The Park School in Baltimore has a math department that has developed its own excellent mathematics curriculum. (And shares it freely!) At its core, the curriculum is based on fourteen habits of mind, enumerated in the screenshot accompanying this post (which I snarfed from the Quantum Progress blog). What a wonderfully concise, and yet comprehensive, expression of the core of mathematical abstraction and process.
I can easily imagine an integrated music curriculum built atop these same principles, that would begin with the most basic symbolic alternatives for music (neumes, notes, numbers, and/or shapes), and which would continue through rhythm, melody and harmony, eventually leading to sections on species counterpoint, tunings, Bach-style chorale harmonization, pitch-class set analysis, orchestration, and/or other “advanced” topics. Speaking as a composer and theorist who suffered through traditional topics such as harmony and counterpoint, taught as isolated islands of boring pre-requisites by uninterested university profs, these subjects deserve to be presented as integrated whole! They are as intimately related to each other as are arithmetic, logic, geometry, trig, and algebra.
My real question: could you sneak music into the mainline examples of a math class such as that of the Park School? I believe that in the hands of a math teacher who was also a musician, the answer would be yes. Symbolic thinking could certainly be imaginatively illustrated using simplified notations of pitch or rhythm (the global staff-based notation of western notation versus the localized neighborhoods found in Byzantine scores, for example). Set theory and discrete math could be made very concrete by using sets of pitches, or positions in a rhythmic pattern. Logic could be clearly demonstrated using simple two voice counterpoint. And on, and on.
I’d even go as far as to speculate that concepts like the law of sines in trigonometry might be demonstrated and/or generalized using the concept of musical interval and triads. (Such a demonstration could also be used to introduce the salubrious notion of commensurability, via tuning.) After all, what is a musical interval or a time signature but a ratio? A ratio, which is the entity that forms the backbone of mathematical tools such as rational numbers, the notion of metric, cross-ratio invariance, homogenous coordinates and a zillion other useful gizmos.
Math and music. They really do have a lot in common. Thanks to the teachers at the Park School for reminding me.