Dear William,
In another thread Mohajira, where Eric and I got into some historical
aspects, you asked a question which may have a simpler if quite rich answer:
Arab or more generally Near Eastern modes that might be found within
Mohajira as realized in 24-ED2. Since I know that 24-ED2 is your main
interest, I'll focus on that, rather than other kinds of just or tempered
versions of Mohajira.
In 24, of course, Mohajira has a simple structure, whether viewed as a type
of scale or a method for building a 7-note scale or larger tuning set. We
simply use the neutral third of 350 cents (quite close to 11/9 at 347 cents)
as a generator -- 7 steps of 24.
From 7 such generators, we get the basic Mohajira scale in 24:
C-Dv-Ev-F-G-Av-Bv-C 0-150-350-500-700-850-1050-1200 cents. This can be viewed
as two tetrachords of 3-4-3 steps or 150-200-150 cents, with a middle tone
at 200 cents: JTJ-T-JTJ, where "T" is a 200-cent tone and "J" the Arab
mujannab or neutral second, here always 150 cents (a single size).
By the way, a 24-ED2 Mohajira mode is close to the JI tuning
1/1-12/11-11/9-4/3-3/2-18/11-11/6-2/1.
So the basic form of Mohajira, showing its two Mohajira tetrachords (written
Mohajira-4 to show that their tetrachords, as Eric Ederer does) is, with "T"
showing a tone either joining two tetrachords as here, or completing the
octave:
Mohajira-4 T Mohajira-4
|-----------|----|------------|
C Dv Ev F G Av Bv C
0 150 350 500 700 850 1050 1200
150 200 150 200 150 200 150
J T J T J T J
From this point of view we can try each of the rotations, and see if it
matches up with any Near Eastern modes (or, at least, the ones I think I
know -- when in doubt, trust traditional Near Eastern musicians as the best
source!).
Our first rotation would be at Dv, the neutral second step, and it turns out
that we do get a Near Eastern mode, or one form of a mode, known in Turkey
as Penchgah, although the tuning would be a bit different:
Penchgah-5 Rast-4
|---------------|-------------|
Dv Ev F G Av Bv C Dv
0 200 350 550 700 900 1050 1200
200 150 200 150 200 150 150
T J T J T J J
This is very much like a standard Rast, except that the fourth step is at
around 11/8 (in a 24-ED2 tuning) rather than 4/3 -- or, more generally, at
around a 9/8 tone above the usual neutral third step of Rast. In Turkish,
both the third and fourth steps might be higher by 10-35 cents; but the
basic structure is the same.
In fact, the raised fourth of Penchgah could occur as a routine variation of
Rast in an Arab style. In Rast, the fifth is the ghammaz or note of
emphasis after the final or resting note -- for example, G in a usual
Arab placement of Rast, C-D-Ev-F-G-A-Bv-C. To emphasize the Ghammaz, it is
often approached by a kind of leading tone -- which means raising the fourth
note to bring it closer to the goal of the ghammaz. In modern Arab music,
this leading tone is often raised by a full semitone, thus F#-G. However, a
more subtle inflection would be to raise it by about a quarter or third of a
tone -- or, in 24, more specifically a quartertone of 50 cents. Thus would
give a momentary form of C-D-Ev-F^-G-A-Bv-C, the same as Penchgah here.
Note that an idiomatic Turkish Penchgah also needs a 4/3 step, but there's
no reason we can't experiment with a 7-note or heptatonic version having the
raised fourth (a bit like the medieval European Lydian mode, F-F, where the
natural fourth step at a tritone F-B, often remained unaltered when rising
to C, but tended to become Bb when descending toward F.
Now let's look at our next rotation:
Mohajira-4 T Bayyati-4
|-----------|-----|-------------|
Ev F G Av Bv C Dv Ev
0 150 350 500 700 850 1000 1200
150 200 150 200 150 150 200
J T J T J J T
Jacques Dudon has called this mode Moha-Baya, short for the lower Mohajira
tetrachord and upper Bayyati tetrachord of 150-150-200 cents or JJT. It is
one form of a medieval mode called Hijaz -- now a name that implies a
tetrachord with a middle step considerably larger than 9/8, and in 24-EDO
typically at either 250 or 300 cents. In the 13th century, however, the term
Hijaz could relate to the general form J T J, as well as to the meaning of a
middle step larger than a usual tone -- the "augmented" step or "plus-tone"
of modern Near Eastern theory.
Note that this Moha-Baya mode also has a Rast tetrachord on the 4/3 step:
Av-Bv-C-Dv, which allows for yet more variety. While I'm not aware of this
as a modern Near Eastern mode -- and ready to be further educated on this --
it's a very beautiful medieval mode. Jacques Dudon has a JI tuning of it
called Ibina: 1/1-13/12-11/9-4/3-3/2-13/8-16/9-2/1, which has the property
of "differential coherence" involving the relationship of difference tones.
Our next rotation, on the fourth of the original Mohajira mode, is:
Penchgah-5 Mohajira-4
|----------------|-------------|
F G Av Bv C Dv Ev F
0 200 350 550 700 850 1050 1200
200 150 200 150 150 200 150
T J T J J T J
This mode has a lower Penchgah pentachord and upper Mohajira, so that a
version of this mode in the Scala archive is called rast_moha.scl. While I'm
not sure if there's any historical precedent, it's a fascinating mode.
Moving now to G, the 3/2 of our original Mohajira, we get
Mohajira-4 Mohajira-4 T
|-------------|-------------|---|
G Av Bv C Dv Ev F G
0 150 350 500 650 850 1000 1200
150 200 150 150 200 150 200
J T J J T J T
This is a fascinating example of a "twin" mode with two identical conjunct
tetrachords G-Av-Bv-C and C-Dv-Ev-F, so that the 4/3 is both the highest
note of the first tetrachord and the lowest note of the second one. In
medieval theory, this mode with its JTJ tetrachords could be one
interpretation of Hijaz (the variety with a middle step of a usual 9/8 tone
rather than an augmented step such as 250 or 300 cents in 24).
Our next rotation is to Av, the 850-cent step of our original Mohajira:
Rast-4 T Mohajira-4
|-------------|-----|--------------|
Av Bv C Dv Ev F G Av
0 200 350 500 700 850 1050 1200
200 150 150 200 150 200 150
T J J T J T J
This form is what Jacques Dudon has called a "folk Rast," and in fact is the
standard Rast of some modern Turkish theory, with an upper tetrchord
including neutral sixth and neutral seventh. It thus may be counted as a
form of Rast.
Finally, we move to the neutral seventh step of Mohajira, Bv:
Bayyati-4 Mohajira-4 T
|--------------|--------------|-----|
Bv C Dv Ev F G Av Bv
0 150 300 500 650 850 1000 1200
150 150 200 150 200 150 200
J J T J T J T
This is Safi al-Din al-Urmawi's interpretation of Hijaz, having in modern
terms a lower tetrachord of Bayyati (JJT) and a conjunct upper tetrachord of
Mohajira. It is a very rich mode, and well worth exploring. While the
absence of a 3/2 degree may make things a bit different, exploring the
tetrachords as pure melody is a very productive approach.
In short, the rotations on the second step (Dv) and sixth step (Av) produce
Rast variations recognized in modern Near Eastern theory: Maqam Penchgah and
Jacques Dudon's "folk Rast" taken as the standard Rast in one 20th-century
Turkish view.
The main Mohajira mode itself is synonymous with medieval Awj, while the
rotations on the third step (Ev), fifth step (G), and seventh step (Bv) all
relate to medieval variations on a type of mode called Hijaz with a middle
step which could be around 9/8 -- as opposed to the wider middle step
associated with a different interpretation of Hijaz known by 1300 and now
standard.
The rotation on the fourth step (F), Penchgah-Mohajira, has Rastlike
qualities, and provides another Rast family variation, having the lower
pentachord of Penchgah and the upper Mohajira tetrachord of Dudon's "folk
Rast."
Note if we desire some more typical modern "textbook" modes, a set of
Mohajira 10 will supply these more familiar forms.
C Dv D Ev F F^ G Av A Bv C
0 150 200 350 500 550 700 850 900 1050 1200
150 50 150 150 50 150 150 50 150 150
Note that in 24-ED2, 10-note Mohajira has only two step sizes: either 150
cents, a near-just 12/11 (151 ccents) or 50 cents, close to 33/32 (53
cents).
What we might quickly notice is a "textbook" Rast at C: C-D-Ev-F-G-A-Bv-C.
this means that all the familiar modern rotations of Rast are also
available.
Best,
Margo Schulter
mschulter@calweb.com
First draft September 22, 2013