Practice by Traditional Masters
Dr. Can Akkoç
Institue of Applied Mathematics (IAM), The Middle East Technical University (METU), Ankara, Türkiye (Turkey)
The `Universal` Scale
Pitch measurements on performances by indisputable masters of Traditional Turkish Music have presented maqam scales in the from of pitch clusters , referred to as “perde”s in the Turkish lexicon, rather than single pitches, as in the 12t-ET deterministic scale in Western music. Sample improvisations used in this study were taken from traditional masters with a very special request for staying within the declared maqam for the entire piece, and not straying into neighboring maqams, common practice in Turkish Maqam Music, so that characteristic musical patterns for the host maqam could be determined without contamination through seepage from other maqams. Furthermore, since these
1 E-mail: firstname.lastname@example.org , written correspondence may be sent to ODTÜ,
Uygulamal? Matematik Enstitüsü,
Ankara, Türkiye (TURKEY).
2  C. Akkoç, Non-Deterministic Scales Used in Traditional Turkish Music,
Journal of New Music Research, Vol.31, No.4, December 2002, pp.285-293 (in English).
 Kemal Karaosmanoglu ve Can Akkoç, Türk Musikisinde Icra-Teori Birligini Saglama Yolunda bir Girisim,
(Müz-Dak 10. Sempozyumunda Sunulan Bildiri; T.C. Kültür Bakanligi yayini) Folklor Derlemeleri ve Müzik Arastirmalari, Istanbul
Teknik Üniversitesi Türk Musikisi
Devlet Konservatuvari Mezunlari Dernegi, Istanbul, Türkiye, 3-5 Aralik 2003 (in Turkish).
masters were trained in the traditional mesk3 system, it is our thinking and hope that the structure of the perdes making up the underlying non-deterministic scales for various maqams shaped4 in their musical memory were not influenced heavily, if at all, by theoretical models coming down the centuries, thus devoid of the conflict addressed in this article.
Given these circumstances and the associated assumptions, it appears reasonable to attempt construction of a theoretical model for Turkish Maqam Music based entirely on pitch measurements from recorded/live performances of the masters that will not clash with practice.
Observations on the acoustic structures of perdes for a limited number5 of sample improvisations can be summarized as follows:
- Pitch clusters pertaining to the main perdes for a given maqam presented pitch distributions that were steep and compact compared with those of minor perdes playing a secondary role in the maqam structure (Figure 1)6.
- Minor perdes, on the other hand, displayed pitch distributions that were wide spread, scattered, sometimes showing local clustering underneath the main distribution (Figure 1).
- Pitch distributions pertaining to certain minor perdes appear to be changing personalities or character, in the statistical sense, depending upon their role in the underlying scale of the maqam they are used in. As an example, statistical7 structure of the perde Çargah, as it appears in the underlying scale for the Ussak maqam 8(Figure 2), presents a totally different picture when viewed in the underlying scale of the Sabã maqam (Figure 3).
3[transliteration~meshk] In this grueling master-apprentice type learning mode music is passed on by endless repetitions involving reciting/performing/disecting/etc.
compositions, melody and rythm, from traditional composers, ultimately followed by attempts, under the watchful eyes and never ending criticisms of the master,
on improvisations in various maqams. Improvs are considered to be the pinnacle of musical practice in traditional Turkish music.
4 As a typical example, Neyzen Niyazi SAYIN, featured in this article, is a legendary representative of the 5th generation of master musicians after the great Hammamizade
Ismail Dede (1778-1846). Because of the mesk system he was brought up with, we believe, Niyazi SAYIN’s musical memory embodies acoustic details of
the perdes, going all the way back to the Great Dede, presenting an incredible treasure of musical information and know how.
5 Samples referenced in this article were taken from one particular master, Neyzen Niyazi SAYIN, at a certain day and time. Statistically they bear no significance
in drawing inferences on the perdes. However, the intent in this article is not to present a complete perde structure for Turkish music. Instead, we try
to set the parameters needed for ultimately getting to the said acoustic structure of the perdes in all maqams.
6 For all figures: Coordinates along the horizontal axis are musical distances, measured in cents, from the tonic, while vertical coordinates represent the
durations, in % of the entire piece, of the individual pitches used in the improvisation.
7 General shape, range (width), compactness, etc.
8 Per the, so called, Arel-Ezgi system, a theoretical model currently used in schools of music.
Figure 1. Pitch cluster at perde “Dugah” is, relatively speaking, steep and compact, while the perde in the “Kurdi_dik Kurdi” zone is wide and scattered. Both are legitimate perdes of the Ussak maqam.
Figure 2. Perde “Cargah” is 18 cents wide when used in the underlying scale for the Ussak maqam, and has the shape shown in the histogram.
Figure 3. Perde “Cargah”, as it appears in the scale of the Sabã maqam, is now 8 cents wide and has the distinct shape observed in the histogram. One could speculate that the perde Cargah is being pressured from the right by the “nim Hicaz-Hicaz” group, which is a major perde for the host makam, and does not have much room to spread(?).
Conclusions and Suggestions [universal scale]
In order to construct a universal scale of statistical significance for Turkish music, sufficient number of reliable samples need to be processed. Sample pieces9 representing different maqams collected into a massive database, categorized and analyzed statistically after standard smoothing procedures, then followed by musical analysis targeting the maqam structures, probably calls for a major project of grotesque proportions. Such analysis should also include statistical character determination of each perde appearing in more than one maqam.
9 We suggest that samples be sought that were not contaminated by motifs or fragments belonging to other maqams, as would be the case if the performer had been wandering around visiting neighboring maqams. This, we realize is a tall order. Requesting a pure improvisation from a master musician that does not delve into other maqams could be viewed as absurd, almost like handcuffing the performer. However, if the master could be communicated that the proposition is entirely for purposes of academic research, it may be received favorably.
Theoretical Model [universal scale] There appears to be a pressing need for the development of a theoretical model capable of accomodating mathematical profiles of all perdes used in Turkish music for all maqams. Creators of such a model are going to be burdened, among others, with the following crucial problem: Given, the universal scale is formed by a sequence of pitch clusters, rather than discrete pitches, as in deterministic systems, what sort of mathematical representation would be satisfactory?
If a deterministic system is attempted, as widely accepted in musicological circles globally, which particular pitch is going to be selected from each cluster to characterize the location of the clusters? Furthermore, some clusters present smaller accumulations, clustrettes, under the umbrella of a wide and scattered cluster recognized as a single perde in practice? Are such clustrettes going to be accounted for in the deterministic system in any form or shape?
This inconvenient issue could be swept under the rug by a Hamparsum10 type fuzzy tuning model, where the score lays out the road map for the journey between pitch clusters, and says nothing about individual pitches to be selected from each cluster visited during the performance. It is expected that the mesk system will fill in the gaps. This, however, could be viewed as a major deterrent for beginners today who may not have the dervish patience in their souls, needed to survive such a challenging and grueling experience.
Given the bleak picture skethed here, and until such time a stochastic tuning system can be conceived, a deterministic system of adequate resolution, capable of accomodating the profiles of pitch clusters to mimic the maqam structure with reasonable accuracy and compromise would be our recommendation at this point in time.
An alternative approach that might prove to be effective, in particular for training beginning students in schools of music, would be to construct a multi-level theoretical model with several layers of sophistication. However, the perde structure, intrinsic to Turkish music should never be ignored, even at the least sophisticated level. It may be plausible to build a cascade of models starting with a quasi-deterministic system, ultimately leading to a full stochastic model11 for the master classes.
10 Hamparsum Limonciyan (1768-1839) was an Armenian Ottoman composer of Armenian church music and Turkish classical music and musical theorist who developed the Hamparsum notation system. The system was the main music notation for Turkish classical music until modern times and is still used by the Armenian Apostolic Church. 11 It is conjectured that the seyir issue (discussed below) is going to be an integral part of the said stochastic model since the actual pitches to be selected from the perdes are thought to be dictated by where the music has been for the past several steps (short term memory) before getting to a given cluster.
The Seyir12 Issue/Problem
We strongly believe the genetic structure, so to speak, of Turkish Maqam Music lies heavily in the seyir issue. If such a conjecture carries any merit, then the most urgent theoretical issue would be to get to the bottom of this musical structure. On the other hand, there are strong indications13 that this might be a formidable task, if not a mission impossible.
If seyir is viewed, in the abstract, as a “sequence of pitches” the melody line follows, then the seyir problem can be formulated asconstructing mathematical characterizations for all admissible seyirs14 for every maqam in Turkish Maqam Music.
Durations of pitches in the seyir sequence is another critical parameter that demands special consideration. In fact, the sequence of durations of pitches may turn out to be the centerpiece of the seyir problem down the road15 . In glissandos the durations are short and brief, whereas in slow moving passages they are longer and more persistent.
Most authors of theoretical works in Turkish music invariably suggest generic recepies for seyir in the maqamat discussed in their book. On the other hand, one sometimes runs into recordings or live performances of masters where the said recipes are violated just about every step of the way, and the seyir is still acceptable for the maqam on hand. In fact, such out of the box seyirs are sometimes viewed as a stroke of a genius, with lasting influences on the generations to follow.
Seyir over Pitch Clusters
From a different animated perspective, seyir can also be viewed as a journey between pitch clusters forming the underlying scale of the host maqam. The itinerary for the said journey — which clusters are going to be visited in which sequential order, and how long will each visitation last(?) — is believed to be determined by the inspirational mood/mode of the composer for formal compositions, and of the individual performer in free improvisations.
However, such itineraries can never be totally random walks. On the contrary, certain flow patterns, no matter how fuzzy they sometimes might get, pertaining
12 A term of Arabic origin “sayr~path” used in Turkish lexicon for the melody line, or the path the melody follows.
13 For Western music the only study we are aware of, resembling somewhat the seyir problem in Turkish Maqam Music, is Dmitri Tymoczko’s work at Princeton University, referenced at the Internet address .
14 It is conjectured that the set of all admissible seyirs for a given maqam is a countably infinite set. Then its complement, the set of all inadmissible seyirs, would also be a countably infinite set.
15 This statement is sheer speculation on our part at this point. Our apologies for such a bold venture.
15 Plural form for maqam.
to the host maqam must be observed. The mesk system, after all, can be viewed as a setting where the master transmits such wealth of stochastic patterns to the apprentice, with possibly some room allowed for fuzziness at a higher level to stimulate the future master’s creative faculties.
This introduction immediately brings up a crucial question: Given, the perdes in the underlying scale are made up of pitch clusters17 , how does a performer decide exactly which particular pitches should be selected from each cluster as the music weaves its way between the perdes?
This question could be neutralized and swept under the rug by suggesting that the mesk system has evolved precisely to address this extremely critical issue. An aspiring apprentice accumulates and assimilates such stochastic patterns after an almost never ending ordeal under constant repetitions. Yet, in this article we are interested in getting down to the genetic code of this matter in mathematical terms, no matter how fuzzy things might get.
What we have been able to conceive so far in the way of a thinking model is that of a “short term memory”18 , plausibly embedded in pitch sequences forming the melody line. This model suggests that, what directs the flow of music to a particular pitch within a given cluster, about to be visited, is the sequence of pitches selected just before getting to that cluster.
We strongly believe, the genetic code for such musical phenomena, if this turns out to be the case, is embedded entirely in the seyir problem and will begin unravel itself if and when the admissible set of seyirs are completely characterized for the maqamat in Turkish Maqam Music.
Conclusions and Suggestions [seyir]
Although the problem, as posed and partially formulated here, appears to be a tall order for the author, new mathematics coming down the pipeline during the past few decades, particularly those intended for infrastructuring the computers,
17 Such stochastic structures are not necessarily unique to Turkish music. A theoretician in the West, Willam A.
Sethares, suggests “Similar observations have also been made concerning Western performances.” in his book “Tuning, Timbre, Spectrum, Scale”,
Springer, 2nd Edition, 2005, page 71.
18 When the manuscript for the article referenced above in footnote #2 (Akkoç) was reviewed, one of the referees
was reported to have commented, “I consider especially useful the finding of two different tunings: the micro and the macro level “grand space”. The
suggestion that micro level is strongly connected to the concept of the short term memory gives good possibilities to connect the study of scale systems
to cognitive processes. Actually, the term “cognitive reference point” could be easily related to the finding of sound clusters, that were called in
the paper as “anchor notes”. For the future steps I believe that the author would benefit a lot if she/he
would have time to be more familiar with the main concepts and approaches in cognitive psychology of music.”
might generate a stimulating setting for a mathematical resolution to the seyir problem.
A few examples of such mathematics would be hidden Markov models, machine learning, neural networks, Kohonen maps (self organizing maps_SOM), Farey series, symbolic dynamics, formal languages, and data mining techniques.
The search and the expedition continues. However, the only progress report we are able to present at this time, in terms of concrete results, is none other than that of the five blind men and the elephant. We are sorry for not being able to end this report in a more positive note.