---------------------------------------------
Peppermint 24: A HEP neo-medieval temperament
Part IIA: Near Eastern and mixed styles (1)
---------------------------------------------
In discussing the use of the Peppermint 24 temperament for Near
Eastern styles of music, or "mixed" styles combining Near Eastern
elements with medieval European ones, we may find it helpful to
emphasize early and often the intonational flexibility of Near Eastern
ensemble music.
Outstanding studies of Eastern Arabic ensemble music by Scott Marcus[1]
document ensemble practices favoring flexible intonation even where
academic models favor a fixed tuning system, for example the 19th-20th
century division of the octave into 24 equal quartertones.
While it is tempting to show how Peppermint can nicely approximate
some tuning ratios and scales specified in medieval Near Eastern
treatises, and I will yield to this temptation to a certain degree
later in this article, the themes of flexibility and choice in tuning
a given scale or mode may thus be most relevant.
As Marcus discusses[2], some modern fixed-pitch instruments do strive
for an equal temperament with 24 quartertones per octave, but stringed
instruments often favor setting some notes and intervals in
Pythagorean tuning with pure 3:2 fifths or 4:3 fourths. Ensemble or
solo vocal intonation is flexible: a step with the same location in
the theoretical model of equal quartertones might be tuned a bit
lower, or higher, depending on the _maqam_, a term more or less
equivalent to the European "mode."
The Arab understanding of a given _maqam_, plural _maqamat_, brings
into play such aspects as the arrangement of steps or tetrachords; the
location of the concluding note, somewhat analogous to a European
modal final; and characteristic patterns for finding a _sayr_,
literally a "path" or "road," through the maqam, or as Marcus puts it,
"a common path or progression for movement through the mode's various
regions."[3]
For example, the maqam Bayyati we are about to discuss has a central
octave of D-D, or more specifically of D4-D5 (with C4 as middle C)
within the Arabic two-octave "fundamental scale" as Marcus terms it
(in some ways analogous to a medieval European gamut) of G3-G5.[4] It
is a characteristic procedure in Bayyati to start a composition or
improvisation in the middle portion of this central octave, say
somewhere around F4-Bb4, "and then proceed down to the tonic D."[5]
Marcus cites an 1840 treatise by Mikhail Mashaqa, a leading exponent
of the equal quartertone model, that if one instead started at or
below the tonic, "this would be a characteristic feature of a
different mode and should not be considered _bayyati_, although he
admits that `most knowledgeable Syrians' would still have considered
it to be _bayyati_."[6]
Thus while the following discussion of Peppermint intonational
alternatives for Bayyati and other maqamat or modes focuses mainly on
the aspect of mode as an arrangement of scale steps, the aspect of
mode as _sayr_ or "procedure" is also an important part of the
traditional understanding in practice and theory. As Marcus observes,
this latter aspect has typically been learned "by osmosis, beginning
in early childhood," as one listens to and practices improvisations in
different modes; modern conservatories or other academic institutions
in places like Cairo often treat the definition of a mode mainly or
exclusively in terms of the tonic pitch, interval structure, and
tetrachords.[7]
--------------------------------------------
1. Maqam Bayyati in Peppermint: Some options
--------------------------------------------
According to the modern Arabic intonational model of 24 equal
quartertones per octave, which in European terms might be called
24-tone equal temperament or 24-tET for short, tetrachords and modes
are mostly formed from the following melodic steps, as identified by
the size of the step in equal quartertones:
--------------------------------------------------------
Interval Quartertones Cents Generic Type
--------------------------------------------------------
Semitone 2 100 S
Neutral 2nd 3 150 M
Tone 4 200 T
Augmented 2nd 6 300 A
--------------------------------------------------------
Although Marcus notes that modern Arabic theory does not itself use
the measure of cents (1200 to an octave, and 50 to a quartertone),
he finds cents helpful for giving an idea of some of the nuances of
ensemble tuning in practice, and of course they will also be helpful
here for measuring the sizes of tempered intervals in Peppermint.
Additionally, I have suggested some possible abbreviations for the
four usual "generic types" of melodic steps in this system:
(S)emitone; (M)iddle second; (T)one; and (A)ugmented second.
Following the equal quartertone model, the basic structure for the
central D4-D5 octave of Maqam Bayyati is as follows, with note
locations and step sizes shown in quartertones and cents:
Bayyati Nahawand
|-----------------|---------------|
M M T T S T T
D4 Ed4 F4 G4 A4 Bb4 C5 D5
0 3 6 10 14 16 20 24
3 3 4 4 2 4 4
150 150 200 200 100 200 200
The ASCII notation "Ed4" stands for "E half-flat," a step located a
middle or neutral second above D, and thus about midway between Eb and
E. The modern Arabic term, interestingly, is _mi nisf bimul_, also
translated as "E half-flat" -- with the solmization syllable _mi_
designating the note E in a modern French manner, and _nisf bimul_
meaning "half flat." Here _bimul_, like the 16th-century Spanish
_bemol_, refers to any flat -- a term derived from the medieval
European B-molle or "soft-B" sign showing the upper note of a mi-fa
semitone.[8]
As our diagram shows, this central octave is formed from two patterns
of dividing a fourth, or tetrachords (D4-G4, G4-C5), plus a tone to
complete the octave (C5-D5). The lower Bayyati tetrachord at D4-G4 has
a pattern of 3-3-4 or M-M-T, with two middle or neutral second steps
plus a tone; the upper Nahawand tetrachord at G4-C5 is 4-2-4 or T-S-T.
The Arabic term for tetrachord is _jins_ (plural _ajnas_), from the
Greek _genus_; a Greek genus or Arabic jins is a "kind" of division of
the fourth (or sometimes, in Arabic theory, another interval such as a
third or "trichord," or fifth or "pentachord"). Like Bayyati, Nahawand
is the name both of a mode (discussed below) and of a tetrachord which
may occur in various modes.
Here the two tetrachords are _conjunct_, sharing the note G4, the
highest note of Bayyati (D4-G4) and the lowest of Nahawand (G4-C5),
with an upper tone (C5-D5) filling out the octave. Another common
arrangement of the octave has _disjunct_ lower and upper tetrachords,
with a bridging tone in the middle (e.g. D4-G4, A4-D5), as in Maqam
Nahawand which we will meet below. If we consider the larger range of
about a twelfth (G3-D5) typical for Maqam Bayyati, extending about a
fifth below the central octave, we can observe another point of scale
structure:
|---------- Main modal octave ----------|
T M M T M M T T S T T
G3 A3 Bd3 C4 D4 Ed4 F4 G4 A4 Bb4 C5 D5
-14 -10 -7 -4 0 3 6 10 14 16 20 24
-700 -500 -350 -200 0 150 300 500 700 800 1000 1200
4 3 3 4 3 3 4 4 2 4 4
200 150 150 200 150 150 200 200 100 200 200
Note that while the main modal octave D4-D5 has Bb4, a 2-quartertone
or semitone step above A4, the lower range has Bd3, a 3-quartertone or
neutral second step above A3. To emphasize this element of asymmetry,
that the notes of Maqam Bayyati "do not duplicate at the octave"[9],
I have used negative values in quartertones or cents to show the
locations of notes below the tonic of D4.
Seeking to approximate this equal quartertone model for the tuning of
Maqam Bayyati in Peppermint, we might arrive at this solution, with
measurements given in rounded cents, and note names given both as
above, and as locations on the 24-note Peppermint keyboard, with the
asterisk (*) showing a note on the upper keyboard raised by the
quasi-diesis of about 58.68 cents:
|----------- Main modal octave ----------|
T M M T M M T T S T T
G3 A3 Bd3 C4 D4 Ed4 F4 G4 A4 Bb4 C5 D5
G*3 A*3 B3 C*4 D*4 E4 F*4 G*4 A*4 Bb*4 C*5 D*5
-704 -496 -346 -208 0 150 288 496 704 784 992 1200
208 150 138 208 150 138 208 208 80 208 208
To consider some subtle distinctions between the equal quartertone
model, or EQT for short, and Peppermint, we might start by focusing on
the Bayyati tetrachord of the mode at D4-G4, or Peppermint D*4-G*4.
Here, as in EQT, the third D4-F4 is divided into two middle or neutral
seconds, D4-Ed4-F4, realized in Peppermint as D*4-E4-F*4. The lower
step D*4-E4, a major second less a quasi-diesis or "Maj2-QD" for short
at ~149.51 cents, is almost identical to the EQT step of 150 cents;
but the step E4-F*4, a minor second plus quasi-diesis or "min2+QD," is
somewhat smaller at ~138.20 cents.
Thus in comparison to the EQT division of 150-150 cents for D-F, we
have an unequal division of 150-138 cents; note that the Peppermint
minor third at around 288 cents (or more precisely ~287.71 cents) is
somewhat narrower than the 300 cents or 6 quartertones of EQT.
The Bayyati tetrachord also includes the neutral third Ed4-G4, in EQT
a 7-quartertone interval of 350 cents, approximated in Peppermint as
E4-G*4, a minor third plus quasi-diesis (min3+QD) at a slightly
narrower 346 cents (or more precisely ~346.39 cents).
In EQT, Ed4-F4-G4 or M-T is a division of 3-4 quartertones or 150-200
cents; in Peppermint, we have a division E4-F*4-G*4 of around 138-208
cents.
Thus while the EQT Bayyati tetrachord of D4-Ed4-F4-G4 has steps of
150-150-200 cents, our Peppermint realization of D*4-E4-F*4-G4 is
150-138-208 cents, a similar _jins_ or type of pattern (M-M-T), but
with subtle differences of intonation.
------------------------------------------------------------------
1.1. Maqam Bayyati and common practice: An alternative realization
------------------------------------------------------------------
In comparing the EQT model for Maqam Bayyati with "performers'
knowledge," Marcus points to some intonational nuances and
"understandings" of "common practice."[10]
For example, the notes F4 and Bb4 "are generally played lower than
their equal-tempered positions of 300 cents and 800 cents,
respectively, above the tonic" D4. He suggests the possibility in each
case "that the position is best represented by the Pythagorean
interval," that is by the "Pythagorean minor third" at 32:27 or
~294.135 cents and the "Pythagorean minor sixth" at 128:81 or ~792.18
cents. Marcus notes that "the open strings of the 'ud are tuned in a
series of perfect fourths and thus create Pythagorean intonation for
many of the pitches."[11]
From this viewpoint, Peppermint differs from the theoretical EQT in
the same direction as the Pythagorean intonation which Marcus suggests
as one possibly congenial model -- only more so, indeed by almost
exactly twice as much (actually a tidge more).
In EQT, the 700-cent fifths (14 quartertones) are tempered about 1.955
cents narrow; in Pythagorean, they are pure at ~701.955 cents; in
Peppermint, they are about 2.14 cents wide. Thus EQT and Peppermint
differ from Pythagorean by almost exactly the same amount, but in
opposite directions, for intervals such as these in rounded cents:
--------------------------------------------------
Interval EQT Pythagorean Peppermint
--------------------------------------------------
minor 2nd 100 90 80
Major 2nd 200 204 208
minor 3rd 300 294 288
Major 3rd 400 408 416
4th 500 498 496
5th 700 702 704
minor 6th 800 792 784
Major 6th 900 906 912
minor 7th 1000 996 992
Major 7th 1100 1110 1120
--------------------------------------------------
Thus in comparison to the EQT values for the minor third and sixth
D4-F4 and D4-Bb4 at 300 and 800 cents, and the narrower Pythagorean
values at around 294 and 792 cents suggested by Marcus as one possible
approximation of common practice, Peppermint has yet narrower values
of around 288 and 784 cents -- offering, as with medieval European
music, an "accentuated Pythagorean" quality.
Marcus discusses two other nuances of note in the common practice
tuning of Maqam Bayyati. There is a "widely held understanding" that
the step Ed4 above the tonic "is tuned slightly lower" than in EQT.
While remarking that apparently no studies have sought to quantity
"the exact pitch (or range of pitch) for this note," he would suggest
"a lowering from the 150 cents of the tempered half-flat of between 5
and 15 cents."[12]
Additionally, "many hold" that the Bd3 of Maqam Bayyati below the
tonic D4 "also differs from its equal-tempered position by being
raised slightly above that position."[13]
In Peppermint, a transposition reflecting these nuances places the
tonic for the mode at F#4 on the lower keyboard, with usual note names
for the untransposed Arabic version again given for reference:
|----------- Main modal octave ----------|
T M M T M M T T S T T
G3 A3 Bd3 C4 D4 Ed4 F4 G4 A4 Bb4 C5 D5
B3 C#4 Eb4 E4 F#4 G*4 A4 B4 C#5 D5 E5 F#5
-704 -496 -337 -208 0 138 288 496 704 784 992 1200
208 159 129 208 138 150 208 208 80 208 208
Here the step D4-Ed4 is realized as F#4-G4 (min2+QD), around 138
cents, about 12 cents narrower than in EQT, implementing the
suggestion by Marcus for an adjustment of around 5-15 cents.
For the Bayyati tetrachord of D4-G4, here F#4-B4, we have steps of
138-150-208 cents, as compared to 150-138-208 cents in the previous
Peppermint version: a different arrangement of the same sizes.
Also, the step Bd3 is realized as Eb4, located at a Peppermint
augmented second or supraminor third below the tonic F#4, an interval
of about 337 cents -- about 13 cents higher than the EQT position of
350 cents (7 quartertones) below the tonic. In our previous version,
this step was about 346 cents below the tonic, quite close to EQT.
This last nuance affects the sizes of other intervals in the lower
fifth below the tonic or final of G3-D4, with a T-M-M-T pattern, here
realized as B3-C#4-Eb4-E4-F#4, with steps of 208-159-129-208. The
Peppermint diminished third C#4-Eb4, at around 159.04 cents, leans
somewhat toward a small tone, and the chromatic semitone Eb4-E4 at
around 128.67 cents somewhat toward a large semitone, giving this
division a melodic cast or flavor a bit different from one closer to
the EQT values. Compare the 208-150-138-208 division of the previous
version, with both middle or neutral second steps in a central
"neutral" region more distinct from a regular semitone or tone.
These distinctions between our two realizations for the lower fifth
can also affect the flavor of the upper octave when a typical move in
the _sayr_ or "procedure" for Maqam Bayyati comes into play: the
inflection or substitution of a neutral sixth above the tonic in place
of the standard minor sixth degree.
-------------------------------------------------
1.2. Maqam Bayyati and tetrachord transformations
-------------------------------------------------
This manuever provides an opportunity to illustrate the important
concepts of degree inflection, octave transposition, and tetrachord
transformation, factors which can make the implications of a given
Peppermint realization for a maqam more intricate, since the procedure
of a composition or improvisation may branch out into other
tetrachords or modal patterns.
For Maqam Bayyati, Marcus suggests that a usual procedure is to start
in the middle range of the mode, say the F4-Bb4 region in the standard
placement of the central modal octave at D4-D5, followed by a descent
settling on the tonic and some exploration of the lower region
including the step Bd3 a neutral third below the tonic, followed by
ascending motion to the middle and eventually the upper region. In the
middle region, G4 is often an important note because of its role as
the lowest step of the Nahawand tetrachord (G4-C5). Interestingly,
Bb4, the minor sixth above the tonic, often plays a role in the
descending figure Bb4-A4-G4-F4, which some theorists take as defining
a tetrachord known as _'ajam_ or _jaharka_ at F4-G4-A4-Bb4 with a
pattern of 4-4-2 quartertones or T-T-S.[14]
Marcus discusses as another "common movement" the one we are
considering: "a held A [A4] followed by a fall to A from c [C5] using
B-half-flat [Bd4]," for example A4 C5 Bd4 C5 A4.[15]
Comparing the intonation for this figure in our two Peppermint tunings
may suggest some of the melodic nuances possible:
Standard Arabic gamut A4 C5 Bd4 C5 A4
EQT cents 300 150 150 300
Peppermint version 1 A*4 C*5 B4 C*5 A*4
288 138 138 288
Peppermint version 2 C#5 E5 Eb5 E5 C#5
288 129 129 288
In comparison with the EQT version, either Peppermint version has a
transposed neutral or "semi-neutral" second step equivalent to Bd4-C5
somewhat smaller than 150 cents. The 138-cent step of the first
version might be taken as a quite subtle nuance of shading bringing
these two notes a bit closer together; the rather narrower 129-cent
step of the second version might be heard as more decidedly leaning
toward a semitone.
This figure might be connected with two themes in Arabic music
illustrated by Maqam Bayyati. One of these themes is "an understanding
of characteristic accidentals for each mode" -- a theme with rich
ramifications in medieval and Renaissance European practice and theory
also.[16]
Thus, again using the customary untransposed placement for Maqam
Bayyati with the central octave at D4-D5, Marcus notes that the upper
Nahawand tetrachord at G4-C5 "might rarely have a momentary B-natural
functioning as a replacement for the B-flat and serving as a
discontinuous lowering neighboring tone to the note c." He gives this
example, with the raising of the sixth above the tonic or final D4
from minor to outright major somewhat reminiscent of the neutral sixth
Bd4 in the above figure C5-Bd4-C5[17]:
G4 A4 Bb4 C5, C5 B4 C5, B4 C5 D5 C4 Bb4 A4 G4
Here it might be interesting to compare the intonation in EQT, a
Pythagorean model suggested by Marcus as one possible approximation of
common ensemble practice, and our two Peppermint versions:
G4 A4 Bb4 C5, C5 B4 C5, B4 C5 D5 C5 Bb4 A4 G4
EQT 200 100 200 100 100 100 200 200 200 100 200
Pyth 204 90 204 90 90 100 200 200 200 100 200
Pmint1 G*4 A*4 Bb*4 C*5, C*5 B*4 C*5, B*4 C*5 D*5 C*5 Bb*4 A*4 G*4
208 80 208 80 80 80 208 208 208 80 204
Pmint2 B4 C#5 D5 E5, E5 Eb*5 E5, Eb*5 E5 F#5 E5 D5 C#5 B4
208 80 208 70 70 70 208 208 208 80 208
In comparison to the EQT semitone at 100 cents, the Pythagorean model
has regular semitones at a narrower 90 cents, and the first Peppermint
version at yet more compact 80 cents, with these sizes applying
consistently. The second Peppermint version introduces the nuance that
while the usual modal semitone C#5-D5 (untransposed A4-Bb4) is a
regular 80 cents, the inflected semitone Eb*5-E5 (untransposed B4-C5)
is a yet narrower 70 cents, which might make this figure especially
incisive and telling.
The nuance of two different semitone sizes in this last version was
not a planned feature, but rather an incidental ramification of a
Peppermint transposition striving to approximate widespread
understandings about the fine tuning of standard steps in Maqam
Bayyati as described by Marcus (untransposed Ed4 a bit lower, and Bd3
a bit higher, than in the EQT model). As it happens, in this
realization the untransposed inflected semitone B4-C5 maps to Eb*5-E5,
a Peppermint chromatic semitone less quasi-diesis of ~69.90 cents.
Thus the "pull" to a given degree of a mode can motivate accidental
inflections, such as the substitution for Bb4 in untransposed Maqam
Bayyati of Bd4 or even B4 in the figure C5-Bd4-C5 or C5-B4-C5. The use
of Bd4, however, can also illustrate another process of Arabic music:
octave transposition.
As we have noted, the usual steps of Maqam Bayyati include an upper
Bb4 a minor sixth above the tonic D4, and a lower Bd3 a neutral third
below it. However, as Marcus explains, after a scenario such as the
one he has sketched of starting in the middle region of the mode,
descending to the final and some of the range below it, and then
ascending back to the middle region, an improvisation or composition
"might commonly rise to the octave region."[18]
"In order to effect this shift in focus, the note Bb [Bb4] is generally
replaced by B-half-flat [Bd4]." Marcus suggests that this might best
be understood as "a full octave shift of the overall scale" of Maqam
Bayyati, with Bd3 becoming Bd4, and the tonic itself temporarily
shifted from D4 to D5.[19]
This octave shift transforms the usual upper Nahawand tetrachord of
Maqam Bayyati (G4-A4-Bb4-C5), with a pattern of 4-2-4 quartertones or
T-S-T, to G4-A4-Bd4-C5: a Rast tetrachord pattern of 4-3-3 or T-M-M.[20]
As Marcus notes, in older theory a movement or figure of the type
A4-C5-Bd4-C5-A4 "was once considered a characteristic feature of a
distinct mode," Maqam Husayni; but in the branch of current Arabic
practice that he surveys, it has virtually fallen into disuse as a
distinct mode, and rather "the movement is now generally subsumed"
within Maqam Bayyati.[21]
For some theorists, either the use of this figure with Bd4 invited by
a "pull" to C5, or the octave shifting of lower Bd3 to Bd4, represents
the introduction of Maqam Husayni with conjunct lower Bayyati and
upper Rast tetrachords plus an upper tone forming the central octave,
with steps and intervals shown in the EQT model:
Bayyati Rast
|-----------------|---------------|
M M T T M M T
D4 Ed4 F4 G4 A4 Bd4 C5 D5
0 3 6 10 14 17 20 24
0 150 300 500 700 850 1000 1200
3 3 4 4 3 3 4
150 150 200 200 150 150 200
Practical musicians, however, regard these uses of Bd4 in place of Bb4
as part of the usual procedure of Bayyati, which can involve
transformations of the mode's G4-C5 tetrachord from Nahawand to Rast
or, as we shall see, to another pattern called Hijaz. Transformations
of this kind are distinguished from modulation to a different mode,
which also commonly occurs and is part of "a performer's knowledge."[22]
However we analyze it, the introduction of a Rast tetrachord at G4-C5
illustrates some intonational nuances of our two Peppermint
transpositions:
T M M
Arabic gamut G4 A4 Bd4 C5
EQT: 4 3 3
200 150 150
Peppermint version 1 G*4 A*4 B4 C*5
208 150 138
Peppermint version 2 B4 C#5 Eb5 E5
208 159 129
While the 150-138 cent division for the two middle or neutral second
steps A4-Bd4-C5 in the first Peppermint version (A*4-B4-C*5) is a
rather subtle variation on the EQT tuning, the 159-129 division of the
second version (C#5-Eb5-E5) gives somewhat more "semi-neutral" or
"polarized" step sizes which might be heard as leaning a bit
respectively toward tone and semitone.
Another tetrachord transformation regarded by practical musicians as a
usual procedure within Maqam Bayyati rather than a modulation to
another mode is the introduction of a Hijaz tetrachord at G4-C5 in
place of the standard Nahawand: G4-Ab4-B4-C5, in EQT a 2-6-2 or S-A-S
pattern with the augmented second Ab4-B4 at 6 quartertones or 300
cents (the same size as the regular minor third). The EQT values are
shown along with representations in our two Peppermint versions:
S A S
Arabic gamut G4 Ab4 Bd4 C5
EQT: 2 6 2
100 300 100
Peppermint version 1 G*4 G#*4 B*4 C*5
129 288 80
Peppermint version 2 B4 C*4 Eb*5 E5
138 288 70
These Peppermint options for a Hijaz tetrachord share a feature of the
EQT model: the augmented second step is the same size as a regular
minor third, in Peppermint around 288 cents. However, in contrast to
the two equal 100-cent semitones of the EQT model, the first
Peppermint version has an enlarged lower step of 129 cents and a
compact upper step of 80 cents; the second version carries this
disparity yet further with steps of 138 cents and 70 cents.
These versions, especially the second, lean toward a medieval Near
Eastern description of Hijaz as having a lower neutral second, middle
augmented second, and small upper semitone, with one treatise
reporting ratios yielding steps of around 150-267-81 cents, to be
further discussed in a coming portion of this article.[23]
As explained by Ali Jihad Racy, an acclaimed performer, composer, and
ethnomusicologist from Lebanon, this introduction of a Hijaz
tetrachord on the fourth degree of Maqam Bayyati is a "precadential"
movement leading to a resolution to the tonic or final of D; this
resolution, as Marcus notes, occurs "commonly after replacing the
_hijaz_ tetrachord on G with a _rast_ or _nahawand_ tetrachord and
often after descending below the tonic at least to the extent of
presenting the subtonic note C."[24]
----------------------------------------
2. Other modern modes: Rast and Nahawand
----------------------------------------
To conclude our illustrations of how Peppermint 24 can provide various
interpretations of certain modern Arabic maqamat or modes, let us
consider nuances of tuning which Marcus mentions for two other modes,
Rast and Nahawand. He cites these nuances as examples of performers'
"understandings" not transmitted in academic music education, where
the equally tempered quartertone model prevails.[25]
The first understanding he mentions concerns Rast, a mode with
untransposed tonic or final on C4, with two disjunct Rast tetrachords
(C4-F4, G4-C5) having a pattern of 4-3-3 or T-M-M, and the bridging
tone between them of F4-G4, as follows in EQT:
Rast Rast
|-----------------| |-----------------|
T M M T T M M
C4 D4 Ed4 F4 G4 A4 Bd4 C5
0 4 7 10 14 18 21 24
0 200 350 500 700 900 1050 1200
4 3 3 4 4 3 3
200 150 150 200 200 150 150
Marcus reports that the E-half-flat or Ed4 in this mode "is commonly
understood to be higher" than the Ed4 in Maqam Bayyati[26], forming
respectively the third degree of Rast and the second of Bayyati.
As discussed above, many performers understand the step D4-Ed4 in
Bayyati as somewhat smaller than the EQT value of 150 cents, with
Marcus suggesting a lowering of Ed4 by about 5-15 cents. For Rast, we
might make D4-Ed4 somewhere around a full 150 cents.
The following Peppermint tuning very closely approximates this size,
with untransposed steps of the Arabic gamut indicated above their
Peppermint representations, as in some previous diagrams:
Rast Rast
|-------------------| |-------------------|
C4 D4 Ed4 F4 G4 A4 Bd4 C5
C*4 D*4 E4 F*4 G*4 A*4 B4 C*5
0 208 358 496 704 912 1062 1200
208 150 138 208 150 138
Here the Peppermint step of D*4-E4, a whole-tone less quasi-diesis
(Maj2-QD) at around 149.51 cents is almost identical to the 150-cent
step of the EQT model -- or, as it happens, also the step in ancient
Greek and medieval Near Eastern theory defined by the ratio of 12:11,
or about 150.64 cents.
Another understanding places Eb4 as the third degree of Maqam Nahawand
"especially low," and specifically lower than the Eb forming the
second degree of Maqam Hijaz.[27]
Let us first consider Maqam Nahawand, and then compare one Peppermint
tuning of Maqam Hijaz.
Like Maqam Rast, Maqam Nahawand is constructed from two disjunct
tetrachords which bear the same name as the mode, each with a pattern
4-2-4 or T-S-T, and also has its untransposed final or tonic on C4,
thus placing the tetrachords at C4-F4 and G4-C5, with a bridging tone
at F4-G4. Here are the steps and intervals in EQT:
Nahawand Nahawand
|-----------------| |------------------|
T S T T T S T
C4 D4 Eb4 F4 G4 A4 Bb4 C5
0 4 6 10 14 18 20 24
0 200 300 500 700 900 1000 1200
4 2 4 4 4 2 4
200 100 200 200 200 100 200
The performers' understanding that Eb4 is "especially low" suggests
tuning the minor third C4-Eb4 at considerably smaller than the EQT
value of 300 cents, or the semitone D4-Eb4 at considerably smaller
than its EQT value of 100 cents.
Suppose we play this scale in Peppermint using regular intervals on a
single keyboard: then we will get the following tuning, here shown
with notes on the lower keyboard:
Nahawand Nahawand
|------------------| |------------------|
C4 D4 Eb4 F4 G4 A4 Bb4 C5
0 208 288 496 704 912 992 1200
208 80 208 208 208 80 208
The temperament of Peppermint fifths at about 2 cents wide gives us
regular minor thirds at around 288 cents, and diatonic semitones
around 80 cents -- considerably narrower than the EQT values, or even
the Pythagorean values of around 294 cents and 90 cents suggested by
Marcus as one possible model of general intonational practice.
Here, however, an "especially low" position for Eb4 might suggest
tuning the third C4-Eb4 or the semitone D4-Eb4 rather narrower than
the general norm for a given tuning system or model: EQT, Pythagorean,
or Peppermint.
Let us consider two degrees of adjustment available in Peppermint. In
one approach, we might place the final or tonic at Bb*3 on the upper
keyboard:
Nahawand Nahawand
|------------------| |------------------|
C4 D4 Eb4 F4 G4 A4 Bb4 C5
Bb*3 C*4 C#4 Eb*4 F*4 G*4 G#4 Bb*4
0 208 278 496 704 912 982 1200
208 70 218 208 208 70 218
This approach yields a minor third above the transposed final of
Bb*3-C#4, an augmented second less quasi-diesis (Aug2-QD) at around
278.18 cents, or about 10 cents smaller than the usual Peppermint
minor third; and a semitone step of C*4-C#4, a chromatic semitone or
apotome less quasi-diesis (apotome-QD) at around 69.90 cents, again
about 10 cents narrower than the usual Peppermint diatonic semitone.
One consequence of this approach is to produce slightly unequal
whole-tone steps within each tetrachord: a regular step of around 208
cents, plus a somewhat enlarged step of C#4-Eb*4 or G#4-Bb*4, a
diminished third plus quasi-diesis (dim3+QD) at around 217.72 cents,
yielding a tetrachord division of around 208-70-218 cents.
Another approach offers yet smaller sizes for these intervals, here
shown with the final or tonic in its untransposed position at C4:
Nahawand Nahawand
|------------------| |------------------|
C4 D4 Eb4 F4 G4 A4 Bb4 C5
C4 D4 D*4 F4 G4 A4 A*4 C5
0 208 267 496 704 912 971 1200
208 59 229 208 208 59 229
Here we have the minor third above the final or tonic of C4-D*4, a
whole-tone plus quasi-diesis (Maj2+QD) at around 266.87 cents, and a
semitone step of D4-D*4, the 58.68-cent quasi-diesis -- both intervals
a bit more than 20 cents narrower than their regular Peppermint
counterparts.
An effect of this tuning is that, as with the previous approach, we
have unequally sized whole-tones within each tetrachord, here the
usual 208-cent step and a considerably enlarged step of D*4-F4 or
A*4-C5, a minor third less quasi-diesis (min3-QD) at around 229.03
cents. Thus our tetrachord division is approximately 208-59-229
cents.
Let us now compare one possible Peppermint tuning for Maqam Hijaz,
placed in its untransposed position with the final on D4. This mode,
with an untransposed final or tonic on D, has two conjunct
tetrachords: lower Hijaz (D4-G4) with a 2-6-2 or S-A-S pattern, and
upper Nahawand (G4-C5) with a 4-2-4 or T-S-T pattern, plus the upper
tone C5-D5. In the EQT model, we have:
Hijaz Nahawand
|--------------------|------------------|
S A S T S T T
D4 Eb4 F#4 G4 A4 Bb4 C5 D5
0 2 8 10 14 16 20 24
0 100 400 500 700 800 1000 1200
2 6 2 4 2 4 4
100 300 100 200 100 200 200
Here the semitone D4-Eb4 above the final or tonic has a standard size
of 100 cents. For our Peppermint tuning, let us take the same
untransposed position for the mode, using the regular 80-cent
semitone:
Hijaz Nahawand
|--------------------|------------------|
D4 Eb4 F#4 G4 A4 Bb4 C5 D5
0 80 416 496 704 784 992 1200
80 337 80 208 80 208 208
The process of rounding interval sizes in cents causes a slight
anomaly: the semitones are around 79.52 cents, and the augmented
second step Eb4-F#4 around 336.86 cents, so that rounded values
suggest a Hijaz tetrachord division of 80-337-80, with a fourth D4-G4
at around 497 cents (actually ~495.91 cents) and a major third D4-F#4
of around 417 cents (actually ~416.38 cents). In this diagram I have
shown the nearest rounded value for each indicated interval, producing
these incongruities of addition.
Here the step representing Eb4 is a bit more than 20 cents higher than
the D*4 of our second Peppermint version of Maqam Nahawand, thus
fitting with the performers' understanding reported by Marcus.
Another point of interest is that in the EQT model, either the minor
third C4-Eb4 in Maqam Nahawand or the augmented second Eb4-F#4 in
Maqam Hijaz is an interval of 6 quartertones, or 300 cents. In these
two Peppermint representations, however, Nahawand C4-Eb4 as
represented by C4-D*4 is only 267 cents, while Hijaz Eb4-F#4 is almost
337 cents, or about 60 cents larger. In Section 1.2, we encountered
forms of the Hijaz tetrachord using the regular Peppermint minor third
at around 288 cents, somewhere between these values.
In these examples, following the outlook of Marcus, I have tried to
focus on nuances of intonation in different modes rather than matching
or modelling as closely as possible precise tuning ratios or interval
sizes. For example, as it happens, the 267-cent Peppermint version of
the minor third in Maqam Nahawand (C4-D*4 representing C4-Eb4) has a
just ratio of 7:6, also featured in some medieval Near Eastern theory.
Our interest here, however, is not so much in the exact ratio as in
the availability of this interval as one option for the category of an
"especially low" minor third. Another option, as we have seen, is a
278-cent interval like Bb*3-C#4, which happens to approximate the more
complex just ratio of 27:23 (~277.59 cents).
In the next portion of this article, I will consider some medieval
Near Eastern accounts of tuning systems based on integer ratios which
can be closely approximated in Peppermint. However, first considering
some of the flexibility and variability in practical ensemble tunings
might help to put fixed tuning schemes, often rich with mathematical
elegance and musical beauty, in perspective as guides to tuning
certain instruments or general models rather than exhaustive
descriptions of the performer's art.
-----
Notes
-----
1. In this presentation I rely specifically on his article "The
Eastern Arab System of Melodic Modes in Theory and Practice: A Case
Study of _Maqam Bayyati_," in Virginia Danielson, Dwight Reynolds, and
Scott Marcus, eds., _The Garland Encyclopedia of World Music, Volume 6:
The Middle East_ (New York: Garland Publishing or Routledge, 2002),
pp. 33-44. This article gives an excellent and very readable
introduction to some aspects of the maqam or modal system and the
tuning of modal scales in modern theory and practice. For a more
comprehensive account see, for example, Scott Marcus, "Arab Music
Theory in the Modern Period," Ph.D. dissertation, University of
California at Los Angeles (1989),
2. Ibid. pp. 34-36, 38-39.
3. Ibid. pp. 39-41.
4. Ibid. p. 34 and Figure 1. In Arabic theory, as in medieval European
theory the modes have standard or untransposed locations within the
gamut, plus certain commonly recognized transpositions, for example by
a fourth, fifth, whole-tone, or octave, ibid. at p. 36. For example,
Maqam Rast, discussed in Section 2 below, has a standard location or
"root position" for its final or tonic at C4, and "also occurs
commonly" on G3, G4, and C5, and "less commonly" on D4 "and a few
other positions," see ibid.
5. Ibid. p. 39.
6. Ibid. pp. 39-40.
7. Ibid. p. 34. See also p. 38, where a contrast is drawn between
academic theory focusing on "scales with a recognized tonic pitch,"
analyzed "in terms of intervals and tetrachords," and "a more complex
and dynamic definition" based on performers' knowledge and practice,
"involving such considerations as the notes not duplicating at the
octave, specific nontempered intonation for some notes, a common
progression or path for moving through the various regions of a mode's
scale, additional tetrachords beyond those indicated in the simple
scalar presentation of the mode, characteristic accidentals, and a set
of standard modulations to other modes."
8. Arabic theory also uses the term _nisf diaz_ for a "half-sharp"
inflection raising a note by a quartertone, see Marcus, ibid. at
pp. 35-36; I suspect that this term might derive from _diesis_ in its
European usage as a name for a sharp sign or inflection. An
interesting essay might be written on the different definitions of
diesis as an interval smaller than a semitone (sometimes in ancient
Greek theory at or around half of a semitone, or a quartertone); a
regular diatonic semitone (in Pythagorean intonation, 256:243 or
~90.22 cents); the difference between three pure 5:4 thirds or four
pure 6:5 thirds and a pure octave (respectively the "lesser" and
"greater" dieses); the difference in a regular tuning system such as a
meantone between 12 fifths and 7 pure octaves; or an accidental
raising a note by some kind of semitone. The last definition would
seem the most relevant one if my guess as to _nisf diaz_ is correct.
9. Ibid. p. 39, and Figure 13.
10. Ibid. p. 39.
11. Ibid. p. 39; on tendencies toward Pythagorean sizes for many
intervals on instrument such as the 'ud (or oud, also source of the
European lute) and violin tuned in "perfect fourths or fifths," see
also p. 36.
12. Ibid. p. 39.
13. Ibid. p. 39.
14. Ibid. p. 40.
15. Ibid. p. 40.
16. For example, medieval European theorists observe that the use of
Bb is most common in protus and tritus (the first and third families
of modes, with untransposed finals respectively on D and F), and less
common in deuterus and tetrardus (the second and fourth families, with
untransposed finals on E and G). In 14th-16th century polyphony,
additionally, certain cadential inflections to obtain progressions
such as major sixth to octave are "understood" as a norm in
conventional performance practices, and often supplied by performers
when left unindicated in a manuscript or printed edition, with
diverse interpretations documented, for example, by 16th-century
tablatures giving instrumental versions of vocal compositions.
See, e.g., , my
presentation on hexachords and _musica ficta_ at the Early Music FAQ
Site of the Medieval Music and Arts Foundation, Todd McComb, editor.
17. Ibid. p. 41. One might draw a parallel between the options for
performances in Maqam Bayyati and the tendencies noted in certain
styles of medieval European music, for example, to use Bb3-A3-G3-F3
as a descending figure (with Bb or B-fa) and B3-C4 as an ascending
semitone "pulling" toward C (with B-natural or B-mi).
18. Ibid. p. 40.
19. Ibid. p. 40. One might possibly make a certain comparison with the
medieval European hexachord system where, in the standard gamut, the
degree Bb/B or BfaBmi is flexible in the range of Bb3/B3 or higher,
but only B-natural is recognized at B2 as part of the lowest regular
hexachord on G2 or _gamma ut_, with B2 as _mi_ in this hexachord. An
inflection Bb2 is thus regarded as _musica ficta_, a note outside the
regular or _musica recta_ gamut, since a Bb2 or B-fa at this location
would imply its role as _fa_ or the fourth degree in a hexachord on
F2, a note lower than the lowest note G2 of the regular system.
20. Ibid. p. 40. See also pp. 36-37 and Figure 8 for a discussion with
diagrams of some common tetrachords in modern Arabic music.
21. Ibid. p. 40; Figure 14 at p. 41 gives a diagram of some modes
including Husayni.
22. Ibid. p. 41.
23. For this medieval description of the Hijaz tetrachord, also known
as Hijazi, by Qutb al-Din al-Shirazi in the section on music of the
encyclopedia _Durrat al-taj_ (c. 1300) as having steps with the ratios
of 12:11-7:6-22:21, or about 150-267-81 cents, see O. Wright, _The
Modal System of Arab and Persian Music A.D. 1250-1300_, London
Oriental Series, Vol. 28 (Oxford: Oxford University Press, 1978),
p. 51.
24. Ibid. p. 40.
25. Ibid. p. 36; Marcus describes the situation specifically in
"present day Cairo."
26. Ibid. p. 36.
27. Ibid. p. 36.
Most appreciatively,
Margo Schulter
mschulter@calweb.com