Axis-Fan Pedagogy: A Tritone-First Harmony Sequence

0. Premise

Traditional harmony pedagogy introduces the tritone last: as the interval to resolve, the note in the scale you’re taught to handle carefully. This document specifies an inverted sequence, built directly from PPT’s own generative grammar (see Du-Fractal DuTri Closure), that introduces the tritone first — not as an exception to a scale, but as the founding structural fact a student encounters before any scale exists to measure it against.

The governing principle: expression before recitation. A student is not handed a pre-completed collection (a major scale) that already contains a built-in tonal hierarchy. Instead, the collection is built outward, in public, from a single axis — and the student is present for, and responsible for, every decision about where “home” is along the way.

1. The compressed operation

The entire sequence reduces to two operations, alternated once:

Draw an axis.
Fan it (dress both ends with the Tri operator, ±3/2 and ±4/3).
Draw the perpendicular axis (Du-fractal bisection of the arcs the first axis created).
Fan it.

On the pitch circle (1200¢ = 360°), the Do–Fi axis sits at 0°/180°. The Me–La axis, reached by bisecting each half of the first axis, sits at 90°/270° — a literal quarter-turn from the first. “Perpendicular” is exact, not metaphorical.

2. The four stages

Stage 1 — The Axis: Octave and Tritone

Only two pitch classes: Do (tonic) and Fi (tritone). With D as Do, this is D and Ab.

Introduces tension and duality without any scale context to resolve into.
Either point can “surrender” to reinforce the other as tonal centre — the student asserts the centre, the collection does not supply one.
On guitar: on any string-pair spaced a fourth apart, the tritone is a same-string 6-fret span, or a single-fret diagonal to the adjacent string. It also bisects the octave shape on a single string (6 frets to tritone, 12 to octave) — alternating pitch classes every 6 frets.
Practical effect: with only two notes and no scale to noodle inside, students are pushed toward register exploration (spread voicings, moving along the instrument) rather than staying clustered in one hand position — a scarcity effect, not a note-count effect. This addresses a common failure mode directly: guitarists who rarely leave open position, pianists who rarely venture beyond middle C. Abundance within easy reach removes the incentive to look further; scarcity restores it.

Stage 2 — Fan the Axis: Tri Family in Cast Space

DuTri-dress both Do and Fi: Fa/So flank Do (G, A against D); Ra/Ti flank Fi (Db, Eb against Ab — i.e., D moving to Db or Eb). Result: a six-note symmetric set, e.g. Db, D, Eb, G, Ab, A.

Three tritone pairs now available: (D, Ab), (Db, G), (Eb, A).
Non-tertial but genuinely consonant harmony: sus4 on the tonic (D-G-A), sus2 read from the subdominant (G-A-D — same three notes, different functional root).
Major thirds/minor sixths appear early via the Do/Fi-adjacent notes: A→C#, G→Eb.
Single-point tritone relaxation (move one note only) always lands on a P4 or P5, regardless of which note or direction moves.
Double-point relaxation (both notes move) always lands on a M3 or m6 — contracting motion gives M3, expanding motion gives m6, and the two are octave inversions of each other. The outcome interval is fully determined by the gesture (how many voices move, which direction), not by which specific pitches were chosen.
On guitar: same-fret, adjacent-string = P4 on every string pair tuned a fourth apart. This is the tuning interval itself, made audible as a harmonic move rather than a memorized shape (cf. the standard power chord, which requires a two-fret jump and a string skip).
One instrument fact worth teaching directly: standard guitar tuning is fourths between every adjacent string pair except G–B, which is a major third. Both the tritone diagonal and the P4 same-fret shape break at this seam, for the same underlying reason — a single fact about the instrument’s geometry, not two separate exceptions to memorize.

Stage 3 — The Perpendicular Axis: Me and La

A Du-fractal bisection (not Tri-dressing) of each existing half gives Me (300¢) and La (900¢) — with D as Do, this is F and B. Together with Do and Fi, these four points form the complete symmetric diminished-7th skeleton.

Where Stage 2 dressed existing anchors with Tri (3-limit) ratios, this stage instead locates the perpendicular axis itself, via Du-fractal bisection. The axis is found, not fanned — a distinct operation worth naming as such to students, since it is the pivot on which the whole four-stage sequence turns.
Mirrored intervals appear across the two Do/Fi-based clusters: F sits a M2 above Eb, m3 above D, M3 above Db; B sits a M2 above A, m3 above Ab, M3 above G — an identical interval fingerprint reflected across the tritone axis, forced by the symmetric construction rather than chosen.
With F and B added, the eight-note set (C#, D, Eb, F, G, Ab, A, B) contains five notes diatonic to C major (scale degrees 2, 4, 5, 6, 7) — a first taste of conventional tonal material, arrived at from the opposite direction to how it’s normally taught.

Stage 4 — Fan the Perpendicular Axis: Closure

DuTri-dress Me and La. Result: full 12-tone chromatic closure, with genuine major triads now available (e.g. G major, Db major — a tritone apart, mirroring the axis the whole sequence was built from).

On the interval from Do: the M2 reached at this stage (Ra, ~102¢ from Do when derived via Fi’s Tri-dressing) is not expressible as any simple JI ratio — see Du-Fractal DuTri Closure, §2, for the proof that this point is provably irrational relative to Do. This is a genuine departure point from JI-based teaching: by the time students reach this interval, the ratio-based framing they may know from elsewhere no longer applies, and the PPT-native description takes over.

3. Why constraint over completeness

A diatonic heptatonic scale contains exactly one tritone (between scale degrees 4 and 7), and that single asymmetry is most of what generates the scale’s sense of a unique tonal centre — the collection hands the performer a home before they’ve made a single choice.

The stage-2 hexatonic set contains three symmetrically-arranged tritones and no structural asymmetry favoring any one note as tonic. Home is not given; it must be asserted — through emphasis, rhythm, register, and repetition. This reframes “constraint” as a transfer of responsibility: the diatonic scale does some interpretive work for the student before they play a note; the axis-fan set does not, and pushes that work onto the performer from the first phrase.

This is also why the sequence resists “modal” framing in the conventional sense. Relative modes (shared collection, shifting tonic) are hard to teach because the reference point moves under a chord progression. The axis-fan set has no independent existence apart from the axis it was fanned from — there is no relative reading available even in principle, because the collection doesn’t exist without the anchor. Fixed-tonic-first is not a pedagogical choice layered on top; it is the only mode of existence the object has.

4. The core reframe

Conventional pedagogy treats the tritone as unstable relative to a scale that has already handed the student a tonic elsewhere — “avoid” only makes sense once a hierarchy (structural notes vs. ornamental ones) has already been accepted. This sequence removes the hierarchy at the outset: there is no scale-given tonic when the tritone is introduced, so it cannot be dissonant relative to anything. It is simply the first structural fact on the instrument.

The resulting frame for students: the tritone is not the note to avoid — it is the note to adjust, if you want release. Resolution becomes a choice of gesture (who moves, which direction) rather than a rule (resolve up, resolve down), and on guitar, a shape already latent in the instrument’s own tuning rather than an interval to fear crossing.