In music theory, a harmonic function is based on three high-level categories. These categories are traditionally called tonic (T), subdominant (SD— also called predominant (PD)), and dominant (D). Each function has associated diatonic scale degrees and more or less perceived tension.
The tonic is the least dissonant state of diatonic harmony. Tonic chords feel as though they are at rest and complete. The tonic scale degrees are I (the strongest), iii, and vi. In the key of C, these would be CΔ7, E-7, and B-7.
The subdominant (sometimes also called predominant) has a medium level of dissonance. Subdominant chords feel as though they have a certain level of tension, but are not intolerably tense or dissonant. The subdominant scale degrees are IV and ii. In C, these are FΔ7and D-7.
The dominant has the maximum level of dissonance in the diatonic framework. This is due to the interval of a tritone between the 4th and 7th scale degrees (F and B in the key of C). Dominant chords feel like they need to resolve and have a high level of unrest. The dominant scale degrees are V and vii (some theorists do not include this chord). In C, these are G7 and Bo7.
SD — D — T
The most fundamental cadence in traditional, tonal jazz is the subdominant-dominant-tonic progression. This primarily takes the form ii-7, V7, IΔ7. This progression goes from a state of medium tension to maximum tension to rest. Functionally, the progressions IVΔ7, V7, IΔ7, and ii-7, V7, vi-7, as well as other instances of tonic (such as iii-7) will retain the same functional sound.
In this publication, these functions will be listed below the staff underneath the roman numeral analysis.
The following image lists the diatonic progression with function marked.