Modal systematics is a new theory of modal music based
on a model inspired by maqâm theories of the 2Oth century:
the author expands the quarter-tone representation of modal structure
developed mainly by Erlanger (1930ies) and integrates
it into a global combinatory theory which allows numerous theoretical
projections. The intervals of modal music are considered to be approximate
multiples of the quarter-tone interval, within a range from one
semi-tone to one and a half tone.
Specific studies of pentatonic and heptatonic systems
are carried, and a general model of scale generation is proposed
in the thesis; the model brings new explanations of some fundamentals
of music, particularly concerning the constitution of pentatonic
and heptatonic scales: synoptical tables of modal scales of Arabic
music are included, allowing further exploration of maqâm structure;
a number of intrinsic criteria of the building of the arabic scales,
never described before in specialised literature (to the knowledge
of the author), are discovered using the modal systematics method
- this specific research uses a combined genre-scale approach
allowed by the modal systematics theory.
Complementary studies of the maqâm by the method emphasize
the possibilities of traditional modulation and present new, alternative
scales, for composers. The appendix proposes an exhaustive list
of potential modal combinations of musical intervals (4795 modal
scales), allowing a comparative study between different types of
music. The theory of modal systematics still has many domains to
explore, including specific characteristics of maqâm music and extensions
to other musics of the world: it may be used in such various domains
as ancient Greek music, Indian music or even music with less obvious
connection to Arabic maqâm ; the model based on quarter-tone approximation
can also be extended to 1/8 tone or any other integral division
of the octave.
Keywords: systematics,
mode,
music, arabic, maqâm,
analysis, theory
download
the extended abstract (pdf format)
thesis
excerpts : access to downloading page (pdf format)
Music Faculty, University Paris IV - Sorbonne
1
rue Victor Cousin, F-75230 Paris Cedex 05 - France
Extended
Abstract : Modal Systematics by Amine Beyhom
A.
Part 1: Understanding maqâm structure
B.
Part 2: Theoretical and statistical study ("Modal Systematics")
C.
Part 3: Systematics of the maqâm
D.
Conclusions
The thesis consists of three volumes, two of which contain
appendices. Audio examples are provided on an accompanying CD-R, with
quick-reference cards supplied for the graphics.
Volume one is divided into three parts:
1. Understanding
the maqâm structure
2. Theoretical
and statistical study («Modal
systematics»)
3. Systematics
of the maqâm
The Appendices contain complementary graphics to the Part
2 of Volume One, full scores, synoptical tables of the Arabic modal
scales and Arabic genera. There is also a complete description
of the 4,795 potential modal scales that can be generated with intervals
(multiples of the quarter-tone) ranking from a semi-tone to a tone
and a half.
Key words: systematics, mode, music, Arabic, maqâm,
analysis, theory.
Part one: Understanding
maqâm structure
This section reviews different Arabic and European language
theories of the maqâm, published in the 20th Century.
Throughout, I highlight the enduring desire of Arabic musicologists
to incorporate maqâm analysis into the realm of the Occidental
theory of tonality, including the use of key signatures with half-flat
(or half-sharp) notes or with neutralized accidentals.
This approach to the maqâm structure creates profound
contradictions in the writings of those Arabic authors whose works
have been reviewed in the thesis. The use of Occidental theory appears
to arise from a desire to "legitimize" Arabic music and
it gives a certain reputation-enhancing cachet to the "master"-musicologist.
Apparently, the more the tonal theory is used to uphold the explanations
of Arabic authors, the less consistent these theories, some of which
deny musical practice and adapt it accordingly, become. These tend
to incorporate the maqâm theory into an adapted concept of
the ancient Greek modal theory and sometimes do not even refer to
Occidental notation or tonal theories (these authors tend to stress
the superiority of theory over practice).
On the other hand, a more synthetic approach to modal analysis
can be detected in some of the theories reviewed in Part 1. By and
large, these result in a simplistic description of the maqâm
structure, such as a single, one-octave maqâm description,
presenting it as a sub-scale generated by displacement of the finalis,
or a one-sided description of modal scales, considered to be composed
of two genera, connected by a disjunction whole tone
interval.
At least one of the authors uses a general assembly method
of genera with a "just" fourth and a connecting whole
tone to generate normalized scales that supposedly conform to traditional
Arabic music. Surprisingly, none of the authors subsequently applies
their theories coherently throughout their "exposé" and
seem to be reluctant to directly oppose the traditional Arabic representations
of the maqâm structure.
As a result of this approach, new features, sometimes very
odd and often incoherent, have been added to the maqâm structure.
These include the creation of new scales – supposedly traditional
and "natural" – which bear no relation to Arabic scales,
or genera seldom if ever found in traditional Arabic scales.
A general feature of "innovative" theories is the
systematic lack of any explanation or justification of new scales,
genera or modal analysis, highlighted in the theoretical works of
these authors. In fact, some authors seem to have succeeded in a more
pragmatic approach to the maqâm, but none has been able to
surpass the major contribution of Rodolphe D'Erlanger in his six-volume
book on Arabic music. Reviewing the writings of Erlanger and the 1995
work of his French compatriot Jean-Claude Chabrier, we notice a mutual
concern for a detailed and exhaustive description of the different
maqams and a desire to relate as accurately as possible a tradition
that both consider worthy and respectable.
Nevertheless, the theories of these two authors differ profoundly
over the description of the intervals used in Arabic music.
Erlanger is the author whose writings, reviewed in Part 1, mainly
stress on the intervallic Representation as a Suite (or RS - scales
or genera represented as a suite or intervals), as a complementary
notation method to the Western standard notation. Erlanger has also
translated into the French some of the most important Medieval manuscripts
on Arabic music by Al Fârâbî, Ibn Sînâ (Avicennus) and Al Urmawî,
including sections commenting on the detailed tuning of the`Ûd
and genera sub-divisions in regard to ancient Greek music.
He concluded that there was a fundamental difference between Arabic
and Greek music due to the existence of the Sîkâ (Ehalf-flat)
note in Arabic music, the Sîkâ being a major constituent of the Arabic
maqâm Râst scale, but disregarded by Chabrier, who prefers
a pythaghorian Limma-comma description of Arabic music intervals.
Erlanger and Chabrier represent the two schools of Arabic
music theory, whose roots can be traced back to the 8th
Century. In its substance, Chabrier's Limma-comma notation is related
to the Systematist theory, founded by Safiyuddîn Al Urmawî in the
14th Century and later developed by the main Turkish theorists,
such as Ezgi-Arel and Yekta Bey. However, despite all these theoretical
developments, the Limma-comma scales appear to be far removed from
current musical practice as Karl Signell and other researchers or
musicians describe it. Practiced intervals seem to be much closer
to the 24 fourth-tone intervals scale as reviewed in Middle-East Folk
music (Detailed discussions about the interval sub-division of Arabic
music scales are covered in Part 2 of the thesis, which is devoted
to Modal Systematics theoretical and statistical studies).
Part
1's conclusions stress upon the many differences of approaches
of the various authors, and on inconsistencies found within the writings
of a single author. As an alternative, I propose a multiple definition
of the maqâm, including multiple scales to characterize one
single maqâm. This pragmatic approach, the premises of which
can be seen in some of the reviewed literature, is fully disclosed
in Part 2 of the thesis and examined further in Part 3, which is devoted
to the practical application of modal systematics of the maqâm
structure.
Top
Part two: Modal
systematics
As
a preamble to Part 3, I explain the reasons for constructing a model
based on equal quarter-tone subdivisions of the octave, in preference
to other possible subdivisions such as 1/3, 1/6, 1/8 tone subdivisions
or even combinations of Limma and Comma subdivisions.
A
review of the theories in Part 1, coupled with my personal experience
as well as various suggestions from Arab musicians and musicologists,
seem to converge, resulting in a wide acceptance of the quarter-tone
concept as representative of the Arabic general scale. The quarter-tone
is not an existing interval in Arabic music but an approximate increment,
a basic sub-divider, which allows for the definition of a 24 interval
grid with 25 notes (the 25th being the higher octave note
of the basic – 1st – note of the normalized scale), all
of which are known to Arabic theorists and musicians.
The
25 notes must be seen as approximate renderings of those in current
practice. Furthermore, Arabic music intervals are rarely fixed but
vary within a certain range – an interval in Arabic music varies constantly
due to aesthetic concerns, transpositions, different maqâms
or different structures of instruments, and depend on transpositions
or finalis positioning.
In
fact, modal systematics will not give any exact rendering of any interval
of Arabic music, but an indication of the existence
of this interval which, qualitatively, is different from other intervals,
and expands, to a certain extent, around the exact dimensions provided
by the 24 quarter-tone intervals grid.
In
short, Modal systematics deals with the existence of
musical intervals, not with their absolute and exact value.
As a major application of the principle of quarter-tone approximation,
I expose the method of the creation of modal scales and its different
possible statistical and theoretical applications, including the filtering
of these scales through musical critera (e.g. the presence of a "just"
fifth or fourth, exclusion of semi-tone paired – or more – intervals,
or of coupled "greater than one tone" intervals), and pursue
a general comparative study on those scales, including quarter-tone/semi-tone
comparison. These applied criteria should not be seen as absolute,
but behavioral hints that are especially helpful in parallel studies
concerning the optimum number of modal scales that can be created
with a precise number of intervals in an octave.
As a whole, this process corresponds to a parallel statistical
research on what I have called the "systems" of modal music,
resulting in a hierarchic ranking of the scales into hyper-systems,
systems and sub-systems:
- Hyper-systems indicate the type and number
of intervals in scales.
- Systems are the modern equivalent of "scales",
corresponding to a certain arrangement of the intervals of a hyper-system
in a regular, distinct suite.
- Sub-systems are considered "extensions"
or "aspects" of Systems, with each Sub-system starting
at a different note (or interval) of the original System or Scale.
The schematic relationship between the three hierarchic components
of the Modal Systematics classifying concept can be seen in the picture
below, where the Arabic figures express interval values in increments
of a half-tone (1 == half-tone, 2 == one whole tone), with Hyper-systems
and Systems ranking from the lowest integer value of the concatenated
interval values to the highest.
Picture
N°1. Arborency example between the three types of scale classifications,
the relationships between Hyper-systems, Systems and Sub-systems
As an example of typical theoretical studies by the
Modal Systematics method, the Pentatonic structure undergoes a thorough
examination with an extensive creation of scales with intervals ranking
between a semi-tone and one and a half tone that can be found in the
Appendix.
Further theoretical studies on the reasons of
existence of the Heptatonic scales seem to indicate that, given the
musical aesthetics of modal musics, the number of seven intervals
per octave corresponds to an optimum of scale generation.
The mere existence of the three quarter-tone intervals,
an intermediate interval between the occidental semi-tone and whole
tone, carries in itself an explanation of the modal consistency of
non-Western music, that includes Arabic music. This single additional
interval paves the way for a considerable number of additional scales
(of which only a small tranche is used in Arabic music) and explains
the persistency of the modulation aesthetics in the latter. These
theoretical results can be considered a strong indicator that the
"cycle of fifth" theory is an irrelevant method of modal
scale construction.
A comparison between the numbers of generated "semi-tone-like"
scales and "quarter-tone-like" scales results in a difference,
which, regardless whether aesthetic criteria are applied, is at least
20 times in favor of the quarter-tone scales.
As a special dedicated study to the semi-tone scales, a few
19th and 20th Century innovations are reviewed
through the Modal Systematics method. Chromatism, dodecaphonism and
other tentatives that try to go beyond the seven interval optimum
and barrier result in a lesser number of created and usable scales.
These, coupled with the phenomenon of increasing redundant scales
among those generated for more than seven intervals to the octave,
result in the abandoning of tonal and modal aesthetics.
A further study tends to show that the limit of a one and
a half tone for the biggest possible interval in a scale may well
also be a result of combinatory systematical modal creation. The use
of bigger possible intervals leads to a radically different aesthetic
of music, along with stressed constraints on musical criteria (perfect
fourth and fifth, "chromatism" seen as a succession of semi-tones).
The existence though of one or more (in general two) intervals
of one and a half tone in the created scales seem to be a main motor
for modal scale generation. In fact, it is more specific for semi-tone
scales, as the results give us a new explanation of the widespread
use of the "augmented second" in most European folk music.
As a closing study to Part 2 of the thesis, a research on
scales created by the modal systematics method resulted in a certain
number of additional scales conforming to
the traditional criteria of Arabic music. To my knowledge,
these are not found in contemporary specialized literature.
To determine the structure of existing Arabic scales, I carried
out a systematic study and relevés of scales in modern literature.
The results gave an overall presentation of Arabic scales through
a series of synoptical tables. These are the main tools used in studying
the Arabic maqâm through the Modal Systematics method carried
out in Part 3.
Top
Part three : Maqâm
systematics
The
synoptical tables of Arabic modal scales showed in the Appendix of
Part 3 are the result of an extensive study of Arabic maqâmât.
These tables are the result of applying the Modal Systematics theory
to the maqâm, presenting Arabic maqâmât synoptically
as a subset of the exhaustive ensemble of potential scales in quarter-tone
intervals.
In
these tables, Arabic modal scales are arranged under their corresponding
entries in the general scale data basis, created in Part 2 of the
thesis (including all interval combinations to the octave for seven
intervals ranking from a semi-tone to a tone and a half).
Each
scale may have one or more names (sometimes quite a few) or finalis,
depending on the different sources used to establish the tables (references
to the manuals and books from which the scales are taken are also
included in the tables as well as remarks – and special internal codes
– on the viability of each scale or author).
As
a first overview of these tables, there appears to be some 50 scales
(considered here as systems, i.e. paradigmatic scales generating each
seven sub-scales – or sub-systems – by a rotational process of the
finalis, or – here – of the first interval) and some 140 sub-systems
(or modal scales) described on over 220 finalis.
I
did not try to establish the exact (not to mention considerable) number
of different denominations for these scales. Nevertheless, a brief
comparison with the total number of potential modal scales seems to
indicate that Arabic music, using a maximum 10 percent of the scales
of the database, is still some way from realizing its intrinsic evolutionary
potential. In fact, the vast gaps in the meta-structure of Arabic
scales, including those internal to systems (missing sub-systems),
contribute significantly to the explanation of the unique aesthetics
of Arabic music and to its internal structure.
In
Part 3, I pursued a thorough examination of Arabic scale structure,
including an alternative approach to some of the most controversial
scales, such as the Awj-Ârâ scale and the establishment
of a series of new intrinsic rules for the building of these scales,
never before, to my knowledge, expressed in specialized literature.
A
second examination of the Arabic scale structure is initiated by a
combined genera-scale approach, including genera assembly
methods reviewed in Part 1. Further investigations of modal scale
creation by the assembly method allowed me to verify and correct common
errors in previous theories by creating a step-by-step hierachy from
those with very restrictive criteria to those with open boundaries
(the exceptions concerning those criteria are the general semi-tone
to one and a half tone restriction on the interval size).
As
a result, the open-boundaries creation outlined in Part 3, proves
the consistency the Modal Systematics theory developed in Part 2.
In both cases, the scales created are identical including those scales
the intervals of which are multiple of the semi-tone interval.Consequently,
a genera-type approach by the systematic method would have
given the same results as the general scale-approach developed in
Part 2, resulting in a similar theoretical formulation.
A
third study stresses on the frequency of intervals in the scales of
the appended database of potential modal scales. A few of the seemingly
"innovated" genera listed in recent theoretical works
on the maqâm gain, through this study, a certain legitimacy,
while others now appear to fall short when trying to meet the traditional
criteria of Arabic music.
These
results cannot be considered definitive due to a lack of information
on Arabic scales in general, while common practice seems to have disregarded
a growing number of old or rarely-used scales, which nonetheless are
cited in specialized literature.
Despite
these uncertainties, the complementary genera-scale approach
offers interesting results concerning some gaps existing in the general
construction scheme of Arabic music. These include restrictions on
the use of the Hijâz tetrachord (2, 6 and 2 quarter-tone intervals
in a row), which seems to carry a rule of integrity preventing scale
constitution whenever the tetrachord can be "broken".
Further
investigations of the Hijâz tetrachord, coupled with acoustical
analysies and studies of the `Ûd organology seem to prove that
the Awj-Ârâ tetrachord (3/4, 6/4, 1/4 – Erlanger), the
Awj-Ârâ-Şaghîr (3/4, 5/4, 2/4) and the Zîrkûlâ
(2/4, 5/4, 3/4) tetrachords
are, from the point of view of Arabic music aesthetics, equivalent.
A
number of other criteria concerning the intrinsic aesthetics of Arabic
scale building are discovered in the study process. Examining the
modulation process by the combined genera-scale approach also
gives a number of results, including many alternative musical "routes",
unforeseen in classical Arabic music theories. Ûd organology
and tuning, despite being a leading factor in the framework of Arabic
music, prevent Arabic music and its aesthetics spreading beyond its
traditional boundaries.
Furthermore,
the modulation techniques described by contemporary authors appear
to rely heavily on the concept of the two variable pitches between
two fixed (border) notes of a tetrachord. A number of different modulations
are examined, some of which relate to `Ûd organology, and are
reproduced in a small program, creating "parent" scales.
Most
of the "alternative" scales of one particular maqâm
can be detected with this method, as can a number of new, alternative
scales not found in the reviewed literature. Furthermore, some specific,
pentacordal modulations are also detected, clearly demonstrating gaps
in the concept of modulation using a purely tetrachordal approach.
The
final
filtering of all the created scales found in the modal scale database,
in regard to all the explicit or implicit criteria reviewed all along
the thesis, allows the establishment of some 81 "traditional-like"
scales not previously described in reviewed literature.
Top
Conclusions
Modal
Systematics is a new theory of modal non-tempered scales, which is
based on a qualitative combinatory approach of musical intervals.
This theory represents a coherent alternative to existing methods
of analysis for modal music, with results seemingly never before achieved.
Some musical fundamentals (i.e. heptatonism and pentatonism) find
a new explanation through this theory, which appears to have a range
of domains that still require exploration including modal verticality
and modulations with displacement of the finalis as well as the study
of Indian, Persian and ancient Greek music. The exhaustive systematical
tables of Arabic maqâmât (with intervals ranking from half-tone
to one and a half tone) provided in the thesis may be used as the
basis for a future comparative study between these different types
of music.
Top
_left_en.gif){kind=small}
.gif){kind=small}
