Piano Triangle Notation

Status

This is an experimental, actively developing notation layer within Prime Period Theory. It explores physical keyboard topography and hand-shape ergonomics rather than replacing the abstract algebraic layers of the framework. Several design areas — including non-heptatonic set encodings, multi-octave skip markers, and chord shape taxonomies — remain open questions under active investigation.

The problem this solves

The standard 12TET piano keyboard is fundamentally non-isomorphic: its black-key layout alternates between groups of two and three keys across the twelve-tone octave. Consequently, no single geometric glyph shape can represent all twelve pitch classes isomorphically while remaining legible at a glance regarding which specific keys are black or white under the hand.

Uniform Solfège addresses the algebraic and intervallic challenge by rotating a single glyph family across four quadrants, ensuring strict geometric isomorphism across pitch, rhythm, and timbre.

Piano Triangle Notation solves a complementary problem: piano-hand legibility and physical topography. Where Uniform Solfège trades immediate physical legibility for structural isomorphism, Piano Triangle Notation trades isomorphism for direct, tactile legibility on the keyboard. These are two distinct tools designed for different purposes, sitting side by side within the broader PPT framework.

The four triangles

The twelve chromatic pitches from C# to C are partitioned into four consecutive three-key groups, anchored centrally around D as the primary tonic axis (as developed in Default Do (12TET Keyboard)):

Name	Alias	Pitches	Geometric Shape	Apex Direction
Down	D	C#, D, D#	Inverted equilateral triangle	Points down (into the central white key)
Left	L	E, F, F#	Right triangle (right angle on right)	Points up
Up	U	G, G#, A	Equilateral triangle	Points up
Right	R	A#, B, C	Right triangle (right angle on left)	Points up

Geometric and topological rationale

The geometry of these four groups reflects the physical topography of the keyboard and its mapping onto the chromatic pitch circle:

Equilateral axes (Down and Up): The Down (C#, D, D#) and Up (G, G#, A) triangles sit on the vertical axis of the chromatic circle (see Chromatic Clock Geometry). On the physical keybed, each consists of one raised black key symmetrically flanked by two flat white keys.
Truncated right triangles (Left and Right): Left (E, F, F#) and Right (A#, B, C) occupy the horizontal positions. On the piano, E–F and B–C are the two locations where no black key intervenes. The black-key-flanked-by-white symmetry is truncated at these seams; Left and Right are geometrically “half” of what would otherwise be equilateral clusters.
Continuous zigzag contour: The hypotenuse of each right triangle runs at the exact same slope as the adjacent edge of the neighbouring equilateral triangle. Across the entire twelve-tone octave, the four-triangle chain traces a single continuous zigzag contour across the keybed tops.
Apex orientation: The apex of each triangle corresponds to the raised black key (or the truncated black-key corner). “Up” and “Down” designate orientation along the pitch clock’s primary vertical axis.

Prior art: US Patent US4054079

In 1976, US Patent US4054079 (“Keyboard and Notation System”, invented by Clinton S. Beman, filed as a continuation of abandoned 1975 application Ser. No. 557,592) independently proposed grouping the twelve piano keys into four symmetrical three-key groups (each comprising one raised key flanked by two flat keys).

While addressing the same fundamental asymmetry of the traditional keyboard, there are fundamental structural differences:

Physical re-engineering vs notational overlay: US Patent US4054079 redesigns the physical instrument by manufacturing custom, uniform-width keys. Piano Triangle Notation requires no physical modification, functioning as an ergonomic mental map and notational layer on standard keyboards.
C-anchor vs D-anchor / tritone symmetry: The 1976 patent anchors its groups to C for traditional staff-notation compatibility. Piano Triangle Notation anchors centrally to D and the vertical tritone axis, integrating directly with PPT’s tetrachord-pair generation and generative grammar.

Point numbering

Each triangle contains three points, numbered 1, 2, and 3 in ascending pitch order within that triangle:

Triangle	Point 1	Point 2	Point 3
Down (D)	C#	D	D#
Left (L)	E	F	F#
Up (U)	G	G#	A
Right (R)	A#	B	C

A single note is written as the triangle letter followed by the point digit (e.g., D2 = D, U1 = G, R3 = C).
A cluster of notes within the same triangle is written as the triangle letter followed by concatenated digits in ascending pitch order (e.g., D12 = C# + D, L13 = E + F#).

Scale encoding: tetrachord chaining around the tonic

Scale encoding directly renders the principles established in Tetrachord-Pair Generation of Heptatonic Scales. Rather than listing pitches as an arbitrary set, a heptatonic scale is constructed by chaining the dominant tetrachord into the tonic tetrachord around the tonal centre:

[5, 6, 7] + [1] + [2, 3, 4]

To encode a scale:

Write the scale in tonic-centred order: [5, 6, 7] (dominant fragment), [1] (tonic), and [2, 3, 4] (tonic fragment).
Transcribe each note into its triangle-point representation.
Merge adjacent notes that fall within the same triangle into a single concatenated segment.

Worked examples

D Major

Pitches: D (1), E (2), F# (3), G (4), A (5), B (6), C# (7)
Tonic-centred sequence: [A, B, C#] + [D] + [E, F#, G]
Unmerged segments: A (U3), B (R2), C# (D1), D (D2), E (L1), F# (L3), G (U1)
Merged encoding: U3R2D12L13U1

Here, C# (D1) and D (D2) merge into D12, while E (L1) and F# (L3) merge into L13.

C Major

Pitches: C (1), D (2), E (3), F (4), G (5), A (6), B (7)
Tonic-centred sequence: [G, A, B] + [C] + [D, E, F]
Unmerged segments: G (U1), A (U3), B (R2), C (R3), D (D2), E (L1), F (L2)
Merged encoding: U13R23D2L12

Here, G and A merge into U13, B and C merge into R23, D stands alone as D2, and E and F merge into L12.

Emergent structural properties

Whether the tonic merges with an adjacent leading tone (as C#+D forming D12 in D major) or remains an isolated segment (as D2 in C major) is an emergent structural property of the scale’s position relative to the keyboard’s triangle boundaries. The notation visually reveals the topographical “hand-feel” of the scale without requiring secondary fingering annotations.

Chord encoding and voicings

Chords follow the same triangle-point syntax. However, chord shapes are not isomorphic across transpositions on the piano; the notation reflects exact physical topography rather than abstract chord quality.

Triad examples

C major triad (C, E, G): R3L1U1 (three distinct triangles, skipping Down).
Observation on triad distributions: Major and minor triads typically engage three separate triangles while skipping one. Diminished triads (whose notes are spaced by minor thirds, 3 semitones) frequently repeat or cluster within triangles.

Voicings and inversions

Segment ordering and adjacency encode register and voicing layout:

Dm7 close position (D, F, A, C): D2L2U3R3 (strict ascending sequence across all four triangles).
Dmaj7 root position (D, F#, A, C#): D2L3U3D1 (C# appears at the end, indicating it sits in the upper register of the Down triangle).
Dmaj7 with 7th in the bass (C#, D, F#, A): D12L3U3 (C# and D merge into D12 at the front of the voicing).

A strict D -> L -> U -> R sequence with no repeated triangle indicates close four-part voicing. A repeated or merged triangle identifies intervals within a minor third, constraining voicing register without requiring separate octave tags.

Octave-skip marker

When a voicing or melody skips an octave across a triangle boundary, a bare triangle letter (with no trailing point digits) denotes an octave leap. This directly adapts the axis-diacritic and token-extension conventions established in Rhythmic Grammar.

Worked example: L3DD1 encodes F# (L3), followed by an octave skip at the Down triangle (D), landing on C# (D1) in the higher octave.

Self-disambiguation property

Because every substantive pitch segment contains at least one numeric digit (1, 2, or 3), a bare alphabetic character (D, L, U, R) is unambiguously parsed as an octave skip marker.

Naming systems and collision warning

Two naming schemes exist for the four triangles:

Primary / Pedagogical Scheme: Down / Left / Up / Right (D, L, U, R) Prioritises tactile hand-shape and keyboard orientation. Recommended for all introductory teaching and practical execution.
Alternate / Mnemonic Scheme: Do / Me / Fi / La (D, M, F, L) Semantically grounded in solfège syllables spaced at equal minor-third intervals (0, 3, 6, 9 semitones from tonic D).

[!WARNING]

Letter Collision Warning

In the primary scheme, L represents the Left triangle (E, F, F#). In the alternate solfège scheme, L represents La (the Right triangle, A#, B, C).

These two letter sets must never be silently mixed within the same document or dataset. When both schemes are presented together, the alternate solfège syllables must be written in lower-case (d, m, f, l) or spelled out in full (Do, Me, Fi, La).

(Note: As an incidental mnemonic, D aligns pleasantly with Do, Down, and Dorian when anchored on D.)

Encoding grammar summary

The formal grammar for Piano Triangle strings is defined as follows:

encoding        := (segment | skip-marker)+
segment         := triangle-letter point-digits
triangle-letter := "D" | "L" | "U" | "R"
point-digits    := [1-3]+  /* in strictly ascending pitch order */
skip-marker     := triangle-letter

Open questions

Non-heptatonic scale sets: Whether pentatonic, whole-tone, and octatonic scales fit naturally into tetrachord chaining syntax or are better expressed as raw lit-point sets.
Chord family taxonomy: Systematic verification of triangle skipping and repetition patterns across extended chord families (9ths, 11ths, altered chords).
Multi-octave skip markers: Formalising the syntax for skips greater than one octave (e.g., whether DD represents a two-octave skip or collapses to a single skip).