----------------------------------------
Tetrachord permutations and alliances:
Reflections on Ibn Sina and Erv Wilson
----------------------------------------
In recent discussions with Kraig Grady, I have been exposed to the
concept of viewing an octave mode or cycle, for example, as a
coalition or alliance of genera each with its own division of the
fourth or fifth, for example, in seeking out arithmetic or harmonic
series. Two different paths may lead logically to this approach, which
I will illustrate with a few examples.
One pathway is the traditional maxim of Near Eastern musics such as
the Arab, Turkish, and Kurdish variations on the system of the maqamat
or modalities (_maqamat_ is the Arab plural, singular _maqam_), and
the closely related Persian dastgah system which evidently developed
from maqam sometime around the 16th-18th centuries: "Follow the
ajnas," i.e. the genera as connected in various modalities (with
Arabic _jins_ a loanword from the Greek _genus_, and _ajnas_ the
plural form).
Another pathway is the studies of Erv Wilson focusing on various
arithmetic (or subharmonic) divisions like those of Kathleen
Schlesinger's _harmoniai_, and these divisions may be used to form
genera joined together in larger cycles such as octave modes. These
cycles are known as "diaphonic" if they join two divisions of series
(e.g. one of a 4/3 fourth, and the other of a 3/2 fifth); "triaphonic"
if they involve three such divisions (e.g. for a lower 4/3 fourth, a
middle 9/8 tone, and an upper 4/3 fourth), etc.
The same approach may be applied to harmonic divisions, or indeed to
combinations of arithmetic and harmonic divisions in the same cycle.
Others such as John Chalmers have also, like Wilson, explored this
approach to scale or modal construction which focuses on the
tetrachord, or more generally on the genus or jins, as a basic unit.
Here I will explore this Wilsonian and Chalmersian focus in
conjunction with the Near Eastern advice to "Follow the ajnas," with
one very subtle treatment of certain ratios and tetrachords being
provided by the great philosopher, physician, and music theorist Ibn
Sina (c. 980-1037), also known in Latin Europe as Avicenna. Ibn Sina
followed in the tradition of his predecessor al-Farabi (c. 870-950),
seeking out a range of tetrachords and modes and also discussing the
practical tuning of the `oud, an instrument one of whose offshoots is
the European lute.
While the ratios of al-Farabi and Ibn Sina (who addresses some of the
tetrachords of his predecessor as well as introducing his own favored
divisions) will be central to my discussion, I should also mention the
influence of the Systematist school around 1250-1300 as exemplified by
the two great theorists Safi al-Din al-Urmawi (c. 1216-1294) and Qutb
al-Din al-Shirazi (1236-1311). These authors sought to catalogue the
many octave cycles or modalities in use, and to suggest tuning ratios
for many of them, themes also pursued by their commentators over the
following two centuries or so.
An important caution is that while this discussion focuses on octave
cycles, also an important theme of theory from al-Farabi through Qutb
al-Din, actual maqam and dastgah music is rather more complex. As the
adage "Follow the ajnas" itself suggests, the focus is often on an
individual jins and where it may lead rather than a fixed "octave
scale." Further, octave equivalence or repetition is _not_ assumed:
this becomes especially clear on instruments such as the Persian
santur, with a limited number of steps which may be tuned differently
depending on the octave (e.g. a 13/12 step in the lower octave of Shur
Dastgah, one common tuning of the "textbook note" a Zalzalian second
above the final or 1/1 of Shur, but a 512/243 or the like at a minor
rather than Zalzalian ninth in the upper octave above 2/1).
Thus the Systematist approach of combining ajnas into octave cycles or
modes is only one line of investigation, and is presented here with
that modest realization.
--------------------------------------------------------------
1. Arithmetic and harmonic divisions: Two varieties of Nahuft
--------------------------------------------------------------
Here I will be focusing in the first part of this article on a modal
type known as Nahuft in the Systematist literature, or sometimes
Nahuft-Hijazi, and by the modern theorist Jacques Dudon as Ibina in
one of its specific tunings.[1]
---------------------------------
1.1. A Nahuft with a 39/32 third
---------------------------------
The first variety of Nahuft in the Zeta-24 tuning system presents
three tetrachords closely resembling a favorite intonation of Ibn
Sina, but with some variations. Here is the octave cycle, with an
indication of my keyboard mapping for those who may be curious:
`iraq mustaqim or rast 44:39
117:108:96:88 44:39:36:33 tone
|---------------------|-------------------|------|
F# G* A* B C# D* E F#
1/1 13/12 39/32 117/88 3/2 13/8 39/22 2/1
0 138.6 342.5 493.1 702.0 840.5 991.2 1200
J T J T J J T
13:12 9:8 12:11 44:39 13:12 12:11 44:39
138.6 203.9 150.6 208.8 138.6 150.6 208.8
|-------------------|
nawruz
52:48:44:39
Here the Systematist symbols T and J represent a whole tone or
_tanini_ (T) and a middle or Zalzalian second or _mujannab_ somewhere
between a usual semitone (often assumed to be 256/243 or 90.225 cents)
and a usual tone (often assumed to be 9/8 or 203.910 cents). Thus the
cycle of Nahuft is generically JTJ-TJJT, with the Systematist
theorists of the 13th-15th centuries often dividing the octave into a
lower fourth and upper fifth.
From this perspective we have three principal tetrachords: a lower
tetrachord (1/1-4/3) of `iraq or JTJ; a middle tetrachord of TJJ,
associated by Ibn Sina with a mode called Mustaqim, and by the
Systematists with the familiar name Rast (4/3-39/22); and an upper
tetrachord of JJT, sometimes known to Systematist writers as Nawruz
(or Nairuz or Nirz, with transliterations from the Arabic varying).
From a Systematist perspective, we may also think of a lower
tetrachord of `iraq or JTJ, coupled with an upper pentachord of rast
or TJJT, a rast tetrachord (TJJ) plus an upper tone.
---------------------------------------------------------
1.1.l. The lower tetrachord: A variation on the pure 4/3
---------------------------------------------------------
Here our interest is in the tuning of these principle tetrachords,
which each resemble but slightly vary from different orderings or
permutations of a tetrachord which Ibn Sina gives in two forms
involving the same step sizes. The first form he gives invites
comparison with our lower `iraq tetrachord above:
Ibn Sina tuning (13:12-9:8-128:117)
468 432 384 351
1/1 13/12 39/32 4/3
0 138.6 342.5 498.0
13:12 9:8 128:117
138.6 203.9 155.6
Zeta-24 tuning (13:12-9:8-12:11)
117 108 96 88
1/1 13/12 39/32 117/88
0 138.6 342.5 493.2
13:12 9:8 12:11
138.6 203.9 150.6
The lower three notes of the two tunings are identical, the rather
small but significant difference being in the placement of the highest
note and size of the upper interval. In Ibn Sina's 13:12-9:8-128:117,
this highest note is at a pure 4/3 fourth -- a given in classic Greek
and Near Eastern tetrachord theory, where indeed the lowest and
highest notes are taken as fixed at 1/1 and 4/3, with only the middle
two notes and the intervals they form taken as "negotiable." A result
of this situation is that while the intervals 13:12 and 9:8 are both
epimoric or superparticular ratios (i.e. taking the form n+1:n), the
upper step to fill out precisely a 4/3 fourth must be more complex,
here 128:117 or 155.562 cents.
Interestingly, Ibn Sina remarks that 128:117 resembles the simpler
ratio of 12:11 (150.637 cents) -- and in our Zeta-24 version, we have
this ratio for our upper interval, so that all three steps are
epimoric: 13:12-9:8-12:11.
However, a consequence of this "modern" tuning is that our tetrachord
has its outer notes at an interval of 117/88 or 493.120 cents --
narrow of a pure 4/3 by a small difference of 352:351 or 4.925 cents.
If the two notes of this narrow fourth are played and held
simultaneously, then a quite audible beating is heard in harmonic
timbres, letting us know that this fourth is not quite pure, but
rather "virtually tempered" or "tempered by ratio."
In modern Iranian music, fifths tempered by around 5 cents may be
rather common on instruments such as the santur; and some 11th-13th
century fretting schemes reported by al-Farabi and Safi al-Din suggest
that similarly impure fifths may have occurred in that era, also.
However, the formal tetrachord theory generally prefers to take 4/3,
when it serves as the measure of a jins boundary, as pure.
A curious axiom applying to this situation is that there is no way to
divide a pure 4/3 fourth into a precise 9:8 tone plus two mujannab (J)
steps in the range of around 14:13 (128.298 cents) to 11:10 (165.004
cents) which are both superparticular. Another way of viewing this
problem is that a 9/8 tone will leave an interval of 32/27 (294.135
cents), a regular minor third from three pure 4:3 fourths less an
octave, to be divided between the two J steps. However, 32/27 cannot
be exactly divided into two superparticular steps in this range,
although the two nearest approximations indeed come rather close.
Here the relevant approximation would use steps of 13:12 and 12:11,
which add up to 13/11 -- smaller than the desired 32/27 by 352:351. As
Ibn Sina shows, if we wish 32/27 (and the outer 4/3 fourth of the
tetrachord) to be just, then there are two obvious solutions. First,
as he prefers, we may keep the steps of 9:8 and 13:12 just, and
instead of a simple 12:11 for the large J, use 128:117 at 352:351
larger. The other choice, taken earlier by al-Farabi and noted by Ibn
Sina, is instead to take 9:8 and 12:11 as pure, and for a simple 13:12
to substitute 88:81 (143.498 cents), again 352:351 wider, for the
smaller J step.
The other near-solution is a tetrachord with steps of 9:8, 11:10, and
14:13. These steps together would fall short of 4/3 by only 2080:2079
or 0.833 cents, producing a fourth at 693/520 or 497.212 cents. Given
the classic assumption that both 4/3 and 9/8 are pure, the solution is
very slightly to enlarge one of the two J steps. Ibn Sina adopts the
division 9:8-14:13-208:189, with 208:189 at 165.837 cents, larger than
the superparticular 11:10 by 2079:2078. He also discusses al-Farabi's
tetrachord of 9:8-11:10-320:297, where the smaller J step at 320:297
or 129.131 cents "very closely resembles" the simpler ratio of 14:13,
being again larger by only 2080:2079.
For purposes of this article, we are mainly concerned with the
question of tetrachords with steps at or approximating 9:8, 13:12, and
12:11.
From this point of view, our lower tetrachord 1/1-13/12-39/32-117/88
or 13:12-9:8-12:11 represents the case where all three steps are
superparticular, but the fourth is slightly narrow at 117/88.
Curiously, this compromise permits what is in some ways a simpler
division that Ibn Sina's classic 13:12-9:8-128:117. His division with
a pure 4/3 fourth has monochord ratios of 468:432:384:351, whereas the
version with all-superparticular steps has 117:108:96:88.
This is not to say that the altered "modern" version is an improvement
on the classic tuning with 4/3 pure: apart from a possible preference
for a just fourth as aurally desirable as well as intellectually
elegant, one might prefer the slightly greater contrast in the sizes
of the two J steps in Ibn Sina's version at 13/12 and 128/117, a
difference of 512/507 or 16.990 cents. In the variation with a narrow
fourth, the reduced size for the large J at 12/11 makes it differ from
the smaller 13/12 by 144:143 or 12.064 cents, a more subtle
distinction.
-------------------------------------------------------------
1.1.2. The middle mustaqim tetrachord: Altering the 9/8 tone
-------------------------------------------------------------
The middle or mustaqim tetrachord (TJJ) on the fourth step -- here a
"virtually tempered" 117/88 rather than a pure 4/3 -- also slightly
varies a classic tuning of Ibn Sina, and indeed perhaps his most
famous, since it is a basis for his recommended `oud tuning. This
tetrachord, like his previous permutation, has steps of 9:8, 13:12,
and 128:117, with the smaller 13:12 preceding the larger 128:117 --
but here with the 9:8 rather than 13:12 step first. Both of these
permutations have in common a third step at 39/32 (342.483 cents).
Ibn Sina tuning (9:8-13:12-128:117)
468 416 384 351
1/1 9/8 39/32 4/3
0 203.9 342.5 498.0
9:8 13:12 128:117
203.9 138.6 155.6
Zeta-24 tuning (44:39-13:12-12:11)
44 39 36 33
1/1 44/39 11/9 4/3
0 138.6 347.4 493.2
44:39 13:12 12:11
208.8 138.6 150.6
Ibn Sina's name Mustaqim for a mode based on this TJJ tetrachord is
Arabic for "true, regular, usual," with the later term Rast in common
use from the 13th century on having this same meaning in Persian. In
some modern uses, these two synonymous names may have intonational
implications, with Mustaqim placing the smaller J step before the
larger one, as in Ibn Sina's tetrachord, and Rast in a modern Arab or
Turkish manner placing the larger J step first.
In describing his preferred tuning for the `oud, Ibn Sina first
details the placements of the usual frets at the 4/3 fourth and 9/8
tone above the open string. Then he discusses the placement of the
fret known as the _wusta Zalzal_ or middle finger fret of the famous
`oudist Mansur Zalzal in 8th-century Baghdad (?-791), credited with
adding this fret to the instrument at some kind of a middle third
between the standard minor third at 32/27 and major third at 81/64
(407.820 cents). While the 4/3 and 9/8 frets were regarded as fixed,
the placement of the Zalzalian third was often regarded as highly
variable, with Ibn Sina noting that some place it higher, and others
lower.
Intriguingly, Ibn Sina gives his own preferred placement as 39/32, so
that this fret is precisely a 13:12 step above the 9/8 fret, and
"approximately" a 12:11 below the 4/3 fret, with this latter ratio
being more precisely 128:117. Given Ibn Sina's earlier observation on
how a 128:117 step "resembles" the simpler 12:11, such a description
of a fretting including a tetrachord at 1/1-9/8-13/12-4/3 is not
surprising.
Our modern variation of 1/1-44/39-11/9-4/3 or 44:39-13:12-12:11 shows
what would happen if one placed some of the frets in a different
order, and took Ibn Sina's approximation of 12:11 for the interval
down from the 4/3 fret to the Zalzalian fret as the literal ratio. In
this modified fretting, one might first place the 4/3 fret as usual,
then measure an interval of 12:11 down to place Zalzal's middle finger
at 11/9 or 347.408 cents, and then measure down from this fret in turn
by 13:12 to place the fret for the tone at a not-so-standard 44/39 or
208.835 cents in place of the simpler 9/8 at 203.910 cents.
Ibn Sina's fretting: 4/3 and 9/8 first, then 39/32
468 416 384 351
|---------------|--------|---------|
1/1 9/8 39/32 4/3
0 203.9 342.5 498.0
9:8 13:12 128:117
203.9 138.6 155.6
Modified fretting with 12:11 step: 4/3 first, then 11/9 and 44/39
44 39 36 33
|----------------|--------|--------|
1/1 44/39 11/9 4/3
0 208.8 342.5 498.0
44:39 13:12 12:11
208.8 138.6 150.6
Here, both tetrachords have the traditional pure 4/3 fourth, but
setting both J steps at superparticular ratios of 13:12 and 12:11
results in a T or whole-tone step at 44/39, larger than the usual 9/8
by 352:351. An additional consequence is that the wusta Zalzal is also
slightly altered from Ibn Sina's 39/32 to a position higher by this
same ratio of 352:351, 11/9. This follows since the 44/39 tone is
larger by this ratio, while the 13:12 step between it and the
Zalzalian third remains unaltered.
In 13th-century terms, these two tetrachords both with a tone (at Ibn
Sina's traditional 9/8 or a "modernistic" 44/39) and two J steps would
qualify as forms of Rast (TJJ); and both would would qualify more
specifically as Mustaqim in the sense that the first J is smaller than
the second -- at 13:12 and 128:117 (138.6-155.6 cents) or 13:12 and
12:11 (138.6-150.6 cents).
As it happens, the modernistic form of 44:39:36:33 involves a smaller
string length for the division that Ibn Sina's classic 468:416:384:351
with its ratio of 128:117 for the upper step -- but, again, this
"simplification" is simply a variation or related shading, not
necessarily an improvement. One might prefer the greater contrast
between the J steps at 13:12 and 128:117, or a tone at the traditional
ratio of 9/8. In Zeta-24, Ibn Sina's tetrachord is available at one
location (F-G-Ap-Bb, or 4/3-3/2-13/8-16/9 with C as the 1/1), as well
as the variation with 44/39 and 11/9 in place of 9/8 and 39/32.
------------------------------------------------------------------
1.1.3. The upper nawruz tetrachord: Modernity and the 13/11 third
------------------------------------------------------------------
Unlike our lower and middle tetrachords of this version of Nahuft, the
upper tetrachord of nawruz or JJT, with two middle or Zalzalian
seconds or J steps followed by a T or tone, is not directly addressed
by Ibn Sina in his catalogue of tetrachords or his `oud tuning.
However, this JJT tetrachord is very common in 13th-century theory,
and likely was common in the 11th-century practice described by Ibn
Sina also, where in one reading of his modal descriptions urged by
Cris Forster it is the tetrachord below the 1/1 or final of his mode
Nawa, which as Forster observes would much resemble the modern Persian
Dastgah-e Nava.
Here is this JJT tetrachord based on Ibn Sina's `oud tuning, and in
our Zeta-24 variation:
Ibn Sina tuning (13:12-9:8-128:117)
416 384 351 312
1/1 13/12 32/27 4/3
0 138.6 294.1 498.0
13:12 128:117 9:8
138.6 155.6 203.9
Zeta-24 tuning (13:12-9:8-12:11)
52 48 44 39
1/1 13/12 13/11 4/3
0 138.6 289.2 493.2
13:12 12:11 44:39
138.6 150.6 208.8
In this permutation of the Ibn Sina tuning, we have a small J at 13:12
followed by a large one at 128:117, together forming a 32/27 minor
third, with an upper T step at 9:8 completing the 4/3 fourth.
In the Zeta-24 tuning, this tetrachord is a permutation of the
previous mustaqim tetrachord, with the smaller and large J steps at
13:12 and 12:11, forming a minor third at 13/11 (289.210 cents),
smaller than the classic 32/27 by 352:351, with the octave completed
by a T step at 44:39, larger than 9:8 by this same small comma. This
division features the arithmetic or subharmonic series 13:12:11, which
appears in a modern setting in the harmoniai of Kathleen Schlesinger.
As far as I know, while divisions such as 12:11:10:9 (Ptolemy's
Equable Diatonic with its arithmetic division of a 6/5 third into
12:11:10) and Ibn Sina's "most noble genus" of 16:14:13:12 (with its
like division of a 7/6 third into 14:13:12) are well known in Near
Eastern theory from al-Farabi on, the 13:12:11 division does not gain
recognition in this literature. This is true although Ibn Sina
discusses both 12:11 and 13:12, and in his discussion of `oud frets
(see Section 1.1.2 above) notes that the 39/32 fret is 13:12 from 9/8
and "approximately 12:11" from 4/3, or more precisely 128:117.
Possibly the absence of a 13:12:11 division may reflect the favored
position of the 32/27 minor third in this theory, with 13/11 very
close but not quite equal in size, so that the slightly large and
standard size is preferred to this variant form -- and likewise with
the standard 9:8 step vis-a-vis the altered 44:39 at slightly larger
which in combination with a 32/27 or 13/11 third completes the 4/3
fourth.
Curiously, while 32/27 has an important status from Greek times on as
the minor third from three pure 4/3 fourths less an octave, and the
epimoric ratios of 7/6 (266.871 cents) and 6/5 (315.641 cents) as the
simplest ratios for a minor third, 13/11 is the next simplest ratio
(as the mediant of 7/6 and 6/5). In Zeta-24, 32/27 and 13/11 are used
as similar and congenial shadings, with the permutation of Ibn Sina's
tuning, 13:12-128:117-9:8 with its 32/27 third, found at G-Ap-Bb-C.
----------------------------
1.1.4. A monochord division
----------------------------
Our Nahuft with a 39/32 third, as it happens, has all three principal
tetrachords following an arithmetic order which may be notated with
lower numbers using string divisions rather than a harmonic series.
The monochord division requires a relatively low length of 234:
`iraq mustaqim tone
117:108:96:88 44:39:36:33 44:39
|------------------|------------------|-----|
|-----|------|-----|------|-----|-----|-----|----...
1/1 13/12 39/32 117/88 3/2 13/8 39/22 2/1
234 216 192 176 156 144 132 117
|-----------------|
nawruz
52:48:44:39
It is interesting to compare a monochord for Nahuft in Ibn Sina's
preferred tuning with consistent steps of 9:8, 13:12, and 128:117,
maintaining the 4/3 fourth and 32/27 minor third in their traditional
proportions rather than altering them by 352:351 at certain points:
`iraq mustaqim tone
468:432:384:351 468:416:384:351 9:8
|------------------|------------------|------|
|-----|------|-----|------|-----|-----|------|----...
1/1 13/12 39/32 4/3 3/2 13/8 16/9 2/1
624 576 512 468 416 384 351 312
|------------------|
nawruz
416:384:351:312
The classic Ibn Sina tetrachords and permutations also share an
arithmetic order, providing a theme for modern variations like that
presented here.
---------------------------------
1.2. A Nahuft with a 16/13 third
---------------------------------
The second variety of Nahuft approximates the classic tuning that
would result from permutations of a famous tetrachord of al-Farabi
(c. 870-950), but again with variations relating to the 352:351
comma. In Zeta-24, this Nahuft version is available from D*, that is,
from the note D on the upper keyboard or chain of fifths.
`iraq rast 44:39
48:44:39:36 88:99:108:117 tone
|---------------------|-------------------|------|
D* E F# G* A* B C* D*
1/1 12/11 16/13 4/3 3/2 18/11 39/22 2/1
0 150.6 347.5 498.0 702.0 852.6 991.2 1200
J T J T J J T
12:11 44:39 13:12 9:8 12:11 13:12 44:39
138.6 208.8 138.6 203.9 150.6 138.6 208.8
|-------------------|
nawruz
36:39:44:48
As with the previous tuning, we have a lower `iraq tetrachord at
(12:11-44:39-13:12), here at 1/1-4/3 with the fourth pure; a conjunct
rast tetrachord (9:8-12:11-13:12) at 4/3-39/22, with a narrow fourth
at 117:88; and an upper nawruz tetrachord (12:11-13:12-44:39) at
3/2-2/1, with a pure 4:3 fourth.
The main difference is that in this tuning, for each tetrachord the
larger J step at 12:11 precedes the smaller J step at 13:12. This
ordering is characteristic of al-Farabi, but with some differences in
the precise sizes of these steps.
----------------------------------------------------------------------
1.2.1. The lower `iraq tetrachord: From al-Farabi to Dudon's Mohajira
----------------------------------------------------------------------
Although al-Farabi does not to my best knowledge himself give a tuning
of what will later be called an `iraq or JTJ tetrachord, he does give
a very famous `oud tuning with steps of 9:8, 12:11, and 88:81 (143.498
cents), that when ordered 12:11-9:8-88:81 form a permutation which
the modern composer and theorist Jacques Dudon has adopted for his
Mohajira or "migratory" tuning.
Jacques Dudon's permutation of al-Farabi tuning (12:11-9:8-88:81)
66 72 81 88
1/1 12/11 27/22 4/3
0 150.6 354.5 498.0
12:11 9:8 88:81
150.6 203.9 143.5
Zeta-24 tuning (12:11-44:39-13:12)
48 44 39 36
1/1 12/11 16/13 4/3
0 150.6 359.5 493.2
12:11 44:39 13:12
150.6 208.8 138.6
From an aural perspective, these two tunings of Nahuft are quite
similar, and share in common the lower J step at 12/11. While
al-Farabi's classic tuning maintains the middle step at a just 9:8,
the Zeta-24 version alters this to 44:39, a 352:351 larger, so that
the third note at 16/13 (359.472 cents) is higher than al-Farabi's
27/22 (354.547 cents) by this same small ratio. Since both tetrachords
maintain a pure 4/3 step, this means that upper step in al-Farabi at
88:81 (143.498 cents) is slightly wider than this step in the Zeta-24
version at 13:12, once more by 352:351.
Another nuance is that in al-Farabi's division, the J steps at 12:11
and 88:81, or 150.6 and 143.5 cents, have a subtle difference of only
243:242 or 7.139 cents, or as Near Eastern performers following modern
Turkish or Syrian theory might say, about 1/3 comma (with the comma
often understood to be the Holdrian comma at 1/53 of a 2/1 octave,
about 22.642 cents). In the Zeta-24 version, the J steps at 12:11 and
13:12 have a somewhat greater difference of 144:143 (12.064 cents), or
about 1/2 comma.
While Ibn Sina does not address this specific permutation of
al-Farabi's tuning, he does discuss 9:8-12:11-88:81, noting that the
upper step of 88:81 "resembles" the superparticular ratio of 13:12.
In al-Farabi's division, the two J steps at 12:11 and 88:81 add up to
32:27, thus together with a pure 9:8 step forming a just 4:3 fourth.
In this Zeta-24 variant, both J steps are superparticular, at 12:11
and 13:11, and the fourth is at a pure 4/3, so that the tone is
altered from a standard 9:8 to 44:39.
From this perspective, the al-Farabi/Dudon and Zeta-24 tunings are
closely related shadings. However, mathematically, there is a curious
contrast. While the al-Farabi permutation can be represented with the
lowest numbers as a harmonic series, 66:72:81:88 (12:11-9:8-88:81),
the Zeta-24 tetrachord yields an arithmetic series, 48:44:39:36
(12:11-44:39-13:12).
Either tetrachord can be expressed as either type of series, but with
higher number required for the alternative type. Thus the al-Farabi
permutation results from either the 66:72:81:88 harmonic series of
Dudon (comparing frequency ratios) or an arithmetic series of
108:99:88:81 (comparing string length ratios, again 12:11-9:8-88:81).
Likewise the Zeta-24 permutation of 12:11-44:39-13:12 can be expressed
arithmetically as 48:44:39:36 or harmonically as 429:468:528:572, with
the latter expression in this instance requiring considerably higher
integers than the former.
-----------------------------------------------------------
1.2.2. The middle rast tetrachord: Al-Farabi's `oud tuning
-----------------------------------------------------------
Just as the middle mustaqim or rast tetrachord (TJJ) of our previous
Zeta-24 Nahuft tuning represented a variation on Ibn Sina's classic
`oud fretting (9:8-13:12-128:117), see Section 1.1.2, so this middle
rast tetrachord in the present tuning represents a small variation on
al-Farabi's earlier and equally famous fretting of 9:8-12:11-88:81,
with the lower three notes this time indeed identical:
Al-Farabi tuning (9:8-12:11-88:81)
108 96 88 81
1/1 9/8 27/22 4/3
0 203.9 354.5 498.0
9:8 12:11 88:81
203.9 150.6 143.5
Zeta-24 tuning (9:8-12:11-13:12)
88 99 108 117
1/1 9/8 27/22 117/88
0 203.9 354.5 493.2
9:8 12:11 13:12
203.9 150.6 138.6
Both tunings share a 9/8 tone and a third step at al-Farabi's
preferred 27/22, somewhat higher than Ibn Sina's 39/32 or 342.5
cents -- a difference that may reflect local or regional tastes.
The al-Farabi and Zeta-24 tunings vary only in the sizes of the upper
steps -- 27/22-4/3 or 88:81 in al-Farabi, and 27/22-117/81 or 13:12 in
Zeta-24, so that the latter has a fourth narrow of 4/3 by 352:351.
An interesting mathematical detail is that while the al-Farabi and
Zeta-24 tetrachords are very similar, the first has an arithmetic form
of 108:96:88:81, and the second has a harmonic form of 88:99:108:117,
as the simpler expression.
----------------------------------------------------------
1.2.3. The upper nawruz tetrachord: 32/27 and 13/11 again
----------------------------------------------------------
In our previous version of Nahuft, the upper tetrachord was
13:12-12:11-44:39 or 52:48:44:39. Here it is another permutation with
the same step sizes, but with the larger J step at 12:11 preceding the
smaller one at 13:12, again forming a 13/11 minor third and an upper
tone at 44:39, which together result in a pure 4/3 fourth. This tuning
could be seen as a variation on a permutation of al-Farabi with the
classical ratios of 32/27 for the minor third and 9:8 for the upper
tone, adopted by Mrad as a likely intonation of the nawruz tetrachord
around 1300:
Al-Farabi permutation of Mrad (12:11-88:81-9:8)
96 88 81 72
1/1 12/11 32/27 4/3
0 150.6 294.1 498.0
12:11 88:81 9:8
150.6 143.5 203.9
Zeta-24 tuning (12:11-13:12-9:8)
33 36 39 44
1/1 12/11 13/11 4/3
0 150.6 289.2 498.0
12:11 13:12 44:39
150.6 138.6 208.8
The al-Farabi/Mrad permutation is simply a rotation of al-Farabi's
famous `oud tuning, 9:8-12:11-88:81 or 1/1-9/8-27/22-4/3, starting on
the second step of this tetrachord.
Likewise, the Zeta-24 tuning of nawruz could be derived from a rast
tetrachord at 39:44:48:52, 44:39-12:11-13:12 or 1/1-44/39-16/13-4/3,
e.g. A*-B*-C#-D*.
Although the al-Farabi permutation and Zeta-24 tunings are quite
similar, it happens that the first involves an arithmetic series,
96:88:81:72, and the second a harmonic series, 33:36:39:44.
----------------------------
1.2.4. A monochord division
----------------------------
Both the Zeta-24 tuning of Nahuft and the classic tuning based on
al-Farabi's `oud fretting feature a mixture of arithmetic and harmonic
divisions for the three tetrachords focused on here. This means that
both versions of the mode have rather complex monochord divisions. In
the Zeta-24 version, a length of 1872 is required in order to express
all three tetrachords in the desired order as arithmetic divisions.
For tetrachords whose simplest divisions are harmonic, these
arithmetic divisions are shown below the diagram.
`iraq rast tone
48:44:39:36 88:99:108:117 44:39
|-------------------|------------------|------|
|------|------|-----|------|-----|-----|------|----...
1/1 12/11 16/13 4/3 3/2 18/11 39/22 2/1
1872 1716 1521 1404 1248 1144 1056 936
|------------------|
nawruz
33:36:39:44
rast tetrachord 351:312:286:264
nawruz tetrachord 156:143:132:117
As it happens, our Nahuft based on al-Farabi's tuning also has mixture
of harmonic and arithmetic divisions, and thus a relatively complex
monochord division with a length of 1296. Again, as with Ibn Sina's
divisions (Section 1.4.1), the classic ratios of 32/27 for minor
thirds and 9:8 for tones are consistently maintained.
`iraq rast tone
66:72:81:88 108:96:88:81 9:8
|-------------------|------------------|------|
|------|------|-----|------|-----|-----|------|----...
1/1 12/11 27/22 4/3 3/2 18/11 16/9 2/1
1296 1188 1056 972 864 792 729 648
|------------------|
nawruz
96:88:81:72
`iraq tetrachord 108:99:88:81
-----------------------------------------------------
2. A variation on al-Farabi's Rast or mode of Zalzal
-----------------------------------------------------
Another Zeta-24 tuning presents a variation on al-Farabi's "Mode of
Zalzal," as Cris Forster terms it, synonymous with a common shading of
the later Rast. This tuning illustrates the cumulative effects
possible with virtual tempering, and the interplay of some small
commas, including the 352:351 we have so far encountered (e.g. in
subtle contrasts of 32/27 vs. 13/11, or 9/8 vs. 44/39).
This Rast begins at G*, with its lowest tetrachord identical to the
middle tetrachord of our last Nahuft tuning (Section 1.2) at
G*-A*-B-C*.
rast rast 112:99
88:99:108:117 273:308:336:363 tone
|-----------------------|---------------------|--------|
G* A* B C* D* E F* G*
1/1 9/8 27/22 117/88 3/2 18/11 99/56 2/1
0 203.9 354.5 498.0 702.0 852.6 986.4 1200
T J J T J J T
9:8 12:11 13:12 44:39 12:11 121:112 112:99
283.9 150.6 138.6 208.8 138.6 150.6 208.8
|----------------------|
nawruz
132:121:112:99
If we compare only the locations of the notes in terms of the 1/1,
then this tuning and al-Farabi's Rast have many ratios in common:
Al-Farabi Mode of Zalzal or later Rast
1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1
0 203.9 354.5 498.0 702.0 852.6 996.1 1200
Zeta-24 variation
1/1 9/8 27/22 117/88 3/2 18/11 99/56 2/1
0 203.9 354.5 498.0 702.0 852.6 986.4 1200
Indeed, from this perspective, only the 117/88 fourth and the 99/56
minor seventh differ from the al-Farabi mode with its classic 4/3 and
16/9.
A closer comparison of tetrachords, and the cumulative effects of
their small nuances of tuning, brings the subtle differences into
clearer relief.
-------------------------------
2.1. The lower Rast tetrachord
-------------------------------
Al-Farabi's Mode of Zalzal is synonymous with the later classic modal
category of Rast, and more specifically with what may now be styled a
conjunct Rast, or in some Arab theory a Nairuz Rast or Nirz Rast
(Scott Marcus documents the latter term). This means that there are
two conjunct rast tetrachords (TJJ), on the 1/1 and 4/3 steps, and
also an upper nawruz (or nairuz or nirz) tetrachord on the 3/2
step. There are various other forms of Rast, as will be addressed in
Section 1.3.5 below.
Here the al-Farabi and Zeta-24 tunings of the lower rast tetrachord
are identical to those of the middle tetrachord in the previous Nahuft
versions (Section 1.2.2):
Al-Farabi tuning (9:8-12:11-88:81)
108 96 88 81
1/1 9/8 27/22 4/3
0 203.9 354.5 498.0
9:8 12:11 88:81
203.9 150.6 143.5
Zeta-24 tuning (9:8-12:11-13:12)
88 99 108 117
1/1 9/8 27/22 117/88
0 203.9 354.5 493.2
9:8 12:11 13:12
203.9 150.6 150.6
As previously noted, al-Farabi has the epimoric steps of 9:8 and
12:11, plus what Ibn Sina describes as a step "resembling" 13:12, but
more precisely at 88:81 (352:351 wider), to complete the 4/3 fourth.
In the Zeta-24 version, all three steps are superparticular,
9:8-12:11-13:12, producing a fourth at 117/88 (narrow of 4/3 by this
same comma of 352:351).
--------------------------------
2.2. The middle Rast tetrachord
--------------------------------
In al-Farabi's Rast, the middle tetrachord on the 4/3 step is
identical to the lower one at 108:96:88:81. However, in the Zeta-24
variation. this tetrachord is subtly different both in starting on the
117/88 step (lower than 4/3 by 352:351), and in itself having a
slightly narrow fourth at 121/91 (narrower than 4/3 by 364:363 or
4.763 cents).
Al-Farabi tuning (9:8-12:11-88:81)
108 96 88 81
1/1 9/8 27/22 4/3
0 203.9 354.5 498.0
9:8 12:11 88:81
203.9 150.6 143.5
Zeta-24 tuning (44:39-12:11-121:112)
273 308 336 363
1/1 44/39 16/13 121/91
0 208.8 359.5 493.2
44:39 12:11 121:112
208.8 150.6 133.8
Here the initial 44:39 step of Zeta-24 is larger than the 9:8 of
al-Farabi's classic tuning by 352:351, while the 12:11 steps are
identical, so that its third note at 16/13 is slightly wider than the
27/22 of al-Farabi. However, the upper step of 88:81 in al-Farabi,
which completes the fourth (27/22-4/3), is replaced in the Zeta-24
version by a notably smaller step at 121:112 or 133.810 cents, about
midway between the epimoric ratios of 13:12 and 14:13, and narrower
than 88:81 by a full 896:891 (9.688 cents). The net effect is to
produce a fourth in the Zeta-24 tuning at 121/91, narrow of 4/3 by
364:363 (the difference between the commas of 896:891 and 352:351).
While al-Farabi's famous tetrachord involves an arithmetic division,
the Zeta-24 variation has as its simplest expression a quite complex
harmonic division, 273:308:336:363.
--------------------------------------------------------------
2.3. The upper nawruz tetrachord: Enter the 33/28 minor third
--------------------------------------------------------------
The upper nawruz tetrachord of al-Farabi's Mode of Zalzal or later
Rast is identical to Mrad's permutation used for the same upper
tetrachord in our previous Nahuft (Section 1.2.3). However, the
Zeta-24 variation introduces some new steps and intervals:
Al-Farabi permutation of Mrad (12:11-88:81-9:8)
96 88 81 72
1/1 12/11 32/27 4/3
0 150.6 294.1 498.0
12:11 88:81 9:8
150.6 143.5 203.9
Zeta-24 tuning (12:11-121:112-112:99)
132 121 112 99
1/1 12/11 33/28 4/3
0 150.6 284.5 498.0
12:11 121:112 112:99
150.6 133.8 213.6
Both tunings begin with a 12:11 step. While the al-Farabi permutation
continues with a classic 88:81 step (forming a 32/27 third) and then a
9:8 tone, the Zeta-24 version has a smaller middle step at 121:112, or
896:891 narrower than 88:81, producing a third at 33/28 (284.447
cents), narrower by this same comma than 32/27. There follows in
Zeta-24 a tone of 112:99 or 213.598 cents, wider than al-Farabi's 9:8
by 896:891, so that the narrower middle and wider upper step balance
out to produce a pure 4/3 fourth, as in al-Farabi's familiar tuning.
Either al-Farabi's permutation or the Zeta-24 variation are most
simply expressed as arithmetic series, 96:88:81:72 or 132:121:112:99.
Curiously, this upper tetrachord of the Zeta-24 version is the only
one to have a traditional 4/3 fourth (3/2-2/1), the others being
framed by narrow fourths: 117:88 or 352:351 narrow for the lower
tetrachord (1/1-117/88), and 121:91 or 364:363 narrow for the middle
tetrachord (117/88-99/56).
--------------------------
2.4. A monochord division
--------------------------
While the celebrated tuning of al-Farabi with its consistent
arithmetic divisions permits a relatively simple monochord scheme, the
Zeta-24 tuning with its mixed harmonic and arithmetic divisions and
slightly impure fourths at both 1/1-117/88 and 117/88-99/56 calls for
a much more complex scheme. Here it may be convenient to give the
simpler classic monochord division first:
rast rast tone
108:96:88:81 108:96:88:81 9:8
|-------------------|------------------|------|
|-------|-----|-----|------|-----|-----|------|----...
1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1
432 384 352 324 288 264 243 216
|------------------|
nawruz
96:88:81:72
Here a length of 432 suffices to express the entire octave mode, by
comparison with the much more complicated situation in Zeta-24:
rast rast tone
88:96:108:117 273:308:336:363 112:99
|-------------------|------------------|------|
|-------|-----|-----|------|-----|-----|------|----...
1/1 9/8 27/22 117/88 3/2 18/11 99/56 2/1
7722 6864 6292 5808 5148 4719 4368 3861
|------------------|
nawruz
132:121:112:99
lower rast tetrachord 351:312:286:264
conjunct rast tetrachord 1936:1716:1573:1456
This more complex tuning requires a string length of 7722.
-------------------------------------------------------
2.5. Varieties of Rast and uses of 99/56 minor seventh
-------------------------------------------------------
A coloristic resource in this Zeta-24 variation on al=Farabi's mode of
Zalzal or conjunct form of Rast is the minor seventh at 99/56 or 986.4
cents, in contrast to the classic tuning at 16/9 or 996.1 cents. While
the classic 16/9 results from a chain of two pure 4/3 fourths,
1/1-4/3-16/9, this narrower minor seventh results from two altered
fourths, at 117:88 (G*-C*) and 121:91 (C*-F*), or 1/1-117/88-99/56.
Thus 99/56 is narrower than 16/9 by an amount equal to the sum of the
commas by which these fourths are narrow, 352:351 or 4.925 cents for
117:88, and 364:363 or 4.763 cents for 121:91 -- adding up to the
appreciable difference of 896:891 or 9.688 cents.
The resulting 99/56 minor seventh is comparable to the size found at
the nearer locations of George Secor's 17-note well-temperament, or
17-WT (secor17wt.scl in the Scala archive) at 985.559 cents, or 0.843
cents narrower, from a chain of two of the smaller fourths in this
tuning at 492.780 cents (5.265 cents narrow of 4/3).
Although such an interval as 99/56 or George Secor's small minor
seventh in 17-WT at
around 986 cents remains considerably closer to 16/9 than to 7/4
(968.825 cents), yet it may have a septimal or "quasi-septimal" color,
suggesting something of the quality of 7/4 or closer approximations.
In the Zeta-24 variation on al-Farabi's Rast, this effect might occur,
for example, in a performance of the mode above a drone when the 99/56
step is sounded for a considerable duration. Additionally, some
related modal forms can also bring this minor seventh into play.
The 13th-century scheme of octave cycles or modes presented by Safi
al=Din al-Urmawi and developed by colleagues and commentators derives
such cycles (Arabic _adwar_ and Turkish _edvar_) by joining a lower
tetrachord with an upper pentachord. Since Safi al-Din lists 7 types
of lower tetrachords and 12 types of upper pentachords, this method
results in 84 cycles or octave modes in all.
These ajnas or genera and modes consists of three types of steps: the
tanini or whole tone T (often at or around 9:8); the bakkiye or
semitone B, often at around 256:243; and the mujannab J, intermediate
in size between 9:8 and 256:243, as with Ibn Sina's steps of 13:12 and
128:117, or al-Farabi's at 12:11 and 88:81. All our modes so far have
featured exclusively T and J steps; but many of the cycles, including
some relevant to Rast, include B steps.
The rast tetrachord, TJJ, is the fourth type, with the fourth type of
pentachord, TJJT, being built from the identical tetrachord plus an
upper tone. Thus the usual 13th-century Rast, identical in structure
(although not necessarily in tuning) to al-Farabi's mode of Zalzal, is
TJJ|TJJT.
Since a family of cycles or modes is defined by the common lower
tetrachord -- in modern terms, a "root" tetrachord -- and there are 12
recognized types of upper pentachords, it follows that each family of
cycles has 12 members. Since Safi al-Din's Rast family, as we might
now call it, is defined by his fourth form of tetrachord, TJJ, its
members make up Cycles 37-48 of his scheme of 84 cycles or octave
modes.
Of these 12 Rast family modes, four seem most relevant to 19th-21st
century Near Eastern maqam theory related to Rast. Three of these
forms occur directly in the scheme of 84 cycles, while the fourth is
found in the 13th-15th century theory of Safi al-Din himself and
others as an alternative interpretation of Cycle 46, Kardaniya. Here
these forms, including both versions of Kardaniya, are shown as tuned
in Zeta-24 with a final of G*.
Cycle 37 (tetrachord IV TJJ + pentachord 1 TTB)
Also modern Arab Suzdular, and standard descending form of Rast
Tetrachord IV TJJ Pentachord 1 TTBT
|---------------------|-----------------------------|
T J J T T B T
G* A* B C* D* E* F* G*
1/1 9/8 27/22 117/88 3/2 27/16 99/56 2/1
0 203.9 354.5 493.2 702.0 905.9 986.4 1200
9:8 12:11 13:12 44:39 9:8 22:21 112:99
203.9 150.6 138.6 208.8 203.9 80.5 213.6
Cycle 40 (tetrachord IV TJJ + pentachord 4 TJJT)
Also modern conjunct Rast or Arab Nairuz/Nirz Rast
|---------------------|-----------------------------|
T J J T J J T
G* A* B C* D* E F* G*
1/1 9/8 27/22 117/88 3/2 18/11 99/56 2/1
0 203.9 354.5 493.2 702.0 852.5 986.4 1200
9:8 12:11 13:12 44:39 12:11 121:112 112:99
203.9 150.6 138.6 208.8 150.6 133.8 213.6
Cycle 46 (tetrachord IV TJJ + pentachord 10 JBTJJ)
|---------------------|-----------------------------------|
T J J J B T J J
G* A* B C* D D* E* F# G*
1/1 9/8 27/22 117/88 81/56 3/2 27/16 24/13 2/1
0 203.9 354.5 493.2 639.9 702.0 905.9 1061.4 1200
9:8 12:11 13:12 44:39 12:11 9:8 128:117 13:12
203.9 150.6 138.6 208.8 150.6 203.9 155.6 138.6
Cycle 46 simplified (tetrachord IV TJJ + pentachord TTJJ)
Also modern disjunct Rast, ascending form
|---------------------|------------------------------|
T J J T T J J
G* A* B C* D* E* F# G*
1/1 9/8 27/22 117/88 3/2 27/16 24/13 2/1
0 203.9 354.5 493.2 702.0 905.9 1061.4 1200
9:8 12:11 13:12 44:39 9:8 128:117 13:12
203.9 150.6 138.6 208.8 203.9 155.6 138.6
Cycle 48 (tetrachord IV TJJ + pentachord 12 TJTJ)
Also a 20th-century Turkish interpretation of Rast
|---------------------|------------------------------|
T J J T J T J
G* A* B C* D* E F# G*
1/1 9/8 27/22 117/88 3/2 18/11 24/13 2/1
0 203.9 354.5 493.2 702.0 852.5 1061.4 1200
9:8 12:11 13:12 44:39 12:11 44:39 13:12
203.9 150.6 138.6 208.8 150.6 208.8 138.6
Cycle 40, the standard 13th-century Rast, can be derived from two
conjunct rast tetrachords plus an upper tone, with that upper tone in
the Systematist approach being included in the upper pentachord TJJT.
It can also be perceived as TJJ+T+JJT, a lower rast tetrachord plus a
middle tone plus an upper nawruz (also nairuz or nirz) tetrachord of
JJT, which may explain the modern Arab name Nirz Rast.
Cycle 37, from a Systematist perspective, is derived as TJJ+TBBT, a
lower rast tetrachord plus the first pentachord, TTBT, including a
tetrachord of TTB or two tones plus a semitone (Ibn Sina's common
diatonic at 9:8-9:8-256:243) plus an upper tone. This is also one
modern Arab interpretation, with a lower rast tetrachord plus some
focus on an `ajam tetrachord on the fourth step, TTB, and an upper
tone to complete the octave. Another Arab interpretation is a lower
Rast tetrachord plus a middle tone, plus an upper TBT or nahawand
tetrachord on the 3/2 step, thus TJJ-T-TBT.
Cycle 46 in its more complex 13th-century form, TJJ-JBTJJ, could be
described as a lower rast tetrachord plus a middle tone which is
divided into a middle or mujannab step and then a semitone (here
117/88-81/56-3/2, with steps of 99:91-28:27 (145.874-62.961 cents).
However, Safi al-Din also documents a simpler form of this cycle,
called Kardaniya (a name evidently related to kirdan in Arabic or
gerdaniye in Turkish, now used as a name for the note in the standard
gamut at a 2/1 octave above the step rast, which in turn is located at
a 4/3 fourth above the lowest note of the system, Arabic yakah or
Turkish yegah). In the simplified form, the middle tone is undivided,
thus producing in effect two disjunct Rast tetrachords, TJJ-T-TJJ.
This simpler version of Kardaniya has become the most common modern
form of Arab or Turkish Rast, especially in ascending.
Cycle 48 has a lower Rast tetrachord and then an upper pentachord of
TJTJ, or TJJ-TJTJ, and could also be interpreted as a lower Rast
tetrachord plus a middle tone and an upper `iraq tetrachord, thus
TJJ-T-JTJ. Rauf Yekta, a 20th-century Turkish theorist, proposed this
as a standard form of Rast, and it is an interesting variation.
In a free improvisation focusing mainly on Cycle 40 or conjunct Rast,
the most important 13th-14th century mode of this family indeed going
back to al-Farabi's famous mode of Zalzal, we might follow the example
of some modern Iranian songs and sometimes, for example in ascending,
raise the sixth degree of the mode from a Zalzalian or middle sixth
(here G*-E at al-Farabi's 18/11) to a major sixth (here G*-E* at a
classic 27/16). While Iranian tunings might tend to prefer a lower
Zalzalian third and sixth than 27/22 and 18/11 -- for example, Ibn
Sina's 39/32 and 13/8 -- this fluidity of the sixth degree can be
charming in a range of shadings.
Form with major sixth (like Cycle 37)
G* A* B C* D* E* F* G*
1/1 9/8 27/22 117/88 3/2 27/16 99/56 2/1
0 203.9 354.5 493.2 702.0 905.9 986.4 1200
9:8 12:11 13:12 44:39 9:8 22:21 112:99
203.9 150.6 138.6 208.8 203.9 80.5 213.6
Form with Zalzalian sixth (like Cycle 40)
G* A* B C* D* E F* G*
1/1 9/8 27/22 117/88 3/2 18/11 99/56 2/1
0 203.9 354.5 493.2 702.0 852.5 986.4 1200
9:8 12:11 13:12 44:39 12:11 121:112 112:99
203.9 150.6 138.6 208.8 150.6 133.8 213.6
In a "fusion" style combining Near Eastern modes with elements of
12th-14th century European polyphony, the occurrence in Cycle 37 of a
major sixth and minor seventh makes possible a beautiful contrapuntal
resolution in two voices, where the minor seventh contracts by
stepwise contrary motion to a fifth:
F* E*
G* A*
While this resolution is very effective with the classic medieval
European or Near Eastern tuning of a minor seventh at 16/9, it takes
on a distinct shading with the 99/56 seventh of our Zeta-24 tuning,
for example as part of a three-voice progression found in 13th-14th
century European music:
F* E*
D* E*
G* A*
Here the outer minor seventh contracts to a fifth and the upper minor
third to a unison, while the two lower voices move in parallel
fifths. In a standard Pythagorean tuning, the unstable seventh
sonority would be at 1/1-3/2-16/9 or 18:27:32, with the two lower
voices moving by 9:8 tones, and the upper voice descending by a
compact and efficient 256:243 semitone at 90.225 cents. The Zeta-24
version has a sonority of 1/1-3/2-99/56 or 56:84:99. with an upper
minor third at 33:28, and a yet more efficient semitonal motion in the
upper voice of 22:21 or 80.537 cents.
This 1/1-3/2-99/56 sonority may have a quality somewhat different than
either 1/1-3/2-16/9 (e.g. C-G-Bb in Zeta-24) or the septimal
1/3-3/2-7/4 or 4:6:7 (e.g. C-G-A* in Zeta-24). This variety of
shadings is one of the advantages of many JI systems, achieved also in
an irregular temperament such as Secor's 17-WT.
For a Zeta-24 variation on Ibn Sina's Mustaqim mode (see Sections 1.1
and 1.1.2) of 1/1-9/8-39/32-4/3-3/2-13/8-16/9-2/1 which also fall into
the 13th-century category of Rast or TJJ-TJJT (Cycle 40), we can find
at E a form with another 99/56 minor seventh, and also with the option
used in some modern Iranian songs of a major sixth step which would be
placed at 27/16 in a classic tuning, and here at the slightly higher
22/13:
Ibn Sina's Mustaqim
1/1 9/8 39/32 4/3 3/2 13/8 16/9 2/1
0 203.9 342.5 498.0 702.0 840.5 996.1 1200
9:8 13:12 128:117 9:8 13:12 128:117 9:8
203.9 138.6 155.6 203.9 138.6 155.6 203.9
Form with Zalzalian sixth (like Cycle 40)
E F# G* A B C* D E
1/1 44/39 11/9 121/91 3/2 13/8 99/56 2/1
0 208.8 347.4 493.3 702.0 840.5 986.4 1200
44:39 13:12 99:91 273:242 13:12 99:91 112:99
208.8 138.6 145.9 208.7 138.6 145.9 213.6
Form with major sixth (like Cycle 37)
E F# G* A B C# D E
1/1 44/39 11/9 121/91 3/2 22/13 99/56 2/1
0 208.8 347.4 493.3 702.0 910.8 986.4 1200
44:39 13:12 99:91 273:242 44:39 117:112 112:99
208.8 150.6 145.9 208.7 208.8 75.6 213.6
Here, if the 22/13 step has been introduced in a given polyphonic
fusion style, there is the possibility of resolutions like
D C#
B C#
E F#
where the lower pair of voices ascend by a tone of 44:39 or the almost
identical 273:243, while the upper voice descends by a compact
semitone at 117:112 or 75.612 cents.
With this version of Mustaqim including a 99/56 seventh, as with the
variation on al-Farabi's Rast likewise including this interval, the
narrower seventh occurs more as a consequence than a primary aspect of
the Zeta-24 design. Specifically, this ratio results from two
successive "virtually tempered" or narrow fourths at 117/88 and
121/91, immediately designed to bring about other ratios such as 13/11
or 14/11. However, these narrow sevenths open the way to a
"quasi-septimal" coloration which presents an intriguing variation on
the classic tunings of al-Farabi and Ibn Sina featuring 16/9.
-----------------------------------------------
3. Variant tunings in context: A systemic view
-----------------------------------------------
To give some perspective on more complex ratios in the Rast and
Mustaqim tunings in Section 2, we may find it helpful to consider the
position of the steps involved in the overall Zeta-24 system. For
example, here are the notes of the al-Farabi Rast variation expressed
as ratios with respect to G*, the 1/1 of this mode, and also to C, the
conceptual 1/1 of the overall system.
|-----------------------|---------------------|--------|
G* A* B C* D* E F* G*
1/1 9/8 27/22 117/88 3/2 18/11 99/56 2/1
0 203.9 354.5 498.0 702.0 852.6 986.4 1200
14/9 7/4 21/11 91/88 7/6 14/11 11/8 14/9
764.9 968.8 1119.5 58.0 266.9 417.5 551.3 764.9
Here some of the complexity is inherent in the system, regardless of
the point of reference: for example, the choice of a step at 91/88,
which can form a fifth within 5 cents of pure with either 11/8
(182:121 at 364:363 or 4.763 cents wide) or 14/9 (176:117 at 352:351
or 4.925 cents wide).
However, the complex minor seventh at 99/56 turns out to be an
interval between two considerably simpler ratios in the overall
system: 14/9-11/8. Tuning both of these ratios just necessarily
involves a ratio between them of 99:56, narrower by 896:891 or 9.688
cents than the classic 16:9.
It is interesting to compare this al-Farabi variation with its
counterpart in Rod Poole's 17-note guitar tuning, which likewise has
available a 99/56 minor seventh at the steps 14/9-11/8.
Al-Farabi Rast in Rod Poole 17 (using Zeta-24 notation)
|-----------------------|---------------------|--------|
G* A* B C* D* E F* G*
1/1 9/8 27/22 297/224 3/2 18/11 99/56 2/1
0 203.9 354.5 488.4 702.0 852.6 986.4 1200
14/9 7/4 21/11 33/32 7/6 14/11 11/8 14/9
764.9 968.8 1119.5 58.0 266.9 417.5 551.3 764.9
Rod Poole's system has the lower tetrachord with a 297/224 fourth, a
full 896:891 narrower than 4/3, and then a conjunct tetrachord with a
pure 4:3 fourth, giving a seventh at 99/56. In Zeta-24 modes, we
have a division of the 896:891 comma, with these two fourths at 117:88
and 121:91.
In systemic terms, the Rod Poole tuning has 14/9-33/32-11/8, while
Zeta-24 has 14/9-91/88-11/8.
A similar pattern holds for the variation on Ibn Sina's Mustaqim --
which in 13th-century terms could be called simply his Rast with a
lower Zalzalian third at 39/32 than al-Farabi's:
|----------------------|---------------------|-------|
E F# G* A B C* D E
1/1 44/39 11/9 121/91 3/2 13/8 99/56 2/1
0 208.8 347.4 493.3 702.0 840.5 986.4 1200
14/11 56/39 14/9 22/13 21/11 91/88 9/8 14/11
417.5 626.3 764.9 910.8 1119.5 58.0 203.9 417.5
Here 99/56 occurs as a ratio between the notes 9/8-14/11, notes also
occurring in Rod Poole's 17-note JI guitar tuning. The difference is
that in Zeta-24, the additional element of virtual tempering
contributes to complexity, with 22/13 chosen in part so as to form
fifths within 5 cents of pure with both 9/8 (176:117) and 14:11
(182:121). In contrast, the Rod Poole tuning has 9/8-27/16-14/11,
making 9/8-27/16 pure, but 27/16-14/11 wide by a full 896:891 or 9.688
cents, at 448:297 (711.643 cents).
As it happens, Poole's 17-note system does not have the extra 56/39
step of Zeta-24, or more generally a step at a tone of around 9:8
above 14/11 (in Zeta-24, a tone of 44:39). However, both systems support variations on this 14/11
step of medieval Nahuft or Jacques Dudon's Ibina mode, thus
illustrating some nuances in the tuning of certain intervals:
Poole 17: Nahuft on 14/11 step (using Zeta-24 notation)
`iraq mustaqim tone
|----------------------|---------------------|-------|
E F* G* A B C* D E
1/1 121/112 11/9 297/224 3/2 363/224 99/56 2/1
0 133.8 347.4 488.4 702.0 835.8 986.4 1200
121:121 112:99 243:224 112:99 121:112 12:11 112:99
133.8 213.6 140.9 213.6 133.8 150.6 213.6
14/11 11/8 14/9 27/16 21/11 33/32 9/8 14/11
417.5 551.3 764.9 905.9 1119.5 53.3 203.9 417.5
Zeta-24: Nahuft on 14/11 step
`iraq mustaqim tone
|----------------------|---------------------|-------|
E F* G* A B C* D E
1/1 121/112 11/9 121/91 3/2 13/8 99/56 2/1
0 133.8 347.4 493.3 702.0 840.5 986.4 1200
121:121 112:99 99:91 44:39 13:12 99:91 112:99
133.8 213.6 145.9 208.8 138.6 145.9 213.6
14/11 11/8 14/9 22/13 21/11 91/88 9/8 14/11
417.5 551.3 764.9 910.8 1119.5 58.0 203.9 417.5
Again, the Poole-17 version has the fourth of the lower tetrachord at
297:224, and the conjunct tetrachord with a pure 4:3, in systemic
terms 14/11-27/16-9/8. In Zeta-24, these fourths are at 121:91 and
117:88, or in systemic terms 14/11-22/13-9/8, both impure by not quite
5 cents each.
A noteworthy feature of Poole-17 is the step at 11/9-297/224 leading
to the fourth of Nahuft, or in systematic terms 14/9-27/16, at the
classic ratio of 243:224 (140.949 cents) found in the Chromatic of
Archytas, 28:27-243:224-32:27 or 1/1-28/27-9/8-4/3.
While Zeta-24 has a Zalzalian sixth step at 13/8 (in systemic terms
14/11-91/88), the more complex 363/224 of Poole-17 at 835.765 cents
(14/11-33/32) is another shading very conducive to Persian tastes.
Indeed Hormoz Farhat's tar or setar tuning arrived at by averaging
some instruments (Scala archive, persian-far.scl) with a number of
middle sixths at 835 cents, and Dariush Anooshfar's related JI tuning
(persian.scl) favoring a ratio of 81/50 or 835.193 cents, nicely agree
with the Poole tuning.
Both the 17-note Rod Poole tuning and Zeta-24, in short, show how JI
systems using primes 2-3-7-11-13 with a focus on Near Eastern music
are likely to involve a mixture of simple and complex ratios. In a
tradition like that of the Near East which much values melodic variety
and subtlety, this feature of JI and also of some irregular
temperaments is an artistic plus.
-----
Note
-----
1. Jacques Dudon's Ibina is a mode based on a differentially coherent
(-c) harmonic series of 72:78:88:96:108:117:128:144, and may be
found on the 1/1 step of Zeta-24.
72 78 88 96 108 117 128 144
C Dp Eb F G Ap Bb C
1/1 13/12 11/9 4/3 3/2 13/8 16/9 2/1
0 138.6 347.4 498.0 702.0 840.5 996.1 1200
13:12 44:39 12:11 9:8 13:12 128:117 9:8
138.6 208.8 150.6 203.9 138.6 155.6 203.9
Dudon's concept of differential coherence refers to a tuning in
which the differences between steps coincide with harmonic ratios
or factors in the tuning. For a fuller discussion, see Jacques
Dudon, "Differential Coherence: Experimenting with New Areas of
Consonance," _1/1_ (Vol. 11, No. 2, Winter, 2003), beginning p. 1.
Margo Schulter
31 March 2014
First revision
1 April 2014