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.
| Mohajira |
81/80, 121/120, 176/175 (11-limit)
11-limit 21-odd-limit: 10.8 ¢
11-limit 21-odd-limit: 24 notes
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.
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~128/105 = 348.4374 ¢ | CWE: ~128/105 = 348.4194 ¢ | POTE: ~128/105 = 348.4149 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~11/9 = 348.5608 ¢ | CWE: ~11/9 = 348.4910 ¢ | POTE: ~11/9 = 348.4767 ¢ |
Notes
- ↑ (/mʊˈhɑːdʒɪɹə, -hæ-/ muu-HA(H)-jirr-ə or /məˈhɑːdʒɪɹə, -hæ-/ mə-HA(H)-jirr-ə, named by Jacques Dudon from Arabic مهاجرة Muhājirah)