Hello, William and all.

This is a note on how to get from Rast, justly known as the Mother of all
Maqamat, to a jazzy 24-ED2 tuning of the related Maqam Hijazkar, both with
standard positions on C in modern Arab notation. An often applicable rule is
that it's permissible directly to modulate between maqamat (Arabic plural of
maqam) sharing the same final or tonic -- the resting note of the mode.
People accustomed to major-minor tonality in Western music often tend to say
"tonic," while those of us accustomed to medieval or Renaissance European
modality may say "final." Either is a possible translation of a very rich
Near Eastern reality.

In its standard modern form, Rast is 

  Rast      Rast
|------|  |------|
C D Ev F  G A Bv C 

or 4-3-3-4-4-3-3 steps of 24 -- 0-200-350-500-700-900-1050-1200 cents. There
are two symmetrical tetrachords on the 1/1 and 3/2 of 4-3-3 steps, with a
200-cent tone, a 350-cent neutral third, and a 500-cent perfect fourth.

These tetrachords, as they're called in classic Greek theory, are in Near
Eastern theory often called ajnas -- the plural of the Arabic term jins,
borrowed from the Greek genus. One maxim of maqan is "Follow the ajnas!"
This means knowing the different tetrachords -- and often also trichords or
pentachords -- that make up a maqam, and often provide doors or gateways to
other maqamat.

One common modulation from Rast is Hijazkar, for which one option is what I
find a very jazzy version in 24: 

  Hijaz     Hijaz
|-------|  |------|
C Db Ev F  G A Bv C 

or 2-5-3-4-3-5-2, at 0-100-350-500 or 100-250-150 cents for the lower jins C
Db Ev F, followed by a middle 9/8 tone at F-G, and an upper jins of G Av B C
or 3-5-2 at 0-150-400-500 cents or 150-250-100 cents.

The lower jins of Hijazkar, C Db Ev F, in this tuning has only one crucial
change from our Rast jins at C D Ev F -- the 9/8 note is lowered to a regular
semitone at 100 cents. That results in a middle interval Db-Ev at 250 cents
or 5 steps, somewhere between a usual 9/8 tone at 200 cents and a regular
minor third at 300 cents -- with the neutral third step Ev at 350 cents
unchanged.

This tetrachord type is called Hijaz, also the name of a maqam with a
standard position at D. While the Hijaz tetrachord has a "textbook" form in
the recent Arab literature of 2-6-2 steps or 100-300-100 cents, sometimes
humorously called a "piano hijaz" because it matches a Western 12-ED2
tuning, many traditional Arab musicians share a taste common among Turkish
and Persian musicians also that the middle interval should ideally be
smaller than a usual minor third, say around 230-275 cents. Often there's a
tendency to tune it at around 7/6 (267 cents), but with lots of variations.
Descriptions such as "slightly augmented second" (Amine Beyhom) or
"plus-tone" (Hormoz Farhat) describe this ideal middle step for Hijaz or
related Persian forms, with 250 cents nicely fitting the category.

Traditionally, Hijaz with its wide middle step is said to represent the
desert of the Hijaz region of the Arabian Peninsula. While tuning this step
at a textbook 300 cents would not be wrong, 250 cents gives it a special
character -- and 24 gives us that "jazzy" option for the Hijaz tetrachord in
various maqamat!

Hijazkar, sharing with Rast a final of C and thus a common modulation
(generally maqamat sharing the same final can modulate to or from each other
well), has the lower Hijaz tetrachord we have tuned at C-Db-Ev-F, then a
middle tone, and then an upper Hijaz tetrachord, here tuned a bit
differently: G-Av-B-C, or 150-250-100 cents. Here the neutral second step
comes first, then the wide step at 250 cents, and then the narrower semitone
up to the octave of the final. One reason for having the semitone as the
upper interval as that it nicely serves as a leading tone pulling up to the
octave.

In "following the ajnas," it's helpful to know the ghammaz for a given
maqam, the note sometimes called in English the "dominant" or "confinal,"
often a focus for melodic emphasis. In Rast, and evidently also Hijazkar,
it's G, the 3/2 -- but in other maqamat it might be the 4/3, or in some
cases the neutral third at around 11/9 or 27/22, 347 or 355 cents, for
example (both mapping to 350 cents in 24-ED2). So G, as the lowest note of
the upper jins, tends to get a lot of emphasis in Rast and Hijazkar.

The Lebanese composer and theorist Amine Beyhom suggests a slightly more
moderate form of Hijazkar with the same lower jins of C-Db-Ev-F, but a
conventional upper jins of G-Ab-B-C or 100-300-100 cents, which also has the
upper semitone as a leading tone to the octave. My 150-250-100 cents is a
bit more adventurous -- although Beyhom recognizes and likes this 3-5-2 form
of Hijaz also, so it's a matter of discretion and taste.

However it's tuned, Hijazkar along with Rast makes a dynamic duo. One
feature of the 2-5-3-4-3-5-2 tuning I've described, with steps above C at
0-100-350-500-700-850-1100-1200 cents, is the neutral second and sixth steps
(close to 12/11 and 18/11). These steps can serve as connecting elements to
another form of Rast, known as conjunct Rast or Nairuz Rast.

This form of Rast has two conjunct Rast or 4-3-3 tetrachords, on the 1/1 and
4/3 steps, plus an upper 9/8 tone -- very slightly tempered in 24, of
course. 

  Rast   Rast
|------|------|
C D Ev F G Av Bb C 
         |-------|
          Bayyati

It's C-D-Ev-F and then F-G-Av-Bb, plus an upper tone Bb-C to the
octave. Note that the Rast tetrachord remains unchanged from a standard
modern or disjunct Rast (i.e. Rast tetrachords at 1/1 and 3/2, with a middle
4/3-3/2 tone of "disjunction"), but the upper tetrachord on the 3/2 has a
neutral sixth and minor seventh, rather than a major sixth and neutral
seventh.

In exploring Nairuz Rast, the advice to "follow the ajnas" can enrich our
enjoyment of this form of Rast. While we can construct this maqam from two
conjunct Rast tetrachords, which was the usual medieval approach (in
medieval times, this was the standard Rast), at least in modern usage, the
ghammaz (dominant or confinal) is the same as for disjunct Rast, on G. 

This means that the tetrachord on the 3/2 takes on central importance:
G-Av-Bb-C or 3-3-4, known as Bayyati, at 150-150-200 cents, with a neutral
second, minor third, and fourth. Dwelling on and delving into this upper
Bayyati jins is thus one of the attractions of Nairuz Rast.

Both the 2-5-3-4-3-5-2 version of Hijazkar and Nairuz Rast have the 850-cent
or 18/11 step, so this is a possible connection. While I'm not sure whether
or how often Arab musicians modulate directly to Nairuz Rast to or from
Hijazkar, it seems an interesting maneuver with these tunings.

The 2-5-3 and 3-5-2 forms of Hijaz are more common in 19th-century Arab
theory, for example Mikhail Mashaqa of Syria, who routinely gives these
forms for various maqamat. Often the 5-step interval can give the impression
of a 7/6 minor third, which is one standard size for the middle interval of
Turkish Hijaz or related Persian forms such as Chahargah.

A bit of intonational politics may be involved here: the 5-step interval
fits nicely with the idea of a middle Hijaz step larger than a 9/8 but
smaller than a regular minor third (often the 32/27 or Pythagorean third
from three pure fourths), but not so well with the influence of conventional
Western music, often based on 12-ED2. Mashaqa, who used the 24-tone system
while noting that an intonation more like the Byzantine theory of the time
based on 68-ED2 could better fit Arab practice, included the 5-steps-of-24
interval in his maqam tunings early and often. But it became less common in
20th-century theory.

Now Amine Beyhom has brought the 5-step interval back into prominence, and
in a way that may promote its use in 24-ED2 as well as unequal tunings of
the kind he especially favors. (He prefers an instrument without any frets,
to permit unlimited freedom.)

Learning maqam is in part a process of becoming aware of choices, and making
them -- not necessarily the choices that another musician would make, but
choices informed both by some knowledge of theory, and by lots of
experience. Asking what the ghammaz of a given maqam is, or what the ajnas
are (and they can be present at many levels and layers!), can be the start
of an interesting melodic direction or modulatory move.

The different tunings of Hijaz in a maqam like Hijazkar -- 2-5-3, 3-5-2, or
the conventional modern 2-6-2 -- illustrate the choices that 24 provides.
Let's make the most of them!