Music & Science Key Center Perception and the Hexatonic Why hearing every note in relation to one fixed tonal center is more efficient than chord-by-chord interval calculation — and how to build that hearing from scratch. Bruce ArnoldMusic & Science There are two ways to hear a note in a piece of music. The first[…]

Music & Science How the Brain Hears Hexatonics The cognitive science of ambiguous tonality — why unresolved harmonic tension keeps the ear engaged, and what neuroscience says about learning to hear from one key center. Bruce ArnoldMusic & Science The ear is not a microphone. It does not passively record frequencies and deliver them to[…]

Music & Physics The Symmetrical Difference How the closest possible relationship between two scales — major and pentatonic a tritone away — becomes a compositional structure covering all 12 pitch classes. Bruce ArnoldMusic & Physics In set theory, the symmetrical difference of two sets is the collection of elements that belong to one set or[…]

Music & Physics Timbre as Trichord How the overtone profile of an instrument shapes which trichords feel natural on it — and why the guitar is uniquely suited to hexatonic pitch class set playing. Bruce ArnoldMusic & Physics The previous article established that the harmonic series is the physical source of musical intervals — that[…]

Music & Physics Overtones and the 026 Trichord Why a trichord built from the 9th and 11th harmonics hovers without resolving — and what the physics of vibrating strings has to do with modern jazz. Bruce ArnoldMusic & Physics Every musical note is a lie. When you hear a single pitch — a C on[…]

Music & Mathematics Symmetry in the 027 Trichord Why the quartal trichord is its own mirror image — and what that mathematical property means for harmony. Bruce ArnoldMusic & Mathematics Most trichords are not symmetrical. The chromatic trichord 012 (C, C#, D), when inverted, produces a different arrangement. Most trichords, when inverted, produce a distinct[…]

Music & Mathematics The 924 Hexatonics How a computer enumerated every possible six-note scale — and what musicians found when they looked at the results. Bruce ArnoldMusic & Mathematics The previous article established that there are exactly 50 distinct hexatonic prime forms — 50 families of six-note scales. But those 50 families generate a much[…]

Music & Mathematics Set Theory and the Hexatonic Scale How mathematics organizes every possible six-note scale into 50 families — and why composers have barely scratched the surface. Bruce ArnoldMusic & Mathematics In 1960, music theorist Allen Forte published a systematic classification of every possible collection of pitch classes using the mathematics of set theory.[…]

Music & Physics The Tritone and Standing Waves Why the most dissonant interval in Western music is physically unstable — and what happens when you pair it with itself. Bruce ArnoldMusic & Physics Every musical interval is a ratio. When you play two notes simultaneously, their sound waves either lock together cleanly or fight each[…]

Music & Physics The Number That Haunts Physics — and Found Its Way Into Music How a mysterious constant of nature became the sonic DNA of a jazz composition. Bruce Arnold Music & Physics · Music & Mathematics There is a number that appears everywhere in physics and comes from nowhere in mathematics. It has[…]