

Latest revision as of 10:19, 8 April 2026

Mohajira
Subgroups 2.3.5.7, 2.3.5.7.11
Comma basis 81/80, 6144/6125 (7-limit);
81/80, 121/120, 176/175 (11-limit)
Reduced mapping ⟨1; 2 8 -11 5]
ET join 24 & 31
Generators (CWE) ~11/9 = 348.5 ¢
MOS scales 3L 4s, 7L 3s, 7L 10s, 7L 17s
Ploidacot dicot
Minimax error 9-odd-limit: 10.8 ¢;
11-limit 21-odd-limit: 10.8 ¢
Target scale size 9-odd-limit: 24 notes;
11-limit 21-odd-limit: 24 notes

Mohajira[note 1] is a meantone temperament that splits 3/2 into two 11/9 generators and the 6/5 into two 11/10's, thus equating 11/10 with 12/11 and 11/9 with 27/22. As a result, eight 11/9's reach 5/4, and five reach 11/8. It tempers out the biyatisma (121/120) and the rastma (along with meantone's syntonic comma). It can be thought of as meantone with quartertones as it merely splits the generator 3/2 into two equally spaced neutral thirds. Among the most common is the 7-note mos Mohajira[7] which consists of the steps LsLsLss, as well as the modmos "Rast" which merely flats the fourth by one quartertone.

The simplest form of this temperament, in 2.3.5.11, is sometimes called mohaha. To extend mohaha into full 11-limit mohajira, 7 is found at 11 neutral thirds down, tempering out the valinorsma such that 14/11 is equated with 32/25. Mohaha is tuned best sharp of 31edo's neutral third. An alternate extension, migration, is optimized for flatter tunings and finds 7 at 20 generators up, the same mapping as in septimal meantone which tempers out 225/224.

As mohajira is a neutral third system, 13 also has a reasonable mapping at −1 generator, equating 11/9 and 16/13. Subsequently, 17 is mapped to −10 generators (the diatonic semitone) and 19 to −6 generators (the minor third).

For technical data, see Meantone family #Mohajira.

Interval chain

In the following tables, odd harmonics 1–11 are shown in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 348.5 11/9, 27/22
2 697.0 3/2
3 1045.5 11/6, 20/11
4 194.0 9/8, 10/9
5 542.5 11/8, 15/11
6 890.9 5/3
7 39.4 33/32, 36/35, 45/44,
55/54, 64/63
8 387.9 5/4
9 736.4 32/21
10 1084.9 15/8
11 233.4 8/7
12 581.9 25/18, 45/32
13 930.4 12/7
14 78.9 22/21, 25/24
15 427.4 9/7
16 775.9 11/7
17 1124.3 27/14, 40/21
# Cents* Approximate ratios
0 1200.0 2/1
-1 851.5 18/11, 44/27
-2 503.0 4/3
-3 154.5 11/10, 12/11
-4 1006.0 9/5, 16/9
-5 657.5 16/11, 22/15
-6 309.1 6/5
-7 1160.6 35/18, 63/32, 64/33,
88/45, 108/55
-8 812.1 8/5
-9 463.6 21/16
-10 115.1 16/15
-11 966.6 7/4
-12 618.1 36/25, 64/45
-13 269.6 7/6
-14 1121.1 21/11, 48/25
-15 772.6 14/9
-16 424.1 14/11
-17 75.7 21/20, 28/27

* In 11-limit CWE tuning, octave reduced

Scales and harmony

Main article: Mohajira/Scales

There are seven modes of mohajira's 7-note mos (named mosh due to being "mohajira-ish") which are structured in an analogous way to the seven modes of the major scale. They can be thought of as quarter tone altered modes in a 24edo setting, but also as microtonal altered scales in any equal temperament. While the Mohajira[7] mos by itself is quite beautiful, the modMOS Rast tends to be more expressive as it contains a few major and minor thirds allowing for more contrast.

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~128/105 = 348.4374 ¢ CWE: ~128/105 = 348.4194 ¢ POTE: ~128/105 = 348.4149 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~11/9 = 348.5608 ¢ CWE: ~11/9 = 348.4910 ¢ POTE: ~11/9 = 348.4767 ¢

Notes

  1. ↑ (/mʊˈhɑːdʒɪɹə, -hæ-/ muu-HA(H)-jirr-ə or /məˈhɑːdʒɪɹə, -hæ-/ mə-HA(H)-jirr-ə, named by Jacques Dudon from Arabic مهاجرة Muhājirah)
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