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'''Mohajira''' (/mʊˈhɑːdʒɪɹə/ ''muu-HAH-jirr-ə'' or /məˈhɑːdʒɪɹə/ ''mə-HAH-jirr-ə'', named by Jacques Dudon from Arabic مهاجرة ''Muhājirah'') is a [[Meantone_family#Mohajira|meantone]] temperament that splits 3/2 into two 11/9's and the 6/5 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.
{{Infobox regtemp
| 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.


= Temperament data (2.3.5.7.11, 24&31) =
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]].
Period: 1\1


Optimal ([[POTE]]) generator: ~11/9 = 348.477
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).  


EDO generators: [[24edo|7\24]], [[31edo|9\31]]
For technical data, see [[Meantone family #Mohajira]].


Scales (Scala files):
== Interval chain ==
In the following tables, odd harmonics 1–11 are shown in '''bold'''.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="display: inline-grid; margin-right: 25px;">
<div style="line-height:1.6;">Interval table (10-note MOS, 2.3.5.7.11 POTE tuning)</div>
{| class="wikitable center-1 right-2"
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5.7.11, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.</ref>
! Approximate ratios
|-
|-
| 0
| 0
| 0.00
| 0.0
| 1/1
| '''1/1'''
|-
|-
| 1
| 1
| 348.5
| 348.5
| 11/9
| 11/9, 27/22
|-
|-
| 2
| 2
| 697.0
| 697.0
| 3/2
| '''3/2'''
|-
|-
| 3
| 3
| 1045.4
| 1045.5
| 11/6
| 11/6, 20/11
|-
|-
| 4
| 4
| 193.9
| 194.0
| 9/8
| '''9/8''', 10/9
|-
|-
| 5
| 5
| 542.4
| 542.5
| 11/8, 15/11
| '''11/8''', 15/11
|-
|-
| 6
| 6
Line 55: Line 65:
|-
|-
| 7
| 7
| 39.3
| 39.4
|  
| 33/32, 36/35, 45/44, <br>55/54, 64/63
|-
|-
| 8
| 8
| 387.8
| 387.9
| 5/4
| '''5/4'''
|-
|-
| 9
| 9
| 736.3
| 736.4
| 32/21
| 32/21
|}
<references/></div></div>
=Modes and MOS=
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 contains 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.
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.
==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
For the modes of the Mohajira[7] MOS, names with "Ice" have been proposed, referring to Hidekazu Wakabayashi's [[Iceface_Tuning|Iceface Tuning]].
{| class="wikitable"
|-
|-
| | Iced Major
| 10
| | C D Ed Ft G A Bd C
| 1084.9
| | M2 N3 S4 P5 M6 N7
| 15/8
|-
| 11
| 233.4
| 8/7
|-
|-
| | Iced Locrian
| 12
| | D Ed Ft G A Bd C D
| 581.9
| | N2 N3 P4 P5 N6 m7
| 25/18, 45/32
|-
|-
| | Iced Fridgian (Iced Minor)
| 13
| | Ed Ft G A Bd C D Ed
| 930.4
| | N2 N3 P4 P5 N6 N7
| 12/7
|-
|-
| | Iced Lydian
| 14
| | Ft G A Bd C D Ed Ft
| 78.9
| | M2 N3 S4 s5 N6 N7
| 22/21, 25/24
|-
|-
| | Iced Mixolydian
| 15
| | G A Bd C D Ed Ft G
| 427.4
| | M2 N3 P4 P5 N6 N7
| 9/7
|-
|-
| | Iced Dark Lydian (Iced Coffee)
| 16
| | A Bd C D Ed Ft G A
| 775.9
| | N2 m3 S4 s5 N6 m7
| 11/7
|-
|-
| | Iced Blizzard (Neutral Scale)
| 17
| | Bd C D Ed Ft G A Bd
| 1124.3
| |N2 N3 P4 P5 N6 N7
| 27/14, 40/21
|}
|}


Strangely, the mode names do not match the typical order of western mode names due to the odd intervalic nature of Mohajira temperament.
</div>
 
<div style="display: inline-grid; margin-right: 25px;">
==Level 1 - Mohajira MODMOS by 1 alteration==
{| class="wikitable center-1 right-2"
Rast - 1 2 v3 4 5 6 v7 1 or C D Ed F G A Bd C
 
==Level 2 - Mohajira MODMOS by 2 alteration==
Altered Melodic Minor - 1 2 b3 4 5 6 v7 1 or C D Eb F G A Bd C
 
Turkish Minor - 1 2 v3 4 5 b6 v7 1 or C D Ed F G Ab Bd C
 
Bayati or [[Screamapillar|screamapillar]] - 1 d2 v3 4 5 b6 b7 1 or C Dd Ed F G Ab Bb
 
==Level 3 - Meantone by 3 alterations==
 
[[File:altered_mohajira_levels.PNG|alt=altered mohajira levels.PNG|altered mohajira levels.PNG]]
 
==Modes of various Mohajira MOS and MODMOS.==
 
===Rast===
 
{| class="wikitable"
|-
|-
| | Mode 1
! #
| | C D Ed F G A Bd C
! Cents*
! Approximate ratios
|-
|-
| | Mode 2
| 0
| | D Ed F G A Bd C D
| 1200.0
| '''2/1'''
|-
|-
| | Mode 3
| -1
| | Ed F G A Bd C D Ed
| 851.5
| 18/11, 44/27
|-
|-
| | Mode 4
| -2
| | F G A Bd C D Ed F
| 503.0
| 4/3
|-
|-
| | Mode 5
| -3
| | G A Bd C D Ed F G
| 154.5
| 11/10, 12/11
|-
|-
| | Mode 6
| -4
| | A Bd C D Ed F G A
| 1006.0
|-
| 9/5, 16/9
| | Mode 7
| | Bd C D Ed F G A Bd
|}
 
===Altered Melodic Minor===
 
{| class="wikitable"
|-
|-
| | Mode 1
| -5
| | C D Eb F G A Bd C
| 657.5
| 16/11, 22/15
|-
|-
| | Mode 2
| -6
| | D Eb F G A Bd C D
| 309.1
| 6/5
|-
|-
| | Mode 3
| -7
| | Eb F G A Bd C D Eb
| 1160.6
| 35/18, 63/32, 64/33, <br>88/45, 108/55
|-
|-
| | Mode 4
| -8
| | F G A Bd C D Eb F
| 812.1
| 8/5
|-
|-
| | Mode 5
| -9
| | G A Bd C D Eb F G
| 463.6
| 21/16
|-
|-
| | Mode 6
| -10
| | A Bd C D Eb F G A
| 115.1
| 16/15
|-
|-
| | Mode 7
| -11
| | Bd C D Eb F G A Bd
| 966.6
|}
| '''7/4'''
 
===Turkish Minor===
 
{| class="wikitable"
|-
|-
| | Mode 1
| -12
| | C D Ed F G Ab Bd C
| 618.1
| 36/25, 64/45
|-
|-
| | Mode 2
| -13
| | D Ed F G Ab Bd C D
| 269.6
| 7/6
|-
|-
| | Mode 3
| -14
| | Ed F G Ab Bd C D Ed
| 1121.1
| 21/11, 48/25
|-
|-
| | Mode 4
| -15
| | F G Ab Bd C D Ed F
| 772.6
| 14/9
|-
|-
| | Mode 5
| -16
| | G Ab Bd C D Ed F G
| 424.1
| 14/11
|-
|-
| | Mode 6
| -17
| | Ab Bd C D Ed F G Ab
| 75.7
|-
| 21/20, 28/27
| | Mode 7
| | Bd C D Ed F G Ab Bd
|}
|}
</div>
<nowiki/>* In 11-limit CWE tuning, octave reduced


=Harmonization of Mohajira=
== Scales and harmony ==
While the structure of mohajira[7] may seem similar to the shape of meantone, because of there being only one size of thirds, it's better to harmonize the scale with a combination of various voicings of 1-5-7-4 and 1-5-7-3 as well as other combinations of chords such as quartal tetrads. The Rast scales generally have more variety as they contain major and minor thirds as well as neutral thirds.
: {{Main| Mohajira/Scales }}


To introduce more interest into harmony involving mohajira, a 10 note MOS can be used to incorporate various synthetic versions of traditional turkish and middle eastern 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.


===Mohajira[10] - The Neutral Superdiatonic Scale===
== Tunings ==
 
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
|-
| | Mode 1
! rowspan="2" |  
| | 3 6 7 10 13 14 17 20 23
! colspan="3" | Euclidean
| | 4 8 9 13 17 18 22 26 30
|-
|-
| | Mode 2
! Constrained
| | 3 4 7 10 11 14 17 20 21
! Constrained & skewed
| | 4 5 9 13 14 18 22 26 27
! Destretched
|-
|-
| | Mode 3
! Tenney
| | 1 4 7 8 11 14 17 18 21
| CTE: ~128/105 = 348.4374{{c}}
| | 1 5 9 10 14 18 22 23 27
| CWE: ~128/105 = 348.4194{{c}}
| POTE: ~128/105 = 348.4149{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
|-
| | Mode 4
! rowspan="2" |  
| | 3 6 7 10 13 16 17 20 23
! colspan="3" | Euclidean
| | 4 8 9 13 17 21 22 26 30
|-
|-
| | Mode 5
! Constrained
| | 3 4 7 10 13 14 17 20 21
! Constrained & skewed
| | 4 5 9 13 17 18 22 26 27
! Destretched
|-
|-
| | Mode 6
! Tenney
| | 1 4 7 10 11 14 17 18 21
| CTE: ~11/9 = 348.5608{{c}}
| | 1 5 9 13 14 18 22 23 27
| CWE: ~11/9 = 348.4910{{c}}
|-
| POTE: ~11/9 = 348.4767{{c}}
| | Mode 7
| | 3 6 9 10 13 16 17 20 23
| | 4 8 12 13 17 21 22 26 30
|-
| | Mode 8
| | 3 6 7 10 13 14 17 20 21
| | 4 8 9 13 17 18 22 26 27
|-
| | Mode 9
| | 3 4 7 10 11 14 17 18 21
| | 4 5 9 13 14 18 22 23 27
|-
| | Mode 10
| | 1 4 7 8 11 14 15 18 21
| | 1 4 9 10 14 18 19 23 27
|}
|}


The intervals of Mohajira and Neutral harmony work in an almost reverse manner from meantone. The most consonant is 3/2 and 4/3 followed by 11/6, 11/9 and 11/8. In context of a chord with a perfect fifth, 11/9's tend to produce a rather rough chord that sounds good in many contexts but is quite rough as a tonic chord. It can be good to think of the neutral seventh 11/6 and 3/2 as the base intervals of a chord with 11/8 and 4/3 acting almost in an analogous way to the major and minor third in meantone. This is why 11/8 can be called the major fourth in 24 ET.
== Notes ==
 
<references group="note" />
See also [[Meantone family #Mohajira]].


[[Category:Temperament]]
[[Category:Mohajira| ]] <!-- main article -->
[[Category:Meantone]]
[[Category:Rank-2 temperaments]]
[[Category:Mohajira]]
[[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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