ETHNO EXTRAS: Zalzalian 12 (Pt. 3) Rotation 4

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Ethno Extras: Zalzalian 12 set
Part 3: Rotation 4
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Hello, all.

To provide a convenient "itinerary" for the Zalzalian 12 set, so that
people can keep track of where we've been and where we're going, I'll
place at the beginning of each part a summary of the rotations adapted
from Scala output. Part 1 included an introduction on Ethno Extras,
the O3 temperament, and Buzurg modes, and started our journey with
Rotation 0. Part 2 addressed Rotations 1-3. Part 3 now addresses
Rotation 4.

Again, my warmest thanks to Jacques and Francois and all
participants in the Ethno2 contest for the inspiration they have
lent, and to Ozan Yarman, Shaahin Mohajeri, and George Secor, as
well as now to Jacques Dudon, for the very special knowledge and
wisdom they have shared in the exploration of these and other
modes.

Here I should also mention a poetic liberty I take below, possibly
less than felicitously, in referring to a tetrachord of 0-162-369-496
cents as a "High Moha/Ishku." I must confess to sometimes falling into
the imprecise colloquialism of using the name "Moha" (short for
Jacques Dudon's Mohajira) for just about any tetrachord with a lower
and upper neutral second step and a middle step not too far from 9:8,
or one might say not so close to 8:7 (or larger) that the name Buzurg
or Hijaz would seem more apt, with 48:52:59:64 (0=139-357-498 cents)
as a form of Mohajira (dudon_mohajira_r.scl, Scala archive) showing
that 59:52 at 218.6 cents is within the limits of the Mohajira type.

My understanding is that "Mohajira" in a proper sense would refer to
such tetrachords where the neutral thirds, unequal or equal, are
somewhere in the "central" range from around 39/32 to 16/13, or about
342-360 cents, with other terms (including possibly Ishku) better for
forms with smaller or larger Zalzalian thirds, such as the tetrachord
resembling that available in Jacques' sumer.scl (Ishku -c) in the
Ethno2 collection at 1/1-79/72-89/72-4/3 (0-161-367-498 cents.

And now for our itinerary, and the continuation of our journey.

Rotation 0: 1/1
0 138.3 207.4 264.8 345.7 472.3 680.9 704.3 842.6 969.1 1050.0 1176.6 2/1

Rotation 1: 138.3 (~13/12)
0 69.1 126.6 207.4 334.0 542.6 566.0 704.3 830.9 911.7 1038.3 1061.7 2/1

Rotation 2: 207.4 (~9/8 or ~44/39)
0 57.4 138.3 264.8 473.4 496.9 635.2 761.7 842.6 969.1 992.6 1130.9 2/1

Rotation 3: 264.8 (~7/6)
0 80.9 207.4 416.0 439.5 577.7 704.3 785.2 911.7 935.2 1073.4 1142.6 2/1

Rotation 4: 345.7 (~11/9)
0 126.6 335.2 358.6 496.9 623.4 704.3 830.9 854.3 992.6 1061.7 1119.1 2/1

Rotation 5: 472.3 (~21/16)
0 208.6 232.0 370.3 496.9 577.7 704.3 727.7 866.0 935.2 992.6 1073.4 2/1

Rotation 6: 680.9 (~77/52)
0 23.4 161.7 288.3 369.1 495.7 519.1 657.4 726.6 784.0 864.8 991.4 2/1

Rotation 7: 704.3 (~3/2)
0 138.3 264.8 345.7 472.3 495.7 634.0 703.1 760.5 841.4 968.0 1176.6 2/1

Rotation 8: 842.6 (~13/8)
0 126.6 207.4 334.0 357.4 495.7 564.8 622.3 703.1 829.7 1038.3 1061.7 2/1

Rotation 9: 969.1 (~7/4)
0 80.9 207.4 230.9 369.1 438.3 495.7 576.6 703.1 911.7 935.2 1073.4 2/1

Rotation 10: 1050.0 (~11/6)
0 126.6 150.0 288.3 357.4 414.8 495.7 622.3 830.9 854.3 992.6 1119.1 2/1

Rotation 11: 1176.6 (~77/39)
0 23.4 161.7 230.9 288.3 369.1 495.7 704.3 727.7 866.0 992.6 1073.4 2/1

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Rotation 4 (11/9): At long last, Buzurg!
----------------------------------------

0 126.6 335.2 358.6 496.9 623.4 704.3 830.9 854.3 992.6 1061.7 1119.1 2/1

We now come to the first appearance of a regular Buzurg mode,
along with some alternatives for filling out the upper tetrchord
to accompany the Buzurg pentachord described by Safi al-Din and
Qutb al-Din. We begin with the most symmetrical choice, an upper
Buzurg tetrachord:

Buzurg "Buzurg"
|-----------------------------|--------------------|
0 127 359 497 623 704 831 1062 1200
1/1 14/13 16/13 4/3 56/39 3/2 21/13 24/13 2/1
127 231 138 127 81 127 231 138

Here I have put quotes around the name of the upper "Buzurg"
tetrachord simply to caution that the idea of a Buzurg
_tetrachord_, as opposed to pentachord, may be a modern concept,
although there is no reason that a medieval theorist might not
have thought in these terms.

An open question is how the 56/39 step, slightly low here at 623
rather than a just 626 cents, might have been used in a medieval
sayr. Personally I am influenced by the modern Shur Dastgah, and
often treat this step as a lowered version of the fifth that
invites descending motion toward the final, for example:

3/2 21/13 56/39 4/3 16/13 4/3 16/13 14/13 1/1
704 831 623 497 359 497 359 127 0

or

4/3 56/39 4/3 16/13 14/13 1/1
497 623 497 359 127 0

Some Buzurg modes of Safi al-Din and Qutb al-Din suggest two
general types of variations for the upper tetrachord which are
supported in this rotation, although the intonational fine
details are sometimes different. The first possibility is what in
modern terms might be called a upper Shur or Arab Bayyati
(Turkish Ushshaq or Arazbar) tetrachord with two neutral second
steps followed by a tone:

Buzurg Shur or Bayyati
|-----------------------------|--------------------|
0 127 359 497 623 704 831 993 1200
1/1 14/13 16/13 4/3 56/39 3/2 21/13 16/9 2/1
127 231 138 127 81 127 162 207

In the medieval Buzurg tunings of this variety, the minor seventh
is at 7/4 (not available in this rotation) rather than 16/9, and
with the upper 8/7 step of 7/4-2/1 further divided into steps of
15:14 and 16:15 (a division of 3/2-13/8-7/4-15/8-2/1).

While somewhat different from this medieval scheme of Qutb
al-Din, our Buzurg-Shur mode does follow the medieval idea of
seeking "consonance," that is notes in the upper genus which form
fourths or fifths with those of the lower genus, here Buzurg.
This is only a general guideline, of course, since it is the
breaking of symmetry which produces variety in maqam/dastgah
music, as in modern physics. Here the we have a fifth at
14/13-21/13, and a fourth at 4/3-16/9.

A second Buzurg mode features almost exactly the intervals
described by Qutb al-Din, the lower Buzurg pentachord being
coupled to this theorist's tuning for Hijaz, one of the most
common forms used in my music:

Buzurg Hijaz
|-----------------------------|-------------------|
0 127 359 497 623 704 854 1119 1200
1/1 14/13 16/13 4/3 56/39 3/2 18/11 21/11 2/1
127 231 138 127 81 150 265 81

Qutb al-Din's Hijaz, a permutation as he notes of the chromatic
genus 22:21-12:11-7:6 (Ptolemy's Intense Chromatic), has the form
1/1-12/11-14/11-4/3 (0-151-418-498 cents), with steps of
12:11-7:6-22:21 (151-267-81 cents). Here the 7:6 step is narrow
by about 2 cents, while 12/11 and 22/21 are virtually just.

Here an element of consonance is the fourth 16/13-18/11,
interestingly at a "virtually tempered" 117:88 or 493 cents in
Qutb al-Din's original JI version (359.5-852.6 cents), where it
is narrow of 4:3 by 352:351 (4.925 cents), and closer to pure in
this irrational temperament where this small comma or kleisma is
dispersed by tempering each fourth narrow by one or two binary
millioctaves (i.e. 1/1024 or 2/1024 octave, 1.172 or 2.344
cents).

A third Buzurg variation uses a minor seventh step like the
first, but with the higher 18/11 rather than 21/13 neutral sixth,
producing an upper tetrachord like a moderate Arab Huseyni or
Turkish Ushshaq at 0-150-288-496 cents (~1/1-12/11-13/11-4/3),
with consonances of 16/13-18/11 (as in the last variant) and
4/3-16/9 (as in the first). The minor seventh at 992.6 cents
could be taken as either 16/9 (996 cents) or 39/22 (991 cents),
the latter equal to a 3/2 fifth plus a 13/11 minor third at 702
and 289 cents (here a tempered 704 plus 288 cents).

Arab Huseyni or
Buzurg Turkish Ushshaq
|-----------------------------|----------------------|
0 127 359 497 623 704 854 993 1200
1/1 14/13 16/13 4/3 56/39 3/2 18/11 16/9 2/1
127 231 138 127 81 150 138 207

Our Buzurg modes invite what I find to be a pleasant modulation to a septimal form of Rast we shall encounter in our next
rotation, and which I will discuss in connection with that Rast.

With the simple symmetrical Buzurg having an upper Buzurg
tetrachord, a wonderful cadence is the following, already
presented in Rotation 1 for the Ottoman-flavored Maqam Sikah:

831 21/13 704 3/2
127 14/13 0 1/1
-138 12/13 0 1/1

Additionally, the present rotation offers a beautiful flavor of
Turkish Maqam Segah, where the structure is somewhat different
from Arab Sikah, and the intonational shading distinct from
either a typical Arab Sikah, which would prefer larger neutral
steps and intervals above the final; or the most usual modern
form of Turkish Segah, which would prefer smaller steps and
intervals of a 5-limit minor nature (e.g. 16/15, 6/5) rather than
Zalzalian ones.

As Ozan and his colleagues have shown, measurements of recorded
Turkish performances fit quite well the model of a lower Segah
tetrachord of 1/1-16/15-6/5-4/3 (0-112-316-498 cents), or steps
of 16:15-9:8-10:9 (112-204-182 cents).

Our gently Zalzalian shade of Segah with a lower tetrachord of
0-127-335-497 cents (~1/1-14/13-17/14-4/3) favors small neutral
or "supraminor" intervals. Applying the seyir or path of melodic
development as described by Ozan in his thesis, we begin with an
ascent exploring this lower Segah tetrachord, and then a disjunct
upper Hijaz tetrachord on the 3/2 step:

Segah tone Hijaz
|---------------------|.......|----------------------|
0 127 335 497 704 831 1119 1200
1/1 14/13 17/14 4/3 3/2 21/13 21/11 2/1
127 207 162 207 127 288 81

As tuned here, the upper Hijaz tetrachord is approximately
1/1-14/13-14/11-4/3 (0-128-418-498 cents), or 14:13-13:11-22:21,
which could be expressed by the arithmetic or string-length
division of 28:26:22:21.

Having reached the 2/1 step of Segah, we descend by an intriguing
path which at this location of our tuning takes on a septimal
quirk, with the notes shown in ascending order but the motion
actually proceeding mostly downward:

Segah 14:13 Mahur 22:21
|---------------------|......|------------------|.....|
0 127 335 497 623 831 1062 1119 1200
1/1 14/13 17/14 4/3 56/39 21/13 24/13 21/11 2/1
127 207 162 127 207 231 57 81

While it is routine for disjunct tetrachords to be connected by a
tone of around 9:8, here we have two striking "disjunctions"
bridged by smaller intervals. From the 2/1 we move down by a
usual 22:21 semitone to 21/11 (the ascending leading tone of the
previous Hijaz tetrachord), and arrive at the top note of a
Mahur tetrachord with a form of tone-tone-semitone (like a
European Mixolydian or Ionian mode), a Persian and Turkish name,
with this genus known in the Arab world as `Ajam.

This tetrachord has steps at 0-207-439-496 cents, or
approximately 1/1-9/8-9/7-4/3, with step sizes at 207-231-57
cents, or ~9:8-8:7-28:27 (204-231-63 cents) -- a septimal
variation on a usual Mahur with two whole tones not far from 9:8,
made yet more striking by the compressed upper step at 57 cents,
considerably smaller than a just 28:27 at 63 cents.

While Safi al-Din describes a tetrachord of 8:7-9:8-28:27, what
we might call one form of septimal Mahur, I must emphasize that
the use of such an entity in Maqam Segah is a combined quirk of
this tuning set and its author! It is certainly possible to find
other locations in the O3 temperament offering a Maqam Segah with
a usual diatonic Mahur tetrachord for this descending seyir, and
such a Segah will be offered, Inshallah, as a part of "Ethno
Extras."

Either this Zalzalzian version of Segah or the more typical
5-limit flavor of modern Turkish practice is intimately related
to a companion form of Maqam Rast, which supplies almost all the
notes needed for the basic ascending and descending seyir we have
just surveyed. Here are the _perdeler_ or notes (Turkish plural
of _perde_, a pitch or note) for our Segah and related Rast, with
Turkish names for these steps, taking the final of this Rast as
perde rast. First, let us consider the notes in the ascending
seyir of Segah:

Segah Hijaz
|-------------------|.....|-----------------------|
231 138 127 209 162 207 127 288 81
21/26 12/13 1/1 14/13 17/14 4/3 3/2 21/13 21/11 2/1 -369 -138 0 127 335 497 704 831 1119 1200
sarp tiz
rast dugah segah chargah neva hisar evdj gerdaniye (sunbule) segah
0 231 369 496 704 866 1073 1200 1488 1569
1/1 8/7 26/21 4/3 3/2 33/20 13/7 2/1 26/11 52/21
231 138 127 209 162 207 127 288 81
|------------------|......|-----------------|
Rast tone "High Moha/Ishku"

All these steps are a regular part of the Rast system except
sunbule at a 21/11 above the final in Segah (the leading tone of
the Hijaz tetrachord pulling up to the 2/1), a step at 1488 cents
above rast, or a near-just 13/11 minor third plus an octave.
While this step is characteristic of Turkish Segah and the
related Sazkar rather than Rast, the Arab Sazkar, regarded as a
member of the Rast rather than Segah (or in Arabic, Sikah)
family, may draw on this Ottoman idiom in its lower tetrachord
with a minor third, neutral third, and fourth above rast. We'll
encounter a variation on this Arab Sazkar, but with a Turkish
flavor of Rast, in Rotation 11.

Since Segah, as it name implies, starts on the "third" step of
Rast, perde segah, its 1/1 is the bright 26/21 Zalzalian third of
Rast, with an opening small neutral step of about 14/13 leading
to Chargah, the 4/3 of Rast. The higher perde segah is placed in
relation to perde rast, or the wider the rast-segah third, the
smaller a step from segah to chargah (here 26/21-4/3 or 14:13),
and likewise a third from segah to neva (here 26/21-3/2, or in JI
63:52 at 332 cents, and in this temperament a slighter wider 335
cents closer to 40:33 at 333 cents or 17:14 at 336 cents).

In much modern Turkish practice, with perde segah yet higher and
at or only slightly below 5/4, we get a segah-chargah step of
about 16:15 (5/4-4/3) and a segah-neva third around 6:5
(5/4-3/2). This creates the interesting situation where a "high
segah" (i.e. the perde or step, in relation to Rast) produces
what might be called a "low Segah" (i.e. the smaller sizes of the
steps and intervals in Maqam Segah with respect to perde segah).

Because of the septimal flavor of the lower Rast tetrachord, the
second or dugah step at 8/7 is called "sarp dugah" in Ozan
Yarman's naming system, distinguishing it from the usual variety
around 9/8. Otherwise, apart from the sunbule of the upper Hijaz tetrachord, the notes of the upward Segah seyir are shared with
the Rast system -- but in a combination that suggests a variant
on Rast not so often recognized.

To complete the lower Segah tetrachord, which starts with perde
segah at 26/21 above rast, with a 4/3, we need a large neutral
sixth at around 33/30 or 104/63 above rast, called perde hisar. To start the next tetrachord of Segah at 3/2, we need a large
neutral seventh at around 13/7 above rast, perde evdj. If we add
the next step of Maqam Segah, gerdaniye at a small 21/13 neutral
sixth above segah and the 2/1 octave of rast, and also fill in
the steps rast and sarp dugah below segah to make a full octave
from rast to gerdaniye, we get this form:

Segah tone Hijaz
|-------------------------|.........|--------...
1/1 14/13 17/14 4/3 3/2 21/13
0 127 335 497 704 831
rast sarp dugah segah chargah neva hisar evdj gerdaniye
0 231 369 496 704 866 1073 1200
1/1 8/7 26/21 4/3 3/2 33/20 13/7 2/1
231 138 127 209 162 207 127
|---------------------------|........|-------------------------|
Rast tone "High Moha/Ishku"

As explained in the next section on Rotation 5, the best-known
forms of Rast often combine a neutral sixth (perde hisar) with a
minor seventh (perde ajem, as it is known in Turkish, or `ajam in
Arabic); and a major sixth (perde huseyni) with a neutral seventh
(perde evdj).

However, the ascending seyir for Segah points us to the
possibility of an upper tetrachord for Rast with a large neutral
sixth and neutral sixth at 866 and 1073 cents. This tetrachord,
in terms of perde rast around 3/2-33/20-13/7-2/1, has steps of
162-207-127 cents (for example 11:10-9:8-14:13), with a large
neutral second, tone, and small neutral second -- a precise
inversion of our Segah tetrachord at 127-207-162 cents!

Since Jacques Dudon's sumer.scl features the Ishku -c series with
an opportunity to form a tetrachord of 1/1-79/72-89/72-4/3 with
intervals of 0-160.6-367.0-498.0 cents and steps of
160.6-206.3-131.1 cents (79:72-89:79-96:89), possibly this might
be called an "Ishku-like" tetrachord, albeit here without the
properties of differential coherence that make Ishku proper
unique.

Let us now briefly consider the steps used in the descending
seyir of our Segah, and also the question of a polyphonic final
cadence.

Segah 14:13 Mahur 22:21
|--------------------|.....|----------------------|......|
127 207 162 127 207 231 57 81
0 127 335 497 623 831 1062 1119 1200
1/1 14/13 17/14 4/3 56/39 21/13 24/13 21/11 2/1
sarp sarp tiz rast dugah segah chargah neva hisar ajem gerdaniye muhayyer sunbule segah
1/1 8/7 26/21 4/3 3/2 33/20 16/9 2/1 16/7 (26/11) 52/21
0 231 369 496 704 866 993 1200 1431 1488 1569
|----------------|---------------------|.......|
septimal Rast Rast tone

Taking the notes in this descending seyir from the 1/1 to the
21/13 of Segah, and filling the steps rast and sarp dugah to form
a Rast octave, we have two conjunct Rast tetrachords, the first
starting with a septimal step of 8:7, and the second the usual
type with a tone at around 9:8. Again, the high sunbule is the
one step "foreign" to a usual Rast system.

Having followed the descending portion of the seyir back to perde
segah, we can explore the lower septimal Rast tetrachord either
before returning to the middle or higher regions, or as a
preparation for a final cadence. As with some modes such as
Buzurg we have explored offering septimal intervals, we can
approach the final (the 26/21 of Rast) from either sarp dugah
(8/7) at 138 cents below, or chargah (4/3) at 127 cents above.
The following cadence combines both motions in a pattern now
familiar

831 21/13 704 3/2
127 14/13 0 1/1
-138 12/13 0 1/1

Having considered this form of Maqam Segah, we now come to our
next rotation which brings us to "Rast Central," with a Rast
system supporting various facets of this "mother of all maqamat."

(Conclusion of Part 3)

Best,

Margo Schulter
mschulter@...

----------------------------------
Ethno Extras: Zalzalian 12 set
Part 4: Rotation 5
----------------------------------

[To preserve formatting as much as possible, please try the Use
Fixed Font Width option if viewing on the Yahoo site.]

Hello, all.

To provide a convenient "itinerary" for the Zalzalian 12 set, so that
people can keep track of where we've been and where we're going, I'll
place at the beginning of each part a summary of the rotations adapted
from Scala output. Part 1 included an introduction on Ethno Extras,
the O3 temperament, and Buzurg modes, and started our journey with
Rotation 0. Part 2 addressed Rotations 1-3; and Part 3, Rotation 4.
We now come to Rotation 5.

Again, my warmest thanks to Jacques and Francois and all
participants in the Ethno2 contest for the inspiration they have
lent, and to Ozan Yarman, Shaahin Mohajeri, and George Secor, as
well as now to Jacques Dudon, for the very special knowledge and
wisdom they have shared in the exploration of these and other
modes.

Rotation 0: 1/1
0 138.3 207.4 264.8 345.7 472.3 680.9 704.3 842.6 969.1 1050.0 1176.6 2/1

Rotation 1: 138.3 (~13/12)
0 69.1 126.6 207.4 334.0 542.6 566.0 704.3 830.9 911.7 1038.3 1061.7 2/1

Rotation 2: 207.4 (~9/8 or ~44/39)
0 57.4 138.3 264.8 473.4 496.9 635.2 761.7 842.6 969.1 992.6 1130.9 2/1

Rotation 3: 264.8 (~7/6)
0 80.9 207.4 416.0 439.5 577.7 704.3 785.2 911.7 935.2 1073.4 1142.6 2/1

Rotation 4: 345.7 (~11/9)
0 126.6 335.2 358.6 496.9 623.4 704.3 830.9 854.3 992.6 1061.7 1119.1 2/1

Rotation 5: 472.3 (~21/16)
0 208.6 232.0 370.3 496.9 577.7 704.3 727.7 866.0 935.2 992.6 1073.4 2/1

Rotation 6: 680.9 (~77/52)
0 23.4 161.7 288.3 369.1 495.7 519.1 657.4 726.6 784.0 864.8 991.4 2/1

Rotation 7: 704.3 (~3/2)
0 138.3 264.8 345.7 472.3 495.7 634.0 703.1 760.5 841.4 968.0 1176.6 2/1

Rotation 8: 842.6 (~13/8)
0 126.6 207.4 334.0 357.4 495.7 564.8 622.3 703.1 829.7 1038.3 1061.7 2/1

Rotation 9: 969.1 (~7/4)
0 80.9 207.4 230.9 369.1 438.3 495.7 576.6 703.1 911.7 935.2 1073.4 2/1

Rotation 10: 1050.0 (~11/6)
0 126.6 150.0 288.3 357.4 414.8 495.7 622.3 830.9 854.3 992.6 1119.1 2/1

Rotation 11: 1176.6 (~77/39)
0 23.4 161.7 230.9 288.3 369.1 495.7 704.3 727.7 866.0 992.6 1073.4 2/1

------------------------------------------------------
Rotation 5 (21/16): Rast abounding, sometimes septimal
------------------------------------------------------

0 208.6 232.0 370.3 496.9 577.7 704.3 727.7 866.0 935.2 992.6 1073.4 2/1

In Arab and Turkish practice alike, Rast is first among the maqamat,
taking pride of place and serving as a cardinal point of reference in
describing the gamut and modal system.

This rotation of the Zalzalian 12 set provides forms and variations
abounding for this premier maqam, while favoring a distinctly Turkish
flavor of Rast with large neutral intervals such as a 370-cent third
(a virtually just 26/21) rather wider than those typically favored in
many parts of the Arab world, where al-Farabi's placement of Zalzal's
_wusta_ or neutral third fret at 27/22 (355 cents) could also serve as
a rough guide for much modern practice.

Another feature of this rotation is that in exploring different forms
of Rast, we are often invited and indeed compelled to seek out
septimal variations of a kind described by Safi al-Din al-Urmawi in
the later 13th century. As we shall see, the step avoilable at 728
cents (a near-just 32/21) makes it possible to realize a permutation
of one of his octave tunings -- and one fitting with an Ottoman
flavor!

A good point of departure for our exploration is a Turkish flavor of
Zalzal's tuning, which consists of two conjunct Rast tetrachords plus
an upper tone at 9/8 -- or here slightly larger because of the
temperament with its extended fifths -- to complete the octave.

Rast Rast tone
|----------------------|-----------------------|.........|
rast dugah segah chargah neva hisar ajem gerdaniye
0.0 208.6 370.3 496.9 704.3 866.0 992.6 1200.0
~ 1/1 44/39 26/21 4/3 3/2 33/20 39/22 2/1
209 162 127 207 162 127 207

Here, in keeping with the Turkish flavor of intonation, the steps are
given their Turkish names. Each of the conjunct Rast tetrachords has a
tone near 9:8 (here, at a rounded 207 or 209 cents, actually closer to
44:39), plus a large neutral second at 162 cents (about three cents
narrow of 11/10) and a small one at 127 cents (less than two cents
narrow of 14/13).

This flavor of Rast at 209-162-127 cents is quite close to a
tetrachord of Safi al-Din: 1/1-9/8-99/80-4/3 or 9:8-11:10-320:297
(204-165-129 cents). His 320:297 might be described as a "virtually
tempered" 14:13 enlarged by the small interval of 2080:2079 (0.833
cents) so as to achieve a pure 4:3 fourth. The neutral third at 99/80
or 369 cents differs by this same minute factor from the simpler 26/21
at not quite a cent larger. Although Dr. Fazli Arslan gives no
indication that Safi al-Din associated this tetrachord with Rast, it
nicely fits a style of historical Turkish Rast where, as Ozan Yarman
has noted (Tuning list #90061), the interval between the steps rast
(the final) and segah (its third) is "at 370 cents or so."

The basic character of the Rast family is established by the lower
tetrachord, known in Arab theory as the "root" tetrachord. This lower
tetrachord may be paired with an upper Rast tetrachord (conjunct or
disjunct), and also with certain other forms of upper tetrachords, to
make up the different recognized members of the Rast family of
maqamat in Turkish or Arab theory and practice.

Here the conjuct upper Rast tetrachord brings into play the steps or
perdeler (Turkish plural of _perde_, "pitch") hisar at a large neutral
sixth above rast, here 866 cents, and ajem at a minor seventh, here
993 cents (tempered a bit narrow of the standard 16/9 at 996 cents).
This variety of Rast is thus often known in Turkish theory as an
"Acemli Rast," since it uses perde acem at the minor seventh.

In the Arab world, this same form is known as Nairuz Rast or Nirz
Rast, possibly by association with a medieval form of tetrachord noted
by the Lebanese scholar Nidaa Abou Mrad called Nawruz with a larger
and smaller neutral second followed by a whole tone. In modern terms,
this would correspond with the tetrachord of Maqam Huseyni, here found
on perde neva, the 3/2 step, here at 162-127-207 cents. This upper
Huseyni tetrachord offers an alternative focus which we may explore in
improvisations.

Rast Rast tone
|----------------------|-----------------------|.........|
rast dugah segah chargah neva hisar ajem gerdaniye
0 208.6 370.3 496.9 704.3 866.0 992.6 1200.0
~ 1/1 44/39 26/21 4/3 3/2 33/20 39/22 2/1
209 162 127 207 162 127 207
|-------------------------|
Huseyni (Nawruz)

In this standard form of Acemli or Nairuz Rast, the position of the
steps corresponding to a Pythagorean tuning is taken as more or less
stable: dugah (9/8), chargah (4/3), neva (3/2), and ajem (16/9) -- in
Arabic, dukah, jahargah, nawa, and `ajam. It is the steps at the third
and sixth, segah (Arabic sikah) and hisar, that are subject to many
shadings of taste and local custom in the Arab world and Turkey
alike.

Thus in al-Farabi's version of Zalzal's tuning, a classic example of
this form of Rast (not yet so named in any known source), segah and
hisar are at 27/22 and 18/11, or 355 and 853 cents. In Safi al-Din and
they are placed at a complex Pythagorean 8192/6561 and 32768/19683, or
384 and 882 cents, only a schisma of 32805/32768 (1.954 cents)
narrower than a just 5/4 and 5/3, a variety of Rast also described by
Qutb al-Din al-Shirazi (c. 1300), and adopted in 20th-century Turkish
theory as the norm.

The tuning of segah and hisar at 370 and 866 cents -- respectively
approximationg 26/21 or 99/80 and 104/63 or 33/20 -- represents a
shading higher than that followed in much of the Arab world, but at
the same time distinctly lower than the 5-limit or quasi-5-limit
flavor of Safi al-Din, Qutb al-Din, and much modern Turkish practice.

While our tuning nicely represents the historical Turkish flavor of
Rast described by Ozan with segah at around 26/21, in certain parts of
the Arab world such as Syria, as well as in Turkey, one encounters a
tuning at around 16/13 (359 cents) or the slightly higher 16 commas
(e.g. 16/53 octave in 53-EDO) at 362 cents. This shading might be
considered in Arab terms a "bright" Rast; but by Turkish standards, as
Ozan notes, 16/13 or 16 commas represents a "low segah," with 26/21
representing more of an "upper range" -- still lower, of course, than
the 5-limit or quasi-5-limit interpretation also popular in Turkey.

All of these Turkish and Arab tunings are shadings of the same basic
structure which has been attested for more than a millennium: two
conjunct Rast tetrachords plus an upper tone.

To move from this Acemli or Nairuz Rast to two other standard modern
forms of Rast, we would need a step not present in this rotation: a
regular major sixth, the classic 27/16 (906 cents) of medieval theory,
realized in our temperament at a somewhat wider 911 or 912 cents, very
close to a just 22/13.

In place of 22/13, we have a step at 935 cents, about two cents wide
of a just 12/7, or septimal major sixth. We also have available a
septimal major second at 232 cents, less than a cent wide of 8/7. Using these septimal steps, we can construct a variation on the most
typical modern form of Rast with _disjunct_ Rast tetrachords joined by
a middle tone:

Rast tone Rast
|-------------------------|.......|------------------------|
sarp sarp
rast dugah segah chargah neva huseyni evdj gerdaniye
0 232.0 370.3 496.9 704.3 935.2 1073.4 1200.0
~ 1/1 8/7 26/21 4/3 3/2 12/7 13/7 2/1
232 138 127 207 231 138 127

This septimal version of Rast closely approximates 1/1-8/7-26/21-4/3
or 8:7-13:12-14:13 at 0-231-370-498 cents (231-139-128 cents), a
tetrachord using the same melodic steps as a tuning of Safi al-Din
where the smaller 14:13 step precedes the larger 13:12, thus
1/1-8/7-16/13-4/3 or 8:7-14:13-13:12 (0-231-359-498 cents or
231-128-139 cents).

More generally, in a disjunct Rast, the kind often assumed in modern
Turkish or Arab theory if the name of the maqam is mentioned without
qualification, we have in the upper tetrachord a major sixth step
huseyni (or here its septimal variation) and a neutral seventh step
evdj (in Arabic, awj). The steps at septimal versions of a tone (8/7)
and major sixth (12/7) are, following a suggestion of Ozan Yarman,
termed sarp dugah and sarp huseyni to show that they are higher by a
septimal comma (in JI, 64:63 or 27.264 cents), here a tempered
difference of about 23 cents.

The neutral seventh step evdj at some 1073 cents, or a tempered 13/7
(1072 cents), very effectively pulls up to the final or its octave by
a step of 127 cents.

As mentioned in the previous discussion of Rotation 4 (Part 3), the
Buzurg modality found in that rotation invites a pleasant modulation
to the Rast found in this, and more specifically to the disjunct
septimal Rast we are now considering. Another version of the previous
diagram will show the nexus between these modal systems:

Rast tone Rast
|-----------------------|..........|------------------------|
sarp sarp
arak rast dugah segah chargah neva huseyni evdj gerdaniye
127 232 138 127 207 231 138 127
-126.6 0.0 232.0 370.3 496.9 704.3 935.2 1073.4 1200.0
~ 13/14 1/1 8/7 26/21 4/3 3/2 12/7 13/7 2/1
Buzurg tone Buzurg
|-----------------------|......!.....|---------------------|
sarp sarp
arak rast dugah segah chargah hijaz neva huseyni evdj
0.0 126.6 358.6 496.9 623.4 704.3 830.9 1061.7 1200.0
~ 1/1 14/13 16/13 4/3 56/39 3/2 21/13 24/13 2/1

From this perspective, taking Rast (the premier maqam) as a point of
reference, our Buzurg mode starts on the step 127 cents below perde
rast, or an octave below perde evdj or a 13/7 with respect to rast, a
step known in Turkish theory as arak, and in Arabic as `iraq. It will
be seen that a Buzurg mode with two disjunct Buzurg tetrachords (here
at a tempered 1/1-14/13-16/13=4/3) shares almost all of its pitches
with our septimal Rast, the exception being perde hijaz at a 3/2 (here
tempered at 704 cents) above Buzurg's final of arak and a small
tritone of 578 cents or around 88/63 above perde rast.

A delightful bridge or pivotal note connecting Buzurg with Rast is
perde chargah, not a note of either Buzurg tetrachord but rather the
extra note of the lower pentachord at 623 cents (~56/39 or 63/44)
which divides the middle tone (segah-hijaz) between these tetrachords
into a small neutral second (segah-chargah) plus a form of semitone or
limma (chargah-hijaz), here at 127-81 cents, and in the corresponding
medieval JI version at 14:13-117:112 or 128-76 cents. As I noted in
the discussion of Buzurg (Part 3, Rotation 4), perde chargah might be
used in the manner of a Persian _moteqayyer_ or variation on the 3/2
fifth apt, for example, as the top note of a phrase descending toward
the final.

To use this note as a bridge to Rast -- or, in the yet more evocative
expression of a modern Arab teacher reported by Scott Marcus, as "the
door" or _al-bab_ providing a modulatory passage to the new maqam --
we might begin at segah (the 4/3 of Buzurg), ascend to chargah, and
from there descend back through segah as if on the way to arak, but
then instead shift the focus to perde rast as the 1/1, with chargah
now serving as the 4/3 of a septimal Rast tetrachord above this new
final. This shift in the role of perde chargah from the "extra" 56/39
or 63/44 step of Buzurg to the 4/3 of Rast, the note providing the
fourth of the all-important lower or "root" tetrachord, is pleasant
indeed.

A third form of Rast which in Turkish theory is sometimes known (like
our first form with conjunct Rast tetrachords) as Acemli Rast, and in
the Arab world by the name of Suzdular. The latter name is a variation
on Suz-i Dilara, understood in a fuller Arab or Turkish appreciation
to be in fact a _tarqib_ or "composite" mixing elements of various
simple maqamat, with the simpler Arab concept of "Suzdular" evidently
referring to one combination of tetrachords that may arise in the
course of Suz-i Dilara:

Rast tone Buselik
|-------------------------|.......|------------------------|
sarp sarp
rast dugah segah chargah neva huseyni ajem gerdaniye
0.0 232.0 370.3 496.9 704.3 935.2 992.6 1200.0
~ 1/1 8/7 26/21 4/3 3/2 12/7 39/22 2/1
232 138 127 207 231 57 207
|-----------------------|
Mahur

In modern Turkish and Arab practice alike, where the most common form
of ascending Rast is the previous type with two disjunct Rast
tetrachords featuring a major sixth and neutral seventh, this general
type with a major sixth and minor seventh is a characteristic
descending form. The upper tetrachord thus presents us with a new type
of interval not present in either the conjunct or disjunct form: a
semitone between the major sixth (perde huseyni) and minor seventh
(perde ajem), in contrast to the tones and neutral seconds that
otherwise prevail. In our septimal flavor, this melodic nuance is
accentuated by the narrowness of the semitone (sarp huseyni-ajem) at
only 57 cents, which might represent an inaccurate 28/27 (the classic
septimal thirdtone of Archytas) or an accurate 91/88 (the difference,
for example, between 12/7 and 39/22, the latter narrower than 16/9 by
a small comma or kleisma of 352:351).

The upper tetrachord above the 3/2 step at 0-231-288-496 cents or
231-57-207 cents, approximately 1/1-8/7-13/11-4/3 (8:7-91:88-44:39),
is quite close to a permutation of a Greek tuning, the Diatonic of
Archytas at 1/1-8/7-32/27-4/3 (0-231-294-498 cents) or 8:7-28:27-9:8
(231-63-204 cents). In my experience, this tetrachord coupled to the
lower Rast tetrachord with its kindred septimal flavor serves as
delightful mode of Rast in its own right as well as nicely
complementing the most common modern ascending form with a neutral
seventh.

In modern Ottoman terms, this tetrachord may be seen as a septimal
variant on Buselik, whose basic pattern is tone-semitone-tone, with
the minor third often around a Pythagorean 32/27 or 294 cents, but
possibly rather narrower, with a septimal flavor around 7/6 (267
cents) possible. Our regular 288-cent minor third, near 13/11 (289
cents), fits this concept. While the Arab term Nahawand might also
fit, the corresponding Turkish name Nihavend would suggest, at least
in theory, a larger minor third typically around 6/5 (316 cents).

This flavor of Acemli Rast offers another tetrachord with a septimal
flavor inviting exploration in improvisations: the tetrachord on the
4/3 step at 0-207-438-496 cents, or 207-231-57 cents. This is a close
to another permutation of the Archytas Diatonic appearing in Safi
al-Din, but with the first two steps in reverse order: 1/1-8/7-9/7-4/3
(0-231-435-498 cents), or 8:7-9:8-28:27 (231-204-63 cents). Here the
regular tone at 207 cents, close to 9:8, comes first, and then the
septimal tone at 231 cents (a virtually just 8:7), and finally the
small 28:27 semitone narrowed in this temperament from 63 to 57 cents.

This general type of tetrachord with tone-tone-semitone is known in
Arab and Persian theory alike as Mahur, and in Arab theory often as
`Ajam.

So far, we have encountered three common forms of Rast sharing in
common their lowest tetrachord, but varying in their upper tetrachords
(3/2-2/1). We might summarize the situation as follows:

Acemli (Nairuz) Rast: neva hisar acem gerdaniye
704.3 866.0 992.6 1200.0
162 127 207

(sarp)
Disjunct Rast: neva huseyni evdj gerdaniye
704.3 935.2 1073.4 1200.0
231 138 127

(sarp)
Acemli Rast: neva huseyni acem gerdaniye
(Arab Suzdular) 704.3 935.2 992.6 1200.0
231 57 207

The variable steps are the sixth, neutral or major (hisar/huseyni);
and the seventh, minor or neutral (ajem/evdj). In addition to the
above standard forms of Rast with neutral sixth and minor seventh,
major sixth and neutral seventh, or major sixth and minor seventh,
there is a fourth possibility highlighted by Jacques Dudon: the
combination of neutral sixth and neutral sixth, or hisar-evdj:

Rast tone Ishku (?)
|-------------------------|.......|------------------------|
rast dugah segah chargah neva hisar evdj gerdaniye
0.0 208.6 370.3 496.9 704.3 866.0 1073.4 1200.0
~ 1/1 44/39 26/21 4/3 3/2 33/20 13/7 2/1
207 162 127 207 162 207 127

Here I have permitted the second step of the lower tetrachord, dugah,
to revert back to its usual position at around 9/8 or 44/39, although
there is no reason why one could not sarp dugah at 8/7 as well. Our
attention is focused especially on the upper tetrachord with its
bright neutral sixth and neutral seventh above the final. This
tetrachord has intervals of 0-161.7-369.1-495.7 cents, with steps of
161.7-207.4-126.6 cents. A fairly close JI equivalent would be another
permutation of Safi al-Din's "Medium conjunct" tetrachord, yielding
1/1-11/10-99/80-4/3 (0-165.0-368.9-498.0 cents), with steps of
11:10-9:8-320:297 (165.0-203.9-129.1 cents).

In the most loose and generic terms, the upper tetrachord might be
called "Moha," short for Dudon's Mohajira, a pattern with lower
neutral second, tone, and upper neutral second. However, in his
concept of Mohajira proper, the neutral third step is generally
somewhere between 39/32 (342 cents) and 16/13 (359 cents), with the
bright Ottoman flavor here at 369 cents (26/21) clearly out of this
range.

One possibly more apt term is Ishku, Dudon's JI sequence with
differential coherence (-c) which makes available a tetrachord of
1/1-79/72-89/72-4/3 (0-160.6-367.0-498.0), or 79:72-89:79-96:89
(160.6-206.3-131.1 cents). See Dudon's sumer.scl (2-Middle East
folder) in his Ethno2 collection. Since our upper tetrachord above is
very similar, the name Ishku may be an auspicious choice.

The mode as a whole we might term Rast-Ishku by analogy with a Scala
archive file named rast_moha.scl for a tuning by Jacques Dudon with a
lower Rast and an upper Mohajira tetrachord.

Having surveyed four varieties of Rast, we might briefly consider a
few polyphonic cadences. Our conjunct or Nairuz Rast with a usual
dugah near 9/8 and a minor seventh step (perde ajem) offers a stately
three-voice cadence identical to a favorite formula of 13th-century
Europe, the only difference being the tempered rather than Pythagorean
intonation resulting from the gently extended fifths and narrowed
fourths. Each voice moves by a regular tone of around 9:8 so that a
minor sixth expands to an octave and a minor third to a fifth, thus
arriving at a complete 2:3:4 sonority.

993 39/22 1200 2/1
497 4/3 704 3/2
209 44/39 0 1/1

Here the minor third at sixth at 288 and 784 cents are close to just
ratios of 13/11 and 11/7 (289 and 782 cents), by comparison to the
Pythagorean 32/27 and 128/81 at 294 and 792 cents. It is curious that
a maqam in a flavor of intonation recalling a tetrachord of Safi
al-Din should invite a polyphonic cadence favored in Europe during the
same century.

In our other form of Acemli Rast with septimal major second and major
sixth steps at around 8/7 (sarp dugah) and 12/7 (sarp dugah), the same
cadence takes on a special color, and if possible has a yet statelier
effect, because of the raised position of dugah at a comma higher,
thus making the minor third and sixth above it a comma narrower, at
around 264.8 and 760.5 cents, close to 7/6 and 14/9 (267 and 765
cents).

993 39/22 1200 2/1
497 4/3 704 3/2
232 8/7 0 1/1

In addition to the vertical septimal color, the wide 8:7 whole-tone
step in the lowest voice may lend this progression an especially
spacious and stately quality.

With our septimal flavor of disjunct Rast, the high position of dugah
at around 8/7 makes possible the use of an especially engaging
Zalzalian sonority in near-just form: 8:12:13 (0-702.0-840.5 cents),
here tempered at 0-703.1-841.4 cents.

1073 13/7 1200 2/1
935 12/7 704 3/2
232 8/7 0 1/1

Harmonically, while the 13:12 neutral second between the upper voices
contributes some notable tension, the outer 13:8 neutral sixth has a
rather "bright" color, an effect which George Secor attributes to the
nature of this interval as the 13th harmonic of the fundamental. While
the two lower voices descend by wide 8:7 steps, the upper voice
ascends by the small Zalzalian second at around 14:13 so typical of
this flavor of Rast. This progression illustrates how Zalzalian
polyphony may arrive at charming results quite different from anything
in the known practice of medieval Europe.

For our Rast-Ishku variation with dugah in its regular position around
9/8, or here closer to 44/39, we have available this characteristic
cadence:

1073 13/7 1200 2/1
497 4/3 704 3/2
209 44/39 0 1/1

While the lower minor third around 13/11 expands to a fifth, the outer
large neutral sixth at around 865 cents, near 28/17 (864 cents) or
33/20 (867 cents), expands to the octave with the lower voice
descending by a tone and the highest ascending by 127 cents. A special
color is lent by the tritone between the two upper voices, about 577
cents (close 39/22 at 574 cents or 88/63 at 579 cents). The effect may
somewhat recall a cadence in Europe around 1200 with a minor third
expanding to a fifth and a major sixth to an octave, with the upper
tritone adding color and tension, but at the same time is unique.

Since this last cadence does not directly involve the sixth degree of
Rast, it may be found not only in our Rast-Ishku with a neutral sixth
step (hisar), but also in the kind of usual disjunct Rast not
available in this rotation with a regular major sixth step (huseyni),
where it is equally characteristic and effective. The steps involved
in the cadence, dugah-chargah-evdj for the first sonority resolving to
rast-neva-gerdaniye, are the same in both these forms of Rast.

Before leaving this rotation, we should note a mode of Safi al-Din
placing two of his septimal tetrachords at 1/1-8/7-16/13-4/3 or
8:7-14:13-13:12 in a _conjunct_ arrangement, here as in our disjunct
septimal Rast with the order of the two neutral seconds reversed so as
to produce a higher neutral third step at around 26/21 (370 cents)
rather than 16/13 (359 cents):

Rast Rast tone
|-------------------------|------------------------|,,,,,,,,|
sarp sarp
rast dugah segah chargah neva hisar ajem gerdaniye
0.0 232.0 370.3 496.9 727.7 866.0 992.6 1200.0
~ 1/1 8/7 26/21 4/3 32/21 33/20 39/22 2/1
232 138 127 231 138 127 207

Melodically, this is a delightful mode, with polyphony somewhat more
delicate because of the imperfect fifth above the final. However,
medieval Near Eastern and European theory suggest one solution: the
generous use of fourths, much favored in the simple polyphonic
practice around the 9th-11th centuries of what was known to theorists
such as Ibn Sina as _tarqib_ (a "composite" or "mixture" of
simultaneous voices), and Europeans such Guido d'Arezzo as _diaphonia_
("singing apart") or _organum_ (an "organized" mixture of voices).
It is interesting that both Ibn SIna and Guido, contemporaries making
their monumental contributions in the early 11th century, both prefer
the fourth as an especially pleasing concord.

In describing how three voices may sing in parallel, Guido likewise
prefers a sonority which could serve as a close above the final,
3:4:6, with a lower 4:3 fourth and an upper 3:2 fifth.

A two-voice texture moving mostly in parallel fourths, judiciously
mixed with fifths as one moves about the mode, would nicely complement
the conjunct tetrachordal structure, with an option for sometimes
adding a third voice at the octave. Such an approach seems in line
with the medieval theory of _tarqib_, a term now more familiar in its
meaning of a "composite" maqam formed as a mixture of simpler maqamat,
rather than a mixture of simultaneous pitches. Traditional practices
favoring polyphony in fourths, as on the kemence or fiddle in Turkish
folk music of the Black Sea region, and also the Pontic lyre, also
provide precedents for such a technique.

Finally, we should quickly mention that a form of Maqam Penchgah is
also available in this rotation, but with some complications resulting
from the lack of a regular as opposed to septimal major sixth. Making
fullest use of this maqam involves an artful alternation between the
two positions of dugah at around 9/8 or 8/7, a nicety likely better
understood after an introduction to the regular Penchgah, which occurs
in the coming Rotation 9.

For now, it may suffice to show the septimal form of this Penchgah,
much like our disjunct septimal Rast but the 4/3 step replaced by a
step at 577.7 cents or around 88/63:

Penchgah Rast
|---------------------------------|------------------------|
sarp sarp
rast dugah segah hijaz neva huseyni evdj gerdaniye
0.0 232.0 370.3 577.7 704.3 935.2 1073.4 1200.0
~ 1/1 8/7 26/21 88/63 3/2 12/7 13/7 2/1
232 138 207 127 231 138 127

To explain why a regular version of perde dugan at around 9/8 (or here
44/39) plays an important role also, let us consider this delightful
Penchgah cadence for three-voice polyphony:

1073 13/7 1200 2/1
578 88/63 704 3/2
209 44/39 0 1/1

Here a large neutral third at 369 cents or around 26/21 expands to a
fifth, while a large neutral sixth at 865 cents between 28/17 and
33/20 expands to an octave, with the lower voice descending by a tone
while both upper voices ascend by approximate 14/13 steps. Placing
dugah at its regular position around 44/39 permits the neutral third
and sixth to have their full size, with a bright and "expansive"
effect so typical of polyphony in Maqam Penchgah, and recalling the
premier final cadence of 14th-century Europe where a major third and
sixth expand to fifth and octave, the lower voice descending by a tone
and the upper voices ascending together semitonally.

This is not to say that a version with dugah at 8/7 would be "wrong,"
but rather that this form should serve as as option supplementing the
above flavor so summing up the flavor of polyphony in Maqam Penchgah:

1073 13/7 1200 2/1
578 88/63 704 3/2
232 8/7 0 1/1

Here the neutral third and sixth at 346 cents and 841 cents
(around 11:9 and 13:8), each a comma smaller than in the previous
example, offer a pleasant flavor of their own. Happily, as we shall
see, Rotation 9 also has available both versions of this cadence.

Most appreciatively,

Margo Schulter
mschulter@...

----------------------------------
Ethno Extras: Zalzalian 12 set
Part 5: Rotation 6
----------------------------------

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Hello, all.

To provide a convenient "itinerary" for the Zalzalian 12 set, so that
people can keep track of where we've been and where we're going, I'll
place at the beginning of each part a summary of the rotations adapted
from Scala output. Part 1 included an introduction on Ethno Extras,
the O3 temperament, and Buzurg modes, and started our journey with
Rotation 0. Part 2 addressed Rotations 1-3; Part 3, Rotation 4;
and Part 4, Rotation 5. We now come to Rotation 6.

Again, my warmest thanks to Jacques and Francois and all
participants in the Ethno2 contest for the inspiration they have
lent, and to Ozan Yarman, Shaahin Mohajeri, and George Secor, as
well as now to Jacques Dudon, for the very special knowledge and
wisdom they have shared in the exploration of these and other
modes.

Rotation 0: 1/1
0 138.3 207.4 264.8 345.7 472.3 680.9 704.3 842.6 969.1 1050.0 1176.6 2/1

Rotation 1: 138.3 (~13/12)
0 69.1 126.6 207.4 334.0 542.6 566.0 704.3 830.9 911.7 1038.3 1061.7 2/1

Rotation 2: 207.4 (~9/8 or ~44/39)
0 57.4 138.3 264.8 473.4 496.9 635.2 761.7 842.6 969.1 992.6 1130.9 2/1

Rotation 3: 264.8 (~7/6)
0 80.9 207.4 416.0 439.5 577.7 704.3 785.2 911.7 935.2 1073.4 1142.6 2/1

Rotation 4: 345.7 (~11/9)
0 126.6 335.2 358.6 496.9 623.4 704.3 830.9 854.3 992.6 1061.7 1119.1 2/1

Rotation 5: 472.3 (~21/16)
0 208.6 232.0 370.3 496.9 577.7 704.3 727.7 866.0 935.2 992.6 1073.4 2/1

Rotation 6: 680.9 (~77/52)
0 23.4 161.7 288.3 369.1 495.7 519.1 657.4 726.6 784.0 864.8 991.4 2/1

Rotation 7: 704.3 (~3/2)
0 138.3 264.8 345.7 472.3 495.7 634.0 703.1 760.5 841.4 968.0 1176.6 2/1

Rotation 8: 842.6 (~13/8)
0 126.6 207.4 334.0 357.4 495.7 564.8 622.3 703.1 829.7 1038.3 1061.7 2/1

Rotation 9: 969.1 (~7/4)
0 80.9 207.4 230.9 369.1 438.3 495.7 576.6 703.1 911.7 935.2 1073.4 2/1

Rotation 10: 1050.0 (~11/6)
0 126.6 150.0 288.3 357.4 414.8 495.7 622.3 830.9 854.3 992.6 1119.1 2/1

Rotation 11: 1176.6 (~77/39)
0 23.4 161.7 230.9 288.3 369.1 495.7 704.3 727.7 866.0 992.6 1073.4 2/1

----------------------------------------------------------------------
Rotation 6 (77/52): In search of medieval (and modern) modes, sans 3/2
----------------------------------------------------------------------

0 23.4 161.7 288.3 369.1 495.7 519.1 657.4 726.6 784.0 864.8 991.4 2/1

The first thing we may note about Rotation 6 is that there is no 3/2
step, the closest approach being 726.6 cents, a tempered 32/21. Unfortunately, it's too easy stop here and say, "Let's move on to the
next rotation" -- at least for me, raised in a European tradition that
"a proper mode must have a proper fifth."

However, medieval and modern Near Eastern practice and theory are more
forgiving, and featuring various modes which do quite well without a
just or approximate 3/2 step.

One fine example is Maqam `Iraq as presented by Safi al-Din al-Urmawi
and Qutb al-Din al-Shirazi, rather different than the modern Arab
maqam of the same name, and here also presented in a different shade
of intonation than that specified by these theorists and explained by
the modern scholars Dr. Fazli Arslan and Owen Wright.

In the following notation, T means a tone, K a large neutral second
around 11/10, and S a small neutral second around 14/13. In Safi
al-Din's 17-note Pythagorean gamut, however, K would represent 10/9
and S, 16/15 -- or, more precisely, 65536/59049 (180.45 cents) and
2187/2048 (113.69 cents), the former a 1.95-cent schisma smaller and
the latter the same schisma larger than the simpler pental or 5-limit
ratios also known in this era.

`Iraq `Iraq tone
|-------------------------|--------------------------|........|
0.0 161.7 369.1 495.7 657.4 864.8 991.4 1200.0
1/1 56/51 26/21 4/3 117/80 28/17 44/39 2/1
K T S K T S T
162 207 127 162 207 127 209

In medieval modes with or without a 3/2 step, this conjunct tetrachord
structure is very common. As I remarked in discussing an instance of
this 162-207-127 cent flavor of tetrachord in the previous article in
this series (Part 4, Rotation 5), while the structure with lower and
upper neutral seconds and a middle tone around 9:8 might suggest
Jacques Dudon's Mohajira or "Moha" for short, the use of a neutral
third step larger than 16/13, and here specifically around 26/21,
suggests the name Ishku, a differentially coherent (-c) JI series used
in his sumer.scl in the Ethno2 collection.

If this charming mode is used as the basis of polyphony, then the
structure of conjunct tetrachords invites a style leaning to parallel
fourths, sometimes mixed at fifths at locations where they are
available.

It should briefly be noted that while the above version with two
conjunct tetrachords and an upper tone is the complete maqam as
specified by Qutb al-Din, Safi al-Din includes an additional step
subdividing the upper tone. Owen Wright places it at 40/21, thus
dividing this tone into a lower 15:14 (119 cents) and an upper 21:20
(85 cents), a division later used, for example, in Kraig Grady's
renowned Centaur "7-cap" tuning. Dr. Arslan, in a Pythagorean
interpretation, places it at a 180-cent diminished third above the
16/9 step, or a Pythagorean comma below the octave. Neither variety of
extra step is available here.

Another kind of conjunct tetrachordal structure invited by this
rotation is a variation the modern Turkish Huseyni, which as described
both by Sulphi Ezgi in 1933 and by Ozan Yarman should have a lower
step around 11/10 or 165 cents, Ezgi's 1/1-11/10-32/27-4/3
(0-165-294-498 cents), or 11:10-320:297-9:8 (165-129-204 cents), as
mentioned in the previous article, being a permutation of Safi
al-Din's 1/1-9/8-99/80-4/3 (0-204-369-498 cents) or 9:8-11:10-320:297.

Our temperament closely approaches Ezgi's tuning of Huseyni, which in
its usual form uses two disjunct rather than conjunct Huseyni
tetrachords:

Huseyni Huseyni tone
|-------------------------|--------------------------|........|
0.0 161.7 288.3 495.7 657.4 784.0 991.4 1200.0
1/1 56/51 13/11 4/3 117/80 11/7 44/39 2/1
K S T K S T T
162 127 207 162 127 207 209

Experience has shown me that this mode is most pleasant not only
melodically but polyphonically, often negotiating the lower tetrachord
in parallel fourths and the upper in parallel fifths, speaking in
terms of the position of the lower voice in a two-voice texture. This
approach may evoke the technique of playing in parallel fourths on a
rebab, which I recall hearing, for example, in a recording of a
performance in Afghanistan, or likewise on the kemence.

As it happens, this mode, while it was suggested to me by the Huseyni
tetrachord of modern Turkish music, matches a maqam of Safi al-Din
with two conjunct tetrachords of a type called Nawruz likewise with a
pattern, using the above modern Turkish sign,s of K-S-T (in a
Pythagorean interpretation, 180-114-204 cents), and likewise named
Maqam Nawruz. By any name, it is a worthy maqam.

So far, we have considered modes featuring two identical tetrachords
arranged conjunctly. Safi al-Din's maqam called Rahawi -- a name with
a range of uses and interpretations in his writings and others! -- as
interpreted by Dr. Arslan involves a conjunct union of a lower
medieval `Iraq tetrachord and an upper medieval Nawruz:

`Iraq Mawruz tone
|-------------------------|--------------------------|........|
0.0 161.7 369.1 495.7 657.4 784.8 991.4 1200.0
1/1 56/51 26/21 4/3 117/80 11/7 44/39 2/1
K T S K S T T
162 207 127 162 127 207 209

Here the contrast between the bright 26/21 neutral third of the lower
tetrachord and the minor sixth of the upper Nawruz adds a pleasing
element.

Improvising in this medieval Nairuz, one option I find attractive is
sometimes using a minor rather than neutral third in the lower
tetrachord, thus transforming it from `Iraq to Nawruz, and modulating
to a medieval Maqam Nawruz (or in modern terms a "conjunct Huseyni").
Another, as realized in this rotation, would be sometimes to raise the
second step of the upper tetrachord from 657 to 727 cents, thus
transforming the genus from Nawruz to that what modern theory would
call a form of Buselik at 0-230.9-288.3.495.7 cents or 231-57-207
cents -- or, in Safi al-Din's terms, Nawa.

`Iraq Buselik (Nawa) tone
|-------------------------|--------------------------|........|
0.0 161.7 369.1 495.7 726.6 784.8 991.4 1200.0
1/1 56/51 26/21 4/3 32/21 11/7 44/39 2/1
162 207 127 231 57 207 209

While the 727-cent step would not form a stable vertical fifth above
the final -- the fourth instead serving this function in polyphony --
I find that it nicely serves melodically as kind of substitute for 3/2
in this upper Buselik tetrachord, which in the course of a seyir
nicely complements the usual Nawruz form.

Another modulation I am drawn to in Maqam Rahawi, after ascending in
my medieval Nawruz or modern "conjunct Huseyni" flavor, curiously
seems to me a septimal variation on a _disjunct_ `Iraq-Nawruz
structure, but with a difference! Here it may be helpful to show again
the Nawruz flavor, identical to the "conjunct Huseyni" above, so that
the transformation may be better seen:

Mawruz Mawruz tone
|-------------------------|--------------------------|........|
0.0 161.7 288.3 495.7 657.4 784.0 991.4 1200.0
1/1 56/51 13/11 4/3 117/80 11/7 . 44/39 2/1
K S T K S T T
162 127 207 162 127 207 209

`Iraq 8:7 tone Shur (21:16)
|-------------------------|..........|------------------------|
0.0 161.7 369.1 495.7 726.6 864.8 991.4 1200.0
1/1 56/51 26/21 4/3 32/21 28/17 44/39 2/1
162 207 127 231 138 127 209

Here, in descending from the octave, the minor sixth at 784 cents
around 11/7 is raised to a large neutral sixth at 865 cents (28/17);
and the very large tritone or "subfifth" at 657 cents (e.g. 117/80) to
a 727-cent septimal fifth near 32/21 -- so that this 32/21 step
becomes the basis for an upper disjunct tetrachord of 138-127-209
cents, or approximately 13:12-14:13-9:8, or 48:52:56:63. This is much
like the 12:13:14:16 version of Ibn Sina's famous tetrachord, viewed
by some as his own original version, which I often term "Shur" because
of its aptness for that Persian dastgah or modal family -- but with
the upper 8:7 tone foreshortened by a comma to 9:8 (or here a
virtually just 44:39), and the tetrachord itself from 4:3 to 21:16,
thus 0-138-265-473 cents or 1/1-13/12-7/6-21/16!

What I should confess is that originally I found this modulation in a
tuning system with a 3/2 step available where the corresponding
tetrachord was something like 132-154-209 cents or 0-132-286-495
cents, another possible sharing of Shur, or perhaps of a popular
Lebanese flavor of the Arab Maqam Bayyati, where Amine Beyhom reports
a taste for a lower step around 130 cents. However, I find a disjunct
upper tetrachord here on the 32/21 step at 0-138-127-209 steps a happy
choice also.

Given Jacques Dudon's boldness in embracing 21/16 fourths and 32/21
fifths, as in his famous article on septimal gamelan tunings where he
speaks of the Surakarta (S) style in its characteristic tuning of using "a wonderful 32:21" above the "tonic," in a scale that "dances
with the wolves."

While the vertical concord of these wide fifths in gamelan music
brings into play the matter of timbre, Dudon's observation that a
viable tonic or final need not have "a 3:2 perfect fifth" holds also
for maqam and dastgah music, where modern as well as medieval modes
without a 3/2 step are well known. Further, Syrian theorists favoring
the Turkish system of 53 commas sometimes describe for certain maqamat
steps at 21 commas (475 cents) or 32 commas (725 cents) above the
final, close to 21/16 and 32/21 at 471 and 729 cents. And measurements
of flexible-pitch performances show that nuances of this kind do occur
in practice.

Above all, I would emphasize that exploring medieval maqamat such as
those discussed here means not only learning the steps and intervals
(itself often a question of varied opinions and usages, medieval and
modern), but learning or rediscovering what is called sayr in Arabic
and seyir in Turkish, the melodic path or procedure of a maqam. With
great humility, I would emphasize that I have only begun to become
acquainted with the Islamic music tradition of a millennium and more
within the last ten years, so that my knowledge and intuition are
alike tentative, making me more all the more indebted to musicians and
scholars immersed in this music from Safi al-Din and Qutb al-Din to
Shaahin Mohajeri and Ozan Yarman.

Most appreciatively,

Margo Schulter
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter wrote:

> ... Rast tone Ishku (?)
> |-------------------------|.......|------------------------|
> rast dugah segah chargah neva hisar evdj gerdaniye
> 0.0 208.6 370.3 496.9 704.3 866.0 1073.4 1200.0
> ~ 1/1 44/39 26/21 4/3 3/2 33/20 13/7 2/1
> 207 162 127 207 162 207 127
>
> Here I have permitted the second step of the lower tetrachord, dugah,
> to revert back to its usual position at around 9/8 or 44/39, although
> there is no reason why one could not sarp dugah at 8/7 as well. Our
> attention is focused especially on the upper tetrachord with its
> bright neutral sixth and neutral seventh above the final. This
> tetrachord has intervals of 0-161.7-369.1-495.7 cents, with steps of
> 161.7-207.4-126.6 cents. A fairly close JI equivalent would be another
> permutation of Safi al-Din's "Medium conjunct" tetrachord, yielding
> 1/1-11/10-99/80-4/3 (0-165.0-368.9-498.0 cents), with steps of
> 11:10-9:8-320:297 (165.0-203.9-129.1 cents).
>
> In the most loose and generic terms, the upper tetrachord might be
> called "Moha," short for Dudon's Mohajira, a pattern with lower
> neutral second, tone, and upper neutral second. However, in his
> concept of Mohajira proper, the neutral third step is generally
> somewhere between 39/32 (342 cents) and 16/13 (359 cents), with the
> bright Ottoman flavor here at 369 cents (26/21) clearly out of this
> range.
>
> One possibly more apt term is Ishku, Dudon's JI sequence with
> differential coherence (-c) which makes available a tetrachord of
> 1/1-79/72-89/72-4/3 (0-160.6-367.0-498.0), or 79:72-89:79-96:89
> (160.6-206.3-131.1 cents). See Dudon's sumer.scl (2-Middle East
> folder) in his Ethno2 collection. Since our upper tetrachord above is
> very similar, the name Ishku may be an auspicious choice.
>
> The mode as a whole we might term Rast-Ishku by analogy with a Scala
> archive file named rast_moha.scl for a tuning by Jacques Dudon with a
> lower Rast and an upper Mohajira tetrachord.

Dear Margo,
Before I get deeper in the lecture of your last wonderful posts, just
a quick comment on what is "Ishku" in my sense - also because
sumer.scl, described as a "Rast scale using Ishku -c variations", was
not very explicit about it ! So now I try to do better :
Ishku is a recurrent sequence (and also a rhythm and a fractal
waveform), based on an -c algorithm that can be generalised that way :
In a triad C : D : E, if you have E - D = C/8 (in frequencies), C :
D : E start a Ishku sequence. Next note will be "F sori" (half-sharp
F) and so on, where each new tone will unvariably tend toward
184.36788 c., that is a slightly large 10/9.
The simplest Ishku series I know, and part of it is in this
sumer.scl, is :
(64) 72 80 89 99 ... (80 - 72 = 8, 89 - 80 = 9, 99 - 89 = 10 ...
and next numbers will jump 3 octaves up, to 881 and 980)

Seen like this, a Ishku series has not much relation with Rast or
Mohajira. But when we look closer, we see they have.
We can explain it from the solution of Ishku's recurrent sequence,
8x^2 - 8x = 1, or (sqrt of 6 + 2) /4 = 1.112372436,
because it is also the arithmetical mean between the semififth (sqrt
of 6 / 2) and 1/1, that means Ishku tones and semififths are always
differentially coherent.
For example in the neutral triad 40 : 49 : 60 that we find in
sumer.scl between D and A,
40 + 49 = 89 = E
and 89 - 80 = 49 - 40 = C.
By coïncidence, but harmony is based on those, three Ishku tones are
very close to 11/8 (we verify it in the series above in the ratio
between 72 and 99).
11/8 or similar have two occurences in Rast, same in Mohajira, it's
another possible bridge with Ishku.
Then the fact that Ishku^3 ~11/8, because of other coherences and
natural reasons suggests musical Ishku-based pentachords, ending with
a fifth,
such as 72 : 80 : 89 : 99 : 108
Here we find that 99 - 72 = 27 and 108 - 99 = 9 (pure 3/2 fifths),
but the Ishku fractal value can also suggest -c corresponding fifths
of 703.592 and 708.4936 c... that I know you may not dislike !
In conclusion, in absence of 80 (or 88, since 88 - 79 = 9 is also of
a Ishku kind) the sumer.scl subset 72 : 79 : 89 : 96 will only be an
uncomplete example of a Ishku tetrachord, nevertheless with 96 - 89 =
79 - 72 = 7 it has some other qualities that need to be explored.

What comes to my mind about this tetrachord is that it could also be
generated by an "Iph" series (that's how I call a generator of 2/
Phi), since 144 the octave of 72 is precisely the following Fibonacci
after 89, and 144/89 ~1,618, same thing with your original 21 : 26
(Fibonaccis 13 > 21), while 11/10 is found at 7 Iphs, or it can be
seen also simply as a 9/8 below the "Iph" third.
Iph (1.236 = sqrtof5 - 1) is actually a very important attractor for
the Buzurg series, as well as the inevitable alter ego of Mohajira's
beginnings of series, because as it appears in a series of three Iph
neutral thirds, Iph - 1 or the difference tone of this neutral third
= Iph^3 / 8
ex. here : 89 - 72 = 17 = 136/8 in the Iph series 72 : 89 : 110 :
136 : 168 : 204 : 256

Just one thing about "rast_moha.scl" : this mode example was sent to
Manuel thanks to a good angel (not me), and rast_moha is a perfect
name for that Mohajira mode, but I never advocated this 24-ET tuning
myself. And there are as many better values of Mohajira generators
than 350c. as there are better meantones than 12-ET, if it is !
- - - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> We can explain it from the solution of Ishku's recurrent sequence,
> 8x^2 - 8x = 1, or (sqrt of 6 + 2) /4 = 1.112372436,
> because it is also the arithmetical mean between the semififth (sqrt
> of 6 / 2) and 1/1, that means Ishku tones and semififths are always
> differentially coherent.

I've asked before and I ask again: could you please define "differentialy coherent"? And I don't mean by citing something in paper and ink and not readily available.

--- In tuning@yahoogroups.com, Gene wrote:

> > (Jacques) : We can explain it from the solution of Ishku's > recurrent sequence,
> > 8x^2 - 8x = 1, or (sqrt of 6 + 2) /4 = 1.112372436,
> > because it is also the arithmetical mean between the semififth (sqrt
> > of 6 / 2) and 1/1, that means Ishku tones and semififths are always
> > differentially coherent.
>
> I've asked before and I ask again: could you please define > "differentially coherent"? And I don't mean by citing something in > paper and ink and not readily available.

Hi Gene,
A few years ago on this list you expressed the differential coherence in perfect mathematical terms, better than I will ever be able to do it myself !
But thanks to ask again - sorry everybody if I did'nt upload my article sooner in the TL files, this is done at present.
This article was published in 1/1 vol.11, number 2 by the Just Intonation Network in 2003, and originally in french by the CNRS in the Acts of the 5th J.I.M. Conference in 1998.
To resume it, with my excuses if my english is not perfect (corrections are welcome) :
Differential coherence is the property of a dyad to have its difference tone(s) in unisson with other tones of the global scale, or whatever musical context it belongs to (it considers of course octaves or other specific periods of the original scale as part of the global scale).
By extension, a scale is said to be "differentially coherent", if many of its dyads are differentially coherent.
In the Ethno collection, most scales are -c.

In the context above of a Ishku series (and considering the octave as the period), any semififth issued from any note will be -c.
ex : 72 : 80 : 89 : 99 ...
80 : 89, 89 : 99, etc... are -c
Also the semififth 80 : 98, whose 1st order difference tone 98 - 80 = 18 = 2 octaves below 72
(and also with the fractal values of Ishku and sqrt of 6 / 2)
- - - - - - -
Jacques

Hi Margo, and all,

Coming back on my answer to your reflexions on these tetrachords :
1/1 - 11/10 - 26/21 - 4/3
or its similar form present in sumer.scl :
1/1 - 79/72 - 89/72 - 4/3

As I said, that it could also be generated by an "Iph" series (a
generator of 2/Phi),
since 144 the octave of 72 is precisely the following Fibonacci after
89 and 144/89 ~1,618,
same thing with your original 21 : 26 (Fibonaccis 13 > 21), while
11/10 is found at 7 Iphs...

The middle tone of this tetrachord, thus resulting of six Phi ratios,
gave me the idea of a great system based on the Iph series !
I have been many times attracted by this quite harmonic coïncidence,
that Phi^6 ~ 18, that we find in the Fibonaccis between 8 and 144 -
(also a common coïncidence between Phi's main series F and P) -
It is in fact one meantone away from 16, that suggests a meantone
fifth of (Phi^6 /8)^1/2 = 1.497676196228...
and a planar temperament based on Iph and this meantone fifth as
generators.
This works remarquably well and allows many versions, from simple
ribbon temperament (one single fifth translation) to planar temperaments, using several fifths by rows.
They all lead to different canvas of 36 tones/octave, so nothing
really new, except these are unequal versions, entirely generated by
Phi, along with specific -c qualities and possible appropriate
applications with Phi timbres.
Luckily, two simple Phi series are existing, who have this meantone
fifth ratio :
The P series (the whole numbers approached by the powers of Phi), and
"Phi 96" :
(7 15 22 37 59) >
96 155 251 406 657 1063 1720 2783 ...
that 8*P completes by one meantone fifth =
144 232 376 608 984 1592 2576 4168 ...
With of course as many versions as you want as you go up in the series.

Here is for example such a "meantone-Iph" version of a scale based on two diatonic tetrachords similar to yours :

! meso-iph7.scl
!
Neutral diatonic variation based on two Iph fractal series
7
!
3072/2783
3440/2783
3712/2783
4168/2783
4608/2783
224/121
2/1
! Iph -c recurrent sequence, x^3 = 8x - 8
! completed by Meso-Iph meantone fifth of (Phi^6 /8)^1/2
! simulated between Phi-series Phi 96 and 8*P

And I propose it here with 5 additionnal notes that complete the Iph
series :

! meso-iph12.scl
!
Partial Meso-Iph fifth transposition of two Iph fractal series
12
!
3008/2783
3072/2783
3184/2783
3440/2783
3712/2783
3936/2783
4168/2783
4608/2783
4864/2783
4960/2783
224/121
2/1
! Iph -c recurrent sequence, x^3 = 8x - 8
! completed by Meso-Iph meantone fifth of (Phi^6 /8)^1/2
! simulated between Phi-series Phi 96 (96 : 155 ... 1720 : 2783)
! and 8*P : 18 : 29 : 47 : 76 : 123 : 199 : 322 : 521
! Dudon 2010

I found the neutral diatonic (7 notes above) very charming and
Malagasy-like
(the mode of F, along with the Bb variation, brings this magnificent
Rast option you mentionned for the first tetrachord),
while the additional tones, close to 13/12 (47), 8/7 (199), 17/12
(123), 7/4 (76) and 9/5 (155) bring more Middle-East-style variations.
I like the very oriental interaction of the "8/7" (199) with E (215)
and F (29) -
and also the "13/12" (47, definitively Buzurg-genre) along with E
(215), F (29), G (521), A (18), and Bb(155).
Of course all these could be expressed in simpler ratios. But it's
interesting to hear the specific color Phi brings in here.
- - - - - - -
Jacques

On September 17 of 2010, Margo wrote :

> Rast tone Ishku (?)
> |-------------------------|.......|------------------------|
> rast dugah segah chargah neva hisar evdj gerdaniye
> 0.0 208.6 370.3 496.9 704.3 866.0 1073.4 1200.0
> ~ 1/1 44/39 26/21 4/3 3/2 33/20 13/7 2/1
> 207 162 127 207 162 207 127
>
> Here I have permitted the second step of the lower tetrachord, dugah,
> to revert back to its usual position at around 9/8 or 44/39, although
> there is no reason why one could not sarp dugah at 8/7 as well. Our
> attention is focused especially on the upper tetrachord with its
> bright neutral sixth and neutral seventh above the final. This
> tetrachord has intervals of 0-161.7-369.1-495.7 cents, with steps of
> 161.7-207.4-126.6 cents. A fairly close JI equivalent would be another
> permutation of Safi al-Din's "Medium conjunct" tetrachord, yielding
> 1/1-11/10-99/80-4/3 (0-165.0-368.9-498.0 cents), with steps of
> 11:10-9:8-320:297 (165.0-203.9-129.1 cents).
>
> In the most loose and generic terms, the upper tetrachord might be
> called "Moha," short for Dudon's Mohajira, a pattern with lower
> neutral second, tone, and upper neutral second. However, in his
> concept of Mohajira proper, the neutral third step is generally
> somewhere between 39/32 (342 cents) and 16/13 (359 cents), with the
> bright Ottoman flavor here at 369 cents (26/21) clearly out of this
> range.
>
> One possibly more apt term is Ishku, Dudon's JI sequence with
> differential coherence (-c) which makes available a tetrachord of
> 1/1-79/72-89/72-4/3 (0-160.6-367.0-498.0), or 79:72-89:79-96:89
> (160.6-206.3-131.1 cents). See Dudon's sumer.scl (2-Middle East
> folder) in his Ethno2 collection.

Jacques Dudon wrote:

> Dear Margo,

> Before I get deeper in the lecture of your last wonderful posts,
> just a quick comment on what is "Ishku" in my sense - also because
> sumer.scl, described as a "Rast scale using Ishku -c variations",
> was not very explicit about it ! So now I try to do better :

[Please note that just now I have seen your new message, which
I want to consider carefully; but it seems to support the idea
that Iphi might be a good name for the tetrachord we are discussing.
Of course, I will want to look closely at your suggested meantone
system based on Iph -c.]

Thank you so much for explaining Ishku -c and also Iph, as well as for
making available your article on differentially coherent (-c) tunings
from 1/1. The point you raise below about various tunings for a
Rast_Moha mode, combined with one of the examples in the 1/1 article,
leads me to share one possible tuning which may have some spectral
coherence, if that concept can apply to tempered systems, although not
differential coherence.

Before responding to your explanations, I should seek your help and
advice in addressing my dilemma, now that you have shown why Ishku is
really not the best name for the kind of tetrachord I want to
describe, something like 1/1-11/10-26/21-4/3 (0-165-370-498 cents).
In O3, this tempers to 0-162-369-496 cents or 162-207-127 cents or so.

This genus differs from Buzurg only in the second step, which is a
high neutral second somewhere around 56/51 or 11/10, rather than a
smaller neutral second around 14/13 or 13/12, the two classic ratios
for medieval Buzurg, of course. Or, we might say that the middle
interval of the tetrachord will be around 8:7 or possibly a bit larger
for Buzurg; but in this genus, it is somewhere around 9:8 or 44:39,
for example.

Since Ishku -c doesn't fit so well, I wonder if there's some other
series that might better fit this kind of tetrachord with a high
neutral second and neutral third. From what you say below about the
Iph -c series, I propose the name Iphi, which, if you agree, I will
use in my Ethno Extras documentation. And maybe I can get through the
last five rotations of Zalzalian 12 by around the end of this month, and move on to other tunings, some of which may not involve this many
articles.

> Ishku is a recurrent sequence (and also a rhythm and a fractal
> waveform), based on an -c algorithm that can be generalised that
> way :

> In a triad C : D : E, if you have E - D = C/8 (in frequencies), C :
> D : E start a Ishku sequence. Next note will be "F sori"
> (half-sharp F) and so on, where each new tone will unvariably tend
> toward 184.36788 c., that is a slightly large 10/9.

A fascinating point here is that Safi al-Din al-Urmawi comes close to
the first two steps in Ishku -c in his tetrachord "doubling" a 10:9
step: 1/1-10/9-100/81-4/3 or 10:9-10:9-27:25 (182.4-182.4-133.2
cents). But in Ishku, as you explain, three slightly wide 10:9 steps
resolt in a just or near-just 11/8, a relationship I hadn't
recognized!

> The simplest Ishku series I know, and part of it is in this sumer.scl, is :

> (64) 72 80 89 99 ... (80 - 72 = 8, 89 - 80 = 9, 99 - 89 = 10 ... and
> next numbers will jump 3 octaves up, to 881 and 980)

> Seen like this, a Ishku series has not much relation with Rast or
> Mohajira. But when we look closer, we see they have. We can
> explain it from the solution of Ishku's recurrent sequence, 8x^2 -
> 8x = 1, or (sqrt of 6 + 2) /4 = 1.112372436, because it is also the
> arithmetical mean between the semififth (sqrt of 6 / 2) and 1/1,
> that means Ishku tones and semififths are always differentially
> coherent.

> For example in the neutral triad 40 : 49 : 60 that we find in
> sumer.scl between D and A,

> 40 + 49 = 89 = E
> and 89 - 80 = 49 - 40 = C.

This 40:49:60 triad is something I didn't notice when looking at the
scale, although now, when I read your message, it's very close:
10/9-49/36-5/3 or 40/36 : 49/36 : 60/36.

> By co?ncidence, but harmony is based on those, three Ishku tones
> are very close to 11/8 (we verify it in the series above in the
> ratio between 72 and 99).

> 11/8 or similar have two occurences in Rast, same in Mohajira, it's
> another possible bridge with Ishku.

Yes, 11/8 or the like will come up, for example, as a neutral third
around 11:9 plus a step around 9:8. This may be especially notable in
the Arab Maqam Sikah, where the lower pentachord may be something like
1/1-88/81-11/9-11/8-3/2 or 0-143-347-551-702 cents, if we follow
al-Farabi's version of Zalzal's tuning (143-204-204-151 cents). In O3,
where the smaller neutral second of this general type of Sikah is
narrower but the tone a bit wider than a just 9:8, we have something
like 0-138-346-554-704 cents.

> Then the fact that Ishku^3 ~11/8, because of other coherences and
> natural reasons suggests musical Ishku-based pentachords, ending
> with a fifth, such as 72 : 80 : 89 : 99 : 108

> Here we find that 99 - 72 = 27 and 108 - 99 = 9 (pure 3/2 fifths),
> but the Ishku fractal value can also suggest -c corresponding
> fifths of 703.592 and 708.4936 c... that I know you may not
> dislike!

Yes, these extended fifths are not unpleasant to me! Curiously, George
Secor's 29-HTT or 29-note version of his "High Tolerance Temperament"
uses a generator of 703.579 cents.

> In conclusion, in absence of 80 (or 88, since 88 - 79 = 9 is also
> of a Ishku kind) the sumer.scl subset 72 : 79 : 89 : 96 will only
> be an uncomplete example of a Ishku tetrachord, nevertheless with
> 96 - 89 = 79 - 72 = 7 it has some other qualities that need to be
> explored.

Thank you for explaining that this tetrachord is related to Ishku but
incomplete in terms of this -c system, so that another name would be
better.

> What comes to my mind about this tetrachord is that it could also
> be generated by an "Iph" series (that's how I call a generator of
> 2/Phi), since 144 the octave of 72 is precisely the following
> Fibonacci after 89, and 144/89 ~1,618, same thing with your
> original 21 : 26 (Fibonaccis 13 > 21), while 11/10 is found at 7
> Iphs, or it can be seen also simply as a 9/8 below the "Iph" third.

Am I right that 11/10 would here be represented by 89/81 (163.060
cents), a bit closer than 161.719 cents in O3?

If this -c system seems to fit -- as it does to me -- then maybe we
could call a tetrachord of 1/1-11/10-26/21-4/3 or the like "Iphi," a
name I find euphonious with the final vowel. Then Iph -c would be the
name of the series, and Iphi the name of this type of tetrachord,
basically like Buzurg with a higher neutral second.

> Iph (1.236 = sqrtof5 - 1) is actually a very important attractor
> for the Buzurg series, as well as the inevitable alter ego of
> Mohajira's beginnings of series, because as it appears in a series
> of three Iph neutral thirds, Iph - 1 or the difference tone of this
> neutral third = Iph^3 / 8

> ex. here : 89 - 72 = 17 = 136/8 in the Iph series
> 72 : 89 : 110 : 136 : 168 : 204 : 256

As an aside, I might mention that this series includes 136:168:204 or
14:17:21, the simplest division of the fifth into two neutral thirds,
as George Secor pointed out to me around nine years ago.

> Just one thing about "rast_moha.scl" : this mode example was sent
> to Manuel thanks to a good angel (not me), and rast_moha is a
> perfect name for that Mohajira mode, but I never advocated this
> 24-ET tuning myself. And there are as many better values of
> Mohajira generators than 350c. as there are better meantones than
> 12-ET, if it is !

Certainly it is possible to choose a range of single generators for
Mohajira, and your analogy with 12-ET and meantone very nicely makes
the point that 24-ET is not the only option!

However, what especially interests me is the possibility of obtaining
Mohajira and the rast_moha mode through what you have nicely and more
generally termed an Entrelacs or "interlace" system where we alternate
two neutral third generators of different sizes.

For example, here is an O3 version of rast_moha inspired by some of
your -c Mohajira tunings, and meant to seek a certain degree of
spectral although not differential coherence, if this is possible in a
temperament. For easier access, I'll also provide a link to the Scala
file:

! O3-rast_moha_Cup.scl
!
Mode of Rast + Mohajira (Dudon) with near-pure 59/48 and 12:11 steps
7
!
207.42188
357.42187
495.70312
703.12500
854.29687
1061.71875
2/1

<http://www.bestII.com/~mschulter/O3-rast_moha_Cup.scl>

A Scala table of steps and intervals might make my spectral coherence
idea, inspired in good part by reading your 1/1 article, clearer,
again with a link to help with possible formatting problems

<http://www.bestII.com/~mschulter/O3-rast_moha_Cup.txt>:

Mode of Rast + Mohajira (Dudon) with near-pure 59/48 and 12:11 steps
0: 1/1 0.000 unison, perfect prime
1: 207.422 cents 207.422
2: 357.422 cents 357.422
3: 495.703 cents 495.703
4: 703.125 cents 703.125
5: 854.297 cents 854.297
6: 1061.719 cents 1061.719
7: 2/1 1200.000 octave
|
Mode of Rast + Mohajira (Dudon) with near-pure 59/48 and 12:11 steps
1/1 : 207.4 357.4 495.7 703.1 854.3 1061.7 2/1
207.4 : 150.0 288.3 495.7 646.9 854.3 992.6 2/1
357.4 : 138.3 345.7 496.9 704.3 842.6 1050.0 2/1
495.7 : 207.4 358.6 566.0 704.3 911.7 1061.7 2/1
703.1 : 151.2 358.6 496.9 704.3 854.3 992.6 2/1
854.3 : 207.4 345.7 553.1 703.1 841.4 1048.8 2/1
1061.7: 138.3 345.7 495.7 634.0 841.4 992.6 2/1
2/1

Here 357.4 cents is a virtually just 59/48, with a difference of 11,
which may fit in with the close approximations of 11:6 or 12:11, and
also 11:8 (found above step 5 at 854 cents). The smaller neutral third
at 345.7 cents is within a cent of 72/59, with a difference of 13,
which may fit in with the near-just approximations of 13:8 or 16:13
and 13:12.

While I would call this basically an Entrelacs system, it is slightly
irregular because while the smaller neutral thirds are all 345.7
cents, the larger ones are either 357.4 cents (~59/48) or 358.6 cemts
(~16/13). The fifths, likewise, are not strictly regular, at either
703.1 or 704.3 cents (more precisely 703.125 or 704.297 cents).

The Entrelacs chain of six neutral thirds would start on step 3 or of
the rast_moha mode (~4/3). Here I'll number the steps in terms of
rast_moha:

358.6 -- 345.7 -- 357.4 -- 345.7 -- 358.6 -- 345.7
495.7 854.3 1200.0 357.4 703.1 1061.7 207.4
step 3 step 5 step 7 step 2 step 4 step 6 step 1

One question, of course, is whether the tempered intervals are
accurate enough to generate difference tone effects: while 357.4 cents
is very close to 59/48, I'm not sure if 345.7 cents at almost a cent
wide of 72/59 would actually produce any audible difference of 13.
While I like these tempered sizes in any event, there is the
interesting question of whether spectral coherence might apply to
tempered systems.

> - - - - - - - - Jacques

With warmest thanks,

Margo

Hi Margo, and all,

These are my last night's thoughts in complement to my precedent
disgressions concerning a "Iph-based" system, almost in real time now !
I thought my last post was perhaps lacking some pedagogy. So in afirst step I'll leave the refinement, still perfectly valid, of the
Meso-Iph temperament aside, and will replace the two "Phi 96" and
"8P" series respectively by a simple Fibonacci or "F" series and a
"3F" series, in other words a Fibonacci series multiplied by 3. This
way it should make it simpler to see how Margo's scale, that I will
call simply "Rast-Iph" at the moment, can be generated by an Iph
series and its transposition by a pure fifth.

Let us start with a grid based on 8 terms of a F series from 8 to 233 :
8 13 21 34 55 89 144 233
And the corresponding 3F series :
24 39 63 102 165 267 432 699

One first "Rast-Iph" (with tonic = 39) will be found here :

8 13 xx xx xx xx 144 233

24 39 xx xx xx xx xx 699

The following one (with tonic = 63) here :

13 21 xx xx xx xx 233 377

39 63 xx xx xx xx xx 1131

and so on, for as many versions we want, from tonics 102, 165, 267,
432 etc.
(this applies also for any other Phi series)

Here are the ratios of the first one :

! rast-iph39.scl
!
Neutral diatonic composed of Rast and Iph tetrachords, based on F and
3F series
7
!
233/208
16/13
4/3
233/156
64/39
24/13
2/1
! Iph -c recurrent sequence, x^3 = 8x - 8
! completed by its 3rd harmonic transposition
! F series 8 : 13 ... 144 : 233
! 3F series 24 : 39 ... 699

And the second one :

! rast-iph63.scl
!
Neutral diatonic composed of Rast and Iph tetrachords, based on F and 3F series
7
!
377/336
26/21
4/3
377/252
104/63
233/126
2/1
! Iph -c recurrent sequence, x^3 = 8x - 8
! completed by its 3rd harmonic transposition
! F series 13 : 21 ... 233 : 377
! 3F series 39 : 63 ... 1131

You might recognize some elements in those now !
Note that 233/156 is tempered by an epimoric comma of 233 : 234 and
377/252 by 378/377
(see Andreas Sparschuh posts #91762, 91446 and others about this)

Now if we check the "xx" notes left aside in both series, we will
find for a start 9 extra possibly -c related notes to be experimented
- but I'll leave it for a further exploration !

- - - - - - - - -
Jacques

On the 20th of September Jacques wrote :
> .../...
> As I said, it could also be generated by an "Iph" series (a
> generator of 2/Phi),
> since 144 the octave of 72 is precisely the following Fibonacci after
> 89 and 144/89 ~1,618,
> same thing with your original 21 : 26 (Fibonaccis 13 > 21), while
> 11/10 is found at 7 Iphs...
>
> The middle tone of this tetrachord, thus resulting of six Phi ratios,
> gave me the idea of a great system based on the Iph series !
> I have been many times attracted by this quite harmonic coïncidence,
> that Phi^6 ~ 18, that we find in the Fibonaccis between 8 and 144 -
> (also a common coïncidence between Phi's main series F and P) -
> It is in fact one meantone away from 16, that suggests a meantone
> fifth of (Phi^6 /8)^1/2 = 1.497676196228...
> and a planar temperament based on Iph and this meantone fifth as
> generators.
> This works remarquably well and allows many versions, from simple
> ribbon temperament (one single fifth translation) to planar
> temperaments, using several fifths by rows.
> They all lead to different canvas of 36 tones/octave, so nothing
> really new, except these are unequal versions, entirely generated by
> Phi, along with specific -c qualities and possible appropriate
> applications with Phi timbres.
> Luckily, two simple Phi series are existing, who have this meantone
> fifth ratio :
> The P series (the whole numbers approached by the powers of Phi), and
> "Phi 96" :
> (7 15 22 37 59) >
> 96 155 251 406 657 1063 1720 2783 ...
> that 8*P completes by one meantone fifth =
> 144 232 376 608 984 1592 2576 4168 ...
> With of course as many versions as you want as you go up in the
> series.

> Hi Margo, and all,
> Coming back on my answer to your reflexions on these tetrachords :
> 1/1 - 11/10 - 26/21 - 4/3
> or its similar form present in sumer.scl :
> 1/1 - 79/72 - 89/72 - 4/3

[Dear Jacques: Thank you for the fine post explaining the simple
Rast-Iph series in two forms, indeed a very helpful pedagogy for
someone like myself not so familiar with these series. Again, as
it happens, I saw your post just after finishing a reply to a
previous message, but I wanted to emphasize how helpful your
new post is. Also, the rast-iph39.scl and rast-iph63.scl show
how these -c series can produce fifths and major seconds, for
example, in the meantone range, a theme tying in with the
more intricate meso-iph family of tunings.]

Dear Jacques,

Thank you so much for giving so much attention to these tetrachords,
which I am calling the "Iphi" genus. It is very exciting to see how
our discussion has led you to a kind of Iph -c meantone, which in turn
has caused me to explore some structures with extended fifths which
might be in some ways similar.

> As I said, that it could also be generated by an "Iph" series (a
> generator of 2/Phi), since 144 the octave of 72 is precisely the
> following Fibonacci after 89 and 144/89 ~1,618, same thing with
> your original 21 : 26 (Fibonaccis 13 > 21), while 11/10 is found at
> 7 Iphs...

An interesting point is that Ibn Sina has 21/13 in his Shur-like
tuning with tetrachords of 12:13:14:16 or 28:26:24:21, but I'm not
sure if he uses 11/10 in some of his tunings, although Safi al-Din
al-Urmawi does some two centuries later. We can get Iphi as a
permutation of Safi al-Din's tetrachord which Ozan nicely terms
"medium sundered," 1/1-9/8-99/80-4/3 or 9:8-11:10-320:297, reversing
the order of the first two intervals (0-165-369-498 cents).

And 89:144 as part of Fibonacci's series and also a fine approximation
of Phi is really neat! Scala shows it is narrow of Phi by 0.06 cents
or so.

> The middle tone of this tetrachord, thus resulting of six Phi
> ratios, gave me the idea of a great system based on the Iph series !

> I have been many times attracted by this quite harmonic co?ncidence,
> that Phi^6 ~ 18, that we find in the Fibonaccis between 8 and 144 -
> (also a common co?ncidence between Phi's main series F and P) -

> It is in fact one meantone away from 16, that suggests a meantone
> fifth of (Phi^6 /8)^1/2 = 1.497676196228... and a planar
> temperament based on Iph and this meantone fifth as generators.

If I am correct, this fifth would be close to 1/8-comma meantone.

> This works remarquably well and allows many versions, from simple
> ribbon temperament (one single fifth translation) to planar
> temperaments, using several fifths by rows.

Here I might quickly comment that my 24-note temperaments with
extended fifths tend to be ribbon temperaments with two 12-note
chains. For example, in O3, the fifths are at 703.125 or 704.297
cents, with a single translation, if I am using your language
correctly, of 57.422 cents. However, as you point out, with something
like your Iph meantone there is no reason we might not have a planar
temperament with several rows or chains of fifths!

> They all lead to different canvas of 36 tones/octave, so nothing
> really new, except these are unequal versions, entirely generated by
> Phi, along with specific -c qualities and possible appropriate
> applications with Phi timbres.

Here I am curious how the 36 tones would be arranged in terms of the
numbers of rows of generators and so forth. I do recognize that some
of the intervals such as Phi, Iph, 8/7, etc., are very nicely served
by 36-EDO, with your meantone generator at about 699.27 cents not too
far from 700 cents; but the differences could be very interesting.

> Here is for example such a "meantone-Iph" version of a scale based on two diatonic tetrachords similar to yours :
! meso-iph7.scl
!
Neutral diatonic variation based on two Iph fractal series
7
!
3072/2783
3440/2783
3712/2783
4168/2783
4608/2783
224/121
2/1
! Iph -c recurrent sequence, x^3 = 8x - 8
! completed by Meso-Iph meantone fifth of (Phi^6 /8)^1/2
! simulated between Phi-series Phi 96 and 8*P

For comparison, here is the same "Iphi" mode in O3:

! O3-quasi-meso-iph7_C#.scl
!
Disjunct tetrachords ~1/1-56/51-26/21-4/3 a bit like Dudon's meso-iph7.scl
7
!
161.71875
369.14062
495.70313
704.29687
866.01563
1073.43750
2/1

> And I propose it here with 5 additionnal notes that complete the Iph series :
! meso-iph12.scl
!
Partial Meso-Iph fifth transposition of two Iph fractal series
12
!
3008/2783
3072/2783
3184/2783
3440/2783
3712/2783
3936/2783
4168/2783
4608/2783
4864/2783
4960/2783
224/121
2/1
! Iph -c recurrent sequence, x^3 = 8x - 8
! completed by Meso-Iph meantone fifth of (Phi^6 /8)^1/2
! simulated between Phi-series Phi 96 (96 : 155 ... 1720 : 2783)
! and 8*P : 18 : 29 : 47 : 76 : 123 : 199 : 322 : 521
! Dudon 2010

> I found the neutral diatonic (7 notes above) very charming and
> Malagasy-like (the mode of F, along with the Bb variation, brings
> this magnificent Rast option you mentionned for the first
> tetrachord), while the additional tones, close to 13/12 (47), 8/7
> (199), 17/12 (123), 7/4 (76) and 9/5 (155) bring more
> Middle-East-style variations.

Please let me confirm a few points to be sure I have understood
correctly, and agree that these are indeed delightful options! The
charming "Malagasy-like" diatonic I take to be this, based on Scala:

0-171.0-366.9-498.7-699.3-873.0-1066.2-2/1

This is what I might term an "equable heptatonic," meaning that it has
some qualities of a division not too far from 7-EDO, as with the steps
of 171 and 174 cents (C-D and G-A).

Zest-24, based on Zarlino's 2/7-comma meantone, has something a bit
like this:

0-171.2-362.8-504.2-695.8-867.0-1058.7-2/1

With the F mode using Bb, I see that we indeed get a Rast_Iphi form,
to use a term analogous to Rast_Moha:

0-200.6-374.3-501.8-701.3-872.4-1068.3-2/1

> I like the very oriental interaction of the "8/7" (199) with E
> (215) and F (29) - and also the "13/12" (47, definitively
> Buzurg-genre) along with E (215), F (29), G (521), A (18), and
> Bb(155).

Oh yes, I agree that, for example, on C, 2783:3008:3440:3712 at
0-134.6-366.9-498.7 cents is quintessential Buzurg! And I see lots of
options for septimal slendros, including a location with 49:32.

> Of course all these could be expressed in simpler ratios. But it's
> interesting to hear the specific color Phi brings in here.

Yes, both Phi itself, and some close approximations like 155/96 and
76/47. These slight variations give this type of -c system some of the
same charms celebrated by George Secor for unequal temperaments
generally. Looking again at the intervals in meso-iph12.scl, I wonder
if your 36-note tuning might be -c variation on 36-EDO.

With an extended fifth "ribbon" tuning like O3, I find that I can get
_some_ of the same Near Eastern genres and modes, but not all of them
or in the same relationships or with the same subtle variety.

Congratulations on the Meso-Iph family of -c tunings!

With many thanks,

Margo

----------------------------------
Ethno Extras: Zalzalian 12 set
Part 6: Rotation 7
----------------------------------

[To preserve formatting as much as possible, please try the Use
Fixed Font Width option if viewing on the Yahoo site.]

Hello, all.

To provide a convenient "itinerary" for the Zalzalian 12 set, so that
people can keep track of where we've been and where we're going, I'll
place at the beginning of each part a summary of the rotations adapted
from Scala output. Part 1 included an introduction on Ethno Extras,
the O3 temperament, and Buzurg modes, and started our journey with
Rotation 0. Part 2 addressed Rotations 1-3; Part 3, Rotation 4;
Part 4, Rotation 5; and Part 5, Rotation 6. We now come to Rotation 7.

Again, my warmest thanks to Jacques and Francois and all
participants in the Ethno2 contest for the inspiration they have
lent, and to Ozan Yarman, Shaahin Mohajeri, and George Secor, as
well as now to Jacques Dudon, for the very special knowledge and
wisdom they have shared in the exploration of these and other
modes.

Rotation 0: 1/1
0 138.3 207.4 264.8 345.7 472.3 680.9 704.3 842.6 969.1 1050.0 1176.6 2/1

Rotation 1: 138.3 (~13/12)
0 69.1 126.6 207.4 334.0 542.6 566.0 704.3 830.9 911.7 1038.3 1061.7 2/1

Rotation 2: 207.4 (~9/8 or ~44/39)
0 57.4 138.3 264.8 473.4 496.9 635.2 761.7 842.6 969.1 992.6 1130.9 2/1

Rotation 3: 264.8 (~7/6)
0 80.9 207.4 416.0 439.5 577.7 704.3 785.2 911.7 935.2 1073.4 1142.6 2/1

Rotation 4: 345.7 (~11/9)
0 126.6 335.2 358.6 496.9 623.4 704.3 830.9 854.3 992.6 1061.7 1119.1 2/1

Rotation 5: 472.3 (~21/16)
0 208.6 232.0 370.3 496.9 577.7 704.3 727.7 866.0 935.2 992.6 1073.4 2/1

Rotation 6: 680.9 (~77/52)
0 23.4 161.7 288.3 369.1 495.7 519.1 657.4 726.6 784.0 864.8 991.4 2/1

Rotation 7: 704.3 (~3/2)
0 138.3 264.8 345.7 472.3 495.7 634.0 703.1 760.5 841.4 968.0 1176.6 2/1

Rotation 8: 842.6 (~13/8)
0 126.6 207.4 334.0 357.4 495.7 564.8 622.3 703.1 829.7 1038.3 1061.7 2/1

Rotation 9: 969.1 (~7/4)
0 80.9 207.4 230.9 369.1 438.3 495.7 576.6 703.1 911.7 935.2 1073.4 2/1

Rotation 10: 1050.0 (~11/6)
0 126.6 150.0 288.3 357.4 414.8 495.7 622.3 830.9 854.3 992.6 1119.1 2/1

Rotation 11: 1176.6 (~77/39)
0 23.4 161.7 230.9 288.3 369.1 495.7 704.3 727.7 866.0 992.6 1073.4 2/1

------------------------------------------------------------------
Rotation 7 (3/2): Magnificent Shur and septimal Ibina-like variant
------------------------------------------------------------------

0 138.3 264.8 345.7 472.3 495.7 634.0 703.1 760.5 841.4 968.0 1176.6 2/1

Just as Rast is the premier maqam in the Arab or Turkish tradition, so
Shur is the premier dastgah or modal family in the repertory of
classic Persian music known as the _radif_.

The radif generally is regarded as consisting of 12 such modal
families, seven including Shur recognized as full dastgah-ha (the
plural of dastgah), and five others as avaz-ha, or "satellite"
dastgah-ha related to one of the seven principal families. Each
dastgah or avaz, in turn, consists of a series or suite of gusheh-ha
or modal themes, so that a performance in a given dastgah or avaz
consists of a sequence of gusheh-ha, many of them performed in a more
or less standard order, starting with a Daramad or opening piece
introducing the main focus of the modal family, and then progressing
through gusheh-ha which may either retain the modal focus of the
Daramad or serve themselves as independent modes, with the sequence
usually returning eventually to the modality of the opening.

While it is possible to speak of "the radif" in the singular, from a
practical perspective one often speaks in the plural of radif-ha, the
"radif-s" of different adept masters and teachers. For example, these
different radif-ha might differ on the question of whether the mode
known as Bayat-e Kord, the "song or verses of the Kurd," should be
considered as a gusheh, for example of Shur, or as an avaz or
satellite dastgah in its own right.

The radif is enriched by the fact that while some gusheh-ha may be
specific to a certain dastgah, others may be performed as part of more
than one dastgah, or "modal suite" as it might be translated in one
sense. From the perspective of the Islamic maqam system out which the
Persian dastgah system seems to have grown around the 17th-18th
centuries, the latter might be seen as a more structured and often
condensed version of the modulations from one maqam to another which
are a vital aspect of the Near Eastern tradition.

The concept of a dastgah as a modal family or "suite of gusheh-ha" is
helpful in appreciating the intricacies of looking for a "scale," or
better a gamut, of pitches for the main gusheh-ha of a given dastgah,
here Dastgah-e Shur ("the dastgah of Shur").

Happily, our Zalzalian 12 set in this rotation includes what might be
called the nine basic steps of Shur, with the caution that there are
significant distinctions between octaves, and that the most relevant
steps may vary from gusheh to gusheh. Here I will give Persian note
names starting from G, one standard note for the final of Shur, and
using ASCII "p" for the sign of the koron, an inflection often
lowering a note by about 50-70 cents, or 1/4 to 1/3 of a tone.

G Ap Bb C Dp D Eb Ep F G
0.0 138.3 264.8 495.7 634.0 703.1 760.5 841.4 968.0 1200.0 ~ 1/1 13/12 7/6 4/3 13/9 3/2 14/9 13/8 7/4 2/1

The lower tetrachord above the final G defines much of the essence of
the Shur family: two neutral second steps, here 138.3-126.6 cents or
very close to 12:13:14, together forming a minor third, with a tone or
major second then completing the fourth. Our division of this fourth
is very close to 12:13:14:16 with a near-just 7:6 minor third followed
by a likewise almost pure 8:7 tone, either the original version or
possibly a permutation of Ibn Sina's tuning in the early 11th century
with steps of 14:13, 13:12, and 8:7.

In practice, the tuning of Shur can and does vary considerably, with
Dariouche Safvate in 1966 reporting measurements showing one tuning of
around 0-136-276-500 cents (136-140-224 cents) for this lower
tetrachord. While minor thirds at or not too far from the septimal
ratio of 7/6 may have been popular in Persian music during at least
the millennium from the writings of Ibn Sina to the present, wider
varieties at around 13/11 (289 cents) or 32?27 (294 cents) seem also
commmon.

In one way, our 12:13:14:16 arrangement may not be the most
characteristic: there is a general preference, at least in theory, for
the smaller neutral second to come first, as also with the Arab Maqam
Bayyati as understood by many traditional performers. However, with
minor thirds of various sizes, one can find measured tunings in which
the smaller and larger neutral seconds occur in either order, as well
as tunings in which they are virtually equal in size, sometimes
closely approaching the 140-140-220 cents recommended as a model for
Shur by the Iranian master Dr. Dariush Tala`i.

Our simple tuning set for the octave above the final of Shur also
shows the flexible nature of two degrees. The fifth of Shur, known in
Persian as a _moteqayyer_ or variable note, has both a 3/2 version (D)
and a lowered version at D-koron or Dp, here around 13/9. One
characteristic use of the lowered or koron version is in descending
passages moving from the middle or upper portion of the octave down
again toward the final. The lowered fifth might serve as a top pitch
in a phrase, e.g. C-Dp-C-Bb-Ap-A, or as a pleasant nuance in a descent
from the more common minor version of the sixth step through the four
to the lower or main tetrachord, e.g. Eb-Dp-C-Bb-Ap-Bb-Ap-A. Note that
the figure Eb-Dp-C or ~14/9-13/9-4/3 has a descending 14:13:12 pattern like that of the lower tetrachord at approximately 16:14:13:12.

Additionally, the Dp or 13/9 step is very common in gusheh-ha of Shur
which effectively transpose the lower tetrachord up a fourth to the
4/3 step: C-Dp-Eb-F. Here a caution is that while the O3 system would
have a 16/9 step available to permit a form of this tetrachord with a
usual fourth, in Zalzalian 12 our available choice is approximately
4/3-13/9-14/9=7/4 with a narrow fourth at 472.3 cents, near 21:16.
Narrow tetrachords do sometimes occur in traditional Persian music, as
with a fourth at 484 cents reported by Safvate for the old version of
Avaz-e Bayat-e Tork (the "verses or song of the Turk," referring not
to the Turks of Anatolia but to the Turkic ethnic communities of
Iran), known as the "old Tork."

The usual "textbook" form of Shur shows a minor sixth, in our Persian
notation Eb at 760.5 cents or a narrow 14/9 (actually closer to the JI
ratio of 273/176 at 760 cents), and in other styles of tuning closer
to 11/7 (782 cents) or 52/33 (787 cents), for example. In fact, this
is the most common form of the sixth degree above the final in modern
Shur, although the neutral sixth form, with a Persian spelling here of
Ep at 841.4 cents or a near-just 13/8, also occurs, with the gusheh of
Gereyli as the notable example I found in some published radif-ha.

As a usual diagram of the octave above the final cannot itself show,
however, the neutral form of this step, or rather its counterpoint an
octave lower in the tetrachord _below_ the final, is standard, a point
especially brought home by the tuning of instruments such as the
traditional Persian santur where for Shur, again using our present
Persian spelling, Ep in the octave below the final but Eb above it
would be the norm. In our septimal version of Shur, the tetrachord
below the final with its "neutral sixth" degree -- or, more precisely,
the degree a neutral third below the final, here about 13/16 -- is as
follows:

D Ep F G
703.1 841.4 968.0 1200.0
-496.9 -358.6 -232.0 0.0
3/4 13/16 7/8 1/1

This Shur tetrachord (here also near 12:13:14:16) below the final
plays a prominent role in the _forud_ or approach to the cadence, with
Hormoz Farhat describing one common formula as moving from the third
below to the second below to the final. In the suite performance that
defines a dastgah, the forud serves as a unifying device reaffirming
the resting note for the dastgah as a whole, in contrast to the often
diverse modalities and modal centers for the various gusheh-ha.

One nuance of Shur not supported in this Zalzalian 12 version is the
tendency in the upper octave to lower the step a second -- or rather
ninth -- above the final from neutral to minor. Again using our
Persian spelling, it would thus be customary on the santur to use the
neutral second Ap but the minor ninth Ab, an option available at this
location in the 24-note O3 system but not this 12-note subset. While
much beautiful music can be played using only the neutral ninth, the
minor version of this step does add an expressive element worth
seeking when it is available. As we shall see, such an extra step,
often in O3 at around 1281 cents or 44/21 (22/21 plus an octave), is
very desirable in a dedicated Shur tuning.

A step not listed in usual presentations of this dastgah, but which I
have found useful in Shur-based polyphony, is found in this rotation
at 472.3 cents or slightly wide of 21/16, forming a perfect fifth
below the 7/4 step.

Here I should emphasize that the dastgah tradition is itself mainly
monophonic, although a drone technique is common in some forms of
Persian music and medieval theorists such as Ibn Sina discuss a
technique of _tarqib_ or "composite" sound in which the performer
plays two notes at the same time, for example at a fourth or a
fifth. (The more common sense of tarqib in the world of maqam means a
"mixture" of two or often several maqamat recognized as a compound
maqam.)

Polyphonic offshoots of dastgah music may range from discreet
arrangements of traditional themes seeking to keep the main focus on
melodic development, using moveable drones for example, to freer
styles where some of the steps and intervals of a given modal system
serve as the basis for an improvisation or composition which may not
necessarily follow any traditional pattern of development. While
styles at various points along this continuum will draw on resources
such as polyphonic cadences involving neutral steps and intervals,
freer styles may showcase "ostentatious" sonorities which might be
considered a bit too bold or striking for contexts where the desire is
more subtly to reinforce and adorn the flow of a traditonal theme.

An example leaning to the freer side is _Prelude in Shur for Erv
Wilson_, a piece illustrating the use of the 21/16 step:

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

Here is an unmeasured JI notation for the opening, showing some uses
of 21/16. My convention is that all notes are sustained until the end
of the passage, or until an "r" sign shows a rest.

2/1 7/4 13/8 7/4 2/1 13/8 7/4 273/176 3/2
3/2 3/2 13/8 3/2 21/16 3/2
3/2 21/16 13/12 r
1/1 1/1 13/12 1/1 7/8 1/1...

As seen, after the initial 2:3:4, a highlight of the opening is the
four-voice sonority 16:21:24:28, used according to Kyle Gann by
LaMonte Young as early as the 1960's, and later advocated by Keenan
Pepper and others. This sonority, which I would rather on the
ostentatious side for a more unassuming style of dastgah-based
polyphony supporting a familiar gusheh, resolves to the fifth
13/12-13/8, the texture for the rest of the passage is in three parts,
with the second voice arbitrarily assigned a rest to show this change
in texture.

The highest voice next moves into a 13/12-13/8-7/4 or 26:39:42
sonority (here a tempered 0-703-830 cents), with a striking 127-cent
small neutral second near 14:13 between the upper voices, and a
near-just 21:13 between the outer voices. The outer neutral sixth
expands to an octave and the upper neutral second to a fourth,
arriving at 1/1-3/2-2/1 on the final. Then the upper voice descends a
large neutral third, presenting us with 1/1-3/2-13/8 or 8:12:13, a
neutral sixth sonority including the rather tense 13:12 between the
upper voices, but somewhat less tense or more blending than the
previous shading of 26:39:42 because of the harmonic 13:8 neutral
sixth. The resolution follows the same basic pattern, with the neutral
sixth expanding to an octave and the neutral second to an upper
fourth, thus arriving at 7/8-21/16-7/4, a 2:3:4 on the step at a
septimal 8:7 tone below the final.

Then the upper voice descends by a tone from 7/4 to a degree which
might have specified as an approximate 14/9 but is more accurately
called 273/176, since the notation 7/4-21/16-273/176 accurately shows
that this minor seventh sonority is very close to a just 22:33:39,
with an upper minor third at 13:11, or here not quite a cent narrower
at 288.3 cents. The minor seventh contracts to a fifth, and the upper
minor third to a unison, thus arriving at 1/1-3/2 and concluding this
opening section.

In a more conservative or less ostentatious style, maybe of the
progressions between the two outer voices, for example, might be
routinely used, as well as a "thickening" of the texture to three
voices for certain important cadences, for example. However, the
balance would be more toward a melody with some polyphonic adornment,
rather than a free polyphonic prelude drawing on the steps of
Dastgah-e Shur.

Another type of mode with a Near Eastern flavor supported by this
rotation is a variation on Jacques Dudon's Ibina series. a
differentially coherent (-c) form of what might be termed
Mohajira-Bayyati or "Moha_Baya" for short. The original form of the
beautiful Ibina, closely approximated in O3 although not in this
subset, is as follows:

72 78 88 96 108 117 128 144
1/1 13/12 11/9 4/3 3/2 13/8 16/9 2/1
0.0 138.6 347.4 498.0 702.0 840.5 996.1 1200.0

In this rotation, we have a variant form which unfortunately does not
share the -c property, but at least to me has its own appeal. Here I
show the frequency ratios and cents for a JI version, followed by the
tempered values present in the Zalzalian 12 version:

72 78 88 96 108 117 126 144
1/1 13/12 11/9 4/3 3/2 13/8 7/4 2/1
0.0 138.6 347.4 498.0 702.0 840.5 968.8 1200.0
0.0 138.3 345.7 495.7 703.1 841.4 968.0 1200.0

The one difference between these two forms is the minor seventh, at
72:128 or 16:9 in the Ibina -c tuning, but at 72:126 or 7:4 in this
variant, a difference of 128:126 or 64:63, the septimal or Archytan
comma (27.26 cents).

In either form, we have a lower tetrachord with one flavor of Dudon's
Mohajira with its general pattern of neutral second, tone, and neutral
second, here 1/1-13/12-11/9-4/3 or 13:12-44:39-12:11. The upper
tetrachord on the 3/2 step is some shading of the Arab Bayyati type
(in Turkey, one type of Maqam Ushshaq), with a pattern of two neutral
seconds, the first often smaller, plus a tone. Ibina proper has a
tetrachord of 1/1-13/12-32/27-4/3 or 13/12-128:117-9:8, while our
septimal variation approximates Ibn Sina's 1/1-13/12-7/6-4/3 or
13:12-14:13-8:7 tuning for a tetrachord which might fit a variety of
modern Persian Shur or Arab Bayyati.

While a near-septimal intonation of Shur has been documented by
Safvate, the Lebanese teacher and theorist Ali Jihad Racy may be
describing a somewhat similar flavor of Bayyati when he demonstrates,
as reported by Scott Marcus, that the minor third of Bayyati should be
played notably lower than would result from using a regular fret at
the Pythagorean third or 32/27 (294 cents). Whether this difference is
often taken as about a septimal comma remains an open question which
measurements by scholars such as Amine Beyhom might help to resolve.

In addition to Dastgah-e Shur and our variation on Jacques Dudon's
Ibina, we have available a couple of forms of septimal slendro. The
first represents a very simple and harmonious form of septimal
pentatonic (e.g. slendro5_2.scl in the Scala archive):

12 14 16 18 21 24
1/1 7/6 4/3 3/2 7/4 2/1
0.0 266.9 498.0 702.0 968.8 1200.0
0.0 264.8 495.7 703.1 968.0 1200.0

Another form (see slendro_7_4.scl) from Lou Harrison and Jacques
Dudon, called NAT, uses 21/16 rather than 4/3:

48 56 63 72 84 96
1/1 7/6 21/16 3/2 7/4 2/1
0.0 266.9 470.7 702.0 968.8 1200.0
0.0 264.8 472.3 703.1 968.0 1200.0

Erv Wilson's 1-3-7-9 hexany uses these same five steps plus a 9/8 step
not present in this rotation of Zalzalian 12, but available in
Rotation 0, as was discussed in the first article of these series.

Most appreciatively,

Margo Schulter
mschulter@...