Geometrical Representations of North Indian Thāts and Rāgs

Year: 2010 Authors: Chirashree Bhattacharya; Rachel Wells Hall

Core claim

Geometrical music-theoretic representations expose when rāgs align uniquely or ambiguously with Bhātkhaṇḍe’s ṭhāṭ system.

Topics

North Indian classical music theory, ṭhāṭ-rāg classification, geometrical music theory, set-theoretic analysis, MATLAB visualization

Domains

geometry, lattice models, set theory, graph representation, music visualization, information graphics, computational art

Methods

five-dimensional lattice embedding, subset-superset analysis, database visualization, MATLAB code

Media

MATLAB-generated images, rāg database, graph figures

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2010: Mathematics, Music, Art, Architecture, Culture

Geometrical Representations of North Indian $Th\bar{a}ts$ and Rāgs

Chirashree Bhattacharya Department of Mathematics Randolph-Macon College Ashland, VA 23005, USA E-mail: cbhattacharya@rmc.edu

Rachel Wells Hall¹ Department of Mathematics Saint Joseph’s University Philadelphia, PA 19131, USA E-mail: rhall@sju.edu

Abstract

In his seminal works on North Indian classical music theory, V. N. Bhātkhaṇḍe (1951, 1954) classified about two hundred rāgs (fundamental melodic entities) by their seven-note parent modes known as ṭhāṭs. However, assigning rāgs to ṭhāṭs is not a straightforward task. Each rāg is defined by a collection of melodic features that guide a performer’s improvisation. Although these features sometimes point to a unique ṭhāṭ, in other situations they either give incomplete information (too few notes) or give conflicting information (too many notes). Our goal in this paper is to construct geometrical models that help us to better understand the relationship between ṭhāṭs and rāgs. Following the principles of geometrical music theory (Callender, Quinn, and Tymoczko 2008), we locate the thirty-two “theoretical ṭhāṭs” in a five-dimensional lattice. Jairazbhoy’s “Circle of Ṭhāṭs” connecting common ṭhāṭs embeds within this lattice (Jairazbhoy 1971). For a given rāg, our geometrical representations show which theoretical ṭhāṭs contain the notes used in the rāg’s various melodic components separately. We have written MATLAB code that produces images of a database containing a number of rāgs. Our models reveal graphically some of the problematic aspects of Bhātkhaṇḍe’s rāg classification system.

1. Introduction

Rāgs are the fundamental melodic entities of North Indian Classical Music (NICM). Rather than being a fixed “tune,” each rāg is a collection of musical features that guide a performer’s improvisation. In his foundational books on North Indian music theory [1, 2], V. N. Bhātkhaṇḍe classified rāgs by seven-note modes known as ṭhāṭs (a mode, such as the major or minor mode in Western music, is a scale with a distinguished tonic). While there are close to two hundred rāgs, Bhātkhaṇḍe assigns each rāg to one of ten ṭhāṭs. Although this assignment is straightforward in some cases, quite a few rāgs have either too many or too few distinct notes to correspond with a unique ṭhāṭ.

Our goal in this paper is to construct geometrical models representing set theoretic relationships between ṭhāṭs and rāgs. While scholars have experimented for centuries with geometrical models for Western modes, including circles of major and minor modes and the Neo-Riemannian tonnetz, geometrical models representing the elements of NICM appeared relatively recently, chiefly in the work of Jairazbhoy [6]. Following the principles of geometrical music theory [3], we locate the thirty-two “theoretical ṭhāṭs” in a five-dimensional lattice. For a given rāg, our geometrical representations show which theoretical ṭhāṭs are supersets of notes used in the rāg’s āroh, avroh, and pakar separately. These reflect the degree to which a rāg is unambiguously identified with its ṭhāṭ. We have written MATLAB code that produces images of a database of rāgs.

The basics of North Indian music theory are as follows. As in Western theory, seven notes, Sa, Re, Ga, Ma, Pa, Dha, and Ni span an octave; this sequence of notes repeats in higher and lower octaves. Of these notes, Re, Ga, Ma, Dha, and Ni have two positions, śuddha (natural) and vikrit (altered), which may either be komal (flat) or tīvra (sharp). The only note among these to have a tīvra position is Ma, while the rest have komal and śuddha versions. Thus the twelve notes in an octave, successively a semitone apart, are: Sa, Re (komal), Re (śuddha), Ga (komal), Ga (śuddha), Ma (śuddha), Ma (tīvra), Pa, Dha (komal), Dha (śuddha), Ni (komal), Ni (śuddha). We will use the abbreviated list {S, r, R, g, G, m, M, P, d, D, n, N} when convenient. We note that Indian note names indicate relative, rather than absolute, pitch; the performer is free to choose the actual pitch identified as “Sa.”

A ṭhāṭ is an ordered collection of the seven notes, where only one version of each note may be selected. Since five of the notes have two positions, it is theoretically possible to create thirty-two $(2^5)$ ṭhāṭs. However, only the ten ṭhāṭs listed in Table 1 are commonly used in NICM. Six of these, including the major (Ionian) and minor (Aeolian) modes, are known in the West as Glarean modes.²

¹Partially supported by a Penn Humanities Forum Regional Fellowship, 2009-2010.

²Glarean modes, named for the sixteenth century music theorist Heinrich Glarean, all belong to the same set class, meaning that, modulo cyclic permutation or reversal, they have the same sequence of intervals between adjacent notes. This set class, the diatonic scale, has quite a few desirable properties, including the fact that it is nearer than any other seven-note collection in twelve-tone equal temperament to the even division of an octave into seven parts. In addition, it is “generated” by a sequence of six perfect fifths modulo the octave (see [4]).

Bhattacharya and Hall

Table 1: The ten common thats of North Indian Classical Music and their Western equivalents.

Table 2: Five rāgs. The column marked “v., s.” indicates the rāg’s emphasized notes (vādī and samvādī). Bold letters within a melodic element indicate prolonged notes; a dot above or below a note indicates transposition up or down an octave, respectively.

While there are only ten thats in common use, there are about two hundred rags. A rag is a melodic theme upon which a performer improvises while staying within the allowable boundaries of note patterns and combinations specific to that rag. Each rag is characterized by its ascending and descending sequences (āroh and avroh), its “catch phrase” (pakar), its emphasized notes (vādī and samvādī), the number of notes it contains (jātī), the octave emphasized, and the time of day it is performed. Rāgs may be pentatonic, hexatonic, or heptatonic depending on the number of distinct notes they use.

Table 2 summarizes five rags from three thats. In theory, a rag is assigned to a parent that largely on the basis of agreement of notes in the rag with those of the that. This is clearly true in the case of rag Åsāvarī: the union of the set of notes in its āroh, avroh, and pakar corresponds exactly with Åsāvarī that. Although both its āroh and pakar are “incomplete” in that they contain less than seven distinct notes, Åsāvarī is the only one of the ten common thats that have the rag’s āroh or pakar as subsets. (For example, its āroh contains the notes S, R, m, P, and d. Four of the thirty-two theoretical thats also contain these notes; of them, only Åsāvarī is a common that.)

Identifying $r\bar{a}gs$ with $t\bar{h}\bar{a}ts$ based on subset relationships is not always straightforward. In particular, the notes of hexatonic and pentatonic $r\bar{a}gs$ are subsets of more than one $t\bar{h}\bar{a}t$ . Bhātkhande mentions these difficulties in his major work, the Kramik Pustak Mālka [2], where he provides brief descriptions for each of about 180 $r\bar{a}gs$ . For example, $r\bar{a}g$ Mālkauns is a pentatonic $r\bar{a}g$ containing only the notes $\{S,g,m,d,n\}$ . On the basis of notes alone, it could equally well belong to $A\bar{s}avart$ or $Bhairavt$ . Bhātkhande notes that $r\bar{a}g$ Mālkauns “is generated from $Bhairavt$ that … some say that it is in $A\bar{s}avarith\bar{a}t$ ” [2, vol. 3, p. 701, translated from Hindi].

The comparison of rāgs Shuddhakalyān and Bhupāli reveals another challenge for the practitioner of NICM. Bhātkhande singles out certain rāgs that are “close” and explains what a performer must do to avoid crossing over to a neighboring rāg. For example, he describes Shuddhakalyān as similar to Bhupāli, but, “unlike Bhupāli, in this rāg the lower octave is used more … In avroh [the note] Ni is used many times and this distinguishes it from Bhupāli” [2, vol. 4, pp. 60-61]. Note that Shuddhakalyān’s heptatonic avroh not only distinguishes it from Bhupāli but also identifies the thāt. (In general, we note that a rāg’s avroh is more likely than its āroh or pakar to signal its thāt.)

In contrast, rāg Kedar has “too many” distinct notes (eight) rather than too few. It belongs to Kalyān thāt, even though subset analysis seems to suggest Bilaval (in particular, its āroh belongs to Bilaval, its avroh contains both Bilaval and Kalyān, and its pakar belongs to Bilaval, Khamāj, or Kāfī). Moreover, Kedar’s vādī (emphasized note) is a natural Ma, while Kalyān thāt has a sharp Ma. Bhātkhande comments that both sharp and natural forms of Ma are used. Ancient writers did not allow use of sharp Ma in Kedar and considered it to be under thāt Bilaval. Presumably, the sharp Ma “trumps” the natural.

Geometrical Representations of North Indian Thāṭs and Rāgs

Figure 1: Circle of Thāṭs after Bhātkhāṇde (left) and Jairazbhoy (right).

2. Geometrical Models

Due to the association between $r\bar{a}gs$ and times of day, the depiction of $r\bar{a}gs$ on a circle is natural. $R\bar{a}gs$ belonging to the same $th\bar{a}t$ are typically performed either at the same time or separated by half a day. On this basis, Bhātkhāṇde proposed to identify $th\bar{a}t$ s with times of day on a twelve-hour cycle. Jairazbhoy [6, p. 63] took the logical next step by arranging $th\bar{a}t$ s on a circle according to Bhātkhāṇde’s time theory, as in Figure 1 (left). Remarkably, nine of the ten $th\bar{a}t$ s, starting with Bhairav and proceeding clockwise to Bhairavī, form a sequence in which each $th\bar{a}t$ is related to its neighbors by a one-semitone alteration in one of its notes. For example, we move from Kalyān to Bilaval by changing Kalyān’s sharp Ma to a natural Ma (F→F‡), while we move from Kalyān back to Mārva by flatting Kalyān’s natural Re (D→D♭). In other words, with the exception of Torī, $th\bar{a}t$ s that are adjacent in time are linked by voice leadings (bijections between collections of notes) which are “efficient” in that only a small amount of chromatic alteration takes place. Roy [7, p. 82] theorizes that the agreement between the ordering of $th\bar{a}t$ s based on efficient voice leading and the ordering based on time theory is probably due to the “tendency of rāgas to follow the line of least resistance in the easy transition from scale to scale … observed to a certain extent by all musicians.” Since moving from one $th\bar{a}t$ to another requires retuning some musical instruments, it is advantageous to arrange the cycle so any two neighboring $th\bar{a}t$ s share as many common tones as possible. In the sequence of six $th\bar{a}t$ s from Kalyān to Bhairavī, one has the added advantage that the new pitch is always a perfect fifth from one of the notes in the original scale. (After the octave, the perfect fifth is the easiest interval to tune.) We also note that typical models of Western modes share the feature that the modes are linked by efficient voice leading [3].

Is there a way the voice leading approach can be made to include $Tor\bar{ı}$ ? And what of the thirty-two theoretical $th\bar{a}ṭs$ : can they be incorporated into a model? We locate theoretical $th\bar{a}ṭs$ as vertices of a graph in Figure 2 (Jairazbhoy depicts a isomorphic graph in [6, p. 184]). Two $th\bar{a}ṭs$ are connected by an edge if and only if they differ by one semitone. Note that, although the graph is a convenient model for local connections between $th\bar{a}ṭs$ , it does not represent distances—each edge in the graph represents a one-semitone alteration, but the edges are different lengths. Moreover, it does not represent all possible pathways between $th\bar{a}ṭs$ .

Bhātkhāṇde’s ten common $th\bar{a}ṭs$ , indicated by ringed circles, define a connected subgraph of the lattice. In order to complete a cycle, Jairazbhoy adds a theoretical $th\bar{a}ṭ$ labelled “A7” (so called because of his classification scheme). This move successfully incorporates $Tor\bar{ı}$ but leaves out $Bhairav$ . Jairazbhoy’s “Circle of $Th\bar{a}ṭs$ ,” as in Figure 1 (right), embeds as a cycle in the graph of theoretical $th\bar{a}ṭs$ . The graph also reveals the problem: $P\bar{u}rv\bar{ı}$ , $Bhairav$ , $Tor\bar{ı}$ , and $Bhairav\bar{ı}$ lie on the vertices of a cube in the lattice, and there is no path that connects them all, using transitions where some note is altered by a single semitone. An alternate to Jairazbhoy’s solution is to allow the path to bifurcate, connecting $P\bar{u}rv\bar{ı}$ to both $Tor\bar{ı}$ and $Bhairav$ , then connecting $Tor\bar{ı}$ , $Bhairav$ , and $Bhairav\bar{ı}$ to the unique theoretical $th\bar{a}ṭ$ that is within a one-semitone alteration of all of them. (Jairazbhoy [6, p. 97-99] cites historical and theoretical reasons for preferring “A7” to this $th\bar{a}ṭ$ , however.)

Bhattacharya and Hall

Figure 2: Lattice of thirty-two theoretical $\text{thats}$ .

Geometrical music theory provides a way of thinking about geometrical representation in general (see [3]): any musical object that can be represented by an $n$ -tuple of pitches corresponds to a point in some $n$ -dimensional Euclidean space. Equivalence relations, such as octave equivalence, define quotient maps on Euclidean space producing a family of singular, non-Euclidean, quotient spaces—orbifolds. Points in these spaces represent equivalence classes of collections of notes, such as chords or scales. Any voice leading corresponds to a line segment or path in an orbifold. In order to represent distances between $\text{thats}$ accurately, we need at most six dimensions (the fact that NICM uses relative pitch means that we lose a dimension—a $\text{thats}$ is really an equivalence class modulo the choice of the pitch Sa). Because all $\text{thats}$ include the pitch Pa, five dimensions suffice, but the number of dimensions is still too great for us to draw a satisfactory representation.

However, we can exploit a feature of $\text{thats}$ here. As with Arab modes (see [5]), each $\text{thats}$ is traditionally considered to be formed from two scalar tetrachords. The lower tetrachord begins with Sa and ends with Ma (or Ma $\text{tivra}$ ) and the upper tetrachord begins with Pa and ends with high Sa. This decomposition suggests a different way of constructing the lattice of theoretical $\text{thats}$ . First, we note that representing tetrachords, modulo translation, requires only three dimensions; in Figure 3 (left), we locate the lower and upper tetrachords on disjoint lattices, where two tetrachords are adjacent if and only if they differ by one semitone. The product of the two tetrachord graphs (Figure 3, right) can be visualized as two nested tori, each corresponding to a different position of Ma. In this picture, each torus has been cut open to form a large square. (This explains why the $\text{thats}$ on the left-hand edge are duplicated on the right-hand edge and the $\text{thats}$ on the bottom edge are duplicated at the top.) $\text{thats}$ with the same first three notes appear in the same vertical plane, while $\text{thats}$ with the same upper tetrachord are in the same horizontal plane. If the edge faces are connected, the resulting graph is isomorphic to the graph of theoretical $\text{thats}$ (Figure 2).

The construction of Figure 3 was first proposed as a tool for representing modulatory relationships between Arab modes, or $\text{maqamaˉt}$ [5]. Figure 4 contrasts the $\text{thats}$ of NICM, the Glarean modes, and the Arab modes. (Since Arab musicians use a quarter-tone scale, there are intermediate modes between lattice points. Only about two-thirds of Arab modes are representable on this lattice—some do not repeat at the octave, and others have a different fifth scale degree.) As previously noted, Glarean modes are a subset of the Circle of $\text{thats}$ . However, there is surprisingly little overlap between the North Indian and Arab modes. In particular, the Arab system uses the diatonic scale sparingly, preferring instead some scales that divide the octave more evenly (this is possible using quarter tones) and others quite a bit less evenly. The fact that the Circle of $\text{thats}$ lies on or near the diagonal of the squares reflects a preference in NICM for what Jairazbhoy calls “balanced” $\text{thats}$ — $\text{thats}$ whose upper and lower tetrachords contain roughly the same scalar intervals.

Geometrical Representations of North Indian Thāṭs and Rāgs

Figure 3: The lattice of thirty-two theoretical thats, configured as nested tori.

3. Examples

Although we have discussed the difficulty of identifying $r\bar{a}g$ s with $th\bar{a}t$ s before, let us see how geometrical methods can help (or at least give us a better visualization). In Figure 5, we contrast $r\bar{a}g$ Åsavari with $r\bar{a}g$ Mālkauns. $R\bar{a}g$ Åsavari belongs to $th\bar{a}t$ Åsavari (indicated by a dotted sphere). Its $a$ roh is a subset of four theoretical $th\bar{a}t$ s and its $pakar$ is a subset of two. (Because the graph is a torus, there appear to be six markers for the $a$ roh—two of them are repeats.) However, its avroh contains exactly the notes of $th\bar{a}t$ Åsavari. In this situation, there is no ambiguity in the classification of the $r\bar{a}g$ . In contrast, the pentatonic (missing Re and Pa) $r\bar{a}g$ Mālkauns is classified under $th\bar{a}t$ Bhairavi (indicated by a dotted sphere). However, due to the omission of Re, its $a$ roh, avroh, and $pakar$ are subsets of two $th\bar{a}t$ s: Åsavari and Bhairavi. This ambiguity agrees with Bhātkhande’s aforementioned comment that theorists differ on whether to assign $r\bar{a}g$ Mālkauns to Bhairavi $th\bar{a}t$ or to Åsavari $th\bar{a}t$ [2, vol. 3, p. 701]. Figure 6 depicts two $r\bar{a}g$ s that have “too many” notes. As previously noted, Kedar contains both sharp and natural versions of Ma; Hamir has this same feature. At present, our models do not distinguish between superset and subset relations: both $r\bar{a}g$ s’ avroh have $th\bar{a}t$ s Kalyān and Bilaval as subsets, rather than supersets.

Our models clearly reflect the fact that the relationship between a $r\bar{a}g$ and its $th\bar{a}t$ is sometimes ambiguous. In terms of pitch class content, $r\bar{a}g$ s belonging to the same $th\bar{a}t$ vary in the degree to which they signal their parent $th\bar{a}t$ and the degree to which they resemble each other. Moreover, a $r\bar{a}g$ ‘s $a$ roh, $a$ vroh, and $pakar$ may convey different (and occasionally conflicting) information. However, there are many features of $r\bar{a}g$ s that are not captured by this geometrical representation. Further analysis is needed to determine which features are most predictive of the assignments of $r\bar{a}g$ s to $th\bar{a}t$ s.

References

[1] V. N. Bhātkhānde, Hindustānī Sangeet Paddhati, Hindi edition, Vol 1-4, Sangeet Kāryālaya, Hathras, India, 1951-57. [2] V. N. Bhātkhānde, Kramik Pustak Mālikā, Hindi edition, Vol 1-6, Sangeet Kāryālaya, Hathras, India, 1954-59. [3] Clifton Callender, Ian Quinn, and Dmitri Tymoczko. Generalized voice-leading spaces. Science, 320:346-348, 2008. [4] Norman Carey and David Clampitt. Aspects of well-formed scales. Mus. Theory Spectrum, 11 (2) 187-206, 1989. [5] Rachel Wells Hall. Geometrical models for modulation in Arab music. Preprint, 2009. [6] N. A. Jairazhboy, The Rags of North Indian Music - Their structure and evolution, Wesleyan University Press, Connecticut, 1971. [7] H. L. Roy, Problems of Hindustani Music, Calcutta, 1937.

Bhattacharya and Hall

Figure 4: North Indian $\text{thats}$ (left), Glarean modes (center), and Arab modes (right).

Figure 5: Rāg Āsāvarī and rāg Mālkāuns (graph generated by MATLAB).

Figure 6: Rāg Kedar and rāg Hamir (graph generated by MATLAB).