As a saxophonist, musical harmony is a subject I employ every day both in practice and performance. However, its relative insignificance in most formal syllabuses baffles me.
Improvisation I believe to be the most engaging and rewarding skill a saxophonist can acquire, and an understanding of musical harmony unlocks a player's potential to achieving this. The great players such as John Coltrane, Grover Washington Jr. and Bob Mintzer all employ harmony to great effect creating incredible and moving performances, one always notable in my mind being the performance of Flamenco Sketches from Miles Davis's album 'Kind of Blue'. This now begs the question, 'what is harmony?'
Harmony occurs whenever more than on note is played simultaneously and the resultant sound achieved. In its simplest form two notes are combined and this is harmonically described by an interval. This idea can be taken formed with the introduction of a third note to create simple chords and then resultant progressions. A n example of a well-recognized chord progression is a I IV V such as Gm Cm Dm7 seen in Shackles (Praise You) by Mary Mary.
These ideas clearly can be further extended and built upon. However, to what end? As a student I found the most daunting aspect of Harmony was its limitless possibilities. Where do I start? What should I learn first? There are many good starting points to go from, however in teaching harmony it is vital to teach the right approach as much as the specific detail. Each player has preferences to the harmonic ideas that they like and incorporate into their playing, the combination of which makes them sound like them. It is therefore important to teach students how to listen and identify harmonic ideas and create exercises to incorporate this into their playing.
Harmony ultimately is very stylistic to the genre you are studying and the preferences you have to the musicians playing in that genre. However, effective understanding of the subject will allow you to unlock ideas you like and bring them into your own sound. Ultimately, as time passes your improvisational style will become a combination of you own harmonic preferences, and the result will be that you will sound like you and no one else.