From the Merriam-Webster Online Dictionary definition of harmony: 2 a : the combination of simultaneous musical notes in a chord 2 b : the structure of music with respect to the composition and progression of chords 2 c : the science of the structure, relation, and progression of chords
These are all very good! However, they do not specify what that structure is & how it relates (i.e. “with respect to”).
The foundations of Tonal Harmonic Theory are based on voice leading & on triadic harmonies built from diatonic scales. I am going to assume the reader has some knowledge of music at this point. Harmonic structure, per se, has a beginning, a middle (some motion) & an end. The goal is the cadence.
The degree to which any piece (or part of a piece) serves to move toward it’s goal (or not) is the degree to which it has successfully integrated it’s harmonic content with it’s other components.
Side note: if a piece or theme does not have a proper cadence, it does not resolve. It is not necessarily “bad” to not resolve or avoid a proper cadence. Sometimes, if done effectively it can add a great deal of tension/suspense (among other possible effects). But the fact remains that if you are going to use tonal tools (scales, chords, etc.) you are by definition using an integrated system of information with a specific nature (i.e. there is a root note, there is a heirarchy of importance among the notes, the cadence is the goal, etc.).
The premise underlying all of harmony is voice leading. Voice leading is regarding all the notes that comprise a chord as one note of a “voice”. As you move to the next chord you regard each note as moving to the next corresponding note in that next chord. “Smooth” voice leading is the “goal” of well-constructed music.
Many theorists & composers also regard voice leading as not merely a chordal analysis tool but an essential way of creating independent & yet interweaving melodies. This is very true in Bach (& Beethoven!). For although at any time you can isolate a chord based on all the notes that are sounding at one time, the more important thing is that each voice carries it own melodic thread. So if you have (like Bach did in many of his classic 4 part chorales) a piece that all the way through uses four note chords it is constructed in a way that if you follow say the top note of every chord (“the top voice”) you will find a complete melody; & likewise for the other 3 voices.
There has been a great deal of nonsense spread about “the rules of music” (usually in regards to counterpoint & harmony in general) being “arbitrary human conventions”. Then, there is a long list of composers that “broke/bent/ignore/re-wrote the rules”. It does not help that some of the very specific classical rules of counterpoint voice leading are not essential to the nature of tonality. These exceptions are frequently held up as examples of why throwing out all the rules is a good idea. Most of the rules exist for a reason; some of them exist for very good reasons; some are non-essential. Beethoven wrote that all the traditional voice leading rules can be distilled into one: “Direct motion into perfect consonances is forbidden.” And even that is merely a convention tied to strict Fuxian contrapunctal writing and traditional classical voice leading.
For any individual with a serious desire to understand the mechanics of music, the study of contrapunctal motion is essential. This wiki site provides a good overview.
The fountainhead of this field is “Gradus ad Parnassum” by Johann Fux. You can find it in full text here.
I’ve worked through this book many times & rework it occasional.
I also find the works of Henrik Schenker very helpful in grasping general, natural principles of music theory. He was, to my knowledge, the first theorist to incorporate the horizontal aspect (melodic line) of music with the traditional vertical aspect (chord function and harmony).
This site is a great resource on Schenkerian Music Analysis.