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