Persian koron/sori notation and 17-note systems

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Dear Jacques,

Please let me try to adddress one question you raised where my
response might conveniently make a separate article, while
emphasizing that I will need to devote a number of articles to
the exciting of discovering your Ethno2 collection and also the
delight we share about the elegant solution you found within the
Soria algorithm itself to a question we are discussing with two
versions of Eb at 252 and 268 cents!

> One question I would like to ask you (and others), is how I
> could write the names of the 17 notes of, lets say more
> generally, a 17-ET, in Persian music (with sori ad koron
> symbols) and in (western) microtonal music ?

Here we should briefly consider that in modern Persian music
theory, musicians and scholars such as Hormoz Farhat and Dariush
Tala`i make a distinction between a regular limma or semitone
(e.g. E-F) at about 80-90 cents, and the smaller amount by which
a koron (p) lowers or a sori (>) raises a note, typically around
45-70 cents.

The recommended 17-note tunings of the tar and similar
instruments presented by these scholars are therefore irregular:
Farhat, for example, uses a limma of 90 cents (a nicely rounded
number equivalent to the Pythagorean 256:243), and koron or sori
steps of 45, 65, or 70 cents. See persian-far.scl in the Scala
archives. The most common division of a whole tone at a rounded
205 cents (equivalent to 9:8), e.g. C-D, is 90-45-70:

C Db Dp D
0 90 135 205
90 45 70

This 90-45-70 division applies in Farhat to each diatonic
whole-tone of his 17-note octave (C-C) except F-G, where a
division of 65-65-70, quite close to 17-ET, applies:

F F> Gp G
500 565 630 700
65 65 70

Thus we have:

205 205 90 200
|------------|-------------|----|-------------|
C Db Dp D Eb Ep E F F> Gp G
0 90 135 205 295 340 410 500 565 630 700
90 45 70 90 45 70 90 65 65 70

205 205 90
|------------|-------------|----|
G Ab Ap A Bb Bp B C 700 790 835 905 995 1040 1110 1200
90 45 70 90 45 70 90

In Farhat's tuning, a 90-cent limma step always receives a
diatonic semitone spelling (e.g. C-Db), while a smaller step is
always spelled as a koron or sori interval. According to Farhat
and Tala`i, a koron or sori interval is _not_ used as a direct
melodic step except in the context of certain ornaments sometimes
described as "microtones."

Jean During reports the results of one survey of tunings used by
some noted Iranian performers (voice, ney, tar, or setar)
suggesting that semitone steps of around 70-75 cents, comparable
to a larger koron or sori interval of Farhat or Tala`i, are not
so uncommon in practice. Another remark of Farhat, who notes that
"chromatic" progressions like Eb-E are not used, suggests that
the theoretical exclusion of even large koron or sori steps from
the usual patterns of melody may reflect the unexpected placement
of these steps as well as their absolute size, since Eb-E would
be a 135-cent neutral second much like the routine D-Ep in Shur,
for example.

Whatever lines may or may not be drawn in modern performances of
classical Persian music, however, many 17-note systems present a
situation where regular semitone or limma steps are often
identical or interchangeable with those used for a koron or sori
inflection. Before delving into the use of koron/sori notation
for such systems, we may find it helpful briefly to consider the
kind of classical modern tar intonation, as it might be
described, suggested by Farhat's recommended tuning and his
musical examples for some of the dastgah-ha.

----------------------------------------------------
1, Some examples of koron and sori in Persian theory
----------------------------------------------------

First, let us consider his Dastgah-e Shur, with the final on D:

Ap
630
135|70
A Bp C D Ep F G A Bb C D
-500 -365 -205 0 135 295 495 700 790 995 1200
135 160 205 135 160 200 205 90 205 205

Often a koron lowers a step by about 70 cents, or a third of a
tone. Shur, as interpreted by Farhat, illustrates a pattern that
I generally seek to follow: when a minor third in Persian music
is divided into two Zalzalian or neutral second steps, the
smaller step often precedes the larger, here in a division of
around 135-160 cents (e.g. A-Bp-C below the final; D-Ep-F).

Not all Persian theorists or performers observe this preference:
thus Tala`i recommends a tetrachord up from the final of about
140-140-220 cents, making no distinction between the two
Zalzalian steps. Farhat's examples show how musicians who do
recognize such a distinction may adjust a fretting scheme to
obtain the desired order of smaller and larger Zalzalian steps at
common locations for some of the dastgah-ha, for example D Shur.

A characteristic trait of Persian Shur as well as Arab Bayyati is
a distinction between a minor sixth above the final in the
"standard" form of the mode (Bb), but a neutral third below it
(Bp). The extra note Ap is a motaqqayer or "mutable note" often
used as a lowered form of the fifth degree A in descents toward
the final, e.g. Bb-Ap-G...

Let's next consider the notation of the basic form of Dastgah-e
Chahargah, rather like that of two conjuct Hijaz tetrachords in
maqam music with the final as the common note where the
tetrachords join, here C:

|-------------------|------------------|
G Ap B C Dp E F
-500 -365 -90 0 135 410 500
135 275 90 135 275 90

Farhat describes the middle interval of a Chahargah tetrachord at
about 275 cents as a "plus-tone" somewhat smaller than a regular
minor third at around 290-295 cents. The spelling, Ap-B or Dp-E,
fits with the idea of a tone plus a koron interval rather smaller
than a limma: thus Ap-A-B or Dp-D-E, with the concept of a
"thirdtone" often fitting nicely, although smaller koron or sori
intervals on the order of 45-60 cents may also occur.

While koron/sori notation readily shows the location of Zalzalian
or "plus-tone" steps (the latter, in Farhat, on the order of 7:6
or 75:64), it is not always so transparent in suggesting finer
distinctions between interval types. For example, Farhat explains
that the tetrachord leading up to the final of Dastgah-e Bayat-e
Esfahan has a small neutral second, tone, and large neutral
second, suggesting something like 135-205-160 cents, which is
indeed one attractive interpretation, known as the "Old Esfahan."
However, his actual notation is as follows:

D Ep F> G

Glancing at the notation, we see that D-Ep and F>-G are Zalzalian
seconds as expected -- but must arrive at the sizes in the
context of a given tuning scheme (for a fixed-pitch instrument),
such as Farhat's:

D Ep F> G
0 135 360 495
135 225 135

We have in fact a tetrachord of the Buzurg type, with two
smallish neutral thirds plus a middle interval around 8:7, indeed
quite close to the classic 14:13-8:7-13:12 at 128-231-139 cents!

If we placed the final on C rather than G, then Farhat's order of
small neutral second, tone, and large neutral second would
obtain:

G Ap Bp C
0 135 340 495
135 205 160

This form of Esfahan may be placed in the wider category of
tetrachords in maqam or dastgah music with the middle interval of
a tone not too far from 9:8, and a lower and upper neutral
second: in medieval and modern Arab theory this genre is known as
`Iraq, while on this list it is often called Mohajira. As you
have noted here, this type of genre using 9:8 should really be
distinguished from Buzurg with a step of around 8:7.

As Farhat himself emphasizes, the notation is meant to be
"flexible," and thus can be congenial to a range of regular or
irregular 17-note systems. Happily this congeniality extends to
systems where limma and koron/sori steps are freely
interchangeable.

---------------------------
2. The simplest case: 17-ET
---------------------------

In 17-ET, a step of 1/17 octave of 70.588 cents identically and
uniformly defines either a regular limma (e.g. E-F) or a koron or
sori inflection (e.g. F-F>, Gp-G). Further, for example, this is
no distinction between a regular minor third such as G-Bb and a
"plus-tone" such as Ap-B in a setting such as Dastgah-e
Chahargah. Both are equal to 4/3-tone, or about 282.35 cents.

Our 17-ET notation is thus straightforward, with equivalent
notations shown for each accidental:

C> Dp D> Ep F> Gp G> Ap A> Bp C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C 0 70 141 212 282 353 424 494 565 635 706 776 847 918 988 1059 1129 1200

Here the rule is that a flat or sharp lowers or raises by two
tuning steps, or 141 cents, while a koron or sori lowers by one
such step or 71 cents -- thus, as it happens, literally serving
as a "half-flat" or "half-sharp." The more general concept of a
thirdtone inflection, also literally realized in 17-ET, is often,
although not always, applicable as an approximate guide in other
17-note systems.

An interesting feature of 17-ET is that the main tetrachord of
Shur, e.g. D-Ep-F-G at 141-141-212 cents, is not too far from
Tala`i's recommended 140-140-220 cents.

628
G#
Ap
A# D# 132|77
A Bp C D Ep F G A Bb C D -494 -353 -212 0 141 282 494 706 776 988 1200
141 141 212 141 141 212 212 77 212 212

Note that while the guideline of Farhat and Tala`i against using
a kori or sori inflection as a usual melodic step (e.g. Ap-A)
does not hold consistently in many 17-note systems and styles
where the same physical interval may be used as a more or less
ordinary semitone (e.g. G#-A) in other transpositions or modal
contexts, their rule is still a very useful guide in the modal
context where the koron or sori notation occurs. Thus D Shur's spelling tells us that Ap does _not_ normally move directly
upward to A, a koron higher. Rather, this inflection often marks
a descent toward the final; and it may also occur when the main
tetrachord of Shur immediately above the final (D-Ep=F-G) is
transposed upward by a fourth, so that we have G-Ap-Bb-C.

The central interval of a Chahargah tetrachord, 141-282-71 cents,
is likewise not very far from Farhat's suggested size around 270
cents, although it is also the regular minor third and so does
not have special narrower size of his "plus-tone." If we place
the final of this dastgah on C, with conjunct tetrachords below
and above, we have:

G# C#
G Ap B C Dp E F
0 141 424 494 635 918 988
141 282 71 141 282 71

A lower Esfahan tetrachord leading up to the final, 141-212-141
cents, is like the Old Esfahan in using a regular middle second
for the middle step; but somewhat different in not having any
large neutral second, say on the order of 12:11 or 128:117. Here,
following Farhat, the final is on G, the fourth step of D Shur:

G# A#
G Ap Bp C
0 141 353 494
141 212 141

------------------------------------------
3. Two other tunings: regular and coherent
------------------------------------------

In 17-ET, the limma cannot be distinguished from a koron or sori
inflection because there is only size, 71 cents, available for
either use. However, the two types of steps may also be treated
as fully or at least partially interchangeable in tuning systems
where varying step sizes are available, as happens in circulating
temperaments such as George Secor's 17-WT, as well as a range of
regular or irregular temperaments where such circulation (at
least in a conventional sense) is not a goal.

Here I will consider first a regular but unequal temperament, my
scheme using a fifth of 704.607 cents; and then a coherent
rational tuning, your soria12.scl which presents a subset of a
forthcoming 17-note system.

In the 704.607-cent temperament, we can produce a 17-note MOS
either simply by using a chain of 16 fifths; or by the "Mohajira"
method of chaining together neutral thirds with alternating sizes
of 341 and 363 cents.

Either method produces a 17-note set with two adjacent step
sizes: a regular limma at 77 cents, and a diesis or small
semitone at 55 cents. Placing the 1/1 so as to match as closely
as possible Farhat's C-C scheme, we find that a keyboard octave
of E-E serves this purpose. Here I show Farhat's spellings (C-C),
and their equivalents in terms of a pragmatic notation to show
keyboard locations, where this 17-MOS is a subset of a 24-note
system using two conventional 12-note manuals (Eb-G#) at a diesis
of 55 cents apart. Note that an asterisk (*) shows a note on the
upper keyboard raised by this 55-cent diesis.

209 209 77 209
|--------------|--------------|-----|----------------|
Keyboard E F F* F# G G* G# A Bb Bb* B
Persian C-C C Db/C> Dp/C# D Eb/D> Ep/D# E F Gb/F> F#/Gp G
0 77 132 209 286 341 418 495 572 628 705
77 55 77 77 55 77 77 77 55 77

209 209 77
|--------------|---------------|-----|
Keyboard B C C* C# D D* Eb*/D# E
Persian C-C G Ab/G> Ap/G# A Bb/A> Bp/A# B C
705 782 837 913 991 1046 1123 1200
77 55 77 77 55 77 77

As the Persian or C-C equivalent spellings show, each diatonic
whole-tone in this C-C spelling scheme is divided into steps of
77-55-77 cents. A flat or sharp lowers or raises a note by 132
cents (e.g. Bb-B, C-C#), while a koron or sori lowers or raises
it by one 17-note step of 55 or 77 cents -- in this keyboard
arrangement, as it happens, generally 77 cents.

Here is a version of Shur Dastgah on Persian D, or keyboard F#:

628
C*
Ap
132|77
C# D* E F# G* A B C# D E F#
A Bp C D Ep F G A Bb C D -495 -363 -209 0 132 286 495 705 782 991 1200
132 154 209 132 154 209 209 77 209 209

As with Farhat's tuning above, we have Bp below the final but Bb
above it in a "textbook" citation of Shur, and the lowered form
of Ap often used in descending toward the final.

As with his tuning, so here, there are two forms of Esfahan. Let
us first consider placing the final on Persian G, or keyboard B,
approached by an ascending tetrachord at D-G (keyboard F#-B):

F# G* Bb B
D Ep F> G
0 132 363 495
132 231 132

Our resulting Buzurg is quite close to a just 128-231-139 or
Farhat's 135-225-135, with either modern version, however, losing
the subtle distinction in Safi al-Din al-Urmawi and Qutb al-Din
al-Shirazi between the initial 14:13 and upper 13:12 steps.

As with Farhat's tuning, if we place the final, for example, on
Persian C, then the "Old Esfahan" results:

B C* Eb E
G Ap Bp C
0 132 341 495
132 209 154

The tetrachord here is 132-209-154 cents, not too far from the
135-205-160 cents that would result in Farhat's tuning, or a just
version at 1/1-13/12-39/32-4/3 or 96:104:117:118, with steps of
13:12-9:8-128:117 (139-204-155 cents) suggested by Ibn Sina's
use of these step sizes in Mustaqim or 1/1-9/8-39/32-4/3, In
fact, as you have suggested, if Dastgah-e Segah is begun on the
third below the modern final -- often C and Ep respectively --
then Ibn Sina's Mustaqim or something quite similar results, as
is shown in segah2.scl in the Scala archives.

With Dastgah-e Chahargah, we face a dilemma using "only" 17 notes
of a system I am accustomed to in a 24-note form. Either we must
use Chahargah tetrachords with ordinary minor thirds at 286 cents
rather than the desired middle step around 7:6 (here 264 cents);
or we must resort to some various curious transpositions. The
reason is that our 704.607-cent temperament produces a near-7:6
third from a chain of 14 fifths up, so that our MOS-17 set has
these thirds at only three locations -- by comparison to the ten
locations available in a 24-note tuning. Following Farhat, we
will place the final of Chahargah on Persian C (keyboard E), the
step at a whole tone below the final of Shur, to demonstrate the
readily available form using a regular minor third:

B C* Eb* E F* G# A
G Ap B C Dp E F
0 132 418 495 627 914 991
132 286 77 132 286 77

To demonstrate a really idiomatic form of Chahargah with the
near-7:6 step, we must choose more unlikely location by usual
Persian modal standards. One curious choice places the final on
Persian E or keyboard G#, a major second above the final of our
Shur as shown above at Persian D or keyboard F#:

Eb*/D# F G* G# Bb C* C#
B C> D# E F> G# A
0 154 418 495 627 914 991
154 264 77 154 264 77

As these two examples show, Persian notation flexibly indicates
the general type of an interval such as a neutral second, often
shown as either a tone less koron (G-Ap or C-Dp) or a limma plus
sori (B-C> or E-F>), while leaving open the specific size, which
happens to be 132 cents in the first example and 154 cents in
the second. In remote transpositions, this notation can become
somewhat obscure, especially if one is not intimately familiar
with a given tuning system.

Thus Ap-B or Dp-E in the first example readily suggests some kind
of "plus-tone" interval as expected for the central step of a
Chahargah tetrachord, here the less telling result of a regular
minor third at 286 cents (i.e. a tone plus a koron here equal to
a usual 77-cent limma). The second example is more musically
idiomatic but notationally obscure, using the spelling C>-D# or
F>-G# to show an augmented second C-D# or F-G# less a sori! Only
when we know that an augmented second in this regular tuning is
equal to 341 cents, and the sori in this context to 77 cents, can
we determine that the resulting step is our desired 264 cents.

In some of the best 17-note systems, including the tar tunings of
adept Persian musicians, step and interval sizes may in fact vary
continually as one moves about the system. An outstanding example
of this variability which is still emerging in its full 17-note
or 18-note form is your Aulos/Soria tuning. While a complete
17-note version would obviously be required to appreciate the
fuller potential of the system, your admirable subset will
suffice to illustrate some Persian notation.

Let us again begin with Shur, taking C as the final, as in your
discussion of the placement of Bb in this version of the tuning
at around 7/4:
631
Gp
137|71
G Ap Bb C Dp Eb F G Ap Bb C
-497 -357 -226 0 137 283 495 703 843 974 1200
140 132 226 137 146 212 208 140 132 226

Given the inherent limitations of a 12-note chromatic scale in
contrast to a set of 17 or more notes, we must choose only one
form of the degree Ap/Ab -- here the koron rather than the flat,
which thus appears both a sixth above and a third below the
final. This choice agrees with Ibn Sina's famous septimal tuning
we have celebrated here, with the G-C tetrachord at 140-132-226
cents nicely approximating one interpretation of his tetrachord
at 12:13:14:16 (0-139-267-498 cents) or 139-128-231 cents. Indeed
your coherent ratio of 12443:10638 or 271.329 cents is very close
to 7:6, and comparable to an interval of 12 commas in Pythagorean
tuning or 53-ET. Further, unlike my tuning, the distinction
between the two subtly unequal Zalzalian steps at 13:12 and 14:13
is here respected!

While with 12 notes we must choose one form for the sixth above
the final (or third below it), nevertheless the motaqayyer of Gp
at 631 cents is very nicely placed; this inflected step could be
used as the highest note of a phrase seeking to descend toward
the final (e.g. F-Gp-G-Eb-Dp-D), or in conjunction with Ap rather
than the Ab more common for a descending gesture in a fuller
tuning set (e.g. Ap-Gp-F...).

While realizing a complete Esfahan taking advantage of the ideal
position for a Buzurg type of tetrachord below the final at G#-C#
(or Ap-Dp) in this version would require a step at D# (a tone
above the final), the tetrachord itself is most noteworthy:

G# C#
Ap Bb C Dp
0 132 357 494
132 226 137

Again, unlike Farhat's 135-225-135 or my 132-231-132, the two
Zalzalian steps are very subtly unequal.

For Dastgah-e Segah, taken as Ibn Sina's Mustaqim with the modern
final placed on the third step, a final of Bb serves best:

F# G# B C# E F#
Gp Ap Bb Cp Dp Eb Fp Gp
0 211 343 494 706 851 989 1200
211 132 151 211 146 138 211

The lower tetrachord at 211-132-151 cents nicely approximates Ibn
Sina's 204-139-154 (F#-B), while the conjunct upper tetrachord at
C#-F# at 211-146-138 cents has a different ordering of the
neutral step sizes. If we are looking to approximate 13/8 more
closely, and are ready to run with a 480-cent fourth (comparable
to narrow fourths mentioned by Near Eastern theorists or measured in
practice), then we might try:

G# C#
F G Ap Bb C Dp Eb F
0 208 348 480 705 842 988 1200
211 132 151 226 137 146 212

Now the tetrachords F-Bb and Bb-Eb both have the smaller neutral
second first: 211-132-151 and 226-137-146. In addition to a
near-just 13/8 at 842 cents, we get the interesting variation of
a near-just 11/9 at 348 cents in the lower tetrachord -- with
this 11/9 from the perspective of Ibn Sina's Mustaqim serving as
the modern final of Segah.

As it happens, the spellings with flats and korons are identical
to those at one placement of the final occurring in Persian
practice, with the modern final at Ap.

Finally, let us consider Chahargah. In soria12.scl, the tuning
structure calls for us to place the final on C, also the final of
Shur:

G# C#
G Ap B C Dp E F
0 140 423 497 634 917 992
140 283 74 137 283 74

The result is generally quite similar to 17-ET. Although Jean
During does not include Dastgah-e Chahargah in his brief table
comparing the intonations of some leading performers, he does
report on a similar tetrachord in Dastgah-e Homayun where Mahmud
Karimi in a vocal performance used a middle step measured at
about 70.7 savarts or 282 cents, very close to a just 28:17 or
17-ET. Other performers often used steps closer to 7:6, thus
fitting Farhat's suggested size of around 270 cents. A 17-note
version of Soria with a larger complement of near-septimal minor
thirds may make this type of interpretation possible at a number
of locations.

Even in this 12-note version, the coherently rational Soria
system reveals a wealth of intervals shadings and nuances; the
17-MOS of my 24-note regular temperament at 704.607 cents offers
more "predictability" but less variety or ability to observe
subtleties such as the distinction between 14:13 and 13:12; and a
tar fretting like Farhat's seeks an apt ordering of intervals
such as smaller and larger neutral seconds to fit the musician's
conception of the qualities each dastgah in its most common
location or locations.

As Tala`i discusses at some length, such schemes bring into play
the reality of "temperament," a term which to him means
especially the everyday situation where one transposes a given
tetrachord to a location where its interval sizes differ somewhat
from the ideal. Thus someone favoring the diverse sizes of a
system like Soria will be in good company.

My main purpose here has been to illustrate some uses of the
koron and sori symbols, while attempting to put these symbols in
some musical context.

Most appreciatively,

Margo Schulter
mschulter@...