Harmony is two sounds or more produced [or perceived] simultaneously, or played in such a way as to give the impression of a unified sound.
In contemporary Western music, harmony is normally described based on musical notes or pitches and the relative spacing between the notes heard.
Any chord or harmonic interval can be represented by letters. Below follows my method of describing musical chords. They are meant to be easily read by most musicians with a reasonable level of music theory.
monads and dyads
One note [termed a monad] is represented by its name in lower case.
Here a random note is specified by r (although of course you could use any other letter that isn't a note name).
Harmonic intervals/dyads are represented as note 1 + note 2, with both note letter names in lower case.
Nothing is said about the spacing between the notes: so the two notes could be played in any octave or register. To specify that one note is to be played lower than the other, the forward slash is used and the dyad is written higher note/lower note.
Alternatively, the interval can be written in superscripted brackets after the lower note. The type of brackets you use is up to you.
Three-note chords (or triads) are based primarily on the semi-descriptive method currently used by many musicians.
Any triad can be represented by writing the tonic in lower-case and then writing the stacked intervals in brackets.
Major triads are shown by the root name in UPPER CASE. It says nothing about what order the notes should be played.
Minor triads have a lower-case m [in the United States, a minus sign (-)] written after the root.
Augmented triads employ an addition sign (+) after the root, or “aug”
Diminished triads have a superscripted circle (°) after the root, or “dim”
Suspended triads use “sus”, and then the number of the suspended interval:
To state an inversion, I put a superscripted number before the root:
A random chord is shown by R.
larger chords: adding extensions
I like to write any alterations/additions as superscript, but you may not. It's up to you. Just be consistent.
A major seventh chord is represented by writing maj7 after the root name.
The dominant seventh chord is shown by simply writing “7” after the main chord.
Added or altered intervals within an octave are written as numbers, superscripted after the root name and triad alteration; and are commonly written in parentheses when a seventh is present.
Larger intervals are always written in parentheses.
More intervals can be added by using commas after each number:
The diminished symbol for the triad is also used as the full diminished chord. But the half-diminished chord (“minor-seven, flat-five”) is denoted by a Ø after the root name:
Chords with an specified bass note: CHORD/bass note
Non-specific polychords: CHORD1 + CHORD2 + ...
Specific stacking of chords: HIGHER CHORD/LOWER CHORD/...
Various combinations are possible. But the aim is always to generate a chord description that can be read quickly.
Sometimes it is necessary to state an implied scale. This is achieved by writing the tonic of the scale in lower case and then an abbreviation of the scale type in superscript:
A scale can be altered by adding the affected intervals after the scale type in brackets.
Personal list of abbreviations
Of course, this list is neither exhaustive nor prescriptive. As long as you are clear and consistent, you can create your own.
Smaller tone clusters are adequately covered using the generic note[stacked intervals] approach:
If the cluster is a scale, then the scale is notated with braces around the scale, and the range of the cluster [in octaves] superscripted outside the parentheses:
Clusters, chords and notes can be then be combined in different ways, as you choose.