The question of whether or how neutral thirds -- also known as Zalzalian
thirds (see below) or middle thirds -- are used in Near Eastern music is a
complex one, with various views expressed by practical musicians and
theorists beginning with al-Farabi (c. 870-950).
First, in practice, the range of these neutral or middle thirds -- somewhere
between a large minor third at 6/5 (316 cents) and a small major third at
5/4 (386 cents) -- is quite wide, with the spectrum of possible sizes nicely
surveyed by the great philosopher, physician, and music theorist Ibn Sina
(c. 980-1037). And this amazing variety continues today in the many local
and regional practices of Arab, Turkish, Kurdish, and Iranian music.
Here I will show how each of the answers proposed in the question may have
some grain of truth, but with qualifications that suggest a more complex
reality over the last 1200 years or a bit more of Near Eastern music in
practice and theory. To sum up in advance:
(1) Theoretical writings of the 10th-15th centuries show a wide range of
middle thirds in use, mostly based on a diverse set of melodic steps using
superparticular ratios of the pattern (n+1:n), e.g. the 9:8 tone and middle
second steps such as 11:10, 12:11, 13:12, and 14:13; as well as more complex
ratios such as 88:81 or 128:117 that inevitably occur in these tunings also.
While 11:9 at 347 cents may sometimes be found (e.g. when adjacent steps of
9:8 at 204 cents and 88:81 at 143 cents occur), these middle thirds cover a
wide spectrum from around 63:52 (332 cents) to 26:21 (370 cents).
(2) The method of deriving middle intervals from very long chains of pure or
virtually pure fifths -- of which a chain of 52 plus a pure octave make a
musical circle for all practical purposes -- would fit the "53-comma" theory
of modern Turkey and Syria. There are indications that some Turkish
performers on flexible-pitch instruments may produce sizes for certain
intervals which are almost identical to those of the 53-comma system.
However, the comma system seems more of a general guide to interval
categories and regions than an explanation of how intervals are tuned in
practice.
(3) The concept of "an intonational inflection not linked to the harmonic
series or the framework of Pythagorean intervals" may reflect the fact that
intonational expressiveness and small nuances are valued for their own sake
in Near Eastern traditions, and that the tuning of the Persian tar, setar,
or santur is very much to taste. This does not mean that some linkage to
other intonational parameters of the music, such as the tuning of regular
fifths and fourths, does not exist; but that indeed the performer has great
freedom and discretion, with a given theoretical step realized in practice
by a "cluster" of pitches at different points in a performance.
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1. Classic Near Eastern Theory: The 10th-15th centuries
-------------------------------------------------------
Beginning with al-Farabi and Ibn Sina, Near Eastern writers of the 10th-15th
centuries sought to document and measure the tunings in use, for example in
fretting the `oud (an instrument of which the European lute is an offshoot
both in name and in design). One starting point was the famous `oudist
Mansur Zalzal of Baghdad (?-791).
Zalzal was renowned for introducing a middle finger or _wusta_ fret at a
middle third above the open string, known in his honor as a Zalzalian third,
and with its fret called the _wusta Zalzal_, or middle finger fret of
Zalzal. However, whatever Zalzal's own preferences may have been in
8th-century Baghdad, later musicians had a variety of placements for this
fret.
Near Eastern music theory of the 9th-15th centuries represented a creative
synthesis between the Classic Greek tradition and the indigenous practices
of the Arab, Persian, and other peoples of the region. Especially
influential were Pythagoras and his followers, who emphasized the basic
concords of the octave (2:1), fifth (3:2), and fourth (4:3); and also
Ptolemy, who favored a wide range of intervals and ratios which had the
property of being superparticular. A superparticular ratio follows the
pattern of (n+1:n): e.g. 3:2, 4:3, 5:4, 6:5, 7:6, 8:7, etc. These favored
ratios may be quite small, for example 28:27 (63 cents) as a semitone or
thirdtone much esteemed by Archytas for his melodic modes (also cited by
Ptolemy); or Ptolemy's 22:21 step (81 cents).
Also, like Classic Greek theory, Near Eastern theory often focused on
tetrachords or divisions of the 4/3 fourth using four notes or three
intervals. These tetrachords could be combined to build scales over an
octave or more; and pentachords, or divisions of the fifth (typically using
five notes or four intervals) sometimes come up both in 10th-15th century
theory and in modern approaches to Near Eastern theory.
For al-Farabi, a tetrachord which became part of his `oud tunings was formed
by the steps 9:8-12:11-88:81 (204-151-143 cents), or 1/1-9/8-27/22-4/3
(0-204-355-498 cents). His Zalzal fret was thus placed at 27/22, or 355
cents, the sum of the two lower superparticular steps in this tetrachord:
the usual 9:8 tone, plus the 12:11 middle or Zalzalian second step.
Note that while 11/9 (347 cents) would be the simplest ratio for a neutral
or Zalzalian third, al-Farabi's 27/22 (355 cents) is a bit higher, a small
difference of 243:242 or 7.139 cents. In melodic terms, note that in his
tetrachord of 9:8-12:11-88:81 or 204-151-143, the larger Zalzalian step at
151 cents comes before the smaller one at 143 cents. This preference is also
found in current versions of Arab and Turkish Rast, which feature a similar
tetrachord with a tone, followed by a larger and smaller Zalzalian step.
Interestingly, al-Farabi's tuning of what would later be known as Rast in
the simple "textbook" form of an octave scale -- Near Eastern modes are far
more intricate, and involve many types of inflections and modulations to
related modalities! -- does include some 11:9 thirds, but not in relation to
the main resting note or _qarar_, which we'll here call the final. This
"Mode of Zalzal" is built from two of al-Farabi's tetrachord plus an upper
9:8 tone to complete the octave. Here, following modern Arab conventions,
we'll take the 1/1 step or final as C.
Lower tetrachord Upper tetrachord 9:8
|----------------------|--------------------|.......|
C D Ed F G Ad Bb C
1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1
0 204 355 498 702 853 996 1200
9:8 12:11 88:81 9:8 12:11 88:81 9:8
204 151 143 204 151 143 204
Here the symbols Ed and Ad use ASCII "d" like a modern Arab half-flat, to
show that a note is lowered by a small interval which might be close to some
variety of "quartertone," but with a range of shadings possible. For
example, the same composition might be performed with notated Ed a bit lower
in Egypt and a bit higher in Syria. As Sami Abu Shumays observes, the fine
shade of tuning preferred in a given locality or region becomes a kind of
mark of the distinctive musical "dialect" for that group of musicians. We
have clues that this may have also been the situation in the 11th-14th
century Near East. Thirds at 11:9 or 347 cents may be found in al-Farabi's
tuning at 27/22-3/2 (Ed-G), and 18/11-2/1 (Ad-C).
In approaching Near Eastern music, it's very important to recognize that the
focus is above all on pure melody, not on complex stable or unstable
vertical sonorities of a kind which by around 1200 were a central feature of
European composition (e.g. Perotin). While Near Eastern theorists had a very
sophisticated concept of relative degrees of consonance and dissonance,
their interest especially concerned melodic grace and subtlety, with the
many shadings of Zalzalian steps and intervals a vital aspect of this focus.
In the early 11th century, Ibn Sina described a local practice that
preferred to place the fret of Zalzal a bit lower, so that the smaller
Zalzalian step comes before the larger one. Like al-Farabi, he started the
tetrachord of Zalzal used in his suggested `oud tuning with a 9:8 tone, here
followed by a 13:12 Zalzalian step at 139 cents (a superparticular ratio,
like al-Farabi's 12:11 at 151 cents), and then a larger Zalzalian step at
128:117 (156 cents) to complete the 4/3 fourth. His Mode of Zalzal, as it
might be called, is like this:
Lower tetrachord Upper tetrachord 9:8
|-----------------------|---------------------|........|
C D Ed F G Ad Bb C
1/1 9/8 39/32 4/3 3/2 13/8 16/9 2/1
0 204 342 498 702 841 996 1200
9:8 13:12 128:117 9:8 13:12 128:117 9:8
204 139 156 204 139 156 204
Here the Zalzal fret is at 39/32 or 342 cents, a bit lower than 11/9 at 347
cents. We may also notice that while al-Farabi's Zalzalian steps of 151-143
cents are not too far from equal, Ibn Sina's 139-156 cents differ by 17
cents, and provide more melodic contrast. In the 11th century or today, the
subtle differences in degrees of contrast between large and small Zalzalian
steps is one dimension of melodic creativity in Near Eastern music.
While al-Farabi and Ibn Sina illustrate differences in taste as to exactly
how high the third of Zalzal should be -- and Ibn Sina notes that some
people place the fret higher, and others lower -- their tetrachords show a
common element that helps us understand how the theories of the Pythagoreans
and Ptolemy both influenced these writers.
The type of mode we are considering was called Mustaqim by Ibn Sina, Arabic
for "right, correct, standard, usual." By the 13th century, the Persian name
Rast had caught on for this type of mode -- also meaning "right, correct,
standard, usual." The name Rast has become the norm, although Mustaqim can
be useful for describing a variety of Rast where the smaller Zalzalian step
comes before the larger, as in Ibn Sina's tuning.
In Rast, as tuned by al-Farabi or Ibn Sina, the lower step is the standard
9:8 tone of Pythagoras and his followers. This tone is equal to two pure 3:2
fifths (each 702 cents) less a 1200-cent octave at 2/1, or 204 cents; and
also to the difference between the 3:2 fifth (702 cents) and the 4:3 fourth
(498 cents).
Since both the pure 4:3 fourth and the 9:8 tone are givens for this type of
tetrachord, the two Zalzalian steps completing the tetrachord must add up to
the difference between 4:3 and 9:8, a usual Pythagorean minor third at 32:27
or 294 cents. In different tunings of al-Farabi and Ibn Sina, including the
two most famous given above, the lower 9:8 tone is followed by a
superparticular Zalzalian step at 11:10 (165 cents), 12:11 (151 cents),
13:12 (139 cents), or 14:13 (128 cents). However, as it turns out, it is
impossible to divide a 32:27 minor third precisely into two superparticular
ratios each within the range from 11:10 to 14:13.
Thus, as Ibn Sina discusses, there must be a less tidy "remainder" interval
to complete the 4:3 fourth which will not itself be superparticular, but
which the ear will accept as part of the total experience of the tetrachord
with its other superparticular or "consonant" steps.
Surveying the different ways that a 9:8 tone can be joined to a
superparticular Zalzalian step, Ibn Sina shows the range of Zalzalian or
middle third sizes known by the early 11th century. Our examples above
fall in the middle of the range, with 9:8 joined to 12:11 or 13:12; here we
add al-Farabi's tetrachord using an 11:10 step (quoted by Ibn Sina), and
another tetrachord of Ibn Sina using a 14:13 step:
(a) Al-Farabi, 9:8-11:10-320:297 (204-165-129 cents)
Zalzalian or neutral third at 99:80 (369 cents)
Lower tetrachord Upper tetrachord 9:8
|----------------------|--------------------|.......|
C D Ed F G Ad Bb C
1/1 9/8 99/80 4/3 3/2 33/20 16/9 2/1
0 204 369 498 702 867 996 1200
9:8 11:10 320:297 9:8 11:10 320:297 9:8
204 165 129 204 165 129 204
(b) Al-Farabi, 9:8-12:11-88:81 (204-151-143 cents)
Zalzalian or neutral third at 27:22 (355 cents)
Lower tetrachord Upper tetrachord 9:8
|----------------------|--------------------|.......|
C D Ed F G Ad Bb C
1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1
0 204 355 498 702 853 996 1200
9:8 12:11 88:81 9:8 12:11 88:81 9:8
204 151 143 204 151 143 204
(c) Ibn Sina, 9:8-13:12-128:117 (204-139-156 cents)
Zalzalian or neutral third at 39:32 (342 cents)
Lower tetrachord Upper tetrachord 9:8
|-----------------------|---------------------|........|
C D Ed F G Ad Bb C
1/1 9/8 39/32 4/3 3/2 13/8 16/9 2/1
0 204 342 498 702 841 996 1200
9:8 13:12 128:117 9:8 13:12 128:117 9:8
204 139 156 204 139 156 204
(d) Ibn Sina, 9:8-14:13-208:189 (204-128-166 cents)
Zalzalian or neutral third at 63:52 (332 cents)
Lower tetrachord Upper tetrachord 9:8
|-----------------------|---------------------|........|
C D Ed F G Ad Bb C
1/1 9/8 63/52 4/3 3/2 21/13 16/9 2/1
0 204 332 498 702 830 996 1200
9:8 14:13 208:189 9:8 14:13 208:189 9:8
204 128 166 204 128 166 204
From this range of tunings and melodic colors, we can learn, as Ibn Sina was
already aware a millennium ago, there is no "One True Size" for a Near
Eastern Zalzalian third, or the melodic steps making up a Rast mode. A small
Zalzalian third like Ibn Sina's 63/52 at 332 cents may have a "supraminor"
quality, and a large Zalzalian third like al-Farabi's 99/80 at 369 cents, or
Ibn Sina's similar 26/21 at 370 cents, may give a "submajor" impression.
Sizes such as al-Farabi's `oud fret at 27/22 (355 cents), or Ibn Sina's at
39/32 (342 cents), may give a range of intermediate impressions.
Also, while al-Farabi's famous 9:8-12:11-88:81 tuning has near-equal
Zalzalian steps of 151-143 cents, Ibn Sina's 9:8-13:12-128:117 with 139-156
cents shows greater contrast. In al-Farabi's 9:8-11:10-320:297 with 165-129
cents, or Ibn Sina's 9:8-14:13-208:189 with 128-166 cents, the difference is
one of over 35 cents.
It's also worthy of notice that ease of measuring frets can sometimes lead
to quite mathematically sophisticated and complex solutions -- which,
however, are very simple to implement in practice! Thus Ibn Sina mentions
and Safi al-Din al-Urmawi, a 13th-century theorist, gives prominence to a
fretting solution in which the third of Zalzal was placed exactly halfway
between the standard Pythagorean frets at the 9/8 tone and 4/3 fourth. Here
it's convenient to assume an open string length of 72:
72 64 59 54
|--------------|----------|----------|
1/1 9/8 72/59 4/3
0 204 345 498
9:8 64:59 59:54
204 141 153
After finding the familiar 9/8 and 4/3 frets, one would simply place the
Zalzal fret so as to divide the distance between these frets (at lengths of
64 and 54) into two equal parts, arriving at a length of 59! Safi al-Din
makes the resulting division of 72:64:59:54 one of his principal
tetrachords, notable from a modern viewpoint for its use of the higher prime
59. It is also in musical terms a very attractive tuning if one prefers that
the smaller Zalzalian step precede the larger (141-153 cents), and likes the
rather subtle contrast between these steps at around 12 cents.
As it happens, the resulting Zalzal third at 72/59 (344.7 cents) is quite
close to the simple ratio of 11/9 (347.4 cents), with a difference of only
649:648 or 2.7 cents. More generally, as the range of 10th-15th century
tunings illustrates, the region around 11/9 is simply one possible taste for
Near Eastern Zalzalian or neutral thirds.
--------------------------------------------------------------------
2. Long chains of fifths and the 53-comma theory of Turkey and Syria
--------------------------------------------------------------------
Near Eastern musicians are often, although not always, influenced by the
Pythagorean technique of tuning stringed instruments, for example, in pure
3:2 fifths or 4:3 fourths. Scott Marcus has suggested that regular minor
thirds on such instruments may often be around 32:27 (294 cents), the minor
third produced by three pure fourths (e.g. D-F from D-G-C-F). This 32:27
minor third, and also the 9:8 tone, are generally assumed in 10th-15th
century theory we have just surveyed.
During those centuries, some musicians experimented with the idea of using
longer chains of pure fifths or fourths to build a tuning system. As
chronicled by Cris Forster in his book _Musical Mathematics: On the Art and
Science of Acoustic Instruments_, al-Farabi described such a style of tuning
on the Tunbur of Khorasan. In the 13th century, Safi al-Din described one
approach to tuning the `oud as a system based on a chain of 17 notes in pure
fifths or fourths.
However, one complication of such an approach is that very large sets --
larger than the 17-note systems explored in both the Near Eastern and
Western Europe around the 13th-15th centuries -- are needed to obtain the
middle or Zalzalian steps and intervals so central to Near Eastern melody.
Rather, middle second steps like 14:13, 13:12, 12:11, and 11:10 are treated
as independent elements of music, not simply as derivatives from a chain of
fifths.
In modern times, however, Turkish and Syrian musicians have carried the
approach of tuning in pure or virtually pure fifths further, at least in
theory, to arrive at an elegant system for categorizing intervals and
conceptualizing their approximate sizes.
From a literal mathematical perspective, no number of pure 3/2 fifths can
ever precisely equal any number of pure 2/1 octaves. However, in practice, 53
pure fifths will exceed 31 pure octave by a small factor called the comma of
Mercator, about 3.615 cents. Thus a 53-note Pythagorean circle is possible,
in which a chain of 52 pure fifths is tuned, followed by a pure 2/1 octave,
leaving the "odd" 53rd fifth (which is not tuned directly) narrow by the
small comma of 3.615 cents. Many historical European temperaments, as well
as measured Near Eastern tunings on fixed-pitch instruments, have fifths at
least this impure; and here the compromise applies to only one fifth out of
a circle of 53.
If this method were followed, then the octave would be divided into 53 steps
of two slightly varying sizes. At 41 locations, we would have the
traditional Pythagorean comma at 531441/524288 or 23.460 cents; this is the
amount by which 12 pure fifths exceed 7 pure octaves. At the other 12
locations, we would have a small comma at 19.845 cents, smaller by the comma
of Mercator.
A different approach, often preferred for the sake of simplicity, is to
regard each of the 53 fifths as tempered by a minute amount equal to 1/53 of
the comma of Mercator, so that all 53 steps are at an identical 22.642
cents. More generally, in this simplified scheme, there will be 53 sizes of
intervals as exact multiples of this standardized comma, named after the
17th-century English theorist John Holder as the "Holderian" (or sometimes
"Holdrian") comma.
From a Near Eastern perspective, the 53-comma system offers middle or
Zalzalian second steps at 135.8 cents (close to 13/12, 138.6 cents); and
158.5 cents (a bit larger than Ibn Sina's 128/117 step at 155.6 cents). In
some styles of Arab and Persian music, favoring steps at around 160 and 135
cents, the 53-comma system gives a rather close approximation to these
sizes.
Indeed for tunings like Ibn Sina's, or modern Persian variations such as
those of Hormoz Farhat, a 53-comma system gives excellent approximations.
However, many musicians understand the comma system not so much as a literal
scheme of measuring interval sizes, as a useful way of categorizing
intervals while leaving the exact intonation up to the performers.
For example, an Egyptian, Lebanese, or Syrian Rast might be expressed as
"9-7-6 commas," meaning that a usual whole-tone at around 9:8 (9 commas in
this system) is followed by a larger and then a smaller Zalzalian step. If
the steps are actually around 151-143 cents, as in the famous tuning of
al-Farabi, then in fact we would have something more like 9-6.7-6.3 commas.
However, 9-7-6 commas is understand more flexibly to say that the larger
Zalzalian step precedes the smaller, whatever the exact tunings.
In contrast, 9-6-7 commas might suggest Ibn Sina's Mustaqim, or a modern
Persian counterpart where the smaller step precedes the larger.
An interesting example where certain Turkish performers do seem very closely
to approximate the precise values of the comma system is documented by Karl
Signell. Here the interval in question is the large or "augmented" step
often featured as the middle interval of Arab or Turkish Hijaz, a type of
tetrachord named for a desert region on the Arabian Peninsula. Around 1300,
Qutb al-Din al-Shirazi reported a tuning of 12:11-7:6-22:21 or 151-267-81
cents, a permutation or rearranged order for the step sizes of a tuning by
Ptolemy, the Intense Chromatic at 22:21-12:11-7:6 or 81-151-267 cents.
While there are many shadings of Hijaz (in a modern Turkish spelling, Hicaz)
and related genera (e.g. Persian Chahargah) in use in the Near East,
Turkish theory calls for a middle step of 12 commas, quite close to 7:6, but
a bit larger (271.7 cents). In measuring the intervals used by the Turkish
musician Necdet Ya[s-cedilla]ar, highly esteemed for his expert and precise
intonation, Signell found that they were consistently in the range of
270-273 cents, with 272 cents as the average -- a veritable 12 commas, and a
bit larger than 7:6 at 267 cents.
This is not to say that the comma system is invariably a guide to exact
Turkish intonation, which can be very flexible: Amine Beyhom, for example,
measured a Turkish Hicaz with a middle step of around 265 cents, very close
to 7:6 and in fact very slightly smaller.
When used to describe middle, neutral, or Zalzalian thirds, the 53-comma
system recognizes two general types. The smaller or 15-comma middle third,
at a literal 339.6 cents, might more generally suggest the region of a
third somewhere around Ibn Sina's 63/52 (332.2 cents) or 39/32 (342.5
cents), or possibly the slightly larger 11/9 (347.4 cents).
In contrast, the 16-comma middle third, literally 362.3 cents, might suggest
Ibn Sina's 16/13 (359.5 cents) or 26/21 (369.7 cents).
In practice, there is an infinite variety of shadings, but the 15-comma and
16-comma categories can often indicate the general type of middle third
expected. For example, a 16-comma third would be the norm in an Arab Rast
(e.g. C-D-Ed-F); while a 15-comma third might be expected in Sika/Segah.
which characteristically starts with the trichord Ed-F-G. Here C-G is
presumably a pure or near-pure fifth at around 31 commas, with C-Ed at
around 16 commas (the Rast third), and Ed-G a smaller third at around
15-commas (the Sika/Segah third).
To sum up, while regular intervals such as major and minor thirds are often
at or close to Pythagorean sizes determined by chains of pure fifths or
fourths, the 53-comma system is more of a theoretical map to approximate
interval sizes and categories than a method of tuning middle intervals. In
that function, it does convey one very interesting side of the picture, but
with personal and regional tastes taking priority over any precise
mathematical scheme.
-------------------------------------------------
3. Musicianly taste, tarab, and "glissando zones"
-------------------------------------------------
In addition to surveying the range of neutral or Zalzalian steps and third
sizes in use around the early 11th century, Ibn Sina tells us that some
people, for example, place the third of Zalzal higher or lower on the `oud
than others, and that the ability of performers to distinguish different
sizes of steps also varies greatly. He says that many musicians do not
discriminate between steps of 14:13 (128 cents) and 13:12 (139 cents), but
that the adept can and do.
Can Akko[c-cedilla], a Turkish theorist, has measured flexible-pitch
performances of Turkish music and demonstrated that the realization of a
given step is in practice not fixed but variable by a comma or more, with a
"cluster" of actual pitches representing such a modal step. Thus, as he
commented at a conference summarized by Eric Ederer, much Western or
Western-derived theory may be applying a "particle" model to what is a
"wave" phenomenon.
Indeed, one recent Turkish concept posits a "glissando zone" in which a
middle step may be placed, with the interpretation highly variable, and the
ability to make this variability as expressive and telling as possible one
of the marks of a true artist.
There is a concept that the middle or Zalzalian steps of a mode -- a middle
third or seventh in Rast, for example, in contrast to the fourth or fifth --
tend to be more fluid, with apt pitches for these steps at different points
in a piece resembling clouds or clusters. Between a regular minor third at
around 32/27 or 13 commas, for example, and a regular major third at around
81/64 or 18 commas, there is a "three-comma glissando zone" which might
range from around 14 commas or a large 6/5 minor third through a small
middle third at 15 commas (e.g. 63/52, 17/14 at 336.1 cents, or 39/32, etc.)
to a large middle third at 16 commas (e.g. 16/13, 21/17 at 365.8 cents, or
26/21) or a small major third at 5/4 or 17 commas.
Where a note is placed at a given moment may depend on personal or regional
tastes, musical context, and sheer inspiration. For example, the Turkish
principle of cazibe or "attraction" suggests that a note will be slightly
raised or lowered in approaching certain cadences. Thus a performer who
places the Rast third at close to 5/4 (386 cents) or 17 commas (a placement
favored in standard 20th-century Turkish theory and approximated in some
modern practice) may lower it to around 16 commas (say 26/21, 370 cents)
when descending to a final cadence. And a performer who favors a 16-comma
Rast third (say 26/21, or perhaps 21/17 at 366 cents) may lower it to
somewhere around 11/9 (347 cents). Such adjustments may be quite subtle, for
example on the order of around half a comma, or 10-15 cents.
As summarized by the Lebanese musician and teacher Ali Jihad Racy, the
20th-century Syrian theorist Tawfiq al-Sabbagh urges that to achieve the
supreme quality in Arab Maqam music of tarab, ecstasy or enchantment, steps
may often usefully be altered or inflected by around a comma.
While al-Sabbagh, as a Syrian musician and an admirer of Turkish practice,
uses the 53-comma system to indicate these approximate tunings and
inflections, Scott Marcus has documented an often very flexible rather than
mathematical understanding in the Arab world of a "comma": any small
interval which makes a musical difference in a given context.
For example, Racy showed Marcus how, in Maqam Bayyati, the third minor step
is a bit lower than the Pythagorean minor third at 32/27 or 294 cents that
would result from the pure fifths or fourths used in tuning the strings.
While a reader oriented to small integer ratios like the 7/6 third, or to
the 53-comma system, might take this to mean that a 7/6 third at 267 cents
is intended, or perhaps the slightly larger third at a precise 12 commas or
272 cents, Marcus clarifies that the intent is a small and unspecified
difference, left to the musician's judgment.
Comparing some measured Persian tunings on tar or setar also leads to the
idea of performer discretion within a certain understood range of
variations, rather than any precise standard for each note, whether based on
just ratios, the comma system, or some other criterion such as a given
precisely equal division of the octave. Thus Nelly Caron and Dariuche
Safvate in 1966 document a tuning of the Dastgah or modal form of Shur
with steps of 136-140-224 cents, and a minor third at 276 cents. This is a
bit higher than 7/6, just a bit wider than 12 commas (272 cents), and
considerably narrower than the common 32/27 (294 cents).
Another Persian musician, Hormoz Farhat, suggests a division for Shur of
around 135-160-205 cents, with a third of 295 cents or a virtually just
32/27. A permutation of Ibn Sina's tetrachord included in his `oud tuning,
using his steps of 9:8, 13:12, and 128:117 in ascending order of size, gives
a very similar 13:12-128:117-9:8 at 139-156-204 cents.
In short, the variability of tunings within a given tradition, and sometimes
within a given piece, require that we regard fluidity, flexibility, and
expressive variation not only as inevitable features of flexible-pitch
performance, but as highly cultivated and desired refinements.
The fact that while the `oud of 10th-15th century practice was generally
fretted, modern `ouds are generally fretless, further accentuates this
element of performerly discretion. Like performers of medieval European
music such as Christopher Page who value a purely vocal performance with the
liberty to tune a given interval aptly on each occasion, Near Eastern
performers such as Amine Beyhom value instruments unconstrained by fixed
tunings.
Furthermore, the complexity and sophistication of fixed-pitch instruments
such as the 79-tone qanun of Ozan Yarman built with a tuning system he
designed intended for Turkish and other Near Eastern musics, or the just
intonation qanun of Julien Jalal Ed-Dine Weiss as documented by the scholar
Stefan Pohlit, also reflects the desire for many finely shaded nuances of
expression.
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4. Conclusion
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As a mainly melodic art, Near Eastern music features many finely graded
sizes of neutral or Zalzalian steps and thirds. This was clearly true by the
early 11th century, when Ibn Sina documented this variety, and has remained
true in the dynamic setting of a constantly evolving set of related but
sometimes divergent modal systems that have developed over the past
millennium and more.
Integer ratios, both simple and complex, play a central role in 10th-15th
century theory. The middle third at 11/9 would sometimes occur, for example
in al-Farabi's Mode of Zalzal between the third step at 27/22 and the 3/2
fifth (1/1-9/8-27/22-4/3-3/2 or 0-204-355-498-702 cents, with 347 cents or
11:9 as the difference of 3/2 and 27/22). However, this ratio does not seem
especially privileged, but simply one possible point on a continuum.
The 53-comma system, based on a 53-note musical circle with pure or
virtually pure fifths, gives an elegant system for mapping regions or
general categories of intervals, with some of the theoretical sizes (like a
smaller or 6-comma middle second at 136 cents) in fact often quite close to
other theoretical or practical values in use, such as Ibn Sina's 13:12 step
at 139 cents, or Farhat's step at 135 cents, or the suggestion by Marcus
that the lower step of an Egyptian Maqam Bayyati might be tuned at around
135-145 cents.
However, variability is the rule, especially in flexible-pitch performance;
and both personal and regional tastes may result in different tunings for
fixed-pitch instruments as well.
Another side of Near Eastern theory is the use, at times, of systems based
on an equal division of the octave. The 53-comma theory, when premised on
precisely equal comma steps at 1/53 octave, falls in this category, although
it also very closely approximates the results of tuning in pure fifths, and
could be applied (as in some Turkish interpretations) to a tuning in pure
fifths, where slightly unequal commas at 23.460 and 19.845 cents would
result if the tuning were carried to a full 53 notes.
Other equal divisions proposed or used -- sometimes with the caution that
in fact the steps should NOT be taken as equal in practice, but variable! --
include the 68-division of Byzantine theory (Chrysanthos of Madytos);
divisions of 36, 72, or 144 based on the late Classic Greek theorist
Cleonides as an interpreter of Aristoxenos (standard in Byzantine music
theory starting in 1881); the 24-division (presented by the Syrian theorist
Mikhail Mashaqa, although he found the 68-division more accurate); and the
17-division suggested as one option by Amine Beyhom.
Sami Abu Shumays has suggested that recognizable local or regional styles of
intonation, like speech communities with distinctive local dialects or
"accents," may be part of the sophisticated musical cultures of the Arab
world. A given tuning of the Rast third, for example, might mark a
musician's style as typical of Egypt or Syria.
The experience of Abu Shumays also fits an Arab musician's impression shared
with Marcus that as you travel east from Egypt to Palestine, Syria, and
finally Turkey, the note sika (the third of Rast) will tend gradually to get
higher and higher. The experience of Julien Jalal Ed-Dine Weiss suggests
that, within Syria, sika tends to be somewhat lower in Damascus (perhaps
centering around 355-365 cents) than in Aleppo (perhaps around 365-375
cents).
In sum, Near Eastern theorists from at least al-Farabi on have provided a
range of sophisticated theoretical concepts, maps, and ratios which
evocatively but imperfectly suggest the sovereign performer's discretion and
taste which inform a yet more sophisticated practice.
Margo Schulter
8 December 2014