‘Thrice the brinded cat hath mew’d’ – Shakespeare: Macbeth (4.1.1548)
From the witches of Shakespeare’s Macbeth to Hollywood’s obsession with trilogies, the number ‘three’ seems to bring about a sense of satisfaction and balance in creativity through the ages. Do ‘patterns of three’ resonate with our thinking in some unique way, offering clues as to how our minds perceive and enjoy structure in art?
One very potent and conceptually intriguing example of how a ‘pattern of three’ can create artistic impact is the TIHAI (pronounced ‘thee-hi’), a rhythmic device that is ubiquitous in the rich tradition of North Indian (Hindustani) Classical music. Examples of tihais can vary from the very simple and innocuous to the incredibly long and complex, but what they all share is an ability to set up tension and provide catharsis upon resolution. A tihai that is both satisfying and exciting will reveal an elegant construction combining aesthetics and mathematics in good measure, perhaps akin to the ethos of Bach’s counterpoint – music at the intersection of art and science. The anatomy, psychology, and raison d’être of the tihai, as a key ingredient in Hindustani music, share much with ‘patterns of three’ in various other contexts, cultural and artistic. All are keys to how our minds enjoy the interplay of expectation and fulfillment, repetition and variation.
‘The cyclic form is a fundamental concept in Indian rhythm. The cadential philosophy of composition is very different from the cyclic. Where the cyclic form is characterized by a sense of balance and flow, the cadence is characterized by tension and imbalance. Such tension and imbalance naturally seeks a resolution.’ – David R. Courtney: Advanced Theory of Tabla (2000)
In Hindustani music, a tihai is a tripartite rhythmic cadence designed to end on a structurally important beat of the prevailing time cycle or tal (pronounced ‘thaal’). Literally translated as ‘three times’, a tihai is comprised of three phrases, almost always identical in length. It is the last beat of the final phrase that will fall, in the majority of cases, on sam (pronounced ‘sum’) which, as the first beat of the time cycle, is the most structurally important. A tihai may also be designed to end instead on a beat that is not sam, provided that context determines this beat to be structurally important in some other, significant way. An example of this would be the first beat of a composed melody known as the gat (instrumental contexts, e.g. featuring a sitar) or bandish (vocal contexts).
ANATOMY – two ways to deconstruct tihais
To begin with, below is an example of a tihai in perhaps its most basic form, pared down to bare essentials. Please click to enlarge any example on this page. A quick word about notation: Indian time cycles aside for one moment, I have chosen Western notation in order to make these examples accessible to Western musicians, and readers familiar with this system. Unsurprisingly, these are in 4/4: the ‘C major of time signatures'(!). However, I also use the syllables of text known as ‘bols’ (literally ‘words’), that come from the extensive drum language of tabla. The vocalization of these bols ‘are done in resemblance to the sounds produced from the drums’ (Chatterjee, 2006). I find that tabla bols allow for an expressivity that fuses the rhythmic with the lyrical – and all within a linguistic context. I hope that my style of notation, combining aspects of both East and West, will present the examples as effectively as possible. I recommend reciting the bols out loud while tapping or clapping the quarter note pulse:
Ex.1: tha dha – tha dha – tha DHA
This tihai begins on the second 8th pulse of the first bar, and ends on the most obvious structurally important beat, which is the downbeat of the next bar. The tripartite construction is evident; the brackets indicate that the motif ‘x’ occurs thrice, separated by the rest ‘y’ twice. The motif ‘x’, then, may be understood as the durational quantity that is one phrase, or palla, of the tihai, while ‘y’ is the durational quantity of silence, or the ‘gap’ that separates the three phrases from each other. However, consider the following alternative to labelling the tihai components:
There are two important differences between examples 1 and 2: firstly, the last pulse of what was ‘x’ is now part of ‘y’. Consequently, it is perhaps easier for us now to think not in terms of the alternation of phrases and gaps, or sounds and silences, but simply in terms of ‘x’s and ‘y’s, where both quantities may be composed of whatsoever. The second important difference between the examples is that the final ‘dha’ has NOT been included or considered as part of the whole tihai. This might be confusing right now, but as tihais (and their calculations thereof) typically become much more elaborate than our user-friendly specimen above, the exclusion of the final pulse makes for much cleaner calculations! Suresh Talwakar, one of today’s great tabla masters, uses this method for his tihai calculations (Hanley, interview 2013).
TINKERING WITH DENSITY
To create more interesting variations on the very basic tihai of example 1, without interfering with its proportions, you could simply increase the density of the content of ‘x’. For instance, with two pulses instead of one:
Ex.3: thake dha – thake dha – thake DHA
Doubling that density again could give you:
Ex.4: thirikite dha – thirikite dha – thirikite DHA
A very basic tihai soon develops a virtuosic appeal in this manner. Acknowledging that density is a parameter that may be manipulated for such effect, you can then turn to the proportional relationship of ‘x’ and ‘y’. By varying ‘y’ while keeping ‘x’ constant, you could create:
Ex.5: thirikite dha thirikite dha thirikite DHA
By varying ‘x’ while keeping ‘y’ constant, you could create the example below. I am using accents and slurs to indicate some aesthetic features of phrasing and articulation:
Ex.6: thi ri ki te dha – thi ri ki te dha -thi ri ki te DHA
This last variation is longer in duration and ‘takes up more space’ within our 4/4 context, requiring two bars (plus one pulse) to complete itself. Using the 1/8th note pulse as the unit of measurement, two bars of 4/4 contain 16 pulses in total. This would be an appropriate segue to thinking now in terms of Indian tala, since the most popular taal in Hindustani music is tintaal (pronounced ‘THEEN-thaal’), a time cycle of 16 beats. The tihai of ex.6 fits neatly between the downbeats of the first bar and the third bar, or correspondingly from sam to sam of one cycle of tintaal.
A BIT OF ALGEBRA
The length of a whole tihai may be expressed as 3x + 2y. For ex.6, the values of ‘x’ and ‘y’ would be 4 and 2 respectively. The calculations for ex.6 would then be as follows:
3x + 2y = 12 + 4 = 16
The length of the tihai in ex.6, therefore, is 16 units, or one time cycle long.
APPEAL & APPRECIATION
‘Informed audiences frequently offer audible appreciation for an ingenious tihai as the melody and rhythm players ‘cadence’ together. It is not unlike the feeling experienced when a trapeze artist makes a ‘catch’ after an elaborate ‘fly’. In fact, musicians can execute the counterpart of a triple somersault […] by inserting a tihai at the appropriate moment. The tihai, then, brings the previous extemporized passage[s] to a satisfying – and often surprising – climax.’ – Thom Lipiczky (1985)
The interactions and audible appreciation of an informed audience during a concert of North Indian Classical music are on joyous display at the 7’30” mark in the following clip, at which point Suresh Talwakar presents, in recitation, an extended and very elaborate form of tihai known as a ‘chakradhar’:
At 7’42” we see that a gentleman in the front row is ‘keeping taal’ with his right hand, which means that he is conscious of the time cycle (tintaal, in this case) as well as how it interacts with the recited composition. At 8’10” (a moment I personally never get tired of delighting in) we see the audience smile, cheer and applaud upon the cathartic resolution of what was a very long and virtuosically executed tihai.
PSYCHOLOGY & AESTHETICS
The sense of release is due to several factors which work in conjunction, a fact which takes us further into what it is specifically about tihais that empower them to elicit such reactions. Firstly (and as observed by Dr. Chintamani Rath in his excellent article of 1989: ‘The Grammar of the Tihai’), the structure of a tihai is such that it comprises sequential components. The very fact that the sequence of phrases is three-fold is one that has long sparked my interest in the tihai, on a psychological level. It is worth considering why, before going any further by delving into the aesthetics and technicalities of how tihai pallas interact within their context. Why is the sequence not two-fold or four-fold? The answer, I believe, is something that burrows deep into the human consciousness.
‘Omne trium perfectum’ (all things that come in three are perfect) – anonymous latin proverb
We perceive time in terms of past, present, and future; perceiving the events of our lives as having a beginning, a middle and an end, or perhaps as ‘being’, ‘becoming’, and eventually ‘disappearing’. In the physical world, we perceive the three-dimensional geometry of objects and space in terms of length, width and height (or depth). We triangulate our positions according to latitude, longitude and altitude (or depth). In literature, from the Hindu Trimurthi and the Buddhist Triple Gem to the Christian Trinity, religious texts the world over abound with symbols grouped in threes. There is much in art through the ages too: from the theatre’s three-act structures of setup, confrontation, and resolution; to musical sonata forms that breakdown into expositions, developments, and recapitulations. We have already met the three witches of ‘the Scottish play’. For those familiar with the canon of Western Classical music, perhaps the example that comes immediately to mind as the ultimate semiotic celebration of ‘all things three’ is Mozart’s operatic masterpiece The Magic Flute, imbued as it is with the solemn symbology of freemasonry. Opening with ‘three knocks at the temple door’ in the form of three weighty and majestic chords in the key of E-flat major (a three-flat key signature), we go on to discover that there are actually three temples, and that the opera’s cast includes three ladies, three priests, three boys and three slaves. Characters in many fairytales stupidly squander two wishes before using their third wish wisely, and there are seemingly no end of jokes that begin predictably with the formula: “An X, a Y, and a Z walk into a bar”.
I do not believe, however, that we appreciate these ‘patterns of three’ in music and art simply because we have been conditioned over time by culture or by the civilizations of antiquity. I believe that there is a linguistic process at work here, and therefore an indication of how a three-fold occurrence is satisfying in a temporal sense. It brings to mind the following observation:
‘A single occurrence is of no significance. A repetition is noticeable, but might easily be the result of coincidence. A third occurrence of the same nature gives the event the impress of law.’ – Vincent Foster Hopper (1938)
While this could be a plausible explanation for why the mind finds a three-fold occurrence (in a temporal context such as poetry, music, theatre, cinema etc.) so satisfying, it assumes that the notion of any kind of ‘law at work’ may not be evident to us upon hearing the second occurrence. Certainly, in the case of many examples of tihais (even those with the simplicity of ex.6) our minds have already perceived a pattern after being exposed to part two of a tripartite sequence. A neuroscientific investigation of this phenomenon is beyond the scope of this essay, but suffice it to say that any comedic writer worth his/her salt will argue for the effectiveness of: saying something, saying it again, and then toying with their audience’s expectations. Twice is usually enough to establish a pattern, while thrice is the minimum that allows for a successful denouement, be it expected or surprising. David Huron, in his book ‘Sweet Anticipation’, categorizes this particular type of surprise as ‘dynamic surprise’ (the term itself could be misunderstood, but his definition is very targeted) – “when the work [or content] itself will set up some work [content]-specific expectation that is then violated.” My interest in the notion of ‘why tihai?’ has led me to analyze some other examples of the ‘psychology of three’, in other artistic contexts.
I find that once the ear and mind have become attuned to the sound and psychology of a tihai, ‘patterns of three’ in other contexts seem to reveal themselves more readily.
Ex.7 – Schubert: Symphony no.9, mvt.1
This threefold passage from the first movement of Schubert’s 9th Symphony D.944 (nicknamed the ‘Great’) is timely in that its cadential quality is put to good use in bringing the exposition to a close. By prolonging a cadential 6/4 progression for twelve bars in three sets of four-bar phrases, the purpose of this passage is to prolong a resolution to the key of G major, which achieves full stability in bar 13, despite the occasional hints in bars 5 and 9. The ear is immediately drawn to the contour of the melody as its basic shape is established in the first phrase (bars 1-4) and then varied twice to create the second and third phrases (b.5-8 and 9-12). The voice leading is smooth and step-wise in the first phrase, but in the second embellished with a small leap (the E in b.6, which acts as an appoggiatura to the D). The third phrase then surprises by beginning with several leaps (b.9-10). The high G in b.10, which creates an exciting ‘spike’ in the contour, is particularly satisfying in that is the tonic pitch. Consequently the whole passage experiences a final surge of energy as it alters the melodic contour beyond linear expectations. One might say ‘exponential’ instead. Place the ‘shapes’ of each phrase side by side and the psychology of this particular sequence perhaps easier to see:
The sense of delayed gratification is managed with skill and subtlety through manipulation of melodic and harmonic parameters alone (it is significant that there is virtually no change in orchestration throughout). The bassline of b.1-4 is also varied twice, discretely becoming increasingly active in b.5 and then in b.9-10 for an arpeggiated run of quarter notes. There is significance however in the composer’s dynamic indications: fortissimo having been established in b.1, he uses hairpins to indicate a ‘swell’, emphasizing the melodic/harmonic variations of the second phrase, but only for the third phrase indicates a fortississimo. Clearly there is psychology behind such choices, and this is what sparks my interest in the detail of this material. This passage is, of course, not a tihai, but I hope that I have elucidated how the psychology of its tripartite construction is undeniably clear both in concept and communication.
MORE PSYCHOLOGY & AESTHETICS
Schubertian sequences aside, tihais create tension and release for reasons beyond their ‘psychology of three’. Consider the following example for a phrase, from which we could construct a tihai:
We could then decide on the following for proportions of x and y:
The calculations would be as follows, taking the 16th note as our basic unit of pulse:
x = 8, y = 2
3x + 2y = 24 + 4 = 28
The length of the whole tihai therefore would be 28 units of 16th notes long, or 7 quarter notes long if we were to group the pulses in fours. In Western notation this would be:
In the context of Hindustani music, this could correspond to rupaktaal or a time cycle of 7 beats.
What the notation reveals is the periodicity of the slurred phrases, or the grouping of the 16th note pulses, against that of the time cycle. The numbers underneath indicate the time cycle in quarter note pulses. But the phrases by themselves suggest a different periodicity altogether:
Put the two together, and tension is created by the ensuing cross-rhythms: in this case the time signatures of 7/4 and 5/8 acting against each other.
Another important aesthetic feature concerning tension and resolution is particular to tihais of this type, in which the middle phrase is ‘placed’ differently with respect to the pulse of the time cycle, or to the overriding perceived pulse. Refer back to ex.11, and this will become apparent as you tap the quarter-note pulse while reciting the tihai. The first palla begins on beat 1 of the cycle, the second palla halfway in between beats 3 and 4, and the third palla begins on beat 6. In even simpler terms, the first tihai phrase sits ON the beat, the second lies OFF the beat, and the final phrase sits back ON the beat. This type of ON-OFF-ON construction is very desirable in tihais. When we experience the second phrase of such a tihai, we are having to process the repetition of material we have immediate memory of (from the first phrase), only now with the considerable tension created by its displacement against a steady pulse, or point of reference. An incredibly memorable quote of Suresh Talwakar‘s that has remained with me, thanks to tabla player Ed Hanley, is that ‘tabla music is reference music’. His simply stated but profound insight is this: without the element of a fixed point of reference (the time cycle, in this case), there would be a whole dimension of aesthetic appreciation missing. We only feel ‘displaced’ after first having an awareness of where and how we felt comfortable!
MORE COMPLEX EXAMPLES
A chakradhar (trans. ‘wheel’/‘circle’) is an example of a more elaborate, extended and exciting form of tihai. While a chakradhar may adopt a cadential function, its appeal as a much ‘grander’ form of tihai will allows it stand alone as a compositional entity, typically in the context of a tabla solo recital (a tabla soloist will most likely present a variety of chakradhars to precede only the virtuosic climax and conclusion of such a recital). The ‘grandeur’ is perhaps evident, for one, in that each of the chakradhar’s three parts itself contains a tihai. But this is not to say that each part is merely a tihai (although it can be, in some simpler examples which are less aesthetic and therefore rare). It is also likely that a chakradhar would be recited to the audience before being played, in order for its complexity to be best understood, and for maximum dramatic impact. One of my favourite examples is from the undisputed tabla master Zakir Hussain’s legendary Ahmedabad recital of 1992:
When discussing what is essentially a tihai nested within another tihai, our terminology may become somewhat confusing at times! I prefer to first analyze one ‘part’ of the chakradhar, which I have transcribed below:
Deciding on the lengths of ‘x’ and ‘y’:
‘x’ is very long in this case! ‘y’, on the other hand, is only one 8th note in duration. Consequently, part 2 of this chakradhar follows immediately and with no silence in between, ‘hot on the heels’ of part 1, hence the apt categorization as ‘bedam’ tihai (‘without breath’, or ‘breathless’). You can in fact hear Zakir Hussain announcing ‘bedam’ to the audience before launching into his recitation.
In analyzing a chakradhar, I think of each part as having a ‘head’ and a ‘tail’ (perhaps a propos if you enjoy the metaphor of a tihai as a small creature that escapes into the wild, only to return later as a beast of a chakradhar that has grown to epic proportions!). I have added some slurs below to indicate possible phrasing and expression. The ‘tail’ is, in fact, a smaller embedded tihai:
The structure of the whole chakradhar may then be expressed in a manner which will hopefully make clear the 3 ‘nested’ tihais within the larger tripartite composition; in essence:
[ HEAD, TAIL (tihai palla 1, palla 2, palla 3) ]
[ HEAD, TAIL (palla 1, palla 2, palla 3) ]
[ HEAD, TAIL (palla 1, palla 2, palla 3) ]
A full transcription follows below. On the top stave is an approximate transcription of the ‘nagma’ or ‘lehra’, the cyclic melody which outlines the tintaal (16 beat) time cycle. In the audio clip, it is performed on sarangi by the late Sultan Khan, one of the finest exponents of his instrument. I have used arrows and highlights to indicate the beginnings of each part of the chakradhar:
I mentioned tihais of this ON-OFF-ON variety before. What makes this particular bedam chakradhar so challenging to perform is the fact that part 2 sits ‘off the beat’, that is, the middle part of the entire composition is displaced from the quarter note pulse by the unit of an 8th note. And as each part is very long, it accounts for the huge tension that is generated, for both performer and listener. Virtuosity not being in question, the middle part is by far the hardest to execute without ‘slipping off’ by accident!
Given this displacement of an 8th note, it would be logical, then, to take the 8th note pulse as our basic unit of calculation. The structure works mathematically as follows:
x = 42, y = 1
3x + 2y = 126 + 2 = 128
The chakradhar is therefore 128 units (of 8th notes) long, and confirms (when you divide 128 by 8) that it would be 16 bars long when notated in a time signature of 4/4.
Incidentally, note that at this tempo there is very little difference between the phrases ‘dheredhere kitethake thakite’ and ‘dhinna kitethake thakite’. One of the features that makes this particular chakradhar interesting is the fact that the phrase which immediately precedes the first palla of the tihai is almost indistinguishable from the palla itself. No doubt, this was designed to create a beautiful ambiguity.
I have sat in rehearsals with some of the great tabla players of the world, watching them silently work out elaborate tihais in their minds and on their fingers. Whenever I asked them to explain their workings, there would be no end to their verbosity. After all, as a composed (or improvised) creative device that is culture-specific, the tihai is very sophisticated. It is an indispensable element in Indian Classical music. ‘Patterns of three’ abound in the music of Brahms but, his predilection for hemiolas notwithstanding, Indian tihais are unique in the way they strive to balance mathematics and aesthetics, regardless of their size or complexity. In this sense, I find that they can possess their own particular ‘magic’.
In studying tihais over the years, even designing my own, I have certainly become more acutely aware of the plethora of ternary structures in various other forms of creativity. These structures are certainly not in short supply in the art and culture of ages; we have already discussed the ‘psychology of three’. For me, the personal relevance of tihais has been the chance to enjoy a greater understanding of the phenomenological effects of other three-fold patterns, in all manner of creative work: for instance, ‘why exactly is the last phrase of that climax in the Schubert symphony so effective?’; ‘why does that last line of Martin Luther King Jr.’s ‘I have a dream’ speech rouse us so? (“Free at last, free at last, great God a-mighty, we are free at last!”); ‘what makes the banana gag in Buster Keaton’s short film The High Sign (1921) funny?’ (he replaces a cop’s gun with a banana, the cop draws the banana on a bad guy but, when its peel is tossed on the ground, Buster fails to slip on it?! Surely, not what we were expecting)*.
Whether or not the design of tihais prescribes to some sort of Hegelian triad of ‘thesis, antithesis, synthesis’, there is more to a satisfying tihai than just its sequential aspect. The tihai goes beyond the sequential, exploring its three-fold musico-linguistic nature in relation to a point of reference – they are context-specific in that they depend upon the ‘time cycle’, taal, to be perceived and appreciated in all respects. However you choose to perceive the taal – be it a ‘grid’ or rhythmic ‘matrix’ – a tihai in this context can create tension and release by using displacement, or by suggesting its own periodicity. Our minds thrive on the reference points, so as to savour the effects of whatever works in opposition. Once the existing status quo has been established, a tihai or any other ‘pattern of three’ has the power to surprise us.
Samir Chatterjee – A Study of Tabla (Chhandayan 2006)
Neil Sorrell & Ram Narayan – Indian Music in Performance (Manchester University Press 1980)
Martin Clayton – Time in Indian Music: Rhythm, Metre, and Form in North Indian Rāg Performance (Oxford University Press 2000)
Robert S. Gottlieb – The Major Traditions of North Indian Tabla Drumming (Musikverlag Emil Katzbilcher 1977)
David R. Courtney – Advanced Theory of Tabla, Volume 2 (Sur Sangeet Services 2000)
Personal interview with Ed Hanley, a Toronto-based tabla player, 2013
*Personal interview with Binnie Brennan, Nova Scotia-based author of Like Any Other Monday (Gaspereau Press 2014), a fictional portrait of Buster Keaton
Vincent Foster Hopper – Medieval Number Symbolism: Its Sources, Meaning, and Influence on Thought & Expression (Columbia University Press 1938)
Annemarie Schimmel – The Mystery of Numbers (Oxford University Press 1994)
Thom Lipiczky – Tihai Formulas and the Fusion of “Composition” and “Improvisation” in North Indian Music (The Musical Quarterly, Vol. 71, No. 2, pp. 157-171, Oxford University Press 1985)
Dr. Chintamani Rath – A Grammar of the Tihai (Ragaculture website 1989)
David Huron – Sweet Anticipation: Music and the Psychology of Expectation (MIT Press 2006)