Rhythm

Tal
In Indian music, the rhythmic organisation is as important as the melodic aspects. Fundamental to the discourse on metric concepts is tal, the root meaning of which has to do with beating with the hands, the basic and original means of keeping control of musical time cycles (Powers [Groves Dictionary] 1990: 118). In the contemporary music tal has twofold meaning; system of rhythm as a whole and a specific metric cycle (Kippen [Garland Encyclopedia] 1999: 110). A tal is a framework which organises and governs rhythm and controls the temporal aspect of performance, both vocal or instrumental music and dance.
Any tal can be primarily described as comprising of certain number of time units known as matra-s (measure /count /beats). The term matra also denotes the duration between two beats (ibid 112). More importantly, a tal is further characterised by vibhag or khand  (sections into which  beats are grouped and stressed). Two tal-s may consist of equal number of matra-s, however they can be distinguished on the basis of vibhag-s. The complete cycle of the tal is called avartan or avart; the root meaning being “rotation”, highlighting its cyclic characteristic. The common North Indian tal-s have 6,7,8,10,12,14 or 16 beats. Tintal having 16 (4-4-4-4) beats is the most common tal.
The first beat of the tal is called sam. The process of improvisation may or may not start from sam but it invariably ends on this beat which is often accentuated. Whereas, the beat which serves as a counterpoint to sam is called khali ( lit. empty). Generally, khali  is in the middle of the tal cycle (except in case of Rupak where it is the very first beat). When the tal is marked by hand claps (which is common during teaching and composing but not in performance), the beginning of every vibhag is marked by clap while the khali is shown by wave of hand.
The identifiable and reproducible sound syllables of tabla (or any instrument for that matter) are known as bol, and articulation of tal using such recognisable drum patterns is known as theka. In other words, the tal is marked by a configurative drum pattern (theka) played on the tabla, which keeps repeating in a cyclical manner.

Thus the pattern for Tintal is:

Matra       1      2       3      4       5       6        7        8       9     10    11    12    13    14     15      16
               sam (clap)             clap                                    khali                       clap

Theka   dha  dhin  dhin  dha  dha  dhin  dhin  dha   dha tin  tin    ta    ta   dhin  dhin  dha
 

Tal-s used in dhrupad are: Chautal (12 beats), Dhamar (14 beats), Sultal (10 beats), Jhaptal (10 beats) and Tevra (7 beats). Khayal mainly uses tal-s like Ektal (12 beats), Tintal (16 beats), Tilvada (16 beats), Jhumra (14 beats), Jhaptal (10 beats) and Rupak (7 beats). Tal-s like Dipchandi (14 beats), Rupak (7 beats), Dadra (6 beats) and Keherva  (8 beats) are preferred in semi-classical genres of thumri and allied forms.

Whilst in dhrupad & dhamar the rhythmic accompaniment is provided by pakhavaj, which is a double-headed barrel drum, the tabla drum pair is widely used in khayal, thumri, instrumental music and kathak dance.
Lay 

Lay is a fundamental term for rhythm in the Western sense of “timing” (ibid 111). In its broad sense lay means ‘time’ of music. The word also has another meaning, “tempo” or the speed at which a composition is performed. In Indian music, lay is specified in relative terms: vilambit (slow), madhy (medium) and drut (fast). Normally, as the performance progresses, the speed of rendition goes on increasing; within a given composition and /or when there is a transition from one composition to the other.
Tihai
Tihai (from ti or tin meaning three) is a rhythmic or melodic cadence in which a phrase is repeated three times. It is a specific kind of improvisation that is used to conclude a melodic or rhythmic passage. It may also stand alone as a self-contained improvisation. In either case, it contains a motif played three times, usually creating a good deal of rhythmic tension that is resolved in a final articulation (ibid 200). Tihai generally ends immediately before the opening section (mukhda) of a composition or on the first beat of the tal cycle (sam).