Note to Readers:
This letter is written to Ozan Yarman, the outstanding Turkish
musician and theorist whose doctoral dissertation has recently become
available on the World Wide Web: _79-Tone Tuning and Theory for Turkish
Maqam Music As A Solution To The Nonconformance Between Current Model
and Practice, Istanbul Technical University, Institute of Social
Sciences, Department of Musicology (June 2008).
The 79-tone tuning presented in the dissertation is a system which, in
simplified terms, is based on 159-EDO, a tuning in which each of the 53
commas of 53-EDO is subdivided into three parts. In the 79-tone system,
an octave is built from 78 steps equal to twice the size of a 159-EDO
step, or 2/3 comma -- plus one step equal to a full comma, the 46th of
the tuning, which has as its upper note a 3:2 fifth above the beginning
of the series. Actually the system is a bit more complicated than this,
since the 2/3-comma step or yarman, as it is appropriately called,
varies minutely in size in order to obtain some pure 3:2 fifths and 4:3
fourths, in contrast to the tiny amount of tempering these intervals
receive in 53-EDO.
These niceties, however, do not significantly divert the system from
its conceptual simplicity as a "79-MOS," that is, a "Moment of
Symmetry" in which adjacent tuning steps have only two sizes, here
basically either 2/3-comma (the yarman), or a full comma (step 46).
In this scheme 33 yarmans form a pure 4:3 fourth, while, as noted
45 of these usual yarmans plus a full comma or approximate 53-EDO step
for a pure 3:2 fifth.
This letter largely focuses on two medieval traditions involving
17-note Pythagorean systems (also MOS tunings, with adjacent steps of a
90-cent limma or 23-cent comma), and compares these Near Eastern and
European tunings with a modern system smaller and less ambitious than
the 79-MOS: a 24-note regular temperament with fifths of 704.607 cents,
or the same system as implemented on a 1024-EDO synthesizer with small
variations in some interval sizes. As observed in the letter, this
24-note system is not itself an MOS, but can be viewed as a union of
two 12-MOS systems, and also includes a 17-MOS inviting comparison with
the medieval 17-MOS tunings.
An appendix proposes a system of interval classification and notation
for Near Eastern music based on some familiar Turkish and Syrian
concepts combined with certain new categories and symbols, with Ozan's
ingenuity and originality a great inspiration in any such effort,
whatever the imperfections of my attempt.
Margo Schulter
22 September 2008
* * *
Dear Ozan,
Please let me share with you my great celebration at now having a copy
of your dissertation, which I was able to have printed at the local
University Library from the PDF file which you have so generously made
available at your Web site.
Your writing in this most noble document, which I am only beginning to
read and digest, together with some of our recent conversations here,
moves me to consider one aspect of my more modest 24-note "e-based"
temperament for Maqam/Dastgah music, a question relating to the
"footing" of this tuning system. The name "e-based," incidentally,
refers to a curious mathematical property of the system: the generator
of 704.607 cents results in a ratio between the logarithmic sizes (for
example as measured in cents) between the regular whole tone and the
limma or diatonic semitone equal to Euler's e, approximately 2.71828.
! eb24n.scl
!
1024-tET/EDO version of e-based tuning (Blackwood R=e, 5th 704.607c)
24
!
55.07813
132.42187
187.50000
209.76563
264.84375
285.93750
341.01562
418.35937
473.43750
495.70312
550.78125
628.12500
683.20312
704.29688
759.37500
836.71875
891.79687
914.06250
969.14063
991.40625
1046.48438
1122.65625
1177.73437
2/1
As you have written, one important goal and feature of the 79-MOS is
that it supports, as a subset, a 12-note chromatic cycle; and this is
also a feature of Yarman24, which indeed is built upon such a cycle as
designed by Rameau, but so artfully augmented as to provide either a
24-note or a 17-note system of the perdeler.
For my 24-note system, which seeks to provide a pleasant and
reasonable subset of maqam/dastgah colors while at the same time
offering a footing in 13th-14th century European styles, a different
but somewhat analogous goal might apply: the availability of a 17-note
MOS like that of the regular Pythagorean tuning (Gb-A#) described and
advocated by Safi al-Din al-Urmavi in the later 13th century, and also
by Prosdocimus de Beldemandis (1413) and Ugolino of Orvieto (writing
sometime around 1425-1440).
Here I will first explore the 17-MOS structure that obtains within
this 24-note regular temperament, and then the division brought about
when all 24 notes are considered.
-----------------------------------------------------------
1. An important distinction: intervals from 7-10 generators
-----------------------------------------------------------
At the outset, I should note an important limitation as well as a
possible advantage of the somewhat different structure of the 17-MOS
in this temperament than in Pythagorean tuning.
While chains of 1-6 generators (fifths or fourths) yield what may be
termed "accentuated Pythagoraen intervals" with major intervals
somewhat larger and minor ones somewhat smaller, chains of 7-10
generators very significantly yield middle or neutral rather than
"schismatic" or 5-limit intervals.
When 5-limit flavors are a priority, as certainly in the tuning system
and maqamat of Safi al-Din as well as much modern Near Eastern
practice, and also in early 15th-century European music where
keyboards often have written sharps tuned as Pythagorean flats, the
way to acquire these flavors is obviously to choose a different tuning
system, with the 79-MOS as one outstanding option.
For a subset of maqam/dastgah music where such flavors may not be so
indispensable, however, our more modest tempered system has a possible
advantage: we can acquire a wealth of neutral as well as "accentuated
Pythagorean" intervals with relatively short chains of fifths, so that
a 24-note tuning makes them available in diverse locations and
patterns.
Indeed, although it happens that some intervals have sizes similar or
even virtually identical to those of the familiar 53-comma theory, this
system might also be seen as a variation on the 17-EDO model of the 17
basic perdeler, since it supports all of the interval types of that
model while, like the 53-comma system, providing a smaller and a larger
size for each type of middle or neutral interval, as well as some
delectable septimal flavors.
Of course, as you well observe in your dissertation regarding the
53-comma system, providing only two sizes for each neutral category is
more the beginning of wisdom than its full consummation, as may be
seen in the subtle variety of shadings realized by musicians in
various parts of the Near East, and much more amply sampled in the
79-MOS. Yet I hope that the more imperfect solution of a 24-note
regular temperament that at least respects the distinction between a
larger and a smaller neutral second step, for example, will offer a
system which, although not ideal, is adequate and pleasant.
------------------------------------------------------------------------
2. Medieval parallels: the Pythagoran 17-MOS in the Near East and Europe
------------------------------------------------------------------------
In showing the 17-MOS within this system, I will take two approaches.
The first is that of Prosdocimus and Ugolino, who define their 17-note
Pythagoraen tunings as having a range of Gb-A#. If we take C as the
1/1, this means a tuning with 6 fifths down and 10 fifths up. Then we
shall compare this with the tuning of Safi al-Din, based as you have
explained on a series of 4 fifths up and 12 fifths down.
-----------------------------------------------------------
2.1. Prosdocimus and Ugolino: In search of regular cadences
-----------------------------------------------------------
Since the e-based tuning actually uses, by custom, two 12-note
keyboards each tuned in a 12-MOS with a range of Eb-G#, with the notes
on the upper keyboard raised by the diesis of about 55.28 cents, to
get the Prosdocimus/Ugolino arrangement we must pick a note as 1/1
which will accommodate the requisite number of fifths in each
direction. Here, for simplicity's sake, I choose the note A.
l d l l d l l l d l
77 55 77 77 55 77 77 77 55 77
0 77 132 209 286 341 418 495 572 628 705
A ---- Bb --- A# ---- B ---- C --- B# ---- C# ---- D ---- Eb --- D# ---- E
|---------------------|--------------------|-------|---------------------|
T T l T
209 209 77 209
l d l l d l l
77 55 77 77 55 77 77
705 782 837 914 991 1046 1123 1200
E ---- F ---- E# ---- F# ---- G --- Fx ---- G# ---- A
|---------------------|---------------------|-------|
T T l
209 209 77
As this diagram shows, an octave of the 17-MOS may be diatonically
divided into five 209-cent tones plus two 77-cent limmas, here placed
at A-B-C#-D-E-F#-G#-A. Each of the five tones is subdivided into three
steps: a 77-cent limma plus a 55-cent diesis plus a 77-cent limma,
here abbreviated l-d-l.
The 77-55-77 cent division of the tone resembles 17-EDO with its equal
division at a rounded 71-71-71 cents (actually about 70.588 cents for
each equal step) -- in contrast to Pythagorean tuning with its
division of the 9:8 tone into limma-comma-limma at a conveniently if
not so accurately rounded 90-24-90 cents (actually with a 256:243
limma at 90.225 cents and a comma at 531441:524288 or 23.460 cents).
In the classic Pythagorean division, the ratio of step sizes is close
to 4-1-4, as precisely holds in 53-EDO with 9 Holdrian commas per
tone; in the e-based temperament, the ratio is very close to 109-EDO,
or 7-5-7.
As in either Pythagorean intonation of 53-EDO, however, there is a
small interval or comma which marks the difference in size between
certain steps, here the 77-cent limma and 55-cent diesis: 23.60 cents
in Pythagorean tuning, 22.64 cents in 53-EDO (the Holdrian comma), and
21.68 cents in a strictly regular version of the e-based system.
In our 17-MOS, this comma does not actually appear as a direct step,
but makes its presence known, for example, if we compare the different
sizes of intervals formed from two adjacent MOS steps. A limma plus a
diesis or "l + d" (e.g. A-Bb-A#) forms an apotome or chromatic
semitone such as A-A# at 132 cents, or roughly 68:63, between 14:13
(128 cents) and 13:12 (139 cents) and a bit closer to the smaller of
these superparticular neutral seconds.
The combination of two limmas or "l + l" (e.g. A#-B-C) forms a
diminished third at 154 cents, or twice the size of the limma, a
larger neutral second a bit wider than 12:11 (151 cents), and very
close to Safi al-Din's ratio of 59:54 (153 cents).
The smaller and larger middle or neutral seconds at 132 and 154 cents
-- differing by the 22-cent comma -- are not too far from intervals of
respectively 6 and 7 Holdrian commas in the 53-EDO system, with
rounded values of 136 and 158 cents. Here both steps are somewhat
narrower than their Pythagorean or 53-EDO counterparts, as is the
regular minor third formed from their sum (e.g. A-A#-C) -- rather
smaller at 286 cents than the Pythagorean 32:27 (294 cents); and quite
close to 13:11 (289 cents), and yet more so to 33:28 (284 cents).
Thus in this 17-MOS of the 704.607-cent temperament, as in 17-EDO
(with fifths of 705.882 cents), any two adjacent steps will form a
neutral or middle second, more generally yielding neutral intervals
quite efficiently with a relatively small number of generators.
In addressing the Pythagorean 17-MOS system of Prosdocimus and
Ugolino, we should show how this tempered counterpart can satisfy the
stylistic goals that led these theorists to champion a 17-note
tuning. Unlike Safi al-Din, who obviously sought after schismatic or
approximate 5-limit intervals in ajnas and maqamat such as his Rast
(equivalent to a modern Turkish 5-limit Rast), these medieval European
theorists were interested in obtaining regular Pythagorean intervals
such as major and minor thirds and sixths for directed polyphonic
progressions.
In these progressions, unstable or "imperfect" intervals such as major
or minor thirds and sixths seek resolution to "perfect" or stable
concords: as in the Near Eastern theory of the Mutazilah Era, and in
the early Greek theory shared as a source for both of these medieval
traditions, the Pythagorean concords of the unison (1:1), octave
(2:1), fifth (3:2), and fourth (4:3).
Specifically, it is axiomatic in the standard 14th-century European
style that major thirds often expand stepwise to fifths, major sixths
to octaves, and major tenths to twelfths; while minor thirds often
likewise contract to unisons, with one voice of the interval moving by
a 9:8 tone (204 cents) and the other by a 256:243 limma (90 cents).
Prosdocimus and Ugolino are concerned that when accidentals are
required to obtain a major third before a fifth, or a major sixth
before an octave, these thirds and sixths have their "fully perfected"
size of 81:64 or 408 cents, and 27:16 or 906 cents, so that can
efficiently expand to the fifths and octaves toward which they
"strive."
Likewise, when a third contracting to a unison must be made minor by
an accidental, it is important in Ugolino's language that the third be
fully "colored" or reduced in size to the regular 32:27 (294 cents),
so that it likewise may efficiently reach its goal through motions of
a 204-cent tone and 90-cent limma.
The Pythagorean 17-MOS of these two theorists is specifically designed
to support cadences with these desired intervals and steps of a
classic 14th-century technique, with either ascending or descending
semitonal motion, on each of the six "fixed" steps or perdeler, as we
might also say, of the standard medieval gamut (C, D, E, F, G, A --
with B/Bb regarded as a fluid degree with both forms as elements of
the regular gamut, or _musica recta_).
For example, a piece centered on a finalis of D, in what might be
considered the mode of Protus or Dorian, will typicallly conclude in
the 14th-century with an intensive cadence on this step calling for the
accidentals C# and G#; and feature as part of its seyir or development
remissive cadences on the second degree E or fifth degree A, the latter
involving the "soft" (_molle_) or flat form of the flexible degree
B/Bb, or Bb, for example. These accidental inflections in routinely
occurring cadences are thus a feature of the " polyphonic seyir" of
this mode, much as customary inflections are part of the seyir of a
maqam such as Ushshaq (the Arab Bayyati) with its flexible sixth degree
above the final, or Buselik where the degree at a tone below the
finalis is often raised when ascending to the finalis.
Here are examples of these progressions and inflections using the very
popular cadence where a major third expands to a fifth while a major
sixth expands to an octave, arriving at a complete 2:3:4 sonority
(e.g. D-A-D, E-B-E, or A-E-A) which generally defines the standard of
complete stable harmony in 13th-14th century European music.
Final cadence on D Internal cadences on E or A
C# D D E | G A
G# A A B D E
E D F E Bb A
Indeed, the contrast between polyphonic cadences with ascending or
descending semitones -- here, following one medieval usage, referred
to as "intensive" and "remissive" -- is based to the musical
development or seyir of 14th-century compositions and forms. An
intensive cadence with ascending semitones typically signals a more
conclusive feeling than a remissive cadence with descending semitones,
which often is used as an internal cadence on a step other than the
finalis. In French, one speaks in a 14th-century context of an _ouvert_
or "open" cadence, often remissive; and a _clos_, that is a "close" or
final cadence, generally intensive.
These contrasting cadences, and the accidentals they frequently
require, shape both the short-range and long-range organization of
forms such as the French ballade or virelai and the Italian ballata,
guiding the expectations of an attuned listener, much like the seyir of
a maqam. Scholars such as Richard Crocker and Sarah Fuller have
elucidated these structural aspects and subtleties of 14th-century
European music, just as others such as Ali Jihad Racy have illuminated
features of the maqam system which are likewise implicit to attuned
musicians and listeners.
While many of the typical patterns of the 14th century, as exemplified
by composers such as Machaut in France and Landini in Italy, can be
realized on a 12-note keyboard with a likely common accidental range
of Eb-G#, Prosdocimus and Ugolino wanted to accommodate the more
adventurous composers or improvisers: around 1400, for example, Solage
in his famous _Fumeux fume_ uses a 15-note accidental range (Gb-G#).
Here the use of A as our 1/1 instead of C results in a transposed
basic medieval European gamut of A-B-C#-D-E-F# plus the fluid degree
G/G#, so that to meet the purposes of Prosdocimus and Ugolino, our
17-MOS should support both intensive and remissive progressions on the
first six of these degrees while keeping all simultaneous intervals in
their usual sizes (e.g. 286-cent minor thirds, 418-cent major thirds,
914-cent major sixths), and permitting all melodic steps in these
cadences to be either regular 209-cent tones or 77-cent limmas.
Our examples again feature the very characteristic cadence where a
major third above the lowest voice expands to a fifth while a major
sixth expands to an octave. For each degree, the intensive form of
this cadence is shown first, and then the remissive form.
Cadences on A Cadences on B Cadences on C#
Int Rem Int Rem Int Rem
G# A | G A A# B | A B B# C# | B C#
D# E D E E# F# E F# Fx G# F# G#
B A Bb A C# B C B D# C# D C#
Cadences on D Cadences on E Cadences on F#
Int Rem Int Rem Int Rem
C# D | C D D# E | D E E# F# | E F#
G# A G A A# B A B B# C# B C#
E D Eb D F# E F E G# F# G F#
Here it might duly be added that while Prosdocimus and Ugolino were
motivated in part by a desire to facilitate these familiar 14th-century
cadences in remote locations, their close analysis of intonation may
also reflect another factor: the evidently widespread popularity in the
early 15th century of Pythagorean keyboard tunings in which some or all
of the usual written sharps (F#, C#, and G# in a usual 12-note tuning)
were in fact realized as Pythagorean flats (Gb, Db, and Ab). Such
tunings served to produce schismatic or near-5-limit thirds in
sonorities and cadences involving sharps -- for example, the widely
favored 12-note tuning of Gb-B. Any Turkish or other Near Eastern
musician sitting down at such a European keyboard could have found Safi
al-Din's Rast, or the same 5-limit flavor as often favored in Turkey
today, by the use of a few accidentals (here shown along with modern
perde names):
rast dugah segah chargah neva huseyni evdj gerdaniye
D E Gb G A B Db D
1/1 9/8 8192/6561 4/3 3/2 27/16 4096/2187 2/1
0 204 384 498 702 906 1086 1200
204 180 114 204 204 180 114
commas: 0 9 17 22 31 40 48 53
9 8 5 9 9 8 5
In an early 15th-century European style, however, this kind of
transposition placing schismatic or 5-limit intervals on the regular
steps of a mode is not the norm, however. Rather, there is a contrast
of "modal color," as Mark Lindley has termed it, between sonorities
with regular Pythagorean thirds or sixths spelled with diatonic notes
or flats only (e.g. F-A-D, Bb-D-G), and those with 5-limit flavors of
these intervals spelled with written sharps: e.g. E-G#-C# before D-A-D,
played at E-Ab-Db at 0-384-882 cents in place of the traditional
14th-century intonation at 0-408-906 cents with the major third and
sixth "fully perfected."
While Prosdocimus and Ugolino expressed a preference for the
traditional intonation, their version of the 17-MOS would more
generally offer performers a choice between the regular Pythagorean
and schismatic or 5-limit flavors at many locations in sonoorities
involving accidentals: it includes all the notes of the traditional
12-note Eb-G# tuning as well as the newer tunings such as Gb-B.
By the mid-15th century, in at least some parts of Europe, there was
evidently a preference not merely to have an artful mixture of these
flavors, but to favor a 5-limit flavor wherever possible -- a desire
fulfilled around this epoch by the advent of meantone temperament, an
ingredient of the 79-MOS which makes possible a 5-limit Maqam Rast
with an unbroken chain of fifths.
The tempered 17-MOS with a fifth at 704.607 cents looks at once back
to the love of Prosdocimus and Ugolino for wide major intervals,
narrow minor intervals, and efficient cadential resolutions based on
fifths and fourths as the favored stable intervals -- and beyond the
realm of historical European music to a special treasure of the Near
Eastern tradition: the plethora of middle or neutral intervals, and of
maqamat or dastgah-ha in which they are featured as routine elements
of a transcendent art. Many of these maqamat or dastgah families may
be relished in this 17-MOS or the full 24-note system.
Further, the system supports an approach to polyphonic realizations in
maqam-related or dastgah-related styles based on stable fifths and
fourths, already an element of some medieval and modern performance
traditions (e.g. Ibn Sina's _tarqib_ or "composite sound" of
simultaneous intervals, with fourths especially preferred, a term also
used to describe an especially intricate maqam compounded of others),
and drawing on some European aspects of practice and theory in the
Mutazilah Era in order to mix and contrast these stable concords
pleasingly with the full range of intervals arising in maqam/dastgah
music -- major, minor, and middle or neutral.
Before considering the question of maqam/dastgah polyphony, however,
we must be sure that a system offers agreeable realizations of a
reasonable range or subset of maqamat or dastgah-ha as pure melody,
with enough scope for the seyir to unfold without undo intonational
obstacles or impediments. This is a worthy and indeed precious goal in
its own right, as well as the necessary foundation for any polyphonic
texture aptly ornamenting and adorning the traditional melodies and
intricate modulations.
---------------------------------------------------------------
2.2. Safi al-Din al-Urmavi and Nasir Dede: A tempered variation
---------------------------------------------------------------
Considering the Pythagorean 17-MOS of Safi al-Din based on a tuning
from the 1/1 of 12 fifths down and 4 fifths up provides an
introduction to the somewhat different range of maqam/dastgah colors
offered by the corresponding 17-MOS in our 704.607-cent temperament, a
realm further enriched by the extra steps and intervals of the full
24-note tuning. If we again take Eb as the note at the flat end of the
17-MOS chain, then D#, at 12 fifths or a diesis (less seven octaves)
higher, will here serve as the 1/1. Following your account of the
Abjad system of 17 perdeler based on the perde names of Nasir Dede, I
take 1/1 to be Yegah and 4/3 to be Rast.
209 209 77
T T l
|--------------------|--------------------|-------|
l l d l l d l
77 77 55 77 77 55 77
0 77 154 209 286 363 418 496
D# ---- E ---- F --- E# ---- F# --- G --- Fx ---- G#
Yegah Pes Pes Asiran Acem Arak Gevast Rast
Beyati Hisar Asiran
209 209 77 209
T T l T
|---------------------|--------------------|-------|--------------------|
l l d l l d l l l d
77 77 55 77 77 55 77 77 77 55
495 572 649 705 782 859 914 991 1068 1145 1200
G# ---- A ---- Bb --- A# ---- B ---- C --- B# ---- C# ---- D ---- Eb -- D#
Rast Suri Zirguleh Dugah Kurdi/ Segah Buselik Cargah Saba Hicaz/ Neva
Nihavend Uzzal
While regular diatonic intervals from chains of 1-6 generators
(e.g. major and minor seconds, thirds, sixth, sevenths) correspond to
those of the Pythagorean 17-MOS of Safi al-Din and Nasir Dede -- but
with major intervals rather larger and minor ones rather smaller, as
we have noted -- the schismatic or 5-limit flavors of the original are
transformed here into neutral flavors, with large middle or neutral
intervals abounding above the 1/1. These large neutral intervals
include a middle second at 154 cents (D#-F); a third at 363 cents
(D#-G); a sixth at 859 cents (D#-C); and a seventh at 1068 cents
(D#-D).
We may find it helpful to compare these neutral intervals with their
5-limit counterparts in the original Pythagorean 17-MOS of Safi al-Din
and Nasir Dede, with note spellings given both as in the above scheme
with perde yegah at D#, and in a widely followed modern custom placing
this perde at G:
--------------------------------------------------------------------
Interval Type Pythagorean cents 5-limit e-based Approx JI
--------------------------------------------------------------------
D#-F or dim3 65536/59049 180 10/9 154 59/54
G-Bbb 182 153
--------------------------------------------------------------------
D#-G or dim4 8192/6561 384 5/4 363 21/17
G-Cb 386 366
--------------------------------------------------------------------
D#-C or dim7 32768/19683 882 5/3 859 23/14
G-Fb 884 859
--------------------------------------------------------------------
D#-D or dim8 4096/2187 1086 15/8 1068 63/34
G-Gb 1088 1068
--------------------------------------------------------------------
The small major or 5-limit flavors obtained in the Pythgorean 17-MOS
from these vitally important diminished intervals involving chains of
7-10 generators are thus transformed in the e-based 17-MOS to large
neutral intervals often on the order of a comma smaller. While the
154-cent neutral second is rather close to 12:11 at 151 cents, and so
may be regarded as within a central neutral range, the large neutral
third, sixth, and seventh at 363, 859, and 1068 cents could also be
described as submajor flavors, rather "bright" and outgoing.
Although notably different from the 5-limit flavors of Safi al-Din,
these large neutral flavors also fit nicely into the world of
maqam/dastgah music. A 154-cent middle second above the finalis gives
Maqam Huseyni a pleasant quality, as does the 859-cent sixth, whose
bright colors likewise fit the ethos of Acemli Rast, a form of Maqam
Rast with two conjunct Rast tetrachords also known in the Arab world
as Nirz Rast. In either this form of Rast or the form with disjunct
Rast tetrachords, the submajor third at 363 cents should likewise be
at home, while in the latter form the large neutral or submajor
seventh at 1068 cents should pull nicely toward the octave of the
finalis.
Just as a good test for the Prosdocimus/Ugolino version of the 17-MOS
was to seek out regular intensive and remissive cadences for each of
the six "fixed" steps of the standard medieval European gamut, so one
test for our tempered version of the Safi al-Din/Nasir Dede 17-MOS is
to attempt a tuning of what I term "basic Rast": a 9-note set
combining the steps of Maqam Rast in its forms with disjunct and
conjunct tetrachords. Here I take perde rast at G# as the 1/1, and
give possible modern perde names which may vary somewhat from those of
Nasir Dede:
Rast Rast
|-------------------------| T |-----------------------|
0 209 363 495 705 914 1068 1200
G# A# C C# D# E# G G#
rast dugah segah chargah neva huseyni evdj gerdaniye
209 154 132 209 209 154 132
T Jk Js T T Jk Js
Rast
|------------------------| T |
495 705 859 991 1200
C# D# F F#
chargah neva dik hisar ajem gerdaniye
209 154 132 209
T Jk Js T
Here the disjunct and conjunct versions of the upper Rast tetrachord
are shown separately, with certain symbols borrowed from medieval and
modern Near Eastern notations, and sometimes combined or modified,
used to show step sizes. Thus T signifies a usual tone or tanini at
209 cents; Js a smaller mujannab or middle second, in this tuning
system the apotome at 132 cents; and Jk a larger mujannab or middle
second, here the diminished third at 154 cents.
The desired form for a Rast tetrachord is thus T-Jk-Js or 209-154-132
cents; in a 53-comma notation, this form could be generalized as
9-7-6. In the more specific _and_ diverse -- and therefore more
powerful --79-MOS notation, we would have a pattern of 14-10-9 steps
or 211-151-136 cents.
Combining the 9-note set from these two versions of Maqam Rast into a
single series, we find other types of intervals as well:
T Jk Js T Jk E B B Js
209 154 132 209 154 55 77 77 132
0 209 363 495 705 859 914 991 1068 1200
G# A# C C# D# F E# F# G G#
rast dugah segah chargah neva dik huseyni ajem evdj gerdaniye
hisar
Here B for Turkish bakiye designates the usual limma or diatonic
semitone of 77 cents, while E or "eksik bakiye" stands for the smaller
or "diminished" semitone at 55 cents, actually in this tuning system
the 12-diesis equal to 12 tempered fifths less 7 octaves, for example
from dik hisar to huseyni or F-E#.
In the full 17-MOS, with each adjacent step either a 77-cent limma (B)
or a 55-cent diesis (E), we find another interesting interval of
maqam/dastgah music which is recognized by Qutb al-Din al-Shirazi
around 1300: a step of 7:6, or 267 cents, used as the middle step of a
Hijaz tetrachord with a tuning of 1/1-12/11-14/11-4/3 or 0-151-418-498
cents, with steps of 12/11x7/6x22/21 or 151-267-81 cents. In the
e-basd 17-MOS, this appears above the 1/1, yegah at D#, in a version
quite close to JI:
Jk A12 B
154 264 77
0 154 418 495
D# F Fx G#
yegah dik mahur rast
hisar
Here the septimal or 7:6 flavor of minor third is shown by the Turkish
sign "A12," meaning an "augmented" step, as in Hijaz, larger than a
tone, and equal to about 12 commas. This 7:6 flavor of step also
occurs in one popular form of the Persian tetrachord and dastgah now
known as Chahargah, where Hormoz Farhat describes it as a "plus
second."
This small minor third or "plus second" is formed in this temperament
from two apotome steps (Js) each at 132 cents, here F-F#-Fx. Since
each apotome or small neutral second is formed from 7 generators up,
this interval thus involves a chain of 14 such generators. Within the
17-MOS, another derivation is also possible: a regular 209-cent tone
(T) plus a 55-diesis (E), here F-G-Fx.
Interestingly, corresponding forms of the "eksik bakiye" or reduced
limma (E) and the near-7:6 minor third or plus second (A12) occur in
a 24-note or larger Pythagorean tuning, or its close approximation in
53-EDO, with the 7:6 flavor also occurring in a Pythagorean 17-MOS,
where it results from 15 generators down, or a 32:27 minor third less
a 23-cent comma.
It thus results in the 17-MOS of Safi al-Din and Nasir Dede when an
interval is formed from a chain of three adjacent limmas, each at
256:243 or 90.224 cents. Taking the 1/1 or perde yegah as G in a
common modern spelling, we find these intervals between perdes
gevart-zirgule or G-Cbb (limma steps G-Ab-Bbb-Cbb), and perdes
buselek-hicaz/uzzal (E-Abb) in the Nasir Dede naming system (limma
steps E-F-Gb-Abb).
The eksik bakiye or reduced limma (E) does not quite occur in the
Pythagorean 17-MOS, because it requires a chain of 17 generators, and
is indeed well defined as the "17-diesis," the difference between 17
fifths at a pure 3:2 and 10 pure octaves, or about 66.76 cents (a
usual 90-cent limma or 5 fifths down less a 23-cent comma or 12 fifths
down). A conventional spelling for this small semitone would be, for
example, E#-Gb.
The interval with a not too dissimilar size in the e-based temperament,
and present in a 17-MOS, is the 12-diesis at 55 cents, corresponding
structurally with the Pythagorean comma (i.e. 12-comma). In this
tempered system, the 17-MOS does not include as a direct step the
structural counterpart of the Pythagorean 17-comma at 67 cents: a
17-diesis with a size curiously very close to that of the Pythagorean
12-comma, and yet closer to the Holdrian comma: 22 cents (e.g. Gx-Bb).
The 23-cent Pythagorean comma, of course, does prominently appear as a
direct step in the 17-MOS of Safi al-Din and Nasir Dede, where it
vitally defines the difference, for example, between small or 5-limit
flavor major third at 384 cents and a regular Pythagorean major third
at 408 cents. With perde yegah at G, this distinctions occurs for
example above perde rast at C between segah at Fb or 384 cents and
buselik at E or 408 cents, In relation to yegah, these two steps are
likewise at 882 and 906 cents, forming a 5-limit and a Pythagorean
flavor of major sixth.
To see how the comparably sized 17-comma of our e-based temperament
also plays a vital role in better realizing maqam/dastgah music, we
now move to a full 24-note tuning.
--------------------------------------
3. From the 17-MOS to a 24-note system
--------------------------------------
From one perspective, a 24-note system may be considerably less tidy
than a 17-MOS: it is not itself an MOS. Both in theory and as mapped
to two conventional 12-note keyboards, however, it can be considered as
a union of two 12-MOS systems placed at the distance of the comma or
diesis defined by the difference between 12 fifths up and 7 pure
octaves: here, with the 704.607-cent temperament, the 12-diesis of
about 55.283 cents.
In practice, the larger 24-note set is very helpful both in providing
more intervals in flavors such as septimal (e.g. 7:6, 9:7, 7:4)
requiring longer chains of generators, and in permitting a greater
choice of subtly different maqam flavors from a given perde or step,
so as to support more intricate and sophisticated modulations.
In certain situations, it is also possible to use the 17-comma of
about 22 cents in order to introduce small and expressive nuances to
the tuning of a given perde as a performance proceeds, thus emulating
to a limited degree the kind of practice advocated by theorists such
as al-Sabbagh and scientifically documented by investigators such as
Can Akkoc who have measured the pitches and intervals produced by
performers on flexible-pitch instruments such as the ney.
The qualification "to a limited degree" merits emphasis: at most, from
a given position in the tuning, we will have two steps available which
could fit the category of a "neutral third" (at 341 or 363 cents), for
example. In contrast, the 79-MOS typically provides three subtly
different flavors of neutral thirds from a given step, thus better
modelling the "cluster" of pitches or interval sizes produced in
practice by flexible-pitch performers.
Comparing the 17-MOS and 24-note versions of the perdeler or steps of
maqam music in the octave from perde rast to perde gerdaniye may help
to demonstrate these points, and to illustrate two common musical
situations where the difference of a 22-cent comma can be used to
emulate the "cluster" effect documented by Akkoc and adopted as one
important basis for mapping out steps and intervals in the 79-MOS.
To arrive at a 17-MOS version of the rast-gerdaniye octave, we need
only take the Safi al-Din/Nasir Dede mapping we have just explored,
moving the lower tetrachord from yegah to rast up an octave so that it
becomes an upper tetrachord from neva to gerdaniye. Values in cents
are accordingly revised so as to be measured from perde rast, with
some perde names revised both to reflect current practices and to
accommodate for certain aspects of the 704.607-cent tuning:
209 209 77 209
T T l T
|---------------------|--------------------|-------|--------------------|
l l d l l d l l l d
77 77 55 77 77 55 77 77 77 55
0 77 154 209 286 363 418 496 572 649 705
G# ---- A ---- Bb --- A# ---- B ---- C --- B# ---- C# ---- D ---- Eb --- D#
Rast Suri Zirguleh Dugah Kurdi Segah Buselik Cargah Hijaz Saba Neva
209 209 77
T T l
|---------------------|--------------------|-------|
l l d l l d l
77 77 55 77 77 55 77
705 782 859 914 991 1068 1123 1200
D# ----- E ---- F --- E# ---- F# --- G --- Fx ---- G#
Neva Beyati Dik Huseyni Acem Evdj Mahur Rast
Hisar
Adding the seven extra notes of the 24-note system produces this
mapping, using for the moment conventional note spellings like those
of our mappings so far, with "c" showing a 22-cent comma (the
17-comma):
209 209 77
T T l
|-----------------------------------|---------------------------|---------|
l l d l l d l
|--------------|------------|-------|---------|-----------|-----|---------|
d c d c d l d c d l
55 22 55 22 55 77 55 22 55 77
0 55 77 132 154 209 286 341 363 418 496
G# ---- F#x -- A ---- Gx -- Bb ---- A# ------ B ---- Ax - C --- B# ------ C#
Rast Nerm Suri Nerm Zengule Dugah Kurdi Dik Segah Buselik Chargah
Shuri Zengule Kurdi
209
T
|-----------------------------------|
l l d
|--------------|-------------|------|
d c d c d
55 22 55 22 55
496 551 572 628 649 705
C# ----- Bx -- D ---- Cx -- Eb ---- D#
Chargah Nerm Hijaz Saba Dik Neva
Hijaz Saba
209 209 77
T T l
|-----------------------------------|----------------------------|--------|
l l d l l d l
|--------------|-------------|------|---------|-----------|------|--------|
l d c d l d c d l
77 55 22 55 77 55 22 55 77
705 782 837 859 914 991 1046 1068 1123 1200
D# ----------- E ----- Dx -- F ---- E# ----- F# ---- Ex -- G --- Fx ----- G#
Meva Beyati Hisar Dik Huseyni Acem Nerm Evdj Mahur Gerdaniye
Hisar Evdj
In mapping out this maqam/dastgah system and proposing names for the
perdeler, placing perde rast at G# may be a convenient choice because
this step has a chain of 11 fifths available either up or down -- as
likewise D#, here perde yegah or neva, another customary reference
point for Near Eastern tuning systems and gamuts. As cardinal points of
reference, indeed, yegah and rast may be somewhat analogous to the
steps G and C in the European gamut, with the former as the foundation
of the medieval hexachord system as the lowest regular note gamma-ut
(thus the term "gamut"), and the latter as the lowest tone of the
natural hexachord (C-D-E-F-G-A, the six "fixed" steps in the regular or
_musica recta_ gamut) and the frequently chosen place in medieval and
later times for a clef in staff notation (along with F, and in later
periods also G).
The diagram above divides the 24-note octave from rast to gerdaniye
into two disjunct tetrachords plus a middle tone. We find three types
of adjacent steps: the 77-cent limma and 55-cent diesis familiar from
the 17-MOS, and the new 17-comma step at 22 cents. The diagram also
shows the 17-MOS division of the gamut into limmas and dieses; and a
possible diatonic division, as in the earlier diagram of the 17-MOS,
into five regular tones and two limmas.
Within the first whole tone from rast to dugah (G#-A#), and likewise in
the tone from chargah to neva (C#-D#) that occurs between the two
tetrachords in the diagram, we have a situation where all adjacent
intervals are either 55-cent dieses or 22-cent commas, with each limma
of the 17-MOS thus divided into these two smaller steps. Elsewhere,
however, five limmas from the 17-MOS remain undivided. If this regular
704.607-cent tuning were carried to 29 notes, then we would reach
another MOS, with all adjacent steps throughout the system either
dieses or commas.
The conventional notation above is intended to show the chains of
fifths; but another system especially practical when using two 12-note
keyboards, each with its 12-MOS, is to use a familiar spelling of Eb-G#
for the notes on each keyboard, with a diesis sign (*) showing a note
on the upper keyboard raised by the 55-cent diesis. Thus D# could also
be written as Eb*, and F#x as G#* -- in the last example, a simpler
spelling to read and interpret, especially at the keyboard. Of course,
a more sophisticated system such as Sagittal can be used with great
success; it is interesting to consider whether or how a generalized
keyboard might affect the matter of a convenient notation.
In any event, I must mention that some crude mechanical problems I have
encountered when using two standard MIDI controller keyboards, which
must necessarily be at a large vertical distance apart, make the above
logical scheme of the perdeler often more relevant in theory than in
practice. Specifically, I have found that playing maqam/dastgah music
satisfactorily requires the ability to move between notes on the two
keyboards easily and fluently with a single hand. The great obstacle
to this is a situation where one must move from an accidental on the
upper keyboard to a natural on the lower keyboard.
Thus the practical solution is to choose a perde for the finalis which
will best accommodate the seyir of a given maqam, including likely
modulations; or likewise will best fit a given dastgah and its
gusheh-ha. Either a generalized keyboard, or a solution involving a
more ergonomic arrangement of two standard 12-note keyboards, would
permit a more consistent approach to the perdeler when actually
playing.
------------------------------------------------------
3.1. Septimal flavors: Ibn Sina's soft diatonic tuning
------------------------------------------------------
The matter of F#x or G#* leads us to an important musical aspect of the
full 24-note system: septimal flavors of a kind featured, for example,
in a beautiful soft diatonic tuning, as John Chalmers has described it,
with two possible interpretations. The first, which Chalmers follows in
a Scala scale archive file (avicenna_diat.scl), has a tetrachord of
1/1-14/13-7/6-4/3 (0-128-267-498 cents) with the smaller middle or
neutral second step at 14:13 preceding the larger at 13:12, followed by
a large or 8:7 tone (128-139-231 cents). These disjunct tetrachords
would have string ratios of 28:26:24:21:
1/1 14/13 7/6 4/3 3/2 21/13 7/4 2/1
0 128 267 498 702 830 969 1200
14:13 13:12 8:7 9:8 14:13 13:12 8:7
128 139 231 204 128 139 231
Another reading which you have favored, Ozan, places the larger neutral
second first in these tetrachords, thus producing a frequency ratio of
12:13:14:16, an arrangement also favored by George Secor (whose use of
it led me to look into its possible history and discover Ibn Sina's
tuning):
1/1 13/12 7/6 4/3 3/2 13/8 7/4 2/1
0 139 267 498 702 841 969 1200
13:12 14:13 8:7 9:8 13:12 14:13 8:7
139 128 231 204 139 128 231
Whatever views people might take as to which of these interpretations
is correct, the 704.607-cent tuning neatly offers a diplomatic solution
by using two tempered steps at an equal 132 cents each, as in this
version starting from C on the lower keyboard, a very likely position
if Ibn Sina's beatiful tuning is treated in modern terms as a variation
on the Daramad of Shur Dastgah:
C C# Cx/D* F G G# Gx/A* C
0 132 264 495 705 837 969 1200
132 132 231 209 132 132 231
As the conventional spellings show, the 132-cent neutral second is the
apotome or chromatic semitone, so that the first two intervals of each
tetrachord are formed by successive apotome steps: C-C#-Cx or G-G#-Gx.
On two standard 12-note keyboards, this sequence is conveniently
notated as C-C#-D* or G-G#-A*, and involves a motion up from F to F#,
for example, and then to G* on the upper keyboard. While the
conventional notation shows that the near-7:6 third is equal to two
apotome steps, the keyboard notation shows that it is equal to a
regular tone plus a diesis or small semitone at 55 cents, e.g. C-D-D*
or G-A-A*.
All transpositions of this beautiful tuning in the 24-note system must
place the 1/1 on the lower keyboard, since a chain of 15 fifths up is
required to obtain all of the steps and intervals including the 7:4
minor seventh (virtually just at 969 cents). The most sharpward of
these transpositions assigns F#x or G#*, a note whose spelling we have
been considering, this vital role of a 7/4 step.
B B#/C* Cx/C#* E F# Fx/G* F#x/G#* B
0 132 264 495 705 837 969 1200
132 132 231 209 132 132 231
As noted above, this is not the likeliest transposition in practice, at
least with my present complications of keyboard ergonomics, since the
vertical space between the two 12-note keyboards makes it difficult to
play an interval like C#*-E or G#*-B smoothly and fluently with a
single hand. However, the position above with C as the finalis, taken
as a form or variation on the modern Daramad Shur, is quite practical
and has some interesting ramifications.
132
628 760
F#/Gp G*
C C# Cx/D* F G G# Gx/A* C
0 132 264 495 705 837 969 1200
132 132 231 209 132 132 231
The first nuance involves the flexible nature of the sixth degree when
Ibn Sina's tuning is used as a stimulating variation on the modern form
of Daramad Shur. Since a minor rather than neutral sixth is typically
cited in "textbook" accounts of Shur showing a seven-note octave, these
two forms are very likely to alternate -- with the minor version of
this step, G* at 760 cents or actually about 5 cents smaller than 14/9
(765 cents), colorfully reflecting the septimal flavor of the
tuning. The small semitone or diesis G-G* at 55 cents offers a very
expressive melodic interval at a vital location in this favorite
Persian dastgah known for its spiritual intensity -- although it
remains an open question whether steps quite this small are used in
traditional Persian music, or how Iranian musicians might regard them
in such a context.
Another step vital to the usual practice of Shur is the lowered fifith
degree at a step conventionally spelled F#, but where the functional
Persian spelling Gp (the "p" symbol indicating the koron, lowering a
note often by about 50-70 cents, or something approaching a third of a
tone) is much more communicative. While "F#" might suggest a leading
tone to G, like buselik-chargah or mahur-gerdaniye in maqam music, in a
Persian setting the lowered fifth degree of Shur, here Gp, pulls
strongly downward toward the lower tetrachord and the finalis. The
descending figure G*-Gp-F is especially characteristic, and involves
two small neutral seconds at 132 cents spanning the distance of about a
7:6 minor third between the steps F at a tempered 4/3 and G* at a
tempered 14/9. This touch mirrors the intonational patterns of Ibn
Sina's soft diatonic.
-----------------------------------------------
3.2. Subtle variations and "cluster" techniques
-----------------------------------------------
While the potential for fine variations in intonation are rather modest
in our 24-note regular temperament by comparison to the more ample
79-MOS, still there are some pleasant opportunities for such nuances
which may often occur. One of them involves the tuning of a Hijaz (or
in the Turkish spelling, Hicaz) tetrachord.
As already noted, the classic tuning of this tetrachord by Qutb al-Din
as 12/11x7/6x22/21 seems ideally to fit this tempered system, or vice
versa, with perde names "of convenience" (not the same as in the
earlier diagrams) to fit a very recent situation where I used this
tetrachord as part of the seyir of Maqam Saba with the finalis placed
on D*, which is thus regarded as perde dugah, and C* as rast:
Jk A12 B
154 264 77
0 154 418 495
C* D E* F*
gerdaniye shehnaz tik tik
buselik chargah
In the descending portion of the seyir, as you explain in your
dissertation at page 121, another tetrachord of this general nature,
called Chargah, also occurs, realized in the 704.607-cent tuning as
follows:
Js A13 B
132 286 77
0 132 418 495
F* F#* A* Bb*
chargah saba huseyni acem
In this version of Maqam Saba, as in the 79-MOS version, this Chargah
jins or genus has a middle step equal to about 13 commas or a bit
smaller, here 286 cents or a near-just 13:11 (289 cents) or 33:28 (284
cents), in contrast to the narrower step of about 12 commas or 7:6 in
the previous Hijaz tetrachord at the beginning of the seyir's
descending portion from tik chargah down to gerdaniye. The Turkish
sign "A13" shows this larger middle or "augmented" step, by comparison
to the "A12" in the previous example.
If, in a different musical context, we desired a 13-comma genus at the
first location C*, or a 12-comma version at F*, the 24-note system
makes them available. Here I refrain from giving perde names, since
these could vary with the situation and I do not immediately have any
specific situation in mind:
Js A13 B
132 286 77
0 123 418 495
C* C# E* F*
Jk A12 B
154 264 77
0 154 418 495
F* G A* Bb*
Although users of the 79-tone qanun or ney players might regard the
term "cluster" as somewhat overblown for a tuning system with only, at
most, two versions of a given basic perde separated by less than the
55-cent diesis, one situation where I find a quasi-clustering technique
very apt and pleasant is in realizing the tetrachord below the finalis
of Dastgah Bayat-e Esfahan, often regarded as an avaz or tributary of
Dastgah Homayun, although Hormoz Farhat prefers to regard it as an
independent dastgah or modal family.
The tuning of this lower tetrachord of Bayat-e Esfahan leading up to
the finalis has been reported to show considerable variations in
practice between some of the leading musicians whose intonations have
been measured, and has also been a topic of some discussion among
Persian and other theorists. In what is known as the traditional or
"old" Esfahan, there is a small neutral second, a whole tone of around
9:8 or possibly larger, and then a larger neutral second as the leading
tone to the finalis ("leading tone" meaning here simply the step of
approach, and not implying as in some European and related theory
specifically the interval of a semitone, possibly obtained by an
accidental inflection):
Js T Jk
132 209 154
B C* D* E
0 132 341 495
In what is known as the "new Esfahan," the tetrachord is regarded as
having a structure similar to that of Chahargah Dastgah and also
prominently featured in Homayun Dastgah: a lower small neutral second,
middle "plus second" often somewhere around 11-12 commas, and upper
semitone. Taking recent views of some Persian theorists that the
regular major third is often close to Pythagorean at 81:64 (408 cents),
and a small neutral second might average around 135 cents, we would
have as one type of Chahargah or a similarly tuned Esfahan something
like 135-280-90 cents. However, Dariush Tala`i recommends a Chahargah
_dang_ or tetrachord at around 140-240-120 cents, and some of Jean
Darling's measurements suggest that an upper step of around 120 cents
may indeed fit some recent practice.
As a kind of pleasant intermediate option between the "old" and the
"new," the 24-note system offers what I might term a Buzurg-Hijaz
tetrachord since it is very similar to the lower tetrachord of Qutb
al-Din's Buzurg genus which spans a fifth:
1/1 14/13 16/13 4/3 56/39 3/2
0 128 359 498 626 702
14:13 8:7 13:12 14:13 117:112
128 231 231 128 76
From this just division of the fifth, a tempered version of the lower
tetrachord results as follows:
Js T10 Js
132 231 132
B C* Eb E
0 132 363 495
Since a tone or "T" is often presumed to be a regular diatonic step of
around 9:8 or 9 commas, the explicit sign "T10" identifies the larger
middle interval here as around 10 commas or 8:7; in Qutb al-Din's
Buzurg, as in Ibn Sina's soft diatonic, the subtly different 14:13 and
13:12 steps are both realized by the same tempered step of 132 cents.
The third step Eb at 363 cents above the lowest note of the tetrachord,
and 132 cents or a small neutral second below the finalis E, is at once
somewhat brighter and more strongly directed toward the finalis than in
the "old Esfahan" version with this step a 22-cent comma lower, and
more nuanced than a yet smaller semitone of some kind in the "new"
fashion.
These alternative tempered renditions of Esfahan at 132-209-154 cents
or 132-231-132 cents can illustrate a point about interval notations:
either version could be notated generally in the medieval manner as
JTJ, a lower and an upper mujenneb or neutral second step with the
middle interval of a tone of some kind. Using comma notation, these two
forms would be 6-9-7 and 6-10-6.
While in a given performance of Esfahan one might lean toward either
the 6-9-7 or 6-10-6 form, I have found it very pleasant to alternate
frequently between the two positions for the third step which
distinguish these forms otherwise using the same notes: playing now D*
at 341/154 cents from the lowest note of the tetrachord and the finalis
respectively, and now Eb at 363/132 cents. Apart from a tendency, not
surprisingly, to prefer the higher position for cadences to the
finalis, I find that these steps may freely alternate, the variation
being in good part for its own sake to "humanize" a keyboard rendition
a bit by introducing a touch of the "clustering" which is a pervasive
feature of flexible pitch instruments including the human voice.
Of course, this technique may also be taken as an enthusiastic
affirmation of the reality that there is more than one way to tune
Bayat-e Esfahan.
-------------
4. Conclusion
-------------
In exploring the 704.607-cent tuning both as a 17-MOS which may be
compared with medieval 17-note Pythagorean MOS systems in the Near East
and Europe, and as a full 24-note system, I have highlighted one
salient feature in which it varies dramatically from both types of
medieval antecedents: the pervasive role of middle or neutral
intervals.
This feature results from a technique evidently not on the conceptual
map of the Near Eastern or European theory of the time: temperament,
and more specifically a rather gentle regular temperament in which
fifths are slightly widened and fourths narrowed in order to produce
neutral flavors with chains of only 7-10 generators, rather than the
19-22 generators which would be required with a pure 3:2 fifth.
Of course, the same technical means can be adopted to quite different
artistic ends: and this holds true both with the Pythagorean 17-MOS,
and with the modern technique of temperaments, regular or irregular,
which may involve either widening or narrowing the fifth, or sometimes
may combine both options in the same system, as with the 79-MOS.
For Prosdocimus and Ugolino (Section 2.1), the Pythagorean 17-MOS
provides a gloriously perfected gamut (Gb-A#) expanding the standard
14th-century universe of regular Pythagorean intervals and polyphonic
resolutions to more remote melodic and harmonic outposts. The focus is
above all on regular diatonic intervals of 1-6 generators; for these
authors, neither the augmented and diminished intervals of 7-10
generators with their schismatic or 5-limit flavor, nor the nonexistent
neutral intervals, seem of much consequence.
For Safi al-Din, in contrast, both the regular Pythagorean and 5-limit
flavors of his 17-MOS are of vital interest, as in his version of Rast;
and so also are the middle or neutral intervals absent from this
tuning, but receiving great attention elsewhere in his discussions of
interval categories, tetrachords, and tuning methods for instruments
such as the `ud. It remains an open question whether his emphasis on
the 17-MOS might reflect a view that 5-limit flavors take a certain
priority over neutral ones; or possibly simply a preference for the
elegance of a regular tuning with an unbroken chain of fifths. In the
absence of temperament, such a preference would necessarily result in a
17-note system favoring regular Pythagorean and schismatic flavors
rather than neutral ones, which would require a larger tuning size.
How does the 704.607-cent temperament in its 17-MOS and 24-note forms
tie in with these different medieval worlds?
From the perspective of a 14th-century European style like that
celebrated by Prosdocimus and Ugolino, a tempered 17-MOS provides a
kind of accentuated variation on the Pythagorean Gb-A# gamut, with
regular major intervals yet wider, minor ones yet smaller, and the
compact Pythagorean limma at 90 cents, so important to directed
resolutions, narrowed to a yet more incisive 77 cents. With these
distinctions of color, the regular steps and progressions sought by
these theorists can map smoothly from the Pythagorean to the tempered
system.
While the neutral flavors of augmented or diminished intervals (7-10
generators) would not appear on a period keyboard tuned in pure fifths,
they might well have arisen in an "exuberant" style of intonation by
singers and other flexible-pitch performers described by Marchettus of
Padua (1318) and favored by such a noted modern performer and scholar
as Christopher Page, where cadential sharps may be raised by a comma or
more above their usual Pythagorean positions. It should be added that
these augmented and diminished intervals play mostly an incidental or
ornamental role; but the possibility that a momentary sonority of C#-F
or F-C#, for example, might sometimes have been sung in 14th-century
Europe as a neutral third or sixth is intriguing.
From perspective of Safi al-Din's Pythagorean 17-MOS gamut, however,
the 704.607-cent temperament is not a reasonable equivalent or
variation, because it lacks the 5-limit flavors that are needed for his
tunings of basic maqamat such as Rast. Rather, this temperament would
need to seek its historical footing in his discussions of middle or
neutral intervals, which of course draw richly upon such earlier
musicians and theorists of the Islamic tradition as Zalzal, al-Farabi,
and Ibn Sina.
Expanding either the Pythagorean or tempered 17-MOS to 24 notes gives
us a reasonable number of locations for another family of intervals
important to medieval Islamic theory: the septimal intervals arising
from 14-17 Pythagorean generators or 12-15 tempered generators. Here it
is interesting that Safi al-Din mentions 7:6 as a relatively concordant
ratio; septimal flavors also play an important role in the tunings of
Ibn Sina, and of Safi al-Din's younger contemporary Qutb al-Din
al-Shirazi.
Just as the Pythagorean 17-MOS structure is used for rather different
musical ends by Prosdocimus and Ugolino in Europe, and by Safi al-Din
in the Near East, so the technique of temperament can be used to
produce different kind of maqam tunings.
Whlle the 704.607-cent tuning at once emulates the regular structure of
the Pythagorean 17-MOS while emphasizing the neutral intervals that in
a medieval approach would be generated from two or more chains of
fifths, the 79-MOS system makes it possible to obtain a 5-limit Rast
reasonably close to that of Safi al-Din with a single unbroken chain of
fifths. This latter technique, of course, is similar to that of
European Renaissance meantone, but used as one aspect of a far more
intricate tuning system and intonational style to realize the diverse
musical world of the maqamat.
In its full 24-note version, the 704.607-cent tuning seeks to survey an
interesting and useful subset of two musical worlds, that of medieval
Europe and that of maqam/dastgah music as it has developed through
medieval and modern times. The 79-MOS draws its footing not only from
the maqam/dastgah tradition but from European common practoce of the
18th-19th centuries, offering for example a 12-note chromatic cycle,
and also combining just fifths with wide and narrow ones in order to
permit maximum flexibility in the fine intonation of the maqamat and
dastgah-ha.
Elsewhere I have compared the 79-MOS to a palatial estate, and the
704.607-cent tuning to a much more modest garden. Each scale of
organization may have its own charms. A larger system may place the
traits and preferences of a smaller one in a fuller perspective, while
a smaller one may encourage us more thoroughly and familiarly to
explore a congenial region which may then be better integrated into a
larger view.
-----------------------------------------------------------------
Appendix: Interval categories and symbols for maqam/dastgah music
-----------------------------------------------------------------
In some of the examples above, we have used some categories and symbols
to show step sizes in maqam/dastgah music, based in part on medieval
and modern Near Eastern theory often based in Pythagorean intonation or
53-EDO, an in part on some specific needs of the 704.607-cent tuning.
A particular goal of this proposed synthesis is to retain compatibility
as much as possible with the concepts and symbols of a modern Turkish
approach, while covering the range of intervals and steps from 1 to 13
commas and, especially, adding categories and symbols for the vital
middle or neutral second steps of approximately 6 or 7 commas.
We deal here with a "middle-resolution" system based on 53 commas to
the octave, taken for mathematical convenience to be the equal or
Holdrian commas of 53-EDO. A "lower-resolution" notation, in
comparison, might be based on 17 thirdtones or 24 quartertones to the
octave; while a "higher-resolution" system could, of course, use cents,
or the JI ratio precisely defining or closely approximated by a given
interval, with the 79-MOS and its measurement of the yarman, equal to
approximately 2/3 of a Holdrian comma, tending in this direction.
An advantage of the 53-comma system is that it is reasonably familiar
as developed in Syrian as well as Turkish theory, and at the same time
general enough that it leaves room for describing the nuances of a
given maqam or gusheh, etc., while leaving open precise realizations in
cents. Thus a "679" tetrachord might mean 141-153-204 (Safi al-Din's
division of 64:59:54:48), or 132-154-209 (e-based), or many other fine
shadings; but the pattern of a smaller followed by a larger neutral
second, and then a tone, is clear.
In the following scheme, intervals of 1-2 commas would typically
indicate interval differences or inflections rather than direct melodic
steps. Intervals of around 3 commas, or sometimes a bit smaller,
however, might serve as small or septimal semitones equivalent to a JI
ratio such as 33:32 (53 cents) or 28:27 (63 cents), the latter much
favored by Archytas.
A step of around 4 commas, of course, is the usual limma. Steps in the
range of 5-8 commas might all be considered in a broad sense
"intermediate" between the limma and the regular tone at around 9:8,
although in medieval and modern tunings such as 5-limit Rast, the
5-comma step may serve as a usual semitone, with some of the
complications regarding modern Turkish accidentals stemming from this
fact. The proposed notation proceeds from "small" (5 commas) to
"small-middle" (6 commas) to "large middle" (7 commas) to "large" (the
small whole-tone of 8 commas). This is not the only possible solution,
but, given the existing history, may be the "least astonishing" one, to
borrow a vivid phrase from computer science.
Next follow the regular tone at 9 commas, the large or septimal tone at
10 commas, and "augmented" steps at 11-13 commas of a kind used most
notably in a Hijaz tetrachord, or the corresponding Chahargah or
Chargah genus associated with the Persian dastgah of that name.
Following tradition, these categories are oriented around Pythagorean
or 53-EDO intervals -- but with equivalents here shown, where
applicable, for the 24-note regular tempered at 704.607 cents, since
these two styles of tuning have been the main focuses of this article.
----------------------------------------------------------------------
Middle-resolution categories based on 53-comma system
----------------------------------------------------------------------
53-EDO type commas cents symbol e-based type 53-commas cents
----------------------------------------------------------------------
12-comma 1 22.64 F 17-comma 0.96 21.68
......................................................................
demi-limma 2 45.28 D -------- ----- -----
......................................................................
reduced limma 3 67.92 E 12-diesis 2.44 55.28
----------------------------------------------------------------------
limma 4 90.57 B limma 3.40 76.97
----------------------------------------------------------------------
apotome 5 113.21 S limma + comma 4.36 98.65
......................................................................
small neut 2nd 6 135.85 Js apotome 5.84 132.25
......................................................................
large neut 2nd 7 158.49 Jk dim 3rd 6.80 153.93
......................................................................
dim 3rd 8 181.13 K reduced tone 8.28 187.53
----------------------------------------------------------------------
tone 9 203.77 T tone 9.24 209.21
......................................................................
large tone 10 226.42 T10 large tone 10.20 230.90
----------------------------------------------------------------------
hemifourth 11 249.06 A11 ---------- ----- ------
......................................................................
small min 3rd 12 271.70 A12 small min 3rd 11.68 264.50
......................................................................
min 3rd 13 294.34 A13 min 3rd 12.64 286.18
----------------------------------------------------------------------
A note is in order regarding the 2-comma interval, here carrying the
symbol "D" -- for "demi-limma," or for the "diaschisma" of Philolaus
and Boethius equal to half of a limma, not to be confused with the
smaller interval of 2048:2025 (19.55 cents) also called a diaschisma in
later theory. One could also call this in English a "half-limma." The
term "hemifourth" may likewise fit an interval or around 11 commas,
about midway between 8:7 and 7:6, and equal in 53-EDO precisely to half
of the 22-comma fourth.
Following a strategy that could appy to some other temperaments also,
e-based categories are mapped based on functional equivalence with the
Pythagorean or 53-EDO types rather than on a simple mathematical
rounding to the nearest 53-comma. Thus "E" in either system is the best
representation of 28:27, and "B" the regular limma. When maqamat or
dastgah-ha calling for regular Pythagorean or septimal flavors are
mapped from one system to the other, this approach may achieve "least
astonishment" -- rather as the Sagittal system of staff notation seeks
to do with considerable success.
For the sake of completeness, I should add that the above table
includes two equivalent categories present at rare and remote locations
in a 24-note version of the regular e-based system: S at 98.65 cents,
and K at 187.53 cents. While not especially accurate approximations of
their schismatic Pythagorean or 53-EDO counterparts, these intervals
can be used to produce a variation on Safi al-Din's Rast:
Rast T Rast
|--------------------|........|-------------------|
Bb C C#* Eb F G G#* Bb
0 209 397 495 705 914 1101 1200
209 188 98 209 209 188 98
T K S T T K S
This tuning, with some intervals comparable to meantone or 12-EDO, is
certainly the exception rather than the rule for the 24-note system.
However, exploring some of the remote corners of a system may give us a
better understanding of the whole.
With many thanks,
Margo Schulter
mschulter@calweb.com