Yahoo Tuning Groups Ultimate Backup tuning Maqam tunings

Hello Mike Battaglia and all!

Please let me try to respond, however belatedly, to your post of
September 3 on "Maqamic temperament" which I just read last
night. Apart from raising a few general philosophical questions,
I'd like to offer one 17-note tuning based on a possibility you
mention: two unequal neutral third generators plus a 2/1 octave.

------------------------------------------------------------
1. Philosophical questions: The Maqam/Dastgah "Solar System"
------------------------------------------------------------

As I understand it, the Regular Mapping Paradigm was designed in
good part as a tool for discovering and classifying tuning
systems previously unknown or not fully recognized. I am tempted
to compare it to a spaceship surveying the galaxy in order to
identify and gather some basic information on planetary systems.

With the millennium-old tradition of Maqam music, and its Persian
offshoot of Dastgah music, we have arrived at a planet, or rather
an extensive solar system, inhabited by a very advanced
civilization indeed, from which those of us who are newcomers can
learn much. Here "microtonality" is not a minority movement but
an integral aspect of theory and performance art.

Let us consider the question from a different perspective: how
would we, as musicians coming from a Near Eastern or other
tradition, design a "Europic" tuning intended to sum up the last
millennium or so of European composition?

One quick answer is that we would likely design more than one
kind of "Europic" tuning: the historical patterns of medieval
Pythagorean intonation, Renaissance meantone, irregular 12-note
circulating schemes of the 18th-19th centuries, and 12-ED2 all
call for recognition and exploration.

The incomplete and evocative rather than definitive nature of
even the most complex and subtle fixed-pitch system need not
deter us from making the most of the recorded historical details
that have come down to us. For example, I would guess that a
colorful 19-note meantone temperament might nicely illustrate the
contrasts of chromatic keyboard music around 1600, giving us a
good basis to consider possible refinements of vocal or other
flexible-pitch performances in this era (e.g. adaptive JI).

With the rich literature of Near Eastern fixed-pitch tunings
going back to al-Farabi (870-950), we likewise have a good
foundation for adopting, modifying, or designing systems which
embrace such characteristic traits as more or less subtly unequal
neutral steps and intervals, septimal shadings, and with larger
systems also steps at a comma apart to permit a crude emulation
of flexible-pitch fluidity.

In my view, an "alternative tuning" approach to Maqam/Dastgah
music may seek to realize characteristic traits in new ways, for
example by regular or near-regular systems where the fifths are
tempered in the wide direction.

There is a good argument that especially for smaller systems, an
unabashedly irregular temperament may be the best choice: Persian
tar tunings in 17 notes, for example, fit this model, as well as
the yet more parsimonious tunings for the santur, typically with
about 10 pitch classes tailored to a given dastgah or modal
family (not all found in the same octave).

The rank-3 tuning I present here may err on the side of regularity;
others are possible, as I discuss, which are yet more simple and
regular (rank-2) while still covering some of the intonational
bases which you mention in your thoughtful post.

------------------------------------------------
2. A 17-Note Matrix (with thanks to Erv Wilson!)
------------------------------------------------

What are some general goals for a relatively simple and
attractive Maqam and/or Dastgah tuning system, say in 17 notes
per octave? Here are a few:

(1) We want fifths either at or quite close to a pure 3/2,
with rather less impurity than in 12n-ED2 (1.96 cents)
as one guide to what is "quite close." I thank Jacques
Dudon for really impressing this point upon me, which
has altered a bit how I usually tune.

(2) We want a variety of Zalzalian or neutral sizes, at
least two and ideally more. With only 17 notes and a
regular or near-regular tuning, we're likely to get
mainly two sizes, e.g. two unequal neutral third
generators which add up to a 3/2 or something close.

(3) We'd like some septimal or near-septimal intervals:
steps near 7:6 are very stylish for Hijaz or Persian
Chahargah, while a 12:13:14:16 division is very
pleasing for a low variety of Persian Shur or Turkish
Ushshaq. With only 17 notes, and fifths quite close to
3/2, we may not get a large number of these septimal
intervals -- also including 9:7, featured by al-Farabi
and Safi al-Din al-Urmawi -- but a few, strategically
placed, are very welcome!

As I'll show in the next section, we can meet (2) and (3) with a
rank-2 system where the regular minor seventh is precisely the
size you mentioned in one of your posts: 990 cents. However,
let's start by considering a rank-3 system which meets all three
of these goals.

Following one of the rank-3 techniques you suggest, we choose as
our generators the 2/1 octave plus two unequal neutral thirds, or
here actually two neutral third generators one of which is
constant (357.4 cents) and one slightly irregular (345.7 or 346.9
cents). The variability of the smaller generator is a quirk of
my 1024-ED2 synthesizer, and results in fifths at 703.1 or 704.3
cents -- a small difference, but one meaning that some fifths are
twice as impure as others!

The larger neutral third at 357.4 cents is almost identical to
59/48 (357.2 cents), a ratio arising in Safi al-Din's `oud tuning
based on a division of 32/27 into 64:59:54 (yes, 59-limit in the
13th century!). For the smaller neutral third, 345.7 cents is
quite close to the 72/59 (344.7 cents) found in this same
division (72:64:59:54), while 346.9 cents is close to 11/9 (347.4
cents).

Here's our 17-note system:

! met24-wilson_rast-bayyati17_Dup.scl
!
Tempering of Wilson's Rast-Bayyati Matrix (17)
17
!
82.03125
150.00000
207.42188
289.45313
357.42188
439.45313
496.87500
564.84375
646.87500
704.29688
785.15625
854.29688
911.71875
992.57813
1061.71875
1142.57812
2/1

This is basically a tempering of Erv Wilson's original
Rast-Bayyati matrix <http://anaphoria.com/RAST.PDF>, which uses
al-Farabi's just thirds at 27/22 and 11/9 to build the matrix.

Why temper? While it certainly isn't required, one advantage is
that we get a few locations with steps near 14:13 (128.3 cents),
actually a bit narrower at 125.4 or 126.6 cents. This makes
possible some 12:13:14:16 divisions, as at Scala steps 9 and 16.
The average impurity of the fifths is about 1.72 cents, a
delicate balance between tempering lightly and getting 14:13
consistently within 3 cents of just.

The position I've chosen for the 1/1 might be one good place to
start exploring, because we have the premier Arab (and Turkish)
maqam: Maqam Rast, with support for both the conjunct and
disjunct form, the first standard in the 10th-14th century era
and the second often considered "fundamental" in modern theory.

Rast Rast
|-----------------------|------------------------|
0 207.4 357.4 496.9 704.3 854.3 992.6 2/1

704.3 911.7 1061.7 2/1
|-----------------------|
Rast

Here both the sixth degree (neutral or major) and the seventh
(minor or neutral) are negotiable. We have a subtle contrast
between larger neutral second steps (150.0 cents) and smaller
ones (138.3 or 139.5 cents), the larger characteristically
preceding the smaller in an Arab or Turkish Rast.

Let's look at the 12:13:14:16 division, available as mentioned
above on Scala step 16:

842.6
0 139.5 264.8 496.9 704.3 761.7 969.1 2/1
622.3

This shading of a Persian Shur has seven "textbook" steps, which
we might analyze as a lower Shur tetrachord (139.5-125.4-232.0
cents) of ~12:13:14:16, a tone, and an upper tetrachord at the
fifth of 57.4-207.4-230.9 cents, or a tempered 28:27-9:8-8:7.
However, when descending and for certain gushe-s or modal themes,
the fifth is often lowered, here to 622.3 cents. Sometimes the
sixth degree is neutral rather than minor, with 842.6 cents quite
close to 13/8, and almost identical to 96/59. While the minor
sixth above the final or note of repose is most common, a neutral
third below the final (here by 357.4 cents) is standard!

In addition to regular intervals (e.g. 288.3 or 289.5 cents near
13/11, and 992.6 cents near 39/22 or 16/9), and the septimal ones
we have just used for this shading of Shur, there are some
"supraseptimal ones" at 275.4 or 276.6 and 979.7 cents, for
example. Such shadings may occur rather often in Persian and
Turkish music, and add to the general variety.

So far, in these Maqam and Dastgah scales, we've seen two types
of tetrachords or pentachords:

(1) "Zalzalian diatonic" forms with a tone (9:8 or
sometimes 8:7) plus two neutral seconds of one type or
another.

(2) Diatonic forms with tones or semitones only
(e.g. 57.4-207.4-230.9 cents for the upper tetrachord
of our septimal Shur taken from the fifth degree, one
permutation of the Diatonic of Archytas with 28:27,
8:7, and 9:8).

A third possibility may be called chromatic, or as John Chalmers
has said "neo-chromatic," in a Greek sense: the Hijaz type, where
the middle interval is often some kind of "augmented second" or
"small minor third," with 7:6 or so an attractive size. One maqam
mixing the Zalzalian and Hijaz types is Maqam Suznak, which we
can think of as a lower Rast tetrachord plus tone, or pentachord,
plus an upper Hijaz, here available on Scala step 4 (289.5 cents):

Rast Hijaz
|--------------------------| |----------------------|
0 207.4 357.4 495.7 703.1 853.1 1118.0 2/1

The Hijaz is quite close to Qutb al-Din al-Shirazi's
1/1-12/11-14/11-4/3 as described around 1300, a permutation of
Ptolemy's Intense Chromatic (1/1-22/21-8/7-4/3). We can also use
another version of Suznak at this location where the upper Hijaz
tetrachord (rather similar to Beyhom's type called Zirkula) has
the small interval or semitone first, then the large middle
interval, with a neutral second as the upper step.

Rast Hijaz (Zirkula)
|--------------------------| |----------------------|
0 207.4 357.4 495.7 703.1 772,3 1060.5 2/1

These two forms of Suznak might routinely occur as modulations
from Rast. The first form ties in nicely with a conjunct Rast,
with only the seventh degree altered from minor to major. The
second form with Beyhom's Zirkula variety of Hijaz, more or less,
is identical to a disjunct Rast apart from the minor rather than
major sixth step. Another difference is that while the large
middle interval of the first form of Hijaz is 264.8 cents, a near
7:6, the second or Zirkula-like form has a regular 288.3-cent
third, close to 13:11. Beyhom's typical Zirkula would prefer a
smaller or "lightly augmented" middle step, say around 7:6.

Our 17-note tuning also includes a tempering of Jacques Dudon's
beautiful differentially coherent (-c) mode called Ibina,
sometimes known around the 14th century as Nahaft, here given as
found on Scala step 2 (150.0 cents):

72 78 88 96 108 117 128 144
0 138.6 347.4 498.0 702.0 840.5 996.1 1200
Dudon JI: 1/1 13/12 11/9 4/3 3/2 13/8 16/9 2/1

Mohajira Bayyati
|--------------------| |--------------------|
0 139.5 346.9 496.9 704.3 842.6 992.6 1200
|------------------|
Mustaqim

This mode offers three tetrachords which might be called Mohajira
after Dudon's modal type and quest (here 139.5-207.4-150.0
cents), Mustaqim as named by Ibn Sina at 9:8-13:12-128:117 (here
207.4-138.3-150.0 cents), and Bayyati (138.3-150.0-207.4 cents).
In each, the smaller neutral second precedes the larger -- a fine
contrast with Rast, where the larger neutral step comes first.

In sum, this 17-note tuning, while somewhat different than a
typical Persian tuning for tar or setar, offers at least a
taste of the variety sought in this and other Near Eastern
traditions. It is one possible starting point for an exploration
of regular or slightly irregular rank-3 systems.

----------------------------------------------------------
3. A Note on Rank-2 Systems With Unequal Neutral Intervals
----------------------------------------------------------

If one is willing to temper fifths and fourths rather more than
in the above example -- where the average degree of temperament,
I should note, is just a tad greater than that of George Secor's
29-HTT and related tunings -- then it is quite possible to design
a rank-2 system with unequal neutral intervals and some close
septimal approximations.

Here I would like to focus on one 24-note system which, although
it requires tempering fifths and fourths by a full 3 cents and
involves some other compromises, also has some special advantages
which I might welcome in my more complicated rank-3 tunings, and
I suspect Jacques might appreciate also, at least as an
intriguing diversion from more precise systems.

This system is engagingly simple, and hopefully also simply
engaging: a chain of 24 notes in fifths at an even 705 cents!
This rank-2 temperament could also be thought of as a subset of a
rank-1 temperament, 80-ED2.

! reg705_24.scl
!
Regular 705-cent temperament, 24 of 80-tET
24
!
60.00000
135.00000
195.00000
210.00000
270.00000
285.00000
345.00000
420.00000
480.00000
495.00000
555.00000
630.00000
690.00000
705.00000
765.00000
840.00000
900.00000
915.00000
975.00000
990.00000
1050.00000
1125.00000
1185.00000
2/1

The neutral intervals at 345 and 840 cents closely approximate
11/9 and 13/8, with the latter virtually just; the 15-cent
difference between neutral seconds (135/150 cents) and thirds
(345/360 cents) offers what might be termed a "medium" degree of
contrast, rather more than half a comma but less than a full
comma.

One compromise is that while 435 cents is a virtually just 9/7,
7/6 is a bit more than 3 cents wide at 270 cents, and 7/4 much
more so at 975 cents -- still, however, a recognizably "septimal"
seventh.

Another compromise that goes with the rank-2 structure is that
the 12:13:14:16 tetrachord is realized with two identical smaller
neutral seconds, 135-135-225 cents. While one might ideally
prefer some contrast between these two steps, and also the
melodic touch of the narrower 14:13 step, some divisions with
equal or near-equal steps (e.g. 135-225-135 cents in one example
of Hormoz Farhat) do occur in Near Eastern practice.

The 15-cent comma steps permit a choice from the 1/1 of this
Scala tuning, for example, of a minor third at 270 or 285 cents
(near 7/6 or 33/28), and likewise of a minor seventh at 975 or
990 cents. This flexibly can be useful in Persian Shur, for
example.

Additionally, they permit a neat solution to a problem which
Jacques Dudon has discussed in this group: the perennial quest to
have the right steps for an Arab Rast at a given location, and a
Bayyati on the step a 9:8 above.

To see how this situation is neatly addressed, let's place our
Rast on Scala step 3 (135 cents), and look at a relevant subset
of steps for Rast and Bayyati:

Rast (disjunct) 0 210 360 495 705 915 1065 2/1
Rast (conjunct) 0 210 360 495 705 855 990 2/1

Bayyati (0) 210 345 495 705 915 990 1065 2/1 1410
0 135 285 495 705 780 855 990 2/1

Rast has a tetrachord of 210-150-135 cents (larger neutral step
first), while a stylish Bayyati on its 9/8 step calls for
135-150-210 cents with the _smaller_ neutral step first! Having
steps above the 1/1 of Rast at both 345 and 360 cents neatly lets
us optimize the tuning of both Rast and Bayyati. We can also
follow the understanding of many traditional Egyptian performers
as reported by Scott Marcus that while the lower tetrachord of
Bayyati has a small neutral step (135 cents) first, a neutral
sixth degree when it is used tends to be on the high side (here
855 cents).

While the standard or "untransposed" position for Bayyati is on
the 9/8 step of Rast (e.g. C Rast and D Bayyati in usual Arab
notation), in a classic Arab style of the kind studied by Scott
Marcus one does not modulate directly from Rast to a Bayyati on
this step. Rather, a standard modulation is from Rast to a
Bayyati a fifth higher, here also happily possible with the
desired intonation for each maqam:

Rast (combined conjunct/disjunct):
O 210 360 495 705 855 915 990 1065 2/1

Bayyati (with alternative minor or neutral sixth degrees):
705 840 990 2/1 1410 1485 1560...
0 135 285 495 705 780 855...

Since Marcus suggests that the lower (and smaller) neutral step
of Bayyati in an Egyptian tradition might tend to the range of
around 135-145 cents, 705 cents is just enough temperament to get
a 135-cent step as the chromatic semitone or apotome from seven
fifths up.

Both the usefulness of the 15-cent steps (17-commas) for
obtaining a greater choice of neutral intervals, and the desire
for more septimal intervals, amply motivate a larger 24-note
system.

-------------
4. Conclusion
-------------

Maqam and Dastgah traditions have a fluidity which fully emerges
in flexible-pitch performances; for the last millennium and more,
fixed-pitch tunings have sought to emulate some features of this
rich reality.

Goals such as pure or near-pure fifths and fourths, a variety of
neutral steps and intervals, and inclusion of septimal colors may
lead to a great variety of tuning systems for this music. I have
here tried to focus on "medium-size" systems of 17 or 24 notes to
suggest some of these aspects of the question.

At least from the perspective of some modern traditions and
practices, these systems might be regarded as somewhat
"xenharmonic," with the small steps at 57 or 60 cents, and
possibly the regular semitones at a rather narrow 75 cents in the
705-cent temperament, standing out somewhat from the norm. Also,
other types of systems might reckon with the 81:80 or other
aspects here left unaddressed.

In short, this is a question with many aspects. Thank you for
your post and invitation to explore it with you and others!

Most appreciatively,

Margo Schulter
mschulter@...

🔗Mike Battaglia <battaglia01@...>

Hello Margo, and thanks for your thoughtful reply. I read the whole
post, and I have a few comments on it. Sorry for the lack of response
on my end, moving + work has meant that I have a lot less time for
music theory than I'd like these past few weeks.

Firstly, I should note that what you're doing here and what I was
doing with maqamic temperament are not the same thing.

My executive summary of what's gone on is as follows:
- Much of the attention of maqam music on this list since I've join
has been focused on finding a fixed-pitch tuning that can handle all
of the expressive nuances of maqam intonation.
- Maqam intonation varies from region to region, different "dialects" exist.
- Maqam intonation is an important part of the enjoyment of maqam
music; it carries a lot of musical information.

I took a different approach: I ignored all of that. I didn't ignore it
because I don't think it exists, or because I don't believe it's
important, but just because I was trying to do something a little
higher level. There are two arguments for why I called the temperament
I did "maqamic," which I'll call the "strong" and the "weak"
arguments. I think that even the "weak" argument is enough, so I'll
start with that.

=WEAK ARGUMENT=

Even minus all of the stuff I mentioned offlist about deliberately
adaptive temperaments and equivalence classes, I just wanted to some
up with a regular temperament that was "like" maqam music. I wanted it
to be a shameless mimicry, a dumbing down of sorts just to fit it in
the regular temperament paradigm, and that was it. It's not supposed
to be a substitute for actual maqam music, it's just a "maqamic"
temperament. I did this because a precedent for this sort of naming
system exists: we have "pelogic" and "slendric" temperaments, which
are openly disclosed to only be mimicries of their original gamelan
counterparts. The "ic" suffix has come to take on this meaning, a
precedent dating back to 2002 or so:

/tuning-math/message/3334?var=1

So consider this an admission that regular temperament theory so far
fails to handle complex, naturally-occuring irregular tunings. It's
the "regular mapping paradigm." To truly appreciate the glory of
complex ethnological tunings, we'll need to develop an "irregular
mapping paradigm." As it stands, Werckmeister and 12-equal are the
same thing in the regular mapping paradigm, which is <12 19 28|. We
kind of just leave it there and let you tune things however you want.

Comparatively speaking, I believe that I put more effort in hashing
maqamic out than I believe was done for some of these other
ethnological mimicries, like "slendric," for example. Keenan Pepper's
mentioned that slendro scales in real gamelans are tuned closer to
something like a quasi-equal version of Father[5]. He also did some
measurements of pelog for one of the gamelan instruments that he owns,
and it wasn't very close to mavila[7] at all. But all right - it's not
actually slendro, just "slendric temperament," and it's not actually
pelog, just "pelogic."

In my case, I analyzed some Bashir recordings with a spectrogram to
see that he'd sometimes turn the minor seventh into 7/4, I noted that
the major thirds were often close to 5/4, I worked out that all of the
maqamat can be viewed as irregularly-tuned MODMOS's of the 3L4s
MODMOS, so I put the pieces together and called it "maqamic." I saw
that it was common for the neutral thirds to be somewhere around 11/9,
so that brought me into the 11-limit. I saw that sometimes people
wanted the neutral thirds to be closer to 16/13, so that gave me a
13-limit extension. So the emphasis from my part was never on
intonational expression, because I just wanted to analyze the cursory,
high-level foundation of maqam music, and throw it into a temperament.
And likewise, if someone wants to complain that the neutral thirds
don't have much to actually do with 11/9 in maqam music, that's fine -
but they do in maqamic temperament.

In short, I'm not going to claim that you can just load up a
fixed-pitch instrument, tune it to the 36/35, 81/80, and 121/120
linear temperament, and that that'll be flawless for maqam music. So
since I'm not making that claim, I do think that there is a purpose
that it can serve, which shouldn't interfere with anything else.

=STRONG ARGUMENT=

Regular temperament theory deals in terms of very abstract
homomorphisms between different lattices/Z-modules/free abelian
groups. It does so in a way that doesn't have anything to do with
specific tunings or intonations at all, not until you want to
calculate tuning error, anyway.

For example, 12-equal and Werckmeister are both "the same temperament"
in the regular mapping paradigm; they're both represented by the val
<12 19 28|, and 81/80 and 128/125 vanish in both tunings. Also,
1/4-comma meantone and 1/3-comma meantone are the same thing, they're
both represented by the vanishing unison vector |-4 4 -1> and all that
that entails. We obviously don't think that they -ARE- the same, it's
just that (as far as I know) nobody has spent much time hashing out
the problem of irregular or circulating temperaments. And it's
-REALLY- true that nobody has spent time hashing out the problem of
tuning systems for deliberately adaptive instruments like the oud. But
we do have the homomorphism piece of the puzzle together, so we can at
least start there.

There's a zillion different types of neutral third, so obviously a
one-size-fits all regime for the neutral third will suck some of the
life out of maqam music. But, mathematically, it's not a crime to look
at all of these differently and expressively intoned neutral thirds as
being adaptively retuned versions of "the same thing," which is "a
neutral third." And isn't the scalar structure of maqam music based
around that concept anyway? For example, when someone plays a neutral
third tuned sharper in one context, and tuned flatter in another, does
the performer really think those are two fundamentally different scale
degrees? Or just expressively retuned versions of the same thing? I'm
no expert, but given all of the notation I've seen, I think the
latter.

So why not, couldn't all of these expressive regional maqam
intonational variants just be viewed as dynamically-retuned
circulating versions of the MODMOS's of 3L4s, which is made up of a
chain of neutral thirds? Just because we have no way to store and
categorize how, specifically they're dynamically retuned doesn't mean
that it's useless information - far from it. But why not just analyze
things as that the chain of neutral thirds is a fundamental concept to
maqam music (being as all of the maqamat line up neatly as MODMOS's of
them!), and then explore a second, deeper layer of how those thirds
are intoned, but still compatible on the high level with the |<1 1 0
...|, <0 2 8 ...|> structure?

-Mike

🔗Keenan Pepper <keenanpepper@...>

🔗Margo Schulter <mschulter@...>

> Hello Margo, and thanks for your thoughtful reply. I read the
> whole post, and I have a few comments on it. Sorry for the
> lack of response on my end, moving + work has meant that I
> have a lot less time for music theory than I'd like these past
> few weeks.

Dear Mike,

Thank you for your prompt and engaging response. Dialogues like this
can be an opportunity for what George Secor has called
"cross-pollination" and learning all around.

I'll try to follow your good example by seeking to be reasonably
concise (for once!).

> Firstly, I should note that what you're doing here and what I
> was doing with maqamic temperament are not the same thing.

This certainly seems true; and one approach, of course, doesn't
exclude another.

> My executive summary of what's gone on is as follows:
> - Much of the attention of maqam music on this list since I've join
> has been focused on finding a fixed-pitch tuning that can handle all
> of the expressive nuances of maqam intonation.

Or, the more modest among those of us on this quest might say,
"many of the expressive nuances of maqam intonation."

> - Maqam intonation varies from region to region, different "dialects" exist.

Absolutely true!

> - Maqam intonation is an important part of the enjoyment of maqam
> music; it carries a lot of musical information.

Again, we're much in agreement.

And I'd add that your measurements and analysis of the performances by
Munir Bashir really merit a thread in their own right! I'd love to
participate in this.

> I took a different approach: I ignored all of that. I didn't
> ignore it because I don't think it exists, or because I don't
> believe it's important, but just because I was trying to do
> something a little higher level. There are two arguments for
> why I called the temperament I did "maqamic," which I'll call
> the "strong" and the "weak" arguments. I think that even the
> "weak" argument is enough, so I'll start with that.

A curious point here is that one person might strive toward a "higher
level" of generality or abstraction, and another for a "higher level"
of resolution or detail -- for example, in developing and applying a
regular temperament theory of a given kind.

To keep this brief for now, I'm responding only on a few points that
might help clarify both your eloquently stated "weak" and "strong"
arguments, and the kind of regular temperament theory I tend to
follow. And I'd add that your whole post merits a careful reading.

> =WEAK ARGUMENT=
> Even minus all of the stuff I mentioned offlist about deliberately
> adaptive temperaments and equivalence classes, I just wanted to some
> up with a regular temperament that was "like" maqam music. I wanted it
> to be a shameless mimicry, a dumbing down of sorts just to fit it in
> the regular temperament paradigm, and that was it.

Maybe I'd use the term "maqamesque" for this kind of temperament, the
suffix -esque, as in burlesque, emphasizing the idea of a
simplification or variation reflecting an artist's experience of a
given musical tradition, for example the things that especially stood
out or brought special pleasure. And I have no problem at all with
this.

What I want to emphasize, however, is that rank-2 and rank-3 systems
can also nicely if imperfectly emulate features of Maqam music such as
unequal neutral steps and intervals, and comma steps at least crudely
evoking some of the fluidity of flexible-pitch performances. And the
theory and practice is already there.

The math isn't that complex, and it's mostly rather concrete. One
great starting point for a 24-note rank-3 system is George Secor's
larger and irregular 29-HTT, which includes two chains of 703.579-cent
generators at 58.090 cents for some pure 7/4 sevenths.

Take more generally two 12-MOS or 17-MOS chains with a generator in
the range of 703.55-703.90 cents, spaced for pure 7/4 sevenths, and we
have one very attractive variety of tuning for Maqam. In 2010, Ozan
Yarman focused my attention on an 11/10 step for a Turkish version of
Maqam Huseyni, and 26/21 as a characteristic third for an historical
Ottoman Rast, leading me to a system with an average fifth at 703.871
cents. This summer Jacques Dudon emphasized the importance in Maqam of
near-pure fifths, talking me down to 703.658 cents. In 1024-ED2, the
two chains are spaced at 57.422 cents apart.

This is actually "quasi-rank-3," or "shaggy rank-3" as I often say,
because a 1024-ED2 synthesizer does not have any 703.658-cent fifths;
rather we have a chain alternating sizes of 703.125 and 704.297 cents,
with six smaller and five larger fifths. Maybe it's at once
"quasi-regular" because this is the most even approximation of a
703.658-cent generator possible, and "subtly irregular" because some
fifths are twice as impure as others!

Anyway, we get 14:13, 13:12, 12:11, and 11:10 all within 3 cents of
just; lots of septimal approximations; and some comma steps useful for
the kinds of expressive intonation you mentioned. For example, I can
play a Turkish Rast with a usual bright third at 370 cents -- but
with this step expressively lowered in a descending cadential approach
to 346 cents.

One thing I've learned is that while the complex irregular tunings and
refinements of natural flexible-pitch performance in Maqam and other
traditions have no substitute, a rank-3 system of 24 notes, for
example, can often have 8-10 note subsets reasonably approximating an
irregular scheme tuned on a Persian santur or the like.

My main point is that while intonational theories, Near Eastern and
otherwise, are constantly evolving, the kind of regular temperament
theory I'm describing is quite adequate to observe lots of commas and
emulate some finer points of Maqam intonation, however imperfectly,
while also getting lots of near-3:2 fifths.

A final point I'd like to clarify, which doesn't in any way exclude a
beautiful 7-note system like Jacques Dudon's Mohajira or Ibina; or a
7-note or 10-note set of Erv Wilson's Rast-Bayyati matrix.

While a maqam like conjunct or disjunct Rast, the latter considered
the "Arab fundamental scale," can indeed be generated from a chain of
seven unequal or equal neutral thirds, some very important maqamat
like Bayyati, Nahawand, and Hijaz require semitone steps as well as
the whole tones and neutral or Zalzalian seconds which are indeed so
characteristic.

When we get up to around 17 generators, then we get enough of a
variety of these whole tones, semitones, and neutral steps to realize
lots of maqamat.

Most appreciatively,

Margo