PDF reply to study on Turkish maqam music pitch measurements

Dear Marcel and all,

Thank you for this fascinating paper, which does look at the
possibilities of a 24-note Pythagorean set as one solution
to Turkish maqam music tuning. What I'd like to add are a
few comments as someone familiar with paper who has used it
to assess how closely (or not so closely!) a couple of my
favorite maqam tunings come to the results in this paper.

First, I should explain that the standard YAEU system
of Turkish intonational theory through much of the 20th
century is based on a 24-note subset of Pythagorean tuning,
although a different one than yours, and with a political
agenda that interferes with its accuracy for maqam-s
favoring neutral intervals in Turkish practice (e.g. Ushshak
or Huseyni).

The idea, as Ozan Yarman explains, was to present Turkish
music as based essentially on the same "diatonic" scale
as the "progressive" West -- in contrast to the "quartertone"
alterations (i.e. neutral intervals) of a "decadent" Arab or
Byzantine style. Some theorists such as Yekta and Ezgi do
at times mention neutral steps like 12:11 or 11:10 -- but
the emphasis is on Western diatonicism in a "pure"
Pythagorean form.

This approach evidently had the effect of permitting Turkish
maqam music to packaged in theory as reflecting a common
"Pythagorean diatonic" heritage with Western Europe. even
though, in practice, performers sang or played many neutral
intervals much as before! In the more flexible ideological
climate of the late 20th century and beyond, it has become
possible frankly to acknowledge the role of neutral intervals
and seek a more accurate theory for maqam-s using them.

So your 24-note Pythagorean approach is rather like YAEU as
it might have been, or as a Syrian theorist such as Tawfiq
al-Sabbagh practiced a similar theory around the same
mid-20th-century era in a setting where he could express
his own evident 5-limit predilections without needing
to downplay the role of neutral steps.

Comparing the YAEU results with yours may show that the
two often coincide -- except that your more pragmatic
and flexible approach to neutral intervals permits a
better fit.

You could make the fit yet better by using a larger
Pythagorean set, such as 29, 41, or 53. Indeed, a
53-note set would be very close to the 53-EDO model
or "comma system" used by many Turkish and some Syrian
theorists to measure intervals, and used in the article
you are replying to also. The main difference is that
in 53-EDO, there is a single comma size, the 22.642-cent
tuning step. In 53-note Pythagorean, likewise a
circulating system, we have 41 steps at a usual
Pythagorean comma of 23.460 cents, and 12 at a "41-comma"
or 19.845 cents. A "9-comma" tone at 9/8, for example,
consists of 7 large and 2 small commas. However, most
discrepancies between the two systems are quite small.

For example, with Maqam Ushshak (p. 5 of your article,
or p. 60 of the published article you reproduce), what
I'd say we clearly want is to have the neutral second
step somewhere around 129-143 cents, say, but the
neutral sixth step around 32 commas (~32/21) higher,
or actually a bit more, based on the peak at 872 cents
or so. Here a 41-note Pythagorean tuning seems to be
the minimum MOS set for this: we can have 137.1 cents
and 862.6 cents for example.

What I'd also emphasize is that lots of intervals
in maqam music are very flexible, so that performers
sometimes speak of the neutral steps, especially,
as representing "glissando zones" rather than fixed
points. This survey is rather like looking at wear
along a staircase to see where people have most
often been stepping -- not necessarily how any
single artful performance might have proceeded.

With many thanks,

Margo

Dear Margo,

Thank you for your reply.
And I agree with every single thing you say!

I was also aware of the particular YAEU scale used in this paper.
I however did not want to bring this up as I've had a heated discussion
about this with Ozan in the past, and plan on steering well clear of
heated discussion on the tuning list. I've had (and caused) my fair share
already ;)
But as far as the YAEU /53tet scale is concerned, a different chosen
subset of 24-tones connected by perfect fifths would give the same result
as the 24-tone Pythagorean scale I used indeed. The difference is almost
negligible.

And you make a very good analogy.
" This survey is rather like looking at wear
along a staircase to see where people have most
often been stepping -- not necessarily how any
single artful performance might have proceeded."
I fully agree. And I'd like to add some more thought in this direction.
I personally do not hold absolute value to the measurements in studies
such as this.
I've done several pitch analysis of recordings myself as well with
Melodyne DNA.
These were of polyphonic common practice classical music by several
trombone quartets, string quartets and choirs.
Even with polyphonic music, which seems to me to give a greater pitch
consensus than monophonic music like Turkish maqam, there was fairly
substantial difference in performance between performing groups.
Some would play what seemed to be mostly in Pythagorean, and yet another
group performing the same piece would play it mostly in 5-limit.
This is a consistent Syntonic comma difference (about the same as a
Pythagorean comma and 53tet comma), yet both pieces express the same music
to my ears. There was no functional difference in the music due to the
different intonations.

Now I personally belief the functional difference between tones is created
along the Pythagorean chain of fifths. Yet that on top of this there is
fairly great flexibility on how it is actually tuned. - The music itself
plays a big role in making clear which interval is which.
Now one could say on one hand that Turkish maqam music is monophonic so
there's no harmony to indicate anything other than the tuning used so one
hears the function of the tones more clearly as they are actually
played/tuned.
On the other hand one could say that the music still makes the
functionality clear to a large degree and even though the actual tuning of
notes conveys their functionality for a large part, there is still a
fairly great flexibility of tuning while still expressing the same
functionality, which is perhaps even greater than the Syntonic comma of
common practice classical music in performance.
I do not know the answer to this, but I tend to see it more in the
direction of the second way.
So where I use the 24 -tone Pythagorean scale to best match the measured
data, I'm not sure if those points represent the actual "pure tuning" of
the functionality of those specific tones of that specific maqam, or if
they include a comma (or perhaps even more) of additional "coloring".
And one sees a strong suggestion in the data as well that different commas
are used, again I do not know if these represent different tone
functionality (similarly like the difference in functionality between an E
and an F for instance, or a G# and an Ab), or if they represent
non-functional coloring.

About the 137.1 cents + 826.6 cents tuning on p.5.
Yes it would indeed require a 41-tone Pythagorean tuning to cover them
both at once.
However I find it a bit large. Could these performances really
functionally indicate such a large chain of fifths?
I don't know, could be. But I am skeptical towards this.
There isn't such a strong peak at 870, more a vague area with a very small
peak which does not represent many recorded notes relative to the
surrounding area it seems to me. And there could be many causes, one or 2
performances that interpret the maqam functionally differently than the
others, or non-functional coloring, or modulations used in some
performances, or I don't know what.
Wish there were many more studies like this so we had even more to go by!
:)

Kindest regards,

Marcel de Velde

>
> Dear Marcel and all,
>
> Thank you for this fascinating paper, which does look at the
> possibilities of a 24-note Pythagorean set as one solution
> to Turkish maqam music tuning. What I'd like to add are a
> few comments as someone familiar with paper who has used it
> to assess how closely (or not so closely!) a couple of my
> favorite maqam tunings come to the results in this paper.
>
> First, I should explain that the standard YAEU system
> of Turkish intonational theory through much of the 20th
> century is based on a 24-note subset of Pythagorean tuning,
> although a different one than yours, and with a political
> agenda that interferes with its accuracy for maqam-s
> favoring neutral intervals in Turkish practice (e.g. Ushshak
> or Huseyni).
>
> The idea, as Ozan Yarman explains, was to present Turkish
> music as based essentially on the same "diatonic" scale
> as the "progressive" West -- in contrast to the "quartertone"
> alterations (i.e. neutral intervals) of a "decadent" Arab or
> Byzantine style. Some theorists such as Yekta and Ezgi do
> at times mention neutral steps like 12:11 or 11:10 -- but
> the emphasis is on Western diatonicism in a "pure"
> Pythagorean form.
>
> This approach evidently had the effect of permitting Turkish
> maqam music to packaged in theory as reflecting a common
> "Pythagorean diatonic" heritage with Western Europe. even
> though, in practice, performers sang or played many neutral
> intervals much as before! In the more flexible ideological
> climate of the late 20th century and beyond, it has become
> possible frankly to acknowledge the role of neutral intervals
> and seek a more accurate theory for maqam-s using them.
>
> So your 24-note Pythagorean approach is rather like YAEU as
> it might have been, or as a Syrian theorist such as Tawfiq
> al-Sabbagh practiced a similar theory around the same
> mid-20th-century era in a setting where he could express
> his own evident 5-limit predilections without needing
> to downplay the role of neutral steps.
>
> Comparing the YAEU results with yours may show that the
> two often coincide -- except that your more pragmatic
> and flexible approach to neutral intervals permits a
> better fit.
>
> You could make the fit yet better by using a larger
> Pythagorean set, such as 29, 41, or 53. Indeed, a
> 53-note set would be very close to the 53-EDO model
> or "comma system" used by many Turkish and some Syrian
> theorists to measure intervals, and used in the article
> you are replying to also. The main difference is that
> in 53-EDO, there is a single comma size, the 22.642-cent
> tuning step. In 53-note Pythagorean, likewise a
> circulating system, we have 41 steps at a usual
> Pythagorean comma of 23.460 cents, and 12 at a "41-comma"
> or 19.845 cents. A "9-comma" tone at 9/8, for example,
> consists of 7 large and 2 small commas. However, most
> discrepancies between the two systems are quite small.
>
> For example, with Maqam Ushshak (p. 5 of your article,
> or p. 60 of the published article you reproduce), what
> I'd say we clearly want is to have the neutral second
> step somewhere around 129-143 cents, say, but the
> neutral sixth step around 32 commas (~32/21) higher,
> or actually a bit more, based on the peak at 872 cents
> or so. Here a 41-note Pythagorean tuning seems to be
> the minimum MOS set for this: we can have 137.1 cents
> and 862.6 cents for example.
>
> What I'd also emphasize is that lots of intervals
> in maqam music are very flexible, so that performers
> sometimes speak of the neutral steps, especially,
> as representing "glissando zones" rather than fixed
> points. This survey is rather like looking at wear
> along a staircase to see where people have most
> often been stepping -- not necessarily how any
> single artful performance might have proceeded.
>
> With many thanks,
>
> Margo
>
>

Dear Marcel,

Please let me agree that often simply looking at average
values in cents for maqam steps leaves the question open as
to whether we are simply seeing more or less free
variations, or some more specific pattern.

Knowledge of the maqam system and an investigation of how
traditional performers understand the fine points of
intonation can help in answering some of these questions.
Since Turkish and other studies do show that a given maqam
step (_perde_ in Turkish, from Persian _pardah_, a pitch or
step) is often more of a cluster or region of pitches, such
an understanding will always be partial.

However, with either a Turkish Ushshak or the more or less
equivalent Arab Bayyati, we can identify an important
pattern, and also offer at least one explanation for it.

That pattern is the use of a rather low neutral second, in
contrast to a rather high neutral sixth. To understand this
contrast, it helps to know a bit about Maqam Ushshaq (or
Ushshak in a Turkish spelling) and how its genera are
formed.

Often, with Ushshaq and some other maqamat, the lower or
"root" genus (typically but not always a tetrachord) tends
to remain more stable than the upper tetrachord, which can
change without altering the "family" of the maqam. Here
there are two common forms relevant for this analysis.

The first has a root tetrachord of Ushshaq or Bayyati, plus
a conjunct upper tetrachord called Nihavent or Nahawand --
again, the Turkish and Arab names for similar genera. Here
I'll use a Turkish style of note spelling, where Ushshaq is
often on A (in an Arab notation, it would often be on D):

Ushshak-4 Nihavent-5
|--------------|------------------|
A B-3 C D E F G A
0 137 294 498 702 792 996 1200
0 6 13 22 31 35 44 53
137 157 204 204 90 204 204
6 7 9 9 4 9 9

Note that I'm following the adage of doing as the Turks when
analyzing Turkish music, and treating Ushshak as having, in
this "textbook form," a lower Ushshak tetrachord and an
upper Nihavent pentachord. Some of the Turkish theorists use
a comma notation (_koma sesler_, comma pitches) which fits
nicely with Pythagorean intonation, and I've used here along
with cents.

Thus Ushshak-4 (the "4" meaning a tetrachord) is A B-3 C D,
with the "B-3" meaning a B lowered by about three commas,
here our 137.1-cent neutral step quite close to 13/12. This
tetrachord is also neatly summed up as 6-7-9 commas, with
six commas as a small neutral step, seven commas as a large
one, and nine commas as our usual 9:8 tone.

The upper pentachord, "Nihavent-5," has a tetrachord of
9-4-9 commas or tone-limma-tone, plus an upper 9:8 tone
(again 9 commas) to complete the octave. An Arab style of
analysis might speak of a Nahawand tetrachord plus this
upper tone, while Turkish theory prefers to merge them into
a single pentachord.

However, you can see that this form of Ushshaq gives us a
minor sixth step somewhere around 35 commas or 792 cents --
in rational terms, of course, the 128/81 minor sixth.

Now we move to our second form of Ushshaq, or variation
within it:

Ushshak-4 Rast-5
|--------------|------------------|
A B-3 C D E F+3 G A
0 137 294 498 702 862 996 1200
0 6 13 22 31 38 44 53
137 157 204 204 161 134 204
6 7 9 9 7 6 9

The Ushskah-4 genus with its low neutral second at 137 cents
or 6 commas is the same as before. However, for the upper
genus, we now have Rast-5, or 9-7-6-9 commas.

Note the contrast between the ordering of neutral steps in
Ushshak-4 and Rast-5. In Ushshak, we have 6-7-9 commas, with
the smaller neutral step coming first.

In Rast, however, the pattern is 9-7-6 commas, with the
larger neutral step coming first. And a fifth at 3/2 or 31
commas, plus that first large neutral step of Rast at 7
commas, results in a large neutral step for which 41-note
Pythagorean provides a step at 38 commas or 862 cents.
The speling D E F+3 G A shows that this neutral sixth is
about 3 commas higher than F (128/81), or 2 commas lower
than F# (27/16).

You raise the question of whether 41 notes is too large a
tuning set. As a gamut, I would say not -- especially if our
purpose is to have a reasonably flexible system for
adequately managing not only nuances like these involving
neutral intervals, but also the 5-limit approximations which
are quite important in Turkish practice as well as theory.
While YAEU may have underplayed the role of neutral steps,
its popularity may stem in part from the fact that 5-limit
colors do occur in Turkish music, and may be traced back as
far as the tunbur of Khorasan described by al-Farabi
(870-950).

If the goal were only to realize the nuances with neutral
intervals, then there are smaller solutions, although ones
involving temperament. But to have the 5-limit and neutral
intervals really accurate, 41 notes of Pythagorean is about
right.

Here I have focused not on the exact averages reported in
the paper to which you are responding, but only the general
pattern in Maqam Ushshaq (or Arab Bayyati) of a low neutral
second but high neutral sixth, which Pythagorean 41 can
nicely realize.

I might add that the intonations shown above would likely be
acceptable in parts of the Arab world as well as Turkey,
adding the guess that some Arab tastes might prefer a
somewhat more "central" neutral sixth (say around 18/11, 852
cents), while some Turkish tastes might go rather higher
than 862 cents, as reflected by the reported peak at 872
cents or so. One possible interpretation of that peak is
that some performers leaned toward somewhere near 38 commas,
and others toward 39 commas (close to 5/3).

That would fit with variations in Turkish Rast, which is
9-8-5 commas in a YAEU approach (close to 9:8-10:9-16:15),
but more like 9-7-6 commas in a more "neutral" flavor. In
the Ushshak-4 + Rast-5 variation of Maqam Ushshaq, the Rast
genus occurs on the 4/3 step, so the third of Rast becomes
the sixth degree of this Ushshak variation.

Best,

Margo

Dear Margo,

Thank you for your in depth reply.

>
> Dear Marcel,
>
> Please let me agree that often simply looking at average
> values in cents for maqam steps leaves the question open as
> to whether we are simply seeing more or less free
> variations, or some more specific pattern.
>
> Knowledge of the maqam system and an investigation of how
> traditional performers understand the fine points of
> intonation can help in answering some of these questions.
> Since Turkish and other studies do show that a given maqam
> step (_perde_ in Turkish, from Persian _pardah_, a pitch or
> step) is often more of a cluster or region of pitches, such
> an understanding will always be partial.

Yes, I understand this.
And it is my personal belief that such small variations (smaller than a Pythagorean or 53tet comma) are additional non-functional coloring.
Such variations I think do not change the deeper tonal expression. A difference in expression as say an Eb makes from an E or an Ab from a G#.

What I'm interested in is the "essence" of a maqam, or at least seeing these 2 things, unique musical expression of a certain interval and additional coloring, as 2 separate things.

What I see in for instance western common practice diatonic music is that often when it is performed by instruments with a free pitch range the major third is aimed at 5/4. Yet my personal belief is that the essence of the major third is 81/64 and that playing it as 5/4 is an additional color (we still hear it as a major third, not as a Pythagorean diminished fourth which is only 2 cents from 5/4).

Now I wonder in certain maqam music what is the essence of the maqam. Simply looking at the tuning in actual practice does not tell me how much is additional coloring and how much is essence.
I have a different technique for this.
I harmonize the maqam!
I've found that western harmony (including jazz / blues harmony) still works for the most part when harmonizing chromatic melodies.
And where for instance a major and minor triad is stable (1/1 32/27 3/2 and 1/1 81/64 3/2), triads with a diminished fourth or augmented second are not.
So where a maqam's tuning goes close to C D Fb F G for instance, I check whether it's not actually C D E F G, if the C E G forms a stable triad to my ear while still keeping the musical expression of this specific maqam intact then it's an E indeed which was simply played in a colored way (as for instance 5/4, just like we often do in western performances).
But, if harmonizing it as an E destroys the unique expression of the maqam, and harmonizing it as a Fb keeps it intact then it is a Fb indeed!
I have found that Turkish and Arabic maqams indeed represent chromatic melodic intervals by the method described above, intervals such as augmented primes and diminished thirds.
I have also found that they also often use additional coloring.
Now harmonizing more distant enharmonics is not easy. So I find it difficult to check for intervals such as the ones giving 137 and 156 cent intervals. I'm unable to reliably tell whether these present coloring on top of for instance an augmented prime or diminished third.
But I'm making progress with this. Last night I succeeded for the first time in harmonizing the double augmented seventh (the 137 cents interval), although with a little trick by using it as an unstable interval which then drops melodically by a Pythagorean comma to harmonize stably with a new chord. For instance a cadence with Fb stable in a chord, like a minor third on Db+Ab, then play E# as unstable interval which drops melodically to F a Pythagorean comma lower which is used stably again in a chord.
Creative isn't it hehe :-) New frontiers in the making here ;-)
Other than harmonizing intervals, I currently don't know of any other way to separate the essence of a maqam from additional coloring.
Looking at where on the steps the most wear is can for instance simply mean that a certain coloring (of perhaps even more than a comma) is deeply embedded in a culture.

>
>
>
> However, with either a Turkish Ushshak or the more or less
> equivalent Arab Bayyati, we can identify an important
> pattern, and also offer at least one explanation for it.
>
> That pattern is the use of a rather low neutral second, in
> contrast to a rather high neutral sixth. To understand this
> contrast, it helps to know a bit about Maqam Ushshaq (or
> Ushshak in a Turkish spelling) and how its genera are
> formed.
>
> Often, with Ushshaq and some other maqamat, the lower or
> "root" genus (typically but not always a tetrachord) tends
> to remain more stable than the upper tetrachord, which can
> change without altering the "family" of the maqam. Here
> there are two common forms relevant for this analysis.
>
> The first has a root tetrachord of Ushshaq or Bayyati, plus
> a conjunct upper tetrachord called Nihavent or Nahawand --
> again, the Turkish and Arab names for similar genera. Here
> I'll use a Turkish style of note spelling, where Ushshaq is
> often on A (in an Arab notation, it would often be on D):
>
> Ushshak-4 Nihavent-5
> |--------------|------------------|
> A B-3 C D E F G A
> 0 137 294 498 702 792 996 1200
> 0 6 13 22 31 35 44 53
> 137 157 204 204 90 204 204
> 6 7 9 9 4 9 9
>
> Note that I'm following the adage of doing as the Turks when
> analyzing Turkish music, and treating Ushshak as having, in
> this "textbook form," a lower Ushshak tetrachord and an
> upper Nihavent pentachord. Some of the Turkish theorists use
> a comma notation (_koma sesler_, comma pitches) which fits
> nicely with Pythagorean intonation, and I've used here along
> with cents.
>
> Thus Ushshak-4 (the "4" meaning a tetrachord) is A B-3 C D,
> with the "B-3" meaning a B lowered by about three commas,
> here our 137.1-cent neutral step quite close to 13/12. This
> tetrachord is also neatly summed up as 6-7-9 commas, with
> six commas as a small neutral step, seven commas as a large
> one, and nine commas as our usual 9:8 tone.
>
> The upper pentachord, "Nihavent-5," has a tetrachord of
> 9-4-9 commas or tone-limma-tone, plus an upper 9:8 tone
> (again 9 commas) to complete the octave. An Arab style of
> analysis might speak of a Nahawand tetrachord plus this
> upper tone, while Turkish theory prefers to merge them into
> a single pentachord.
>
> However, you can see that this form of Ushshaq gives us a
> minor sixth step somewhere around 35 commas or 792 cents --
> in rational terms, of course, the 128/81 minor sixth.
>
> Now we move to our second form of Ushshaq, or variation
> within it:
>
> Ushshak-4 Rast-5
> |--------------|------------------|
> A B-3 C D E F+3 G A
> 0 137 294 498 702 862 996 1200
> 0 6 13 22 31 38 44 53
> 137 157 204 204 161 134 204
> 6 7 9 9 7 6 9
>
> The Ushskah-4 genus with its low neutral second at 137 cents
> or 6 commas is the same as before. However, for the upper
> genus, we now have Rast-5, or 9-7-6-9 commas.
>
> Note the contrast between the ordering of neutral steps in
> Ushshak-4 and Rast-5. In Ushshak, we have 6-7-9 commas, with
> the smaller neutral step coming first.
>
> In Rast, however, the pattern is 9-7-6 commas, with the
> larger neutral step coming first. And a fifth at 3/2 or 31
> commas, plus that first large neutral step of Rast at 7
> commas, results in a large neutral step for which 41-note
> Pythagorean provides a step at 38 commas or 862 cents.
> The speling D E F+3 G A shows that this neutral sixth is
> about 3 commas higher than F (128/81), or 2 commas lower
> than F# (27/16).

Thank you for showing me the combination of Ushshak-4 and Rast-5, I had not realized this.

>
>
>
> You raise the question of whether 41 notes is too large a
> tuning set. As a gamut, I would say not -- especially if our
> purpose is to have a reasonably flexible system for
> adequately managing not only nuances like these involving
> neutral intervals, but also the 5-limit approximations which
> are quite important in Turkish practice as well as theory.
> While YAEU may have underplayed the role of neutral steps,
> its popularity may stem in part from the fact that 5-limit
> colors do occur in Turkish music, and may be traced back as
> far as the tunbur of Khorasan described by al-Farabi
> (870-950).
>
> If the goal were only to realize the nuances with neutral
> intervals, then there are smaller solutions, although ones
> involving temperament. But to have the 5-limit and neutral
> intervals really accurate, 41 notes of Pythagorean is about
> right.

Well I would be very impressed by the brain if it could indeed understand the musical essence of a non modulating melody up with tones up to 40 fifths apart.
Could very well be the case.
But I have no way to check currently. This will surely be very hard to confirm using my harmonizing method unless I come up with some radical new ways to harmonize things.

But if some of the notes are not essence but additional coloring I'll belief it straight away.
As for 41-tone Pythagorean scale. If one's goal is to cover all the shadings and colorings even this large a scale does not seem like enough to me.
One would have to set a limit on human pitch precision, which is very variable under different circumstances. Under some circumstances it may even be as precise as 1 cent, requiring a 1200 tones per octave scale.
Perhaps a 100 tones is precise enough too.. I don't know. But this is not my personal area of interest.
My goal is to first get the essence (this is what I understand under the words "just intonation"), additional coloring can come in later if I ever want to use it.

>
>
>
> Here I have focused not on the exact averages reported in
> the paper to which you are responding, but only the general
> pattern in Maqam Ushshaq (or Arab Bayyati) of a low neutral
> second but high neutral sixth, which Pythagorean 41 can
> nicely realize.
>
> I might add that the intonations shown above would likely be
> acceptable in parts of the Arab world as well as Turkey,
> adding the guess that some Arab tastes might prefer a
> somewhat more "central" neutral sixth (say around 18/11, 852
> cents), while some Turkish tastes might go rather higher
> than 862 cents, as reflected by the reported peak at 872
> cents or so. One possible interpretation of that peak is
> that some performers leaned toward somewhere near 38 commas,
> and others toward 39 commas (close to 5/3).
>
> That would fit with variations in Turkish Rast, which is
> 9-8-5 commas in a YAEU approach (close to 9:8-10:9-16:15),
> but more like 9-7-6 commas in a more "neutral" flavor. In
> the Ushshak-4 + Rast-5 variation of Maqam Ushshaq, the Rast
> genus occurs on the 4/3 step, so the third of Rast becomes
> the sixth degree of this Ushshak variation.
>
> Best,
>
> Margo

Kindest regards,
Marcel

General disclaimer:
I hope people understand where I'm coming from.
These are just my personal findings. If anybody else has a different understanding of "just intonation" that's just fine with me.
And if people are interested more in what I call non-functional additional coloring that's fine with me too :)
I mean no disrespect to other people's ways of seeing things, it's just that I've developed my own views over time.

Dear Marcel,

Following your wise example, I will emphasize that there are
many ways of approaching both intonational issues generally
and the maqam tradition specifically, with different
methodologies yielding quite different results.

First, I regard maqam music above all a melodic art, with
Zalzalian or neutral intervals and their fine shadings an
integral part of that art. Thus my first priority must be
cultivating an aural sensitivity to these intervals as
distinct categories (e.g. a neutral as opposed to major or
minor second), and to their shadings, which are, of course,
subject to infinitesimal gradations with flexible-pitch
instruments. On my keyboard, the typical gradations (not all
available from the same location) involve differences of
about half a comma: e.g. seconds at 126, 139, 150, or 163
cents.

The challenge of hearing maqam primarily as maqam -- for
those of us who learn it as an additional musical language
rather a mothertongue -- is something basic to me.

Of course, it is possible to have various kinds of
derivative polyphony or "harmonization," and some of them
may help to illustrate, for example, the concept of neutral
intervals as a distinct category. However, the caution I
would see is that the "grammatical rules" of any system of
polyphony or harmony will impose, at least in part, an
agenda quite different from that of historical maqam music.

And those rules can vary immensely. For example, you speak
of 1/1-32/27-3/2 or 1/1-81/64-3/2 as a "stable triad," as
might hold in 16th-19th century European music -- with
the caution that timbre might well affect the perception of
stability for 1/1-81/64-3/2.

However, 13th-14th century rules tell me that both of these
sonorities -- the usual tunings for this era -- are in fact
unstable sonorities, both being forms of the _quinta fissa_
or "split fifth," and will sooner or later lead to some
resolution.

In fact, I've found over the last decade and a bit more that
the 13th-century European rules can adopt very nicely to an
expanded vocabulary of neutral as well major and minor
intervals -- but in some kind of hybrid polyphonic music
quite distinct from maqam itself, however much I may relish
it.

One thing I have found, along with Gene Secor (who helped
get me started in exploring maqam), is that sometimes this
kind of hybrid polyphony can be helpful in bringing home how
neutral intervals and cadences can have a quality radically
different from those of familiar 13th-century European
practice. These pieces may illustrate the point:

<http://www.bestII.com/~mschulter/SubArbore.mp3>
<http://www.bestII.com/~mschulter/SubArbore.pdf>

<http://www.bestII.com/~mschulter/Prelude_in_Shur_for_Erv_Wilson.mp3>

But ultimately I'd say the essence of maqam for me is
melodic. There are polyphonic devices such as drones, adding
simultaneous intervals such as fourths on an `ud or rebab
(Ibn Sina's _tarqib_ or "composite" sound in the sense of
polyphonic, rather than the "mixing" of different basic
maqamat to form a more complex maqam), and so on -- but I
would say that the fine melodic nuances are best appreciated
in their own terms.

Best,

Margo

Dear Margo,

>
> Of course, it is possible to have various kinds of
> derivative polyphony or "harmonization," and some of them
> may help to illustrate, for example, the concept of neutral
> intervals as a distinct category. However, the caution I
> would see is that the "grammatical rules" of any system of
> polyphony or harmony will impose, at least in part, an
> agenda quite different from that of historical maqam music.
>
> And those rules can vary immensely. For example, you speak
> of 1/1-32/27-3/2 or 1/1-81/64-3/2 as a "stable triad," as
> might hold in 16th-19th century European music -- with
> the caution that timbre might well affect the perception of
> stability for 1/1-81/64-3/2.
>
> However, 13th-14th century rules tell me that both of these
> sonorities -- the usual tunings for this era -- are in fact
> unstable sonorities, both being forms of the _quinta fissa_
> or "split fifth," and will sooner or later lead to some
> resolution.

Well you have to understand where I'm coming from.
I see modern western music as expressing 81/64 and 32/27 for major and minor thirds, not 5/4 and 6/5.
So I do see these triads as stable.
I also see 7th chords and extended chords as stable (though not as consonant).
I see chords as unstable when there are remote augmented or diminished intervals involved.
So even the Jimmy Hendrix chord of C E G Bb Eb or a C E G# chord is still somewhat stable, but C Fb G is unstable. Not suitable for a chord which has any sort of resolution, when this chord is used in such a place anyhow my brain will switch and interpret it as a C E G chord no matter how it's tuned.
This is the very point of my harmonization, by using resolutions I know for sure which chord I'm exactly hearing. And by combining known and "proven" chords into specific progressions I can build with confidence chromatic melodies and explore their sound.

>
> In fact, I've found over the last decade and a bit more that
> the 13th-century European rules can adopt very nicely to an
> expanded vocabulary of neutral as well major and minor
> intervals -- but in some kind of hybrid polyphonic music
> quite distinct from maqam itself, however much I may relish
> it.
>
> One thing I have found, along with Gene Secor (who helped
> get me started in exploring maqam), is that sometimes this
> kind of hybrid polyphony can be helpful in bringing home how
> neutral intervals and cadences can have a quality radically
> different from those of familiar 13th-century European
> practice. These pieces may illustrate the point:
>
> http://www.bestII.com/~mschulter/SubArbore.mp3>
> http://www.bestII.com/~mschulter/SubArbore.pdf>
>
> http://www.bestII.com/~mschulter/Prelude_in_Shur_for_Erv_Wilson.mp3>

Thank you for the audio examples.
But we have different ears.
I mean the following with all respect, to each their own thing, so please take no offense.
But these audio examples sound very out of tune to me, and I find my brain has a hard time to identify the intervals.
It does not sound like polyphonic maqam to me the way I experience a maqam. I do not feel it.

Here a very simple example to how I hear chromatic melodies.
Play and hold C E G and then an octave higher play the melody C - C# - Eb - E - G and play around with it a bit etc.
Instant exotic maqam like feeling to me. A bit like some jazz / blues but taken further I feel.
And it's not delicate as the brain has no way to interpret it in a diatonic way, there simply is not diatonic way to play this, the enharmonics I give are the simplest option / lowest state.
The simplest way to further harmonize it is with simple further chords of C major, I, ii, iii, IV, V, vi, etc. while still playing a chromatic melody on top.
More complex and interesting ways to harmonize are to have the chords move in chromatic ways too (but here one has to be careful not to play one thing and have the brain interpret it differently, the brain tends to interpret things closest together along the chain of fifths.)
All the above tuned to Pythagorean of course.
I'll try making audio examples soon.

Our methods are very different.
I hear my own as "correct" in the way that it evokes the unique feelings and expressions of maqams to me.
You may have the opposite if your examples do this thing for you.
You may also have the larger audience which agrees with you. I see many use of neutral thirds in triads around here which have never sounded right to me :)
I wish each of us the best of luck with our own paths.

Kindest regards,
Marcel