The tonic is the first note of a scale or mode, and typically serves as one, or often more, of the following— the primary tonal center, the main resolution tone, one of the most common initiating events of a piece, the most important melodic and or harmonic anchor in tonal music, the generator of other tones either harmonically or subharmonically, and or a discourager against the usage of other microtonally nearby pitches.
Given all this, it is the only function that is known to be universal when it comes to tonal music, with the various other functions being collectively defined as nontonic, thus, it should not come as a great surprise that the tonic exerts a very powerful influence on the context of functional harmony regardless of the nature of the tonal music system in question, even helping to define aspects of the other primary functions on the chord level as opposed to the root level.
Relationship to other tonal centers
In pieces that have more than one tonal center throughout their length, as with many Western Classical pieces, the tonic is the most important of the various tonal centers that appear in that piece. For that reason, it is considered the home note, and, as a consequence, is often referenced in the name of certain works, for example, Bach's Fugue in G Minor, BWV 578, otherwise known as the "Little Fugue", in which G is the Tonic.
As an initiating event, anchor, or resolution tone
In many Medieval, Neo-Medieval and Common Practice pieces, the Tonic is often the tone used for resolution in musical phrases, and, as a logical extension of this, can be considered the principal goal tone of a piece. This is especially known for being established through an Authentic cadence. In addition, it should be noted that other chords in diatonic harmony, namely the mediant and submediant, often have tonic-like function in the sense of being resolution tones, with the caveat that they are ultimately nontonic functions, and hence, such resolutions are considered to be either interrupted or deceptive cadences. In Common Practice music, most pieces began in the tonic key, and thus, the tonic can be considered a very common initiating event for a piece— a function which continues to be widespread in Western Classical music. As a consequence of its role as both a common initiating event and the main resolution tone, the tonic often serves as the single most important anchor for both melodies and harmonies in a tonal piece, and as a consequence, it is likely to occur very frequently, even if it is not the most commonly occurring functional element of a piece.
Relationship to other functions
Since the tonic is the generator of other tones through either the harmonic series or the subharmonic series, it is thus the origin and the definer of all other functions and can be considered the 1/1, and, in octave equivalent systems, 2/1. In such systems, the first 3-limit nontonic pitches in both the harmonic series and the subharmonic series give rise to the dominant and subdominant (or, alternatively, superdominant) harmonies, with the exact role of each 3-limit pitch being determined by whether you build your tonality from the low pitches to the high pitches, or from the high pitches to the low pitches—that is, your tonality's chosen direction of construction. From there, the next 3-limit pitches in both the harmonic series and the subharmonic series give rise to the supertonic and subtonic—with the exact functions of each of these also being determined by your chosen tonality's direction of construction—and the next three limit pitches after that in both the harmonic series and the subharmonic series give rise to variations of the mediant and submediant (or, alternatively, the supermediant) depending on your tonality's direction of construction.
In the 5-limit, the first pair of nontonic pitches in both the harmonic series and the subharmonic series give rise to the simplest variations of the mediant and contramediant, while the first combinations of 3-limit and 5-limit give rise to a set of leading tones— the exact role of each individual pitch is again determined by your chosen tonality's direction of construction. Between the 3-limit and the 5-limit, all diatonic functions are generated.
Of course, this same pattern of generation extends to the 7-limit, the 11-limit, the 13-limit, and so on, resulting in pardiatonic functions. As a result of all this giving rise to other functions, the tonic neutralizes the tension between the dominant and the subdominant (or superdominant) as well as their respective "parallels" and "counter parallels".
Repelling the competition
As first put succinctly Flora Canou, in the realm of microtonality, things repel the similar but not identical, and futhermore, a strong tonic admits no competition. As a consequence, the presence of a strong tonic acts as a powerful discouragement against the usage of other microtonally nearby pitches outside of modulation or deliberate dissonance, as well as shimmer and chorus effects. While this effect is not unique to the tonic, being shared by both the dominant and subdominant (or superdominant), it is most consistently and strongly present around the tonic itself, and, furthermore, is one of the main drivers behind the need to temper out and or fudge either small or unnoticeable commas.
- Analysis on the 13-limit just intonation space: episode ii. Flora Canou. Xenharmonic Wiki.