Tertian chords sound like they might be chords that reside on an alien planet somewhere in the universe. While the word is not typically used, tertian chords are the fundamental chords that are used by all popular american music and most music from around the world. Tertian harmony is so common that most musicians assume it is the only type of harmony in existence. Basic guitar and piano chords like C Major, G Major, D7, A minor, and others are all chords that can be grouped into the tertian chord classification. Tertian refers to the number 3 or a musical interval of a 3rd. An example of a 3rd would be playing A and C notes together. Other examples of thirds are C-E, D-F, E-G, F-A, and so on. Thirds can be either minor or major. A minor 3rd is fundamentally 2 notes spaced by 3 semitones (half steps) and a major 3rd is 2 notes spaced by 4 semitones (half steps). Major chords are chords consisting of 3 notes with the first note spaced a major 3rd from the next note. The second note is spaced a minor 3rd from the last not of the chord. Minor chords reverse the order of the 3rds from that of the major chord. The pattern of a minor chord is first the minor 3rd and then the major 3rd. Theoretically the difference is subtle but the aural effect is obvious. Major and minor chords are used together to create a large percentage of the harmonies that make up modern music.
So if tertian chords make up most of the chords played for modern music, what are some examples of non-tertian chords? Tone clusters are chords that use smaller intervals such as major and minor 2nds to make up chords. One example would be playing C, C#, and D all together as a chord. Experimenting with tone clusters and tertian chords and comparing them will certainly help you understand the function of each classification. Most people will experience the tertian chords as generally open and pleasing to the ears while tone cluster harmony will sound dissonant and generally unpleasing. Composers of classical, jazz, popular music, and other music have found uses for tonal cluster harmony. Quartal and Quintal harmonies involve using the interval of a fourth and fifth respectively to build chords. There is a question whether a chords built from fourths and fifths should be interpreted as a quartal-harmony structure, or if it is more meaningful to interpret is as part of the traditional functional (tertian) harmony. In any case, Quartal and Quintal harmonies have been found in Jazz, Classical and modern music.
If you are interested in learning more about the harmony and theory of music then check out Harmony & Theory by Kieth Wyatt.