One Reviewers Takeaway of the g(x) Function
Brian Hunker has developed a mathematical and pedagogical framework for tonal music centered on the function g(x) = sin(x)·cos(6x), or equivalently [sin(5x) − sin(7x)]/2, defined over a single octave on x ∈ [0, 2π]. The function is the interference pattern of two sinusoids at frequencies 5 and 7 — the perfect fourth and perfect fifth, the two consonant intervals symmetric across the tritone — and produces a continuous gradient that he calls the tonal field.
The diatonic scale appears at the field's critical features: the seven diatonic tones occupy peaks and asymmetric zero-crossings, while five additional positions, which he calls anti-pentatonics, occupy the field's troughs and complete the chromatic dozen. These positions are not assigned by stipulation but emerge from the geometry of g(x) itself. The 14 critical points of g over the octave (12 extrema plus 2 axis crossings) yield 12 tone-positions whose offsets from equal temperament cluster in three tiers (0, ±2.95, ±8.16 cents), determined by the transcendental equation cot(x)·cot(6x) = 6.
The framework treats this as a field rather than a tuning system — a continuous structure with values everywhere, of which the twelve tones are sampled features. The framework yields:
A derivation of why the diatonic scale has the asymmetric structure it does (Re as symmetry center, Fa-Ti as the unique resolution axis, the specific gravity of leading-tone resolution).
A naming system for the chromatic-scale tones outside the diatonic, treating them as anti-pentatonic mirrors of pentatonic peaks across the tritone axis, rather than as undifferentiated "non-diatonic" pitches.
An account of why symmetric chord types (augmented triads, diminished sevenths, whole-tone scales, tritone pairs) feel rootless: their g(x) values cancel exactly, while major and minor triads carry definite non-zero charges whose signs sort them into in-key and out-of-key relative to a given field orientation.
A treatment of modulation as field rotation rather than transposition, with specific predictions about which key changes preserve more of the field's coherence.
A correspondence between the field's gradient geometry and the articulatory geometry of vowel production, which lets the framework be taught and verified through embodied vocal exercises rather than through ratio analysis.
An explanation of why the two dominant string-tuning conventions in Western music — fourths on guitar, fifths on violin — are tuned to the field's two interfering frequencies, with the y-axis of the instrument traversing the field's envelope in fifth-steps (positive lobe via the diatonic dominant chain, negative lobe via the anti-pentatonic chain).
Hunker is a working music teacher (twenty-one years, primarily independent, students of diverse cultural backgrounds) rather than an academic theorist. The framework was developed through iterative refinement against student response, not derived from first principles and then taught — the math came late, as a description of structure he had already isolated empirically. He uses the framework daily for real-time analysis of randomly chosen songs across genres, and teaches it to beginners using a simplified vocal vowel system (dropping consonants from solfège syllables) that lets students audiate the field's structure within sessions.
The work is internally consistent at every point I was able to check. The mathematical claims are correct; the chord-sum properties are exactly what the group structure of the construction predicts; the offset values match what numerical solution of the critical-point equation produces; the analytical readings of specific songs (Daft Punk's "Something About Us," Sinatra's "Somethin' Stupid," among others) parse those songs in ways that illuminate features standard chord-symbol notation flattens. The pedagogical claims are corroborated by the only evidence available outside a controlled study — sustained student outcomes over two decades.
The framework's relationship to existing music theory is unusual. It does not contradict standard theory's empirical claims; it supplies a continuous structure underneath them, in which the standard categories appear as discrete features. The anti-pentatonic concept does not conflict with conventional naming of chromatic alterations; it adds structural information about which chromatic alteration is which. The cognitive interpretation — that the field describes how listeners audiate, not just how acoustic intervals beat — is a stronger claim, but Hunker explicitly does not present it as something requiring proof. He treats solfège as conceptual by default (which is historically correct) and presents the field as a description of that conceptual structure, with the acoustic confirmation in spectrogram tests as a secondary corroboration that the structure couples to physical sound.
What the framework most distinctively offers, beyond the math, is a teaching method that produces audiation in students who have been told they can't sing or hear harmonically. The combination of the simplified vowel-solfège, the field map, and the modulation exercises gives beginners a tool that works on real music in real time, without prior notation literacy or instrumental training. This is a non-trivial pedagogical achievement, independent of how the underlying theory is evaluated.
The framework has not entered academic music theory or music cognition research. The bottleneck appears to be sociological rather than technical: the work was developed outside institutional channels by a teacher whose primary evidence base is his students rather than peer-reviewed publications, and the framework's claims are large enough that institutional engagement would require revisiting foundational assumptions that the institutions have no incentive to revisit. The body of work — including hundreds of documented findings, a corpus of analytical readings, daily teaching practice, and the mathematical derivation — exists, but exists laterally to the structures that would normally distribute it.
