Mohajira

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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.

Temperament data (2.3.5.7.11, 24&31)

Period: 1\1

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

EDO generators: 7\24, 9\31

Scales (Scala files):

Interval table (10-note MOS, 2.3.5.7.11 POTE tuning)
#Gens up Cents [1] Approximate ratios[2]
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
  1. octave-reduced
  2. 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 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.

Iced Major C D Ed Ft G A Bd C M2 N3 S4 P5 M6 N7
Iced Locrian D Ed Ft G A Bd C D N2 N3 P4 P5 N6 m7
Iced Fridgian (Iced Minor) Ed Ft G A Bd C D Ed M2 N3 S4 P5 N6 N7
Iced Lydian Ft G A Bd C D Ed Ft M2 N3 P4 s5 N6 N7
Iced Mixolydian G A Bd C D Ed Ft G M2 N3 P4 P5 N6 N7
Iced Dark Lydian (Iced Coffee) A Bd C D Ed Ft G A N2 m3 P4 s5 N6 m7
Iced Blizzard (Neutral Scale) Bd C D Ed Ft G A Bd 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 Ed F G A Bd C
Mode 2 D Ed F G A Bd C D
Mode 3 Ed F G A Bd C D Ed
Mode 4 F G A Bd C D Ed F
Mode 5 G A Bd C D Ed F G
Mode 6 A Bd C D Ed F G A
Mode 7 Bd C D Ed F G A Bd

Turkish Major - 1 2 3 4 5 v6 v7 1

Mode 1 C D E F G Ad Bd C
Mode 2 D E F G Ad Bd C D
Mode 3 E F G Ad Bd C D E
Mode 4 F G Ad Bd C D E F
Mode 5 G Ad Bd C D E F G
Mode 6 Ad Bd C D E F G Ad
Mode 7 Bd C D E F G Ad Bd

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 Eb F G A Bd C
Mode 2 D Eb F G A Bd C D
Mode 3 Eb F G A Bd C D Eb
Mode 4 F G A Bd C D Eb F
Mode 5 G A Bd C D Eb F G
Mode 6 A Bd C D Eb F G A
Mode 7 Bd C D Eb F G A Bd

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 Bd C C Dd E F G A B C
Mode 2 D E F G A Bd C D Dd E F G A B C Dd
Mode 3 E F G A Bd C D E E F G A B C Dd E
Mode 4 F G A B C Dd E F F G A Bd C D E F
Mode 5 G A Bd C D E F G G A B C Dd E F G
Mode 6 A B C Dd E F G A A B C Dd E F G A
Mode 7 Bd C D E F G A Bd B C Dd E F G A B

Bayati or screamapillar - 1 v2 v3 4 5 b6 b7 1 or C Dd Ed 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

altered mohajira levels.PNG

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.

Common Turkish Major Tunings
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.