----------------------------------------------------
The Zalzalian Diatonic Tetrachords of Ibn Sina
Placing his `Oud tuning in a larger context
by Margo Schulter
----------------------------------------------------
As a philosopher, physician, and musician, Ibn Sina has enjoyed the
admiration of the Islamic world for a millennium, being likewise
honored in Latin Europe (under the name of Avicenna). His fascination
with fine shades of intonation, and in his interest in small commas
and near-equivalences between superparticular and more complex forms
of Zalzalian or neutral second steps, evince the same spirit of
artistic refinement and intellectual analysis which may be found in
the qanun tunings of Julien Jalal Ed-Dine Weiss (here JJW).
In attempting this brief overview and survey of Ibn Sina's admirable
and often remarkable observations, I am deeply indebted both to Stefan
Pohlit, who has invaluably documented JJW's qanun tunings and
masterfully nuanced techniques in interpreting traditional maqam music
as well as in composing new music and providing materials for other
musicians and composers to explore, and of course to JJW himself, both
of whom have shown me friendship, generosity, and grace.
This paper is therefore dedicated to them both, with due notice that
while they are responsible for much of whatever virtue it may have,
the fault for any errors or infelicities lies solely with me.
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1. Introduction
---------------
In his _Kitab al-Shifa_, Ibn Sina (980-1037) presents a fascinating
survey of what I will here term "Zalzalian diatonic" tetrachords,
those involving a tone at 8:7 or 9:8 plus two neutral second steps.
Having surveying these tetrachords, including two earlier discussed by
al-Farabi (c. 870-950), he offers some observations about the
intonational practices of his musical contemporaries. These passages
(d'Erlanger 2001-II: 148-150) are critical in understanding his later
remarks on the placement of the wusta Zalzal (wZ) or neutral third
fret on the `oud (ibid. at 235).
For Ibn Sina, as for al-Farabi, the superparticular ratios are of
special importance, a preference which follows the Greek tradition of
Ptolemy, as Ibn Sina himself emphasizes in a remarkable passage
describing the division of the 4:3 fourth into 16:14:13:12, or a large
tone at 8:7 plus Zalzalian or neutral second steps at 14:13 and 13:12
or 231-128-139 cents (d'Erlanger, ibid. at 148). The "very noble jins"
has all superparticular steps, and Ibn Sina regards as musically
equivalent the two permutations with 8:7 as the lowest step.
With a 9:8 tone as the large step of a Zalzalian diatonic tetrachord,
the situation becomes inevitably more complex, because a 4:3 fourth
less 9:8 leaves a minor third at 32:27 to be divided between the two
neutral second steps. In Ibn Sina's tetrachords, one of these steps is
a superparticular neutral second at 11:10, 12:11, 13:12, or 14:13
(respectively 165, 151, 139, or 128 cents); while the other is a more
complex "remainder," which he sees as analogous to the Greek _limma_
(remainder) in the diatonic at 9:8-9:8-256:253 (204-204-90 cents).
Just as the ear accepts the 256:243 semitone in this usual diatonic
(d'Erlanger, ibid. at 148-149), so it may accept the complex neutral
second which necessarily arises when a 32:27 minor third is divided
into two neutral steps, since no division is available where both
neutral steps are superparticular. As he repeatedly observes, however,
the nonsuperparticular or "remainder" step may be taken as closely
approximating a superparticular ratio. His explorations of these
approximations survey some interesting small commas, although he does
not focus on the precise ratios of these commas (ibid. at 149-150).
Before delving into Ibn Sina's remarks about these Zalzalian diatonic
tetrachords, we may find it helpful to list the forms he surveys and
adopts in this main discussion, along with the related tetrachord in
his `oud tuning (ibid. at 235) at 9:8-13:12-128:117 (204-139-156
cents), the basic for his Mustaqim modality with wZ at 39/32 (342
cents). In addition to the tetrachords which he makes a part of his
system, there are two he cites from al-Farabi (without attribution)
but does not include in his own preferred set.
--------------------------------------------------
Table 1: Ibn Sina's Zalzalian Diatonic Tetrachords
Including two from al-Farabi not adopted (B1, B2)
--------------------------------------------------
A. Tetrachords with 8:7 tone ("most noble jins" of Ibn Sina)
A1. 8:7-13:12-14:13 or 104:91:84:78 (231c-139c-128c)
Zalzalian 3rd at 26/21 (370 cents)
A2. 8:7-14:13-13:12 or 16:14:13:12 (231c-128c-139c)
Zalzalian 3rd at 16/13 (359 cents)
B. Tetrachords with 9:8 tone
B1. 9:8-11:10-320:297 or 396:352:320:297 (204c-165c-129c)
Zalzalian 3rd at 99/80 (369 cents) -- al-Farabi
B2 9:8-12:11-88:81 or 108:96:88:81 (204c-151c-143c)
Zalzalian 3rd at 27/22 (355 cents) -- al-Farabi `oud
B3a 13:12-9:8-128:117 or 468:432:384:351 (139c-204c-156c)
Zalzalian 3rd at 39/32 (342 cents) -- Ibn Sina
B3b 9:8-13:12-128:117 or 468:416:384:351 (204c-139c-156c)
Zalzalian 3rd at 39/32 (342 cents) -- Ibn Sina `oud
B4 9:8-14:13-208:189 or 252:224:208:189 (204c-128c-166c)
Zalzalian 3rd at 63/52 (332 cents)
Ibn Sina's two permutations with an 8:7 tone feature larger Zalzalian
or neutral thirds at 26/21 (369.7 cents) and 16/13 (359.5 cents),
while his preferred tetrachords with 9:8, both in his listing of ajnas
and in his `oud tuning, have smaller neutral thirds at 39/32 (342.5
cents), his wZ, and 63/52 (332.2 cents).
However, he does not adopt the two Zalzalian diatonic tetrachords of
al-Farabi which he mentions, with larger neutral thirds at 99/80
(368.9 cents) and 27/22 (354.5 cents), the latter al-Farabi's notable
placement of wZ.
It is easy to note one contrast because the chosen tetrachords of
al-Farabi and Ibn Sina with a 9:8 tone. While al-Farabi prefers to
place the larger Zalzalian second step before the smaller, Ibn Sina
prefers the converse arrangement. Thus their respective wZ positions
at 27/22 (355 cents) and 39/32 (342 cents).
Clearly Ibn Sina is interested in fine shadings of neutral steps,
which he tells us (ibid. at 150) his contemporaries often confuse, for
example using indifferently either 14:13 (128 cents) or 13:12 (139
cents). This discussion of intonational practices may give clues not
only to his own musical outlook, but as to the local tastes of his
time and place. His report that the placement of wZ varied (ibid. at
150, 235) is much in accord with the observations of Safi al-Din
al-Urmawi (c. 1216-1294) some two centuries later.
Ibn Sina's recognition and use of all four superparticular neutral
or Zalzalian seconds is a development still very influential in the
precise and extremely sophisticated just qanun tunings of Julien
Jalal Ed-Dine Weiss, an Ars Intonationis Subtilissima or "most subtle
intonational art" for the 21st century. More modestly, the MET-24
temperament supports these same four shadings of neutral intervals,
albeit imprecisely, by tempering out the small commas which Ibn Sina
notes and the qanun tunings faithfully observe.
From this perspective, let us consider Ibn Sina's treatment of the
individual Zalzalian diatonic tetrachords he surveys.
----------------------------------------------------
2. Tetrachords with an 8:7 tone: A "most noble jins"
----------------------------------------------------
in presenting Zalzalian diatonic tetrachords, Ibn Sina begins with two
"truly consonant genera" which he describes as "very noble."
When, to the interval 8:7, one joins 13:12, the complementary
interval will be 14:13. (d'Erlanger, op. cit., at 148).
He then demonstrates in mathematical terms how this division, with all
superparticular steps, may be elegantly derived on the monochord,
using an approach which I here illustrate using a length of 16 for the
whole string to show his aliquot arithmetic divisions.
In setting this genre, most noble, one has effected the
last partition by a halving of intervals [i.e. string-length
ratios]. The interval of the double octave, partitioned in half,
has, indeed, yielded the octave. (Ibid. at 148)
[Illustration added -- M.S.]
2:1 2:1
|------------------------------------------|-----------------|....
16 8 4
After this first partition, Ibn Sina proceeds by a series of
arithmetic means.
This last interval [i.e. the octave], partitioned by a means, has
given the fifth and fourth. (Ibid. at 148)
[Illustration added -- M.S.]
4:3 3:2
|-----------------------|-------------------|
16 12 8
This partition of the 2:1 octave into the 4:3 fourth and 3:2 fifth is,
of course, one of the basic divisions of the Pythagoreans, here
realized as 16:12:8 (16:12 or 4:3 fourth plus 12:8 or 3:2 fifth).
Partitioned in the same fashion, the interval of the fourth has
engendered the intervals 8:7 and 7:6; the last in turn has given
13:12 and 14:13. (Ibid. at 148)
[Illustration added -- M.S.]
8:7 7:6
|-----------|-----------|-------------------|
16 14 12 8
8:7 14:13 13:12
|-----------|-----|-----|-------------------|
16 14 13 12 8
The partition of 16:12 or 4:3 into 8:7:6 (16:14:12) is the division of
Archytas, as used by Ptolemy. Here Ibn Sina has carried the process a
step further, arriving at a remarkable result: the Zalzalian division
of 7:6 into neutral second steps of 14:13 and 13:12.
Ptolemy prefers the genre at which we proceed to arrive above all
others. The numeric expression is as follows:
[Illustration added by M.S., cf. Cris Forster (2010, at )
231 139 128
8:7 13:12 14:13
|-------------|--------|-------|
104 91 84 78
1/1 8/7 26/21 4/3
0 231 370 498
Although I know of no such tetrachord in Ptolemy, Ibn Sina very
reasonably asserts that this division superbly realizes Ptolemy's
method and ethos of seeking elegant superparticular ratios arranged in
a musically pleasing way! This style of division might be described as
"Zalzalian-Archytan," since it relies both upon the Archytan division
of 4:3 into 8:7 and 7:6, and upon Zalzal's division of a minor third
into two neutral second steps (here 13:12 and 14:13).
From a mathematical perspective, the Zalzalian division of the 7:6
small minor third, here 91:78, into lengths of 91:84:78 is harmonic,
since the differences of adjacent terms have a ratio of 7:6, also the
ratio of the extreme terms, or the interval being partitioned.
[Illustration added -- M.S.]
7 6
|--------------|-----------|
91 84 78
|--------------------------|
7:6
From a musical perspective, this division produces a tetrachord of
1/1-8/7-26/21-4/3 or 0-231-370-498 cents, and steps of 8:7-13:12-14:13
(231-139-128 cents). It has a bright Zalzalian or neutral third at
26/21 or 370 cents, typical of a modern Syrian Rast in the practice of
Aleppo, and of some historical and modern Ottoman practices, as
illustrated for example by a baglama from Istanbul as reported by
Cameron Bobro (who measured a neutral third fret at 370 cents). The
important distinction between Ibn Sina's tetrachord and these
practices, of course, is the higher position of the tone at 8/7 rather
than a usual Rast where it is at or near 9/8 (204 cents).
Ibn Sina then proceeds to the other permutation of this tetrachord
with the 8:7 tone as the first or lowest interval, the form which he
has already presented in his Ptolemaic demonstration of multiple
aliquot divisions (ibid. at 148):
When, on the other hand, one joins to the interval of 8:7, the
interval of 14:13, the complement will be 13:12 and the genre
obtained will be identical to the previous one:
231 128 139
8:7 14:13 13:12
|-------------|-------|--------|
16 14 13 12
1/1 8/7 16/13 4/3
0 231 359 498
Here the large Zalzalian third is at 16/13 or 359 cents, typical of
a Rast third in the Syrian practice of Damascus. Ibn Sina evidently
finds the effect musically equivalent to the 8:7-13:12-14:13 form,
since he described the two tetrachords as "identical." Both could be
described as septimal variations on what is in modern terms a bright
Rast. To some Syrian tastes, for example, both permutations with their
respective neutral thirds at 26/21 and 16/13 might be congenial.
Interestingly, Bozkurt and colleagues (2009, at 46) find that this
genus "is resemblant of quotidian Arabic rendition of the cadence
region of _maqam Segah_" -- that is, the lower Rast tetrachord
rast-dukah-sikah-jaharkah, as often intoned in making a cadence on the
step sikah, the final of this maqam.
In such an Arab practice, this tetrachord with rast-sikah at around
16/13 would have a normal tuning in theory, and likely in practice, at
around 1/1-9/8-16/13-4/3 or 0-204-359-498 cents, with a quite large
neutral second step between dukah and sikah (9/8-16/13) at 128:117 or
156 cents. However, in cadencing on sikah, there is evidently a desire
for a higher leading tone and a smaller neutral step. Thus dukah, as
reported by Bozkurt and his colleagues, may be raised by about a
comma, from 9/8 to Ibn Sina's 8/7, resulting in a small cadential
neutral second at 14:13 (128 cents).
This small adjustment might be related to the concept in Arab theory
of a _dint_, the raising of a leading tone in approaching the final or
another principal note of a given maqam (Marcus 1989, 612-616). While
a _dint_ inflection, at least in recent modern theory, is generally by
a semitone, here it is more subtly by a comma.
The 8:7-13:12-14:13 and 8:7-14:13-13:12 divisions of Ibn Sina are of
special interest for two reasons. First, they are the only divisions
he adopts which have a large neutral third step at 26/21 or 16/13 --
in contrast to the lower neutral thirds he prefers when a tetrachord
has a tone at 9/8. Secondly, these are the only Zalzalian diatonic
tetrachords where all three steps are superparticular: the 8:7 tone
leaves a 7:6 minor third to be divided between the two neutral second
steps, neatly partitioned into 13:12 and 14:13 (or vice versa).
When the tone is at 9:8, this leaves a 32:27 or regular Pythagorean
minor third (294 cents), whose division into two neutral seconds must
inevitably result in a nonsuperparticular interval, introducing the
complication of various small commas into the tuning equation. Ibn
Sina approaches these complications with great insight, and at the
same time may be reflecting some musical tastes of his time and
region.
----------------------------------------------------------------
3. Tetrachords with a 9:8 tone: remainders, tastes, and shadings
----------------------------------------------------------------
For Ibn Sina, the appearance of "dissonant" or nonsuperparticular
steps, often with strikingly complex ratios, is a phenomenon worthy of
exploration. Why is it that these ratios, despite their deviation from
the Ptolemaic ideal of epimoric or superparticular steps, nevertheless
are often acceptable to the ear? His discussion of this problem is an
apt prelude to his survey of the Zalzalian diatonic tetrachords with a
tone at 9:8, where it is impossible to obtain all superparticular
steps (at least as long as one keeps the fourth at a just 4/3).
Here Ibn Sina's method is to present a series of tetrachords combining
a 9:8 tone with another superparticular step ranging downward in size:
thus 9:8, 10:9, 11:10, 12:11, 13:12, and 14:13. In his classification
of intervals, 14:13 is noteworthy as the smallest superparticular step
which, when doubled, is equal to the greater part of a 4/3 fourth --
196:169, or 256.6 cents (leaving a lesser part of 169:147, or 241.5
cents).
---------------------------------------------------------------
3.1. The Ditonic Diatonic (9:8-9:8-256:253) and the "remainder"
---------------------------------------------------------------
Following this method, he begins with the familiar Pythagorean or
ditonic diatonic at 9:8-9:8-256:256 (204-204-90 cents), and offers
some observations providing a framework for his survey of the
Zalzalian tetrachords that follow (d'Erlanger, op. cit. at 148-149).
As for genera having for a basis the interval 9:8, the first
among them results from a repetition of this interval in the
middle of the tetrachord; it is described as diatonic (or
ditonic). This genus is composed of a tone followed by a tone,
and of a _remainder_ (limma), which wrongly is called a
"demitone," and which is not consonant. The dissonance of this
remainder interval is however mitigated by the richness of
sonority of the two intervals of the tone, and also by the
selfsame nature of these two intervals. They belong, indeed, to
the series of intervals whose denominators are a number
twice even [or equal?].
In other words, the simplicity of the genus as a whole, "which is
compromised of two intervals of 9:8," may make the more complex step
of 256:243 (90 cents) acceptable. He explains, ibid. at 149:
The ear has little by little habituated itself to the remainder
interval. It may happen, perhaps, that this is not a genus of
which the consonance of the complementary interval should be
doubtful, and so the ear accepts it since it accepts the
diatonic. We have sufficiently informed the reader to permit an
understanding of why [this genus] has been pleasant to adopt. The
numeric expression for this genus of fourth, which is comprised
of two intervals of 9:8, is the following:
[Illustration from d'Erlanger, ratios and cents for notes added]
204 204 90
9:8 9:8 256:243
324 288 256 243
1/1 9/8 81/64 4/3
0 204 408 498
Thus is established the principle that if a tetrachord includes two
pleasing superparticular steps, felicitously arranged, then a more
complex or "remainder" interval will not offend the ear. He concludes
his discussion of the Pythagorean Diatonic, or Ditonic Diatonic of
Ptolemy, by addressing a problem raised also by Ptolemy: a rational
ratio which may approximate a true "demitone," or half of a 9:8 tone.
The ratio of the remainder interval is then 256:243. If we seek a
number which with 256 gives us the ratio for the demitone, we
find 241 [256:241, 104.5 cents], or yet 240 [256:240 or 16:15,
111.7 cents], makes up a larger half of a tone. These numbers are
both smaller than that which, with 256, constitutes the ratio of
the remainder interval. The remainder interval is thus smaller
than a demitone. (Ibid. at 149)
Ptolemy, faced with the same problem, estimated the demitone at around
258/243 or 86/81 (103.7 cents), as compared to the actual and
irrational value at 101.955 cents.
---------------------------------------------------------------
3.2. An interlude: 10:9-9:8-16:15 (Ptolemy's Syntonic Diatonic)
---------------------------------------------------------------
Ibn Sina now moves to a tetrachord where a 9:8 tone is complemented by
a small tone at 10:9 (182.4 cents), here interestingly placed so that
the 10:9 step is lowest, ibid. at 149:
When the interval of a tone [9:8] is followed by that coming
immediately after it [in the series of superparticular ratios],
which would be 10:9, the complement in the tetrachord will have a
ratio of 16:15. The intervals so obtained are truly consonant.
The numeric expression of this genus is the following:
[Illustration from d'Erlanger, ratios for notes added]
182 204 112
10:9 9:8 16:15
20 18 16 15
1/1 10/9 5/4 4/3
0 182 386 498
In modern Arab terms, at least, this is not a Zalzalian tetrachord;
but for the Systematist theorist Qutb al-Din al-Shirazi (1236-1311),
the permutation 9:8-10:9-16:15 does define the Rast genus, for which
he gives the monochord ratios 180:160:144:135 (204-182-112 cents), see
Wright (1978, ). Modern Turkish theory likewise favors this tuning of
Maqam Rast, and follows Safi al-Din (who used the Pythagorean
diminished third or double limma at 65536:59049 or 180.45 cents, and
the apotome at 2187:2048 or 113.7 cents, as equivalents for the
simpler 10:9 and 16:15) and Qutb al-Din in considering 10:9 and 16:15
as _mujannab_ or "middle" steps, with sizes somewhere between those of
the 9:8 tone and 256:243 limma.
-------------------------------------------
3.3. Two Zalzalian tetrachords of al-Farabi
-------------------------------------------
Without mentioning al-Farabi by name, Ibn Sina next presents his
predecessor's two Zalzalian tetrachords which in modern terms are
forms of Rast, the second better known because it appears also in
al-Farabi's famous `oud tuning with wZ at 27/22 (354.5 cents). He
first, however, considers the tetrachord with a higher neutral third
at 99/80 (368.9 cents), ibid. at 149.
When the interval of a tone is followed by an interval of 11:10,
the intervals of the genre obtained will not be consonant. The
ratio of the complementary interval will be that of 320:297, very
close to 14:13 [at 129.1 and 128.3 cents respectively]. We have
already shown what one ought to think of such a ratio.
[Illustrations added -- M.S.]
204 165 129
9:8 11:10 320:297
396 352 320 297
1/1 9/8 99/80 4/3
0 204 369 498
Ibn Sina observes that "the intervals of the genre obtained will not
be consonant" -- evidently referring to the complex remainder interval
of 320:297 or 129.1 cents -- and adds that "[we] have already shown
what one ought to think of such a ratio." One possible reading is
that, although Ibn Sina does not himself adopt this tetrachord,
neither does he necessarily consider it unacceptable -- given his
previous discussion of 9:8-9:8-256:243, where he explains that a
complex remainder interval need not prevent a genus from pleasing the
ear.
Of special interest is Ibn Sina's observation that al-Farabi's
remainder interval of 320:297 is "very close to 14:13," a difference
in fact equal to the small comma of 2080:2079 (0.833 cents). This is
the amount by which the 26/21 neutral third (369.7 cents) in Ibn
Sina's jins of 8:7-13:12-14:13 is larger than al-Farabi's 99/80 (368.9
cents).
Note that a tetrachord of 9:8-11:10-14:13 (204-165-128 cents), with
all superparticular steps, would produce a fourth slightly narrow of
4/3 by this same small comma, at 693/520 (497.2 cents). However, to
obtain a division with a tone plus a large neutral step where all
intervals are superparticular and the fourth is at a just 4/3, Ibn
Sina's 8:7 tone is necessary rather than al-Farabi's 9:8.
Ibn Sina now presents al-Farabi's famous Zalzalian tetrachord still
closely approximated in many Arab tunings of Rast, ibid. at 149-150:
When the interval joined to the tone is of the ratio 12:11, the
ratio of the complementary interval will be 88:81, which
approximates 13:12 [respectively 143.5 and 138.6 cents]. We have
already shown a similar case.
[Illustrations added -- M.S.]
204 151 143
9:8 12:11 88:81
108 96 88 81
1/1 9/8 27/22 4/3
0 204 355 498
This is al-Farabi's jins with the famous wusta Zalzal or wZ at 27/22.
Ibn Sina notes that the "complementary" or remainder interval at 88:81
(143.5 cents) "approximates 13:12" (138.6 cents); the difference is
352:351, or 4.9 cents. This is the amount by which 27/22 (354.5 cents)
is smaller than Ibn Sina's 16/13 (359.5 cents) in his 8:7-14:13-13:12
jins.
It would be possible to devise a tetrachord where the two neutral
steps are at 12:11 and 13:12, but this would require compromising or
altering one of the basic definitive intervals of medieval Near
Eastern theory: either the 9:8 tone or the 4/3 fourth. One possibility
is 44:39-12:11-13:12 (209-151-139 cents), or 176:156:143:132, where
the fourth is pure, but the 44:39 tone (208.8 cents) is larger by
352:351 than 9:8. Another is 9:8-12:11-13:12 (204-151-139 cents),
where all steps are superparticular but the fourth at 117/88 (493.1
cents) is narrow by the same comma.
-------------------------------------
3.4. Ibn Sina's Zalzalian tetrachords
-------------------------------------
Having presented al-Farabi's two Zalzalian tetrachords, not
specifically recommending them but arguably implying that their
"complementary" or nonsuperparticular intervals may be acceptable, he
turns to the jins which, in another permutation, will form the basis
for the placement at wZ at 39/32 (342.5 cents) in his `oud tuning.
See ibid. at 150:
When the interval joined to the tone has for its ratio 13:12, the
ratio which completes the fourth will not be consonant, but its
ratio [128:117] much resembles 12:11 [respectively 155.6 and
150.6 cents]. This genus is in favor; we give its numeric
expression:
[Illustration from d'Erlanger, ratios for notes added]
139 204 156
13:12 9:8 128:117
468 432 384 351
1/1 13/12 39/32 4/3
0 139 342 498
In this tetrachord, unlike the others, the lowest interval is the
superparticular neutral step at 13:12 (138.6 cents), followed by the
9:8 tone, and finally by the complementary interval at 128:117 (155.6
cents). Ibn Sina points out that this last interval "will not be
consonant" in itself, but notes that "[t]his genus is in favor" --
statements by no means contradictory, as he has explained in
discussing the Ditonic Diatonic with its 256:243 step.
In observing that 128:117 "much resembles 12:11" (150.6 cents), he
again calls our attention to the 352:351 comma earlier noted in the
discussion of al-Farabi's second jins with 12:11 and 88:81. In each
jins, the comma is the difference between the 32:27 minor third (at
294.1 cents) to be divided between the two neutral steps, and the
slightly smaller 13:11 (289.2 cents) which would result from having
both neutral steps superparticular at 13:12 and 12:11. In al-Farabi's
jins, 12:11 is just, with 88:81 thus larger than 13:12 by the comma;
in Ibn Sina's jins, 13:12 is just while 128:117 is a comma larger than
the 12:11 ratio which it "much resembles."
The 13:12-9:8-128:117 arrangement of this jins might serve as an ideal
form of the tetrachord leading up to the final in the Persian Avaz-e
Bayat-e Esfahan, where the intonation as described by Hormoz Farhat
might be around 6-9-7 commas (or, in Farhat's pragmatic approach,
something like 135-205-160 cents). See Farhat (199 , ). This
permutation, like that used in his `oud tuning at 9:8-13:12-128:117,
has a 39/32 neutral third at 342 cents, with the large neutral second
at 128:117 or 156 cents as the highest step of the tetrachord.
Then follows his second Zalzalian diatonic jins with 9:8, ibid. at
150.
If the interval joined to the tone is the smallest of the emmeles
[intervals apt for melody], the remaining interval will have for
its ratio 208:189. The genus is numbered:
[Illustration from d'Erlanger, ratios for notes added]
204 128 166
9:8 14:13 208:189
252 224 208 189
1/1 9/8 63/52 4/3
0 204 332 498
The complementary interval has here a ratio which approximates,
but rather distantly, 10:9 [respectively 165.8 and 182.4 cents];
it has had {? little significance in music ? -- I am not sure of
correct translation}.
In Ibn Sina's system, as already mentioned, 14:13 (128.3 cents) is the
smallest of the emmeles in the "middle" category where twice the
interval is equal to the greater part of a fourth. From a modern
perspective, it may be considered the smallest of the superparticular
neutral or Zalzalian seconds, although in later medieval theory and
modern Turkish theory, 15:14 (119.4 cents) and 16:15 (111.7 cents) are
also regarded as "middle" intervals, with 16:15 (or its Pythagorean
near-equivalent 2187:2048) as the upper step of Rast. In Persian,
Turkish, and certain flavors of Arab music, steps of around 14:13 (say
125-130 cents) seem rather common.
As in Ibn Sina's last tetrachord, the smaller neutral step precedes
the larger, here 14:13 and 208:189 (165.8 cents). The small neutral
third is at 63/52 (332.2 cents), which might be described as a low
Mustaqim, Ibn Sina's name for the mode featuring 9:8-13:12-128:117
(204-139-156 cents), literally in Arabic the "right, correct, usual"
mode -- equivalent to the later Persian term Rast. Possibly it might
be found in modern Persian practice as a tuning of Dastgah-e Afshari,
or Gushe-ye Shekaste of Dastgah-e Mahur, with a very low neutral
third; I much use it in my own peripheral practice.
In discussing the "complementary interval" of this tetrachord, the
large neutral second at 208:189 (166 cents), Ibn Sina presents us with
a surprise. Earlier, in discussing al-Farabi's 9:8-11:10-320:297
(204-151-129 cents), he observed that 320:297 is "very close" to 14:13
(128 cents) -- the difference being the 2080:2079 comma (0.8 cents).
We might likewise expect him to compare 208:189 to al-Farabi's almost
identical large neutral second at 11:10 (165.0 cents), narrower by
this same comma.
However, he finds instead that 208:189 "approximates, but rather
distantly, 10:9" -- a difference of 165.8 vs. 182.4 cents, or 105/104
(16.567 cents). Possibly this comparison reflects the kind of response
I got from one Turkish musician, who suggested to me that a tempered
tetrachord such as 207-127-163 cents (334-cent neutral third) might
represent, not so much a low Mustaqim, as a high Nihavend, a maqam for
which he generally favored a third around 6/5 (9:8-16:15-10:9), but
with the very small neutral third a pleasant variation. Ibn Sina's
statement thus may represent, not a slip of the pen (10:9 where 11:10
was meant), but a kind of categorical perception that a very small
neutral third may serve as an extra-large minor or "supraminor" third,
and thus the upper step completing the fourth as in effect a very
small tone, somewhat like 10:9.
--------------------------------------------------
4. Ibn Sina's comments on musicians and intonation
--------------------------------------------------
Having surveyed his two "most noble" Zalzalian ajnas with steps of
8:7-13:12-14:13 (with the upper steps arranged in either order), the
tetrachords of al-Farabi where 9:8 is followed by a larger neutral
second at 11:10 or 12:11, and his own ajnas where 9:8 is preceded by
13:12 (a genus "in favor") or followed by 14:13, he turns to the
intonational practices of some of his contemporaries, ibid. at 150.
The musicians of our day confuse the complementary intervals, the
intervals of relaxation, and the smallest intervals in the series
of large emmeles [ranging from 5:4 to 14:13]; they play the one
for the other, without perceiving the differences by which they
are constituted. Thus, they use indifferently a tone augmented by
the interval of 13:12 or 14:13.
These remarks about the frequent imprecision of intonation around a
millennium ago include a possible hint as to Persian tastes in the
earlier 11th century: "they use indifferenctly a tone augmented by the
interval of 13:12 or 14:13." Ibn Sina does not mention the larger
superparticular steps favored in the Zalzalian diatonic tetrachords of
al-Farabi at 11:10 and 12:11, the latter, of course, the basis for his
mode of Zalzal (9:8-12:11-88:81) and his `oud tuning with wZ at 27/22.
The discussion of 13:12 and 14:13, incidentally, brings into play
another comma: their difference of 169:168 (10.274 cents).
His next comment moves from the confusion of 13:12 and 14:13 to the
tuning of the `oud, ibid. at 150:
When they set the place of the middle finger of Zalzal, some
indeed fix the fret higher, and others lower, and certain people
[set it] halfway between the index finger [9/8] and ring finger
[4/3], as will be seen below.
This description tells us, first, that tastes vary in placing wZ, so
that there is not a single generally accepted algorithm, although
"certain people" favor the aliquot division of 72:64:59:54 or
9:8-64:59:54 (204-141-153 cents) which places wZ at 72/59 or 344.7
cents, just a tad higher than Ibn Sina's 39/32 (342.5 cents) -- a
comma of 768:767 (2.256 cents).
Taken together, these comments tell us that people indifferently use
"a tone augmented by the interval of 13:12 or 14:13" -- that is, a
smaller neutral third of 39/32 or 63/52, as in his own two Zalzalian
diatonic tetrachords with 9:8; and that some place wZ higher or lower,
but with 72/59 or 345 cents as one practice, where a tone presumably
around 9:8 would be followed by a step (64:59, 140.8 cents) very
slightly larger than 13:12.
Thus suggests the hypothesis that Ibn Sina -- who is well familiar
with al-Farabi's tetrachords joining 9:8 with 11:10 or 12:11, as well
as his own "very noble" Zalzalian ajnas where 8:7 is followed by a
high neutral third at 26/21 or 16/13 -- has made an artistic choice in
favor of a lower neutral third, and more specifically a lower wZ, when
9:8 is the tone rather than 8:7. The fact that his contemporaries vary
in the placement of wZ, but evidently often lean to a lower position
such as 72/59 -- and that their reported confusion involves 14:13 and
13:12, rather than al-Farabi's 11:10 and 12:11 -- suggests that Ibn
Sina and his Persian contemporaries may have a different preference
than either al-Farabi or the many modern Arab musicians likewise
favoring a higher wZ somewhere between around 27/22 (355 cents) and
99/80 (369 cents), as is illustrated by the range of Syrian
practices.
One factor influencing al-Farabi's placement of wZ at 27/22 may be the
tuning of two other frets that makes it possible to find this interval
by an aliquot division:
wZ
355
27/22
|------------------|----|----|
81 68 66 64
1/1 81/68 81/64
0 303 408
wF ditone
Here wZ may be placed very conveniently midway between the wusta Fars
or "Persian third" at 81/68 (302.9 cents) and the Pythagorean major
third or ditone at 81/64 (407.8 cents) -- giving it a location at
27/22 or 354.5 cents.
The absence of wF in the `oud tuning described by Ibn Sina might make
this placement less obvious or convenient -- but he is fully aware of
the 9:8-12:11-88:81 tetrachord of al-Farabi, and nevertheless
expresses a preference for a smaller neutral second step of 13:12, and
thus a lower position for wZ at 39/32. Since he mentions how some of
his contemporaries use 14:13 and 13:12 interchangeably -- the former
not obviously so close to the 64:59 step after 9:8 that results from
the 72:64:59:54 procedure -- it appears that there is considerable
variation in intonation, but generally tending to smaller neutral
second intervals and neutral third steps in tetrachords with 9:8.
The closing portion of this discussion on intonation in practice might
tend to confirm this general impression, ibid. at 150:
They confuse also the intervals of the remainder (limma) and that
which separates the two frets of the middle finger, a quartertone
[Ibn Sina's favored scheme places these frets at 32/27 and 39/32,
a difference of 1053:1024 or 48.3 cents], and they play
indifferently the one for the other. It is not, however,
impossible to meet artists whose ears are quite refined in
distinguishing these differences.
If we assume that Ibn Sina's contemporaries likewise placed the minor
third fret at around 32/27 (294 cents), then a "quartertone" up to wZ
would imply a placement at somewhere around 39/32 (1053:1024, 48.3
cents); or 72/59 (243:236, 50.6 cents); or possible 11/9 (33:32, 53.3
cents).
In contrast, a higher placement such as 16/13 (27:26, 65.3 cents) or
26/21 (117:112, 75.6 cents) would produce more of a thirdtone,
comparable to the 28:27 of Archytas (63.0 cents) and indeed somewhat
larger. This may be another clue that Ibn Sina and his contemporaries
generally favor a lower wZ, and smaller neutral steps following 9:8,
rather than al-Farabi's 11:10 or 12:11.
-----------------------------------------
5. Ibn Sina's placement of wZ on the `oud
-----------------------------------------
Having established a context by surveying Ibn Sina's approach to
Zalzalian diatonic tetrachords and his description of contemporary
intonational practices -- often marked by the kind of imprecision or
confusion we see alleged by European theorists when dealing with other
aspects of practice -- let us now consider his statement about the
placement of wZ, ibid. at 235:
The moderns have fixed another fret for the medius, about halfway
between the index and the auricular. Some place it lower, others
higher, obtaining therefore diverse genera of the fourth. But in
our day they do not distinguish these differences.
As in the general discussion on contemporary practice, Ibn Sina notes
the popularity of a position "about halfway between the index [9/8]
and the auricular [4/3]," with some placing it lower, others higher.
The aliquot division was evidently popular, because Safi al-Din both
reports it as a much-favored choice some two centuries later, and
makes the resulting Rast tetrachord of 9:8-64:59:59:54 one of his
principal ajnas -- almost identical to Ibn Sina's Mustaqim.
The greater part do this so that the interval between the index
and medius is 13/12. The approximate interval between the medius
and the auricular will thus be 12/11, and the real interval
128/117, which permits the formation of all of the genres/genera
we have cited. (Ibid. at 235)
Here Ibn Sina, who himself has found a jins of 13:12-9:8-128:117 "in
favor," reports that the permutation 9:8-13:12-128:117 is the basis
for for the practice of "[t]he greater part." He then repeats his
point made in the tetrachord discussion that 128:117 "much resembles"
the simpler superparticular step of 12:11 -- the latter and tidier
ratio being the "approximate interval," and 128:117 "the real
interval" found between the wZ at 39/32 and the 4/3 fourth.
While his language about the aliquot division indicates that some
musicians indeed use 72/59 as the placement of wZ, I do not see tha
language about an "approximate interval" of 12:11 between wZ and the
4/3 fourth as indicating a practical tuning with neutral steps at
13:12 followed by 12:11 -- as excited as I might be by such a
practical use in the 11th century of 13:11, the basis of my favorite
maqam tunings! It may be helpful to compare the 72/59 aliquot tuning
later reported as prevalent by Safi al-Din, Ibn Sina's 39/32 tuning he
reports is favored by the "greater part" of musicians, and an
interpretation of a tuning literally implementing neutral second steps
of 13:12 and 12:11 by tuning down from the 4/3 fret -- something I
intuit would be unlikely in practice, except perhaps by indirection.
Aliquot tuning -- wZ midway between 9/8 and 4/3 (72/59, 344.7 cents)
204 141 153
9:8 64:59 59:54
0 204 345 498
1/1 9/8 72/59 4/3
72 64 59 54
Tuning of "greater part" -- wZ at 9:8 + 13:12 (39/32, 342.5 cents)
204 139 156
9:8 13:12 128:117
1/1 9/8 39/32 4/3
0 204 342 498
468 416 384 351
Hypothetical tuning with upper 13:12 + 12:11 (11/9, 347.4 cents)
204 139 151
44:39 13:12 12:11
0 209 347 498
1/1 44/39 11/9 4/3
44 39 36 33
The hypothetical 44:39-13:12-12:11 tuning actually has simplest string
ratios at 44:39:36:33. However, it requires making the traditional
tone at 9/8 larger by 352:351 -- the comma which Ibn Sina addressed,
although without specifying its ratio, in his discussion of
tetrachords -- so as to arrive at 44:39 (208.8 cents), and an upper
minor third at 13:11 rather than 32:27, which neatly divides 13:12:11
(139-151 cents).
Such a tuning might happen by serendipity or creative indirection
either on an 11th-century `oud or on a modern Persian tar, setar, or
santur -- in the modern setting, at least, we know through actual
measurements that tones a bit larger than 9/8, minor thirds at around
13/11 rather than the slightly larger 32/27, and also fourths slightly
smaller than 4/3 are all within the usual range of variation. However,
Ibn Sina himself seems simply to be following his bent for often
noting how a complex "complementary interval" or remainder such as
128:117 may "much resemble" a nearby superparticular ratio such as
13:12. The idea of actually placing the usual 9/8 fret a few cents
higher at 44/39 does not seem to be part of his practical agenda,
although, given the variations of intonational practice he reports, it
might well happened at various points during the centuries when both
he and Safi al-Din report tunings close to 44:39:36:33 much in favor.
-------------------------------------------
6. Summary and some tetrachord permutations
-------------------------------------------
Ibn Sina presents to us a rich variety of Zalzalian diatonic
tetrachords, two "very noble" ones using 8:7-13:12-14:13 (upper steps
in either order), derived from the mathematically elegant and
musically beautiful neo-Ptolemaic division 16:14:13:12. These
tetrachords may be described in later terms as forms of "septimal
Rast" with large or bright neutral thirds at 26/21 (370 cents) or
16/13 (359 cents).
He then, after briefly addressing the Ditonic Ditonic at
9:8-9:8-256:243 as an illustration of how a "dissonant" or
nonsuperparticular step such as 256:243 can nevertheless be part of a
pleasing jins, and noting 10:9-9:8-16:15 with all superparticular
ratios, turns to Zalzalian diatonic tetrachords with a 9:8 tone.
He first describes al-Farabi's 9:8-11:10-320:297 and the better-known
9:8-12:11-88:81 with its neutral third at 27/22, not finding these
tunings in favor, but possibly implying that, as with the Ditonic
Diatonic, the complex "remainder" intervals need not make these ajnas
unpleasant.
Then, he addresses a jins "in favor" with 9:8 -- 13:12-9:8-128:117,
a fine example of what might today might be called the "Old Esfahan"
in Persian music with a rather low neutral second, here 39/32 or 342
cents. The permutation of this at 9:8-13:12-128:117, likewise with the
smaller neutral step preceding the larger, serves as the basis for his
`oud tuning, which he reports the "greater part" of musicians use. He
also describes a jins with the smallest "large emmele," also in modern
terms the smallest superparticular neutral third, at 14:13 -- the jins
9:8-14:13-208:189, with a small neutral third at 63/52 or 332 cents.
Ibn Sina is very interested in resemblances between complex
nonsuperparticular intervals which inevitably arise when dividing a
32:27 minor third (4:3 less 9:8) into two neutral steps, and the small
commas involved -- thus al-Farabi's 320:297 as near 14:13, and 88:81
as near 13:12, as well as Ibn Sina's 128:117 as near 12:11. The first
resemblance involves the 2080:2079 comma at 0.8 cents, while the other
two bring into play the 352:351 comma at 4.9 cents.
The discussion is especially fascinating because it presents the idea
of coupling 9:8 with each of the superparticular neutral seconds: thus
four shadings at 11:10, 12:11, 13:12, and 14:13 (165, 151, 139, and
128 cents), producing neutral thirds at 99/80, 27/22, 39/32, and 63/52
(369, 355, 342, and 332 cents).
Turning to the real-world intonational practices of his
contemporaries, he describes how these musicians often confuse or use
indiscriminately the ratios of 13:12 and 14:13, more specifically when
a tone is "augmented" by one of these intervals, evidently resulting
in a neutral or Zalzalian third at 39/32 or 63/52 if a 9:8 tone is
meant. Given his "most noble" jins with an 8:7 tone followed by 13:12
and 14:13 in either order, however, this observation could also refer
to the interchangeable and sometimes indiscriminate use of the two
forms of this jins also, with a Zalzalian third at 26/21 (370 cents)
or 16/13 (359 cents).
What Ibn Sina does mention in this discussion of actual practice is
joining a tone directly to a larger neutral step at 11:10 (165 cents)
or 12:11 (151 cents), as happens with 9:8 in al-Farabi's two Zalzalian
diatonic ajnas. Rather, in his own tetrachords, Ibn Sina always
joins a tone (at 8:7 or 9:8) to a neutral step at 13:12 or 14:13, with
any large neutral step (as in 9:8-13:12-128:117 or 9:8-14:13-208:189)
relegated to the "complementary interval" completing the fourth.
His comments in this discussion on practice describe how different
people tune wZ on the `oud lower or higher, but often it is at or in
the vicinity of 72/59 (345 cents), or midway between 9/8 and 4/3
(72:64:59:54), a placement later reported as much favored by Safi
al-Din, who adopts the 9:8-64:59-59:54 tetrachord as one of his
principal ajnas. A reference to the 32/27 and wZ frets as a
"quartertone" apart (1053:1024 or 48 cents with Ibn Sina's 39/32
placement (342 cents), or 243:236 or 51 cents with the 72/59 placement
he reports some people use) tends to confirm an impression that a
relative low placement of wZ generally prevails. If it were around
16/13 or 26/21, say, as in the two version of his jins with 8:7, then
the 32/27 and wZ frets would be some kind of "thirdtone" apart, rather
than the "quartertone" that Ibn Sina describes.
His remarks focusing specifically on the placement of wZ again note
that a placement at or near 72/59 is common, with "the greater part"
of musicians choosing 39/32 (9:8 + 13:12). As in his discussion of the
ajnas, where he addressed 13:12-9:8-128:117, he notes that the
complementary and complex interval of 128:117 (156 cents) is an
"approximate" 12:11 (151 cents) -- a difference we know as the 352:351
comma at 4.9 cents. Thus in his 9:8-13:12-128:117 jins (204-139-156
cents) on the `oud, the basis of his Mustaqim modality, he says that
wZ at 39/32 will be 13:12 from the 9/8 fret, an an "approximate" 12:11
but a "real" 128:117 from the 4/3 fret.
Ibn Sina, therefore, does not himself appear to be proposing or
documenting a tetrachord literally at 44:39:36:33 or 44:39-13:12-12:11
(209-139-151 cents), with a 44/39 fret higher than the usual 9/8 by a
352:351 comma, leaving an upper 13/11 minor third (289 cents) smaller
than 32/27 (294 cents) by this same comma, and permitting a neat
aliquot division of this third into 13:12:11. Nor does he appear to be
suggesting some kind of temperament where 352:351 is distributed
between two or more intervals, e.g. 207-139-150 cents (tone slightly
wider than 9:8 but smaller than 44:39, fourth slightly narrow, and
13:12 and 12:11 very close to just). However, his discussion of how
128:117 resembles 12:11, and likewise with al-Farabi's 88:81 and the
superparticular ratio of 13:12, certainly sets the stage for divisions
such as 44:39:36:33 and their tempered variations.
------------------
In surveying the Zalzalian diatonic tetrachords which Ibn Sina
addresses, including the two from al-Farabi, we may find it helpful to
look at all six permutations of each tetrachord (as later recommended
by Safi al-Din), and to do so in a way that identifies what Ibn Sina
might call "resemblances" between forms differing only by the small
commas which he notes (specifically 2080:2079 and 352:351).
Although not listed in his discussion of tetrachords, the aliquot
division he reports as sometimes favored on the `oud, 72:64:59:54, is
also of interest. This tetrachord, with its neutral third at 72/59,
brings into play the additional commas of 768:767 (2.256 cents) and
649:648 (2.670 cents) -- the amounts by which 72/59 (344.7 cents) is
larger than Ibn Sina's wZ at 39/32 (342.5 cents), but smaller than
11/9 (347.4 cents), the neutral third in a permutation of al-Farabi's
famous 9:8-12:11-88:81 to 9:8-88:81-12:11 (204-143-151 cents).
For each permutation, I give a description both in terms of the later
Systematist classification of Safi al-Din and his followers, with T as
a tone (e.g. 9:8 or 8:7), and J as some kind of neutral or "middle"
second step; and in terms of maqam/dastgah categories.
For Ibn Sina's two forms of the jins with steps of 8:7-13:12-14:13, I
make a comparison with al-Farabi's two Zalzalian diatonic tetrachords
in order to note the comparable sizes of neutral thirds, but produced
in al-Farabi's approach by joining a 9:8 tone to a large neutral step
at 11:10 or 12:11, i.e. 9:8-11:10-320:297 or 9:8-12:11-88:81.
The table compares ajnas of Ibn Sina (IS) and al-Farabi (AF).
The arrangement generally follows the Systematist classification of
tetrachords, as presented by Safi al-Din, whose seven types of
tetrachords include three types relevant to this survey IV, V, VI).
with T (tanini) as a tone; B (bakiye) as a limma; and J (mujannab) as
a neutral or Zalzalian second.
I. T T B (current Arab `Ajam or Persian Mahur, e.g. 9:8-9:8-256:243)
II. T B T (current Arab Nahawand or Persian Nava, e.g 9:8-256:243-9:8)
III. B T T (current Arab or Turkish Kurdi, e.g. 256:243-9:8-9:8)
IV. T J J (current Arab or Turkish Rast, e.g. 9:8-12:11-88:81)
V. J J T (current Arab Bayyati or Persian Shur, e.g. 13:12-128:117-9:8)
VI. J T J (Buzurg, current Persian Segah or Esfahan, e.g. 13:12-8:7-14:13)
VII. J J J B (Systematist Isfahan, e.g. modern 13:12-12:11-14:13-22:21)
Each complete set of permutations takes the order of the two forms of
IV, then V, and then VI. Note that ajnas 1, 4, and 5 are Ibn Sina's
own forms; ajnas 2 and 3 are those he addressed from al-Farabi; and
jins 6 is the `oud tuning of 72:64:59:54 which he mentions, and Safi
al-Din later adopts as one of his principal ajnas.
1. Ibn Sina's "most noble" jins (8:7, 13:12, 14:13)
(Thirds of 1.1 and 2.1 differ by 2080:2079 or 0.833 cents;
Thirds of 1.2 and 3.1 differ by 352:351 or 4.925 cents)
IS gives 1.1, 1.2 AF gives 2.1, 3.1
1.1. Systematist TJJ 2.1 Systematist TJJ
"Septimal Rast, 26/21" "High Rast"
(26/21 third, 369.7c) (99/80 third, 368.9c)
104 91 84 78 396 352 320 297
1/1 8/7 26/21 4/3 1/1 9/8 99/80 4/3
0 231 370 498 0 204 369 498
8:7 13:12 14:13 9:8 11:10 320:297
231 139 128 204 165 129
1.2. Systematist TJJ 3.1 Systematist TJJ
"Septimal Rast, 16/13" "Medium-high Rast"
(16/13 third, 359.5c) (27/22 third, 354.5c)
16 14 13 12 108 96 88 81
1/1 8/7 16/13 4/3 1/1 9/8 27/22 4/3
0 231 359 498 0 204 355 498
8:7 14:13 13:12 9:8 12:11 88:81
231 128 139 204 151 143
1.3. Systematist JJT No comparison made
"Higher septimal Shur,
Bayyati, or Ushshak"
364 336 312 273
1/1 13/12 7/6 4/3
0 128 267 498
14:13 13:12 8:7
128 139 231
1.4. Systematist JJT
"Lower septimal Shur, No comparison made
Bayyati, or Ushshak"
28 26 24 21
1/1 14/13 7/6 4/3
0 128 267 498
14:13 13:12 8:7
128 139 231
1.5. Systematist JTJ No comparison made
"Higher Buzurg, Avaz-e
Bayat-e Esfahan, or
Byzantine Soft Chromatic,
8-13-7 steps of 68 in
system of Chrysanthos of
Madytos"
52 48 42 39
1/1 13/12 26/21 4/3
0 139 370 498
13:12 8:7 14:13
139 232 128
1.6. Systematist JTJ No comparison made
"Higher Buzurg, Avaz-e
Bayat-e Esfahan, or
Byzantine Soft Chromatic,
8-13-7 steps of 68 in
system of Chrysanthos of
Madytos"
112 104 91 84
1/1 14/13 16/13 4/3
0 139 370 498
13:12 8:7 14:13
139 232 128
------------------------------------------------
4. Ibn Sina's jins "in favor" (9:8, 13:12, 128:117)
3. Al-Farabi's 2nd cited jins (9:8, 12:11, 88:81)
(Differences of 352:351 or 4.925 cents)
IS gives 4.1, 4.5 AF gives 3.1
4.1. Systematist TJJ 3.2. Systematist TJJ
"Higher Mustaqim, "Higher Mustaqim or Arab
Dastgah-e Afshari, Rast Jadid, 9-6-7 commas
Gushe-ye Shekaste" (cf. Rast, 9-7-6 commas)"
468 416 384 351 396 352 324 297
1/1 9/8 39/32 4/3 1/1 9/8 11/9 4/3
0 204 342 498 0 204 347 498
9:8 13:12 128:117 9:8 88:81 12:11
204 139 156 204 143 151
4.2. Systematist TJJ 3.1. Arab Rast, Byzantine
"Medium-high Arab Rast, Diatonic (Chrysanthos)
Low Turkish Rast" 9-7-6 commas
144 128 117 108 108 96 88 81
1/1 9/8 16/13 4/3 1/1 9/8 27/22 4/3
0 204 359 498 0 204 355 498
9:8 128:117 13:12 9:8 12:11 88:81
204 156 139 204 151 143
4.3. Systematist JJT 3.4. Systematist JJT
"Moderate Shur, Arab "Moderate Arab Bayyati,
Bayyati, Turkish Ushshak" 6-7-9 commas"
416 384 351 312 352 324 297 264
1/1 13/12 32/27 4/3 1/1 88/81 32/27 4/3
0 139 294 498 0 143 294 498
13:12 128:117 9:8 88:81 12:11 9:8
139 156 204 143 151 204
4.4. Systematist JJT 3.3. Systematist JJT
"Moderate Arab Huseyni, "Moderate Arab Huseyni,
Low Turkish Huseyni, 7-6-9 commas"
128 117 108 96 96 88 81 72
1/1 128/117 32/27 4/3 1/1 12/11 32/27 4/3
0 156 294 498 0 151 294 498
128:117 13:12 9:8 12:11 88:81 9:8
156 139 204 151 143 204
4.5. Systematist JTJ 3.6. Systematist JTJ
"Persian Old Esfahan" "Arab `Iraq, in theory,
6-9-7 commas"
468 432 384 351 88 81 72 66
1/1 13/12 39/32 4/3 1/1 88/81 11/9 4/3
0 139 342 498 0 143 347 498
13:12 9:8 128:117 88:81 9:8 12:11
139 204 156 143 204 151
4.6. Systematist JTJ 4.5. Systematist JTJ
"Medium Persian Esfahan" "Medium Persian Esfahan,
7-9-6 commas"
128 117 104 96 108 99 88 81
1/1 128/117 16/13 4/3 1/1 12/11 27/22 4/3
0 156 359 498 0 151 355 498
128:117 9:8 13:12 12:11 9:8 88:81
156 204 139 151 204 143
------------------------------------------------
5. Ibn Sina's jins (9:8, 14:13, 208:189)
2. Al-Farabi's 1st cited jins (9:8, 11:10, 320:297)
(Differences of 2080:2079 or 0.833 cents)
IS gives 5.1 AF gives 2.1
5.1. Systematist TJJ 2.2. Systematist TJJ
"Low Mustaqim, Afshari, Same as 5.1
or Shekaste; possibly
High Turkish Nihavend"
252 224 208 189 360 320 297 270
1/1 9/8 63/52 4/3 1/1 9/8 40/33 4/3
0 204 332 498 0 204 333 498
9:8 14:13 208:189 9:8 320:297 11:10
204 128 166 204 129 165
5.2. Systematist TJJ 2.1. Systematist TJJ
"High Syrian Rast, or Same as 5.2
Medium Ottoman Rast"
468 416 378 351 396 352 320 297
1/1 9/8 26/21 4/3 1/1 9/8 99/80 4/3
0 204 370 498 0 204 369 498
9:8 208:189 14:13 9:8 11:10 320:297
204 166 128 204 165 129
5.3. Systematist JJT 2.4. Systematist JJT
"Low Shur, Lebanese Folk Same as 5.3
Bayyati, or Turkish
Ushshak"
448 416 378 336 320 297 270 240
1/1 14/13 32/27 4/3 1/1 320/297 32/27 4/3
0 128 294 498 0 129 294 498
14:13 208:189 9:8 320:297 11:10 9:8
128 166 204 129 165 204
5.4. Systematist JJT 2.3. Systematist JJT
"Turkish Huseyni or Same as 5.4
High Arab Huseyni"
416 378 351 312 352 320 297 264
1/1 208/189 32/27 4/3 1/1 11/10 32/27 4/3
0 166 294 498 0 165 294 498
208:189 14:13 9:8 11:10 320:297 9:8
166 128 204 165 129 204
5.5. Systematist JTJ 2.6. Systematist JTJ
Low Arab `Iraq Same as 5.5
(as theoretical jins),
High Turkish Segah,
Low Persian Old Esfahan
252 234 208 189 320 297 264 240
1/1 14/13 63/52 4/3 1/1 320/297 40/33 4/3
0 128 332 498 0 129 333 498
14:13 9:8 208:189 320:297 9:8 11:10
128 204 166 129 204 165
5.6. Systematist JTJ 2.5. Systematist JTJ
Possible High Same as 5.6
Persian Segah
208 189 168 156 396 360 320 297
1/1 208/189 26/21 4/3 1/1 11/10 99/80 4/3
0 166 370 498 0 165 369 498
208:189 9:8 14:13 11:10 9:8 320:297
166 204 128 165 204 129
------------------------------------------------
5. Ibn Sina's jins "in favor" (9:8, 13:12, 128:117)
6. Ibn Sina's jins "some" use (9:8, 64:59, 59:54)
2. Al-Farabi's Zalzal jins (9:8, 12:11, 88:81)
(Differences: 5-6, 768:767 or 2.256 cents;
6-2, 649:648 or 2.670 cents;
5-2, 352:351 or 4.925 cents)
4.1. Systematist TJJ 6.1 Systematist TJJ 3.2. Systematist TJJ
"Higher Mustaqim, Same as 4.1 "Higher Mustaqim or Arab
Dastgah-e Afshari, Rast Jadid, 9-6-7 commas
Gushe-ye Shekaste" (cf. Rast, 9-7-6 commas)"
468 416 384 351 72 64 59 54 396 352 324 297
1/1 9/8 39/32 4/3 1/1 9/8 72/59 4/3 1/1 9/8 11/9 4/3
0 204 342 498 0 204 345 498 0 204 347 498
9:8 13:12 128:117 9:8 64:59 59:54 9:8 88:81 12:11
204 139 156 204 141 153 204 143 151
4.2. Systematist TJJ 6.2. Systematist TJJ 3.1. Arab Rast, Byzantine
"Medium-high Arab Rast, Same as 4.2 Diatonic (Chrysanthos)
Low Turkish Rast" 9-7-6 commas
144 128 117 108 2124 1888 1724 1593 108 96 88 81
1/1 9/8 16/13 4/3 1/1 9/8 59/54 4/3 1/1 9/8 27/22 4/3
0 204 359 498 0 204 357 498 0 204 355 498
9:8 128:117 13:12 9:8 59:54 64:59 9:8 12:11 88:81
204 156 139 204 153 141 204 151 143
4.3. Systematist JJT 6.3 Systematist JJT 3.4. Systematist JJT
"Moderate Shur, Arab Same as 4.3 "Moderate Arab Bayyati,
Bayyati, Turkish 6-7-9 commas"
Ushshak"
416 384 351 312 64 59 54 48 352 324 297 264
1/1 13/12 32/27 4/3 1/1 64/59 32/27 4/3 1/1 88/81 32/27 4/3
0 139 294 498 0 141 294 498 0 143 294 498
13:12 128:117 9:8 64:59 59:54 9:8 88:81 12:11 9:8
139 156 204 141 153 204 143 151 204
4.4. Systematist JJT 6.4. Systematist JJT 3.3. Systematist JJT
"Moderate Arab Huseyni, Same as 4.4 "Moderate Arab Huseyni,
Low Turkish Huseyni, 7-6-9 commas"
128 117 108 96 1888 1724 1593 1416 96 88 81 72
1/1 128/117 32/27 4/3 1/1 59/54 32/27 4/3 1/1 12/11 32/27 4/3
0 156 294 498 0 153 294 498 0 151 294 498
128:117 13:12 9:8 59:54 64:59 9:8 12:11 88:81 9:8
156 139 204 153 141 204 151 143 204
4.5. Systematist JTJ 6.5. Systematist JTJ 3.6. Systematist JTJ
"Persian Old Esfahan" Same as 4.5 "Arab `Iraq, in theory,
6-9-7 commas"
468 432 384 351 576 531 472 432 88 81 72 66
1/1 13/12 39/32 4/3 1/1 64/59 72/59 4/3 1/1 88/81 11/9 4/3
0 139 342 498 0 141 345 498 0 143 347 498
13:12 9:8 128:117 64:59 9:8 59:54 88:81 9:8 12:11
139 204 156 141 204 153 143 204 151
4.6. Systematist JTJ 6.6. Systematist JTJ 4.5. Systematist JTJ
"Medium Persian Esfahan" Same as 4.6 "Medium Persian Esfahan,
7-9-6 commas"
128 117 104 96 236 216 192 177 108 99 88 81
1/1 128/117 16/13 4/3 1/1 59/54 59/48 4/3 1/1 12/11 27/22 4/3
0 156 359 498 0 153 357 498 0 151 355 498
128:117 9:8 13:12 59:54 9:8 64:59 12:11 9:8 88:81
156 204 139 153 204 141 151 204 143
Rough preliminary draft -- not complete
Margo Schulter
June 7, 2013