sent in Tue 2006/05/02 10:24
Dear Shaahin,
Please let me try to respond to your most generous
sharing of the Talayee book by giving a bit more information about how some of Talayee's
tetrachord tunings look at different positions in Peppermint. Also, it's
interesting to compare Talayee's suggested interval sizes with those of Farhat,
who is cited as a source in Talayee's book.
-----------------------------------------
1.
Farhat's theory of five interval types
-----------------------------------------
Farhat's theory is interesting because it seems largely
to agree with Arabic theory based on how traditional ensembles actually tune
and perform (for example in
Farhat's basic approach of five standard melodic
interval sizes isn't too different from what contemporary Arab writers such as
Racy and Touma describing and advocating traditional techniques have
formulated, and also scholars from other parts of the world like Scott Marcus.
In this approach, there are five "basic"
interval types, although the average sizes may vary a bit between Arabic and
Persian traditions:
1. The semitone or limma,
often around Pythagorean (256:243, or 90
cents) --
sometimes a bit smaller in Persian practice, as
Talayee's
80-cent step might suggest.
2. The small neutral second,
maybe typically about 135-145 cents
in Arabic
practice, and often tending to be a bit smaller in
Persian
practice, say 125-140 cents, with 135 cents as an
average.
3. The large neutral second,
which together with the small one
makes a
regular minor third of around Pythagorean size (32:27
or 294
cents -- with about 295 cents a Persian average
according
to Farhat). Thus this step is typically around
150-160
cents in Arabic practice, and 150-170 cents in Persian
practice.
4. The whole tone or major
second, around the just Pythagorean 9:8
(204
cents), and sometimes according to Farhat very slightly
larger or
smaller (e.g. 200 cents or 205 cents). I'm not sure
if
Farhat's steps of 200 and 205 cents reflect fine differences
in tuning
a tar or setar, or whether it might just result from
his
rounding of sizes to the nearest 5 cents. Anyway, both
Arabic and
Persian tunings often seem to take the just 9:8 tone
as a
standard.
5. The "augmented
second" (Arabic) or "plus second" (Persian) used
in Maqam
Hizaj or Chahargah Dastgah, often described in the
literature
on Arabic music as roughly equal to a usual minor
third
(around 32:27 or 294 cents), but according to Farhat
notably
smaller in Persian music, around 270 cents (or 7:6 at
267 cents,
as in the Hijaz tetrachord according to Qutb-al-Din
al-Shirazi).
The Persian understanding of the koron and sori as
being not precisely 1/4-tone, but rather more generally somewhat smaller than a
regular semitone or limma, might be applied to Arabic music in a traditional
style, where performers recognize that the corresponding "half-flat"
and "half-sharp" symbols represent
adjustments which can vary in size depending on the maqam, for example.
------------------------------------------------------
2. Some
tunings of neutral seconds: Shur and Chahargah
------------------------------------------------------
One suggestion on Persian intonation I have read
is that a neutral second above the lowest note of a tetrachord tends to be
smaller when the following interval is another neutral second forming a minor
third then when it is a "plus second" completing a major third. In
the Peppermint temperament, my preferred intonations for Shur and Chahargah
tetrachords does follow this advice. Thus:
129 159 208
Shur C C# Eb F
0 129 288 496 or
138 150 208
E F* G A (EDL
52:48:44:39)
0 138 288 496
150 267 80
Chahargah C* D
E* F* (EDL
84:77:66:63)
0 150
416 496 or
138
278
80
E F*
G# A
0 138
416 496
Two of these tunings are so close to EDL that I've
given the almost identical just ratios. The first Chahargah tuning is almost
identical to Qutb-al-Din's Hijaz tetrachord. More generally, there are four
sizes of neutral seconds in Peppermint, three of which are very close to
relatively small superparticular ratios:
C C# E F* F* G C# Eb
128.67
138.20
149.51
159.04
near 14:13 near 13:12 near 12:11
(128.30)
(138.57) (150.64)
The first two sizes of around 129 and 138 cents
could be considered "small," and the last two sizes of around 150 and
159 cents "large."
In a Persian style, if I understand it correctly,
one reason for using a neutral second step in Chahargah somewhat larger than
129 cents is to follow the tendency for a "plus second" to be rather
smaller than a usual minor third -- instead of the same size, as in this
example.
129
288 80
C C#
E F
0 129 416
496
Nevertheless, one might accept this tuning if a
Chahargah tetrachord is necessary or convenient at C, and this brings me to
Talayee's use of the term "temperament."
3. Talayee's "temperament" concept
When I read what you so kindly sent me, I noticed
that Talayee's use of the term "temperament" is maybe a bit different
than in European theory -- and equally logical.
In a typical European setting, a
"tempered" interval means one deliberately tuned slightly smaller or
larger than a desired just ratio. A classic example is the kind of meantone
temperament widely used in the 16th century, where each fifth is tuned about
five or six cents narrow of 3:2 in order to get regular major and minor thirds
around the sizes of 5:4 and 6:5.
When Talayee refers to a 150-cent neutral second
as a "tempered"
interval, I suspect that he is speaking of a
different kind of temperament, since this size is just about identical to a just
12:11
(150.64
cents), a step in Zalzal's scale according to al-Farabi and indeed the simplest
just ratio for a neutral second.
What I'd take Talayee to be saying is that in
certain positions of the Shur tetrachord, one must use a 150-cent step for the
lower neutral second although something smaller -- he suggests 140 cents, and
others such as Farhat suggest an average of around 135 cents -- would best fit
a Persian style.
The common idea in both these concepts of
temperament is the element of "compromise" required for more freedom
of transposition with a fixed-pitch instrument. While the first concept
involves a compromise in the tuning of a given interval (e.g. a fifth slightly
smaller or larger than a pure 3:2), the second can involve the _arrangement_ of
intervals, either just or tempered.
Either the Egyptian understanding of how to tune
Maqam Bayyati or the Persian understanding of the most idiomatic tuning for
Shur could provide an example. Thus in Peppermint, if we start from B, either
the Egyptian or the Persian idiom calling for the smaller neutral second to be
placed below the larger could be satisfied by this arrangement:
138 150 208
B C* D E
0 138 288 496 EDL: 13:12:11
If for some reason we wanted to place the same
tetrachord at C*, however -- Talayee's step of Sabbabeh -- then one of these
patterns would result:
150 138 208
C* D Eb* F*
0 150 288 496 EDL: 156:143:132 (harmonics
11:12:13)
C* C#* Eb F
0 129 288 496
129 159 288
The first solution of 150-138-208 could be
considered a "temperament"
of either Bayyati or Shur in Talayee's meaning,
because the larger neutral second is placed below the smaller, contrary to the
most favored tuning. Here the interval sizes themselves are identical to those
of Bayyati or Shur on B -- with neutral seconds very close to just ratios of
13:12 and 12:11 -- but conversely arranged.
Curiously, the second available solution of
0-129-208 might be considered also a kind of "temperament" of the
usual Bayyati, where as Marcus suggests the ideal range for the lower neutral
second is around
135-145
cents (in comparison to 129 cents here, C-C#); but a more routine tuning for
Persian Shur, where a range of about 125-145 cents might be customary, with 129
cents within this range.
While this is only my reading of Talayee, I do get
a hint that it could be correct from his remark that tempered intervals"
produce an intonation that "always bothers the best musicians" (p.
19). This musical judgment, rather than any mathematical definition, might be
taken as the essence of temperament: "An expert musician will know that something
is slightly imperfect, but an average listener will accept the result."
--------------------------------------------
4. Some
of Talayee's diagrams and Peppermint
--------------------------------------------
In explaining tetrachords, tuning, and
temperament, Talayee gives a diagram like this:
Start of Zabol of Chahargah
|--------------|
|
140
|......
|--------------|----------------------|------------|
Shushtari | 150 |
230
| 120 |
|--------------|----------------------|------------|
|--------------|--------------|--------------------|
Oj | 150 | 150 |
200 |
|-------------|-----|-------|------|--------------|--------------------|
| 140 | 140 |
220
|
Shur
|-------------|-----|-------|---------------------|
|
200
| 80 |
220
|
Dashti
|-------------|-----|-------|---------|-----------|
| 140 |
240
| 120 |
Chahargah
|-------------|-----|-----------------|-----------|
|
200
| 180 | 120 |
Mahur
|-------------------|-----------------|-----------|
With Peppermint it is possible to make similar
diagrams showing how tetrachords may be tuned at different locations for the
"Motlaq" step.
Here I will give the distance in cents of notes
from this "Motlaq" as well as the sizes of tetrachord steps:
|--129--|---- 288 -----|-80-| Zabol
F F#
A Bb
496
625
912 992
|--138--|------ 278 ------|-80-|
Shushtari
D
Eb*
F# G
288
346
625 704
|--138--|---150---|---- 208 ---|
Oj
D
Eb*
F
G
208
346
496
704
|--129--|---159---|--- 208 ----|
Shur
C C#
Eb
F
0 129 288
496
|---- 208 ---|-80-|--- 208 ----|
Dashti
C
D Eb
F
0
208 288
496
|--129--|------ 288 ------|-80-|
Chahargah
C C#
E F
0 129
416 496
|---- 208 ---|--- 288 ----|-80-|
Mahur
C
D
E F
0
208
416 496
Characteristics of this keyboard
"Motlaq" include a Shur at
129-159-208
with what might be typical Persian neutral second sizes, and an Oj at
138-150-208 which also follows the pattern of placing the smaller neutral step
below the larger. However, Chahargah and its Zabal are obliged also to use this
small 129-cent step, along with a central step of a regular minor third at 288
cents rather than a "plus second" at 267 or 278 cents. One could look
at this as either a "temperament" compromise, or as a desirable
variety of steps and colors.
With Shushtari on D, interestingly, we have a
138-cent neutral third and 278-cent plus second (D-Eb*-F#-G), lending a bit of
contrast with Chahargah on C.
Suppose we make our Motlaq higher by a small
thirdtone of about 59 cents, placing it at C* on the upper keyboard. Some of
the tetrachords, those with notes taken from a single keyboard, will remain the
same, but others will change -- or can change if we so wish, as with Chahargah
and its related Zabol.
|---150--|----
267 -----|-80-| Zabol
F*
G
A* Bb*
496
646
912 992
|---150--|----- 267 -----|-80-|
Shushtari
D*
E
F#* G*
288
358
625 704
|---150--|---138--|--- 208 ---|
Oj
D*
E
F*
G*
208
358 496
704
|--129--|---159---|--- 208 ----|
Shur
C* C#* Eb*
F*
0 129 288
496
|---- 208 ---|-80-|--- 208 ----|
Dashti
C*
D* Eb*
F*
0
208 288
496
|---150--|------ 267 -----|-80-|
Chahargah
C* D
E* F*
0 150
416 496
|---- 208 ---|--- 288 ----|-80-|
Mahur
C*
D*
E* F*
0
208
416 496
Here Mahur, Dashti, and Shur remain as before, but
Chahargah, Shushtari, and Zabol are available in 150-267-80 cent versions
almost identical to the tuning of Qutb-al-Din. (If we wished, we could also use
a 129-288-80 version of Chahargah, C-C#*-E*-F*). However, Oj starting on D*
must place the larger neutral step below the smaller (150-138). This is not so
good if one wants consistently to observe the traditional intonation, but from
another viewpoint possibility a stimulating variation of shading or color.
----------------------------------------------
5.
Another diagram: medieval tunings and taste
----------------------------------------------
To this point, we have tuned the minor third of
the Shur and Dashti tetrachords -- and also Oj (like Shur) -- using the regular
Peppermint interval of 288 cents (a bit smaller than the Pythgorean 32:27, 294
cents). However, al-Farabi (following Archytas) and Ibn Sina include in their
tunings a small minor third at a just ratio of 7:6, 267 cents.
In modern Persian music, this seems used mostly as
a "plus second" in Chahargah (where together with a neutral second
below it forms a major third), but a style influenced by medieval literature
might also use it as a minor third formed from two neutral seconds (as in Shur)
or a whole tone plus a small semitone, also known as a thirdtone.
Ibn Sina's tetrachord of 28:26:24:21 (128-139-231
cents) features a
7:6
minor third divided into neutral seconds of 14:13 and 13:12; while a scale of
al-Farabi has a 7:6 third built from steps of 9:8 and 28:27
(204-63
cents).
In Peppermint these forms are approximated with
step patterns of
129-138
cents (e.g. C-C#-D*) using the two smaller sizes of neutral seconds; and 208-59
cents (e.g. C-D-D*) using a regular whole tone plus a small thirdtone.
While the first form builds the small minor third
from two neutral second steps themselves within the usual melodic range of
modern Persian practice, the second form involves the use of a thirdtone around
28:27 (63 cents), or actually in Peppermint a bit smaller at 59 cents.
This size of step around 60 cents is more typical
of a koron adjustment than a usual melodic interval in modern Persian practice
as reported by Farhat and Talayee. If one accepts such a step, however -- and
also the large thirdtone of Peppermint (e.g. F*-F#) at around 70 cents -- then
a larger set of melodic and polyphonic colors are available.
Consider, for example, a tetrachord diagram like
the ones above with a Motlaq of F:
|--129--|---- 288 -----|-80-| Zabol
Bb
B
D Eb
496
625
912 992
|--129--|------ 288 ------|-80-|
Shushtari
G
G#
B C
208 337
625 704
|--129--|---159---|---- 208 ---|
Oj
G
G#
Bb C
208 337
496
704
|--129--|---138--|---- 229 ----|
Shur
F F# G*
Bb
0 129 267
496
|---- 208 ---|-59|---- 229 ----|
Dashti
F
G G*
Bb
0
208 267
496
|--129--|------ 288 ------|-80-|
Chahargah
F F#
A Bb
0 129
416 496
|---- 208 ---|--- 288 ----|-80-|
Mahur
F
G
A Bb
0
208
416 496
Starting from F, our Shur tetrachord must use a
small 7:6 minor third because this is the only minor third available (G* on the
upper keyboard is the closest substitute for a regular Ab, not present in the
Eb-G# tuning of each keyboard). However, Oj starting from G can use the regular
minor third G-Bb at 288 cents, here done to obtain a certain subtle contrast
with Shur.
We could also tune Oj in the same way as Shur,
using a 7:6 minor third
G-A* in place of the regular G-Bb (G-G#-A*-C). The
difference between Bb and A* is a microtone of about 21 cents.
Dashti on F, like Shur, is obliged to use F-G* at
7:6 as the only minor third available, with a whole-tone plus a small thirdtone
step (F-G-G*, 208-59 cents) forming this interval. The tetrachord is completed
by a large tone G*-Bb at 229 cents, very close to 8:7 (231 cents). Some
commentators have suggested that in modern Persian practice this kind of large
tone may be rather uncommon, although Talayee describes steps of 220, 230, and
240 cents
One characteristic of this set of tetrachords is
that the middle step of Chahargah, Shushtari, and Zabol, the regular minor
third of 288 cents, is actually larger than the minor thirds of Shurand Dashti
at
7:6 or
267 cents. This reverses a typical situation where the "plus second"
of these last two tetrachords is smaller than the regular minor third of Shur
or Dashti.
Using another technique, however, we can also
obtain versions of these tetrachords with smaller plus seconds.
--------------------------------------------
6.
Microtonal shift and intonational choices
--------------------------------------------
Suppose we try a variation of our last example
based on a Motlaq of F, with the location of Chahargah and the related Zabol
very slightly
shifting:
|---150--|---- 267 -----|-80-| Zabol
A* B
C#* D*
475
625
891 971
|--129--|------ 288 ------|-80-|
Shushtari
G
G#
B C
208 337
625 704
|--129--|---159---|---- 208 ---|
Oj
G
G#
Bb
C
208 337
496
704
|--129--|---138--|---- 229 ----|
Shur
F F# G*
Bb
0 129 267
496
|---- 208 ---|-59|---- 229 ----|
Dashti
F
G G*
Bb
0
208 267
496
|---150--|------ 267 -----|-80-|
Chahargah
E* F#
G#* A*
-21 129
395 475
|---- 208 ---|--- 288 ----|-80-|
Mahur
F
G
A Bb
0
208
416 496
Rather than starting on precisely the same F step
as Shur, Chahargah begins on E*, about 21 cents lower, and now can have a form
of 150-267-80 cents, with a plus second at 7:6. This tetrachord shares the note
F# with Shur (F-F#-G*-Bb), providing a possible "bridge"
between Shur and Chahargah.
Likewise Zabol starts on A* rather than Bb, and
likewise has a 150-267-80 pattern (A*-B-C#*-D*).
Shushtari is here left as in the previous example,
and provides a variation on Chahargah or Zabol.
A microtonal interval such as E*-F or A*-Bb at
about 21 cents is thus most typically used, rather like a larger koron or sori
in modern Persian practice, not as a direct melodic step but as an amount by
which a step can be shifted depending on the musical context or mood.
Likewise, in Shur on C, it is possible to use
either Eb (288 cents) or
D* (267 cents) as a third step. Thus choice might
depend on melodic as well as polyphonic factors: the small semitone or
thirdtone in the descending figure D*-D-C (steps of 59 and 208 cents) may help
to "pull" the melody toward a final of C.
Since two steps such as E* and F at only 21 cents
apart might be heard as two "versions" of the same note, discreetly
shifting between F Shur and E* Chahargah might emulate the kind of inflections
that can occur in vocal performance or other situations involving
flexible-pitch instruments.
-------------
7.
Conclusion
-------------
What I have written here about Talayee's
tetrachord diagrams and some analogous diagrams for the 24-note Peppermint
temperament is only a first impression. However, it may at least suggest some
possibilities for keyboard improvisation (monophonic, pedal-based, or more
freely
polyphonic) using tetrachords and modes at various
locations, sometimes with microtonal shifts in the location of a tetrachord or
the sizes of some of its intervals.
I should caution that I am just beginning to
explore Persian music, and welcome suggestions from the more experienced.
Most appreciatively,
Margo Schulter
mschulter@calweb.com