Mohajira
Mohajira (/mʊˈhɑːdʒɪɹə, -hæ-/ muu-HA(H)-jirr-ə or /məˈhɑːdʒɪɹə, -hæ-/ mə-HA(H)-jirr-ə, named by Jacques Dudon from Arabic مهاجرة Muhājirah) is a 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.
For temperament data see Meantone family #Mohajira.
Temperament data (2.3.5.7.11, 24&31)
Period: 1\1
Optimal (POTE) generator: ~11/9 = 348.477
Scales (Scala files):
| #Gens up | Cents* | Approximate ratios** |
|---|---|---|
| 0 | 0.00 | 1/1 |
| 1 | 348.5 | 11/9 |
| 2 | 697.0 | 3/2 |
| 3 | 1045.4 | 11/6 |
| 4 | 193.9 | 9/8 |
| 5 | 542.4 | 11/8, 15/11 |
| 6 | 890.9 | 5/3 |
| 7 | 39.3 | |
| 8 | 387.8 | 5/4 |
| 9 | 736.3 | 32/21 |
* Octave-reduced
** 2.3.5.7.11, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.
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 E 
F 
G A B C
For the modes of the Mohajira[7] MOS, names with "Ice" have been proposed, referring to Hidekazu Wakabayashi's Iceface tuning.
| Iced Major | C D E F G A B C
|
M2 N3 S4 P5 M6 N7 |
|---|---|---|
| Iced Locrian | D E F G A B C D
|
N2 N3 P4 P5 N6 m7 |
| Iced Fridgian (Iced Minor) | E F G A B C D E
|
M2 N3 S4 P5 N6 N7 |
| Iced Lydian | F G A B C D E F
|
N2 N3 P4 s5 N6 m7 |
| Iced Mixolydian | G A B C D E F G
|
M2 N3 P4 P5 N6 N7 |
| Iced Dark Lydian (Iced Coffee) | A B C D E F G A
|
N2 m3 P4 s5 N6 m7 |
| Iced Blizzard (Neutral Scale) | B C D E F G A B
|
N2 N3 P4 P5 N6 N7 |
Strangely, the mode names do not match the typical order of western mode names due to the odd intervalic nature of Mohajira temperament.
Level 1 - Mohajira MODMOS by 1 alteration
Rast - 1 2 v3 4 5 6 v7 1
| Mode 1 | C D E F G A B C
|
|---|---|
| Mode 2 | D E F G A B C D
|
| Mode 3 | E F G A B C D Ed
|
| Mode 4 | F G A B C D E F
|
| Mode 5 | G A B C D E F G
|
| Mode 6 | A B C D E F G A
|
| Mode 7 | B C D E F G A B
|
Turkish Major - 1 2 3 4 5 v6 v7 1
| Mode 1 | C D E F G A B C
|
|---|---|
| Mode 2 | D E F G A B C D
|
| Mode 3 | E F G A B C D E
|
| Mode 4 | F G A B C D E F
|
| Mode 5 | G A B C D E F G
|
| Mode 6 | A B C D E F G A
|
| Mode 7 | B C D E F G A B
|
Level 2 - Mohajira MODMOS by 2 alterations
Altered Neapolitan Major - 1 b2 b3 4 5 6 v7 1 or C Db Eb F G A Bd C
Altered Melodic Minor (Altered Dorian) - 1 2 b3 4 5 6 v7 1
| Mode 1 | C D E♭ F G A B C
|
|---|---|
| Mode 2 | D E♭ F G A B C D
|
| Mode 3 | E♭ F G A B C D E
|
| Mode 4 | F G A B C D E♭ F
|
| Mode 5 | G A B C D E♭ F G
|
| Mode 6 | A B C D E♭ F G A
|
| Mode 7 | B C D E♭ F G A B
|
Turkish Minor - 1 2 b3 4 5 v6 b7 1 or C D Eb F G Ad Bb C
Altered Major - 1 2 3 4 5 6 v7 1 and 1 v2 3 4 5 6 7 1
| Mode 1 | C D E F G A B C
|
C D E F G A B C
|
|---|---|---|
| Mode 2 | D E F G A B C D
|
D E F G A B C D
|
| Mode 3 | E F G A B C D E
|
E F G A B C D E
|
| Mode 4 | F G A B C D E F
|
F G A B C D E F
|
| Mode 5 | G A B C D E F G
|
G A B C D E F G
|
| Mode 6 | A B C D E F G A
|
A B C D E F G A
|
| Mode 7 | B C D E F G A B
|
B C D E F G A B
|
Bayati or screamapillar - 1 v2 b3 4 5 b6 b7 1 or C Dd Eb F G Ab Bb
Altered Phrygian - 1 v2 b3 4 5 b6 b7 1 or C Dd Eb F G Ab Bb
Level 3 - Meantone by 3 alterations
Tuning the Turkish Major scale
Turkish Major is a tempered max-variety 3 scale that is equivalent to a smitonic scale with one of its small steps diminished. This makes a Neapolitan Major scale which does not temper out 36/35. Not tempering 36/35 is actually quite useful, because it's the difference between 4:5:6 and 6:7:9 triads. This is important in a neutral third tone system because the smoothest neutral chord with a perfect fifth is 6:7:9:11. As a result, results of tempering out 81/80 or 64/63 are not as bad, because the scale must detemper one if it tempers out the other. Strangely, the detempering of 36/35 is not evident due to the odd intervalic nature of the Turkish Major scale. Smitonic in a sense does the opposite of what Neapolitan Major does in common practice, exaggerating 36/35 to the point that 4:5:6 and 6:7:9 triads no longer have a recognizable 3/2, and the small step of Turkish Major becomes equal to the medium steps.
| Tuning | L:m:s | Good Just Approximations | Other comments | Degrees | |||||
|---|---|---|---|---|---|---|---|---|---|
| D | E | F | G | Ad | Bd | ||||
| (~)9/8 | 5/4
81/64 |
(~)4/3 | (~)3/2 | ~175/108
~44/27 |
~175/96
~11/6 | ||||
| “Just” | 1.649:1.256:1
2.26:1.63:1 |
Just 5/4
Just 9/8 and 4/3 |
193.157
203.91 |
386.314
407.82 |
503.422
498.045 |
696.578
701.955 |
843.646
849.0225 |
1036.803
1052.9325 | |
| 17edo | 3:2:1 | 25/24 | 211.765 | 423.529 | 494.118 | 705.882 | 847.059 | 1058.8235 | |
| 21edo | 4:2:1 | 228.571 | 457.143 | 514.286 | 742.857 | 857.143 | 1085.714 | ||
| 23edo | 4:3:1 | Neogothic thirds | Mavila | 208.696 | 417.381 | 469.565 | 678.261 | 834.783 | 1043.478 |
| 24edo | 4:3:2 | 4/3 | Mohajira | 200 | 400 | 500 | 700 | 850 | 1050 |
| 25edo | 5:2:1 | 36/35 | Mavila | 240 | 480 | 528 | 768 | 864 | 1104 |
| 27edo | 5:3:1 | 27/25 | 222.222 | 444.444 | 488.889 | 711.111 | 844.444 | 1066.667 | |
| 28edo | 5:3:2 | Antikythera | 214.286 | 428.571 | 514.286 | 728.571 | 857.143 | 1071.429 | |
| 29edo | 5:4:1
6:2:1 |
Neogothic thirds | Score | 206.897
248.276 |
413.793
496.552 |
455.172
537.931 |
662.069
786.206 |
827.586
868.9655 |
1034.483
1117.241 |
| 30edo | 5:4:2 | 13/8 | Mavila | 200 | 400 | 480 | 680 | 840 | 1040 |
| 31edo | 5:4:3
6:3:1 |
5/4
8/7 |
Mohajira | 193.548
232.258 |
387.097
464.516 |
503.226 | 696.774
735.484 |
851.613 | 1045.161
1083.871 |
| 32edo | 6:3:2 | Mavila | 225 | 450 | 525 | 750 | 862.5 | 1087.5 | |
| 33edo | 6:4:1
7:2:1 |
9/7 | 218.182
254.5455 |
436.364
509.091 |
472.727
545.4545 |
690.909
763.636 |
836.364 | 1054.5455
1090.909 | |
| 35edo | 6:4:3
6:5:1 7:3:1 |
Neogothic thirds | Has both “perfect“ fifths of 35edo | 205.714
240 |
411.429
480 |
514.286
445.714 |
720
651.429 754.286 |
925.714
822.857 857.143 |
1131.429
1028.571 1097.143 |
| 36edo | 6:5:2
7:3:2 |
Mavila | 200
233.333 |
400
466.667 |
466.667
533.333 |
666.667
766.667 |
833.333
866.667 |
1033.333
1100 | |
| 37edo | 6:5:3
7:4:1 8:2:1 |
13/10 | Has 5/4 and both “perfect” fifths of 37edo | 194.595
227.027 259.4595 |
389.189
454.054 518.919 |
486.4865
551.352 |
681.081
713.5135 810.811 |
843.243
875.676 |
1037.838
1065.866 1135.135 |
| 38edo | 6:5:4
7:4:2 |
6/5
14/13 |
189.474
221.052 |
378.947
442.105 |
505.263 | 694.737
726.316 |
852.632 | 1042.105
1073.684 | |
| 39edo | 7:4:3
7:5:1 8:3:1 |
Misses 39edo perfect fifth | 215.385
246.154 |
430.769
492.308 |
523.077
461.5385 |
738.4615
676.923 769.231 |
861.5385
830.769 |
1076.923
1046.154 1107.692 | |
| 40edo | 7:5:2
8:3:2 |
13/8 | Has both “perfect“ fifths of 40edo | 210
240 |
420
480 |
480
540 |
690
780 |
840
870 |
1050
1110 |
| 41edo | 7:5:3
7:6:1 8:4:1 |
204.878
234.146 |
409.756
468.296 |
497.561
439.024 |
702.439
643.902 731.707 |
848.7805
819.512 848.7805 |
1053.6585
1024.39 1082.927 | ||
| 42edo | 7:5:4
7:6:2 |
Has both “perfect“ fifths of 42edo | 200 | 400 | 514.286
457.143 |
714.286
657.143 |
857.143
828.571 |
1057.143
1028.571 | |
| 43edo | 7:6:3
8:4:3 8:5:1 |
16/15 | 195.349
223.256 |
390.698
446.512 |
502.326
530.233 474.419 |
697.767
753.488 |
865.116
837.209 |
1060.465
1088.372 | |
| 44edo | 7:6:4
8:5:2 |
Has both “perfect“ fifths of 44edo | 190.909
218.182 |
381.818
436.364 |
490.909 | 681.818
709.091 |
845.4545 | 1036.364
1063.636 | |
| 45edo | 7:6:5
8:5:3 8:6:1 |
27/25 | Golden
Has both “perfect“ fifths of 45edo |
186.667
213.333 |
373.333
426.667 |
506.667
453.333 |
693.333
720 666.667 |
853.333
826.667 |
1040
1066.666 |
| 46edo | 8:5:4 | Misses fifth of 46edo | 208.696 | 417.381 | 521.739 | 730.435 | 860.87 | 1069.566 | |
| 47edo | 8:6:3
8:7:1 |
9/8 | Has both “perfect“ fifths of 47edo, all sizes of 47edo major third | 204.255 | 408.511 | 485.106
434.043 |
689.362
638.297 |
842.553
817.021 |
1046.8085
1021.277 |
| 48edo | 8:7:2 | 200 | 400 | 450 | 650 | 825 | 1025 | ||
| 49edo | 8:6:5
8:7:3 |
Has both “perfect“ fifths of 49edo | 195.918 | 391.837 | 514.286
465.306 |
710.204
661.2245 |
857.143
832.653 |
1053.061
1028.571 | |
| 50edo | 8:7:4 | Mavila, only has one 50edo interval | 192 | 384 | 480 | 672 | 840 | 1032 | |
| 51edo | 8:7:5 | 188.235 | 376.471 | 494.118 | 682.353 | 847.059 | 1035.294 | ||
| 52edo | 8:7:6 | 184.615 | 369.231 | 507.692 | 692.308 | 853.846 | 1038.4615 | ||
Harmonization of Mohajira
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.
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:
Mohajira[10] - The Neutral Superdiatonic Scale
| Mode 1 | 3 6 7 10 13 14 17 20 23 | 4 8 9 13 17 18 22 26 30 |
|---|---|---|
| Mode 2 | 3 4 7 10 11 14 17 20 21 | 4 5 9 13 14 18 22 26 27 |
| Mode 3 | 1 4 7 8 11 14 17 18 21 | 1 5 9 10 14 18 22 23 27 |
| Mode 4 | 3 6 7 10 13 16 17 20 23 | 4 8 9 13 17 21 22 26 30 |
| Mode 5 | 3 4 7 10 13 14 17 20 21 | 4 5 9 13 17 18 22 26 27 |
| Mode 6 | 1 4 7 10 11 14 17 18 21 | 1 5 9 13 14 18 22 23 27 |
| 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.
See also Meantone family #Mohajira.