HARKEN NUMERIC NOTATION

Introduction

Harken Numeric Notation (HNN) is an ASCII-based music notation system designed for precise, human-readable representation of compositions in 12-tone equal temperament (12-TET) and 24-tone equal temperament (24-TET). The system uses tonic-relative pitch class numbers (0–11) combined with duration letters, articulation marks, and structural syntax to create a complete musical language that is simultaneously machine-parseable and musically intuitive.

This document describes the complete HNN specification as implemented in Harken Super Composer (current version v1.1.4), the sole composition application, which consolidates the earlier Harken Composer and Harken Microtonal Composer.

Pitch System

12-TET Pitch Classes

The twelve chromatic pitch classes are represented as integers 0–11, always relative to the current tonic:

Critical principle: These numbers are not degrees of a diatonic scale. They are positions in the chromatic circle, with 0 always representing the tonic regardless of mode or key. The labels (m2, M3, P5, etc.) indicate the interval distance from the tonic measured in semitones, following Western interval terminology purely as descriptive size labels.

Octave Modifiers

Pitch class numbers specify position within the octave but not which octave. Octave modifiers shift the reference octave:

Octave modifiers are placed immediately after the pitch class number and before duration letters:

Span Toggles (Multi-Pitch Displacement)

For displacing multiple consecutive pitches, span toggle syntax provides a more efficient alternative to per-note modifiers:

Span Rules:

• Pitches can have explicit durations (w/h/q/e) or default to 1 slot
• No per-note octave modifiers (^/v) allowed inside spans
• Spans can contain parenthetical chords
• To mix octaves, break into segments: ^0q,0q^,0vq,^0q^

24-TET Quarter-Tone Extension

The + modifier creates quarter-tone pitches, raising a pitch class by exactly 50 cents (half of a semitone):

There is no 12+. The pitch class 11+ (1150 cents) is followed by 0 (1200 cents, the octave). Quarter-tone notation operates within the same modulo-12 framework as chromatic notation.

The + modifier is placed immediately after the pitch class number, before octave modifiers and duration letters.

Rhythm and Duration

Duration Letters

Five duration letters represent standard note values:

Duration letters are placed after the pitch class, octave modifiers, and quarter-tone modifier (if present):

Duration Chaining

Multiple duration letters can be chained to create additive durations:

This mechanism replaces traditional dotted note notation with explicit arithmetic.

Dotted Durations

The dot (.) modifier multiplies a single duration letter by 1.5:

The dot is placed immediately after the duration letter it modifies. When chaining durations, each letter can have its own dot:

Duration chaining and dots can be combined to create precise rhythmic values. The parser evaluates left-to-right, applying each dot to the letter immediately preceding it.

Time Signatures and Subdivision

Time signatures are specified using bracket syntax at the beginning of a measure:

The subdivision value (the optional third parameter) determines how many internal timing slots exist per measure. This affects duration calculations:

• Simple meters (x/4): Default subdivision equals the numerator (beats). One slot = one quarter note.
• Compound meters (x/8): Default subdivision equals the numerator (eighth notes). One slot = one eighth note.
• Explicit subdivision: Overrides the default, typically used for triplet-based or complex rhythmic divisions.

Rests

The asterisk (*) denotes a rest. Duration letters follow the same rules as notes:

Ties and Sustains

The tilde (~) creates a tie/sustain, extending the previous note without re-attack:

The ~ symbol references the most recent pitch class and octave. Duration letters after ~ specify how long the sustain lasts.

Cross-Barline Sustains

Duration letters alone (without a pitch class) at the start of a measure create a sustain from the previous measure:

This is equivalent to |0h,4h|~h| but more concise for measure-crossing sustains.

Articulation

The apostrophe (') marks staccato articulation. It is placed after all duration elements (letters and dots):

Staccato reduces the sounding duration to approximately 20% of the notated duration while maintaining the rhythmic spacing of the full duration.

Parenthetical Chord Notation

Simultaneous attack of multiple pitch classes is notated using parentheses. This syntax enables polyphonic harmony within a single melodic line:

Syntax Rules

• Pitch classes are listed inside parentheses, separated by commas
• Each pitch class can have its own + (quarter-tone) and ^/v (octave) modifiers
• Duration letters follow the closing parenthesis and apply to all pitches simultaneously
• At least one pitch class is required (empty chords are invalid)
• The closing parenthesis ) is mandatory

Complex Examples

Parenthetical vs. Chord Symbol Notation

Parenthetical notation specifies exact voicings—every pitch class is explicitly listed. This differs from chord symbol notation (described later), which specifies harmonic function and allows the performer or renderer to choose voicings.

Parenthetical chords are essential for microtonal composition, where precise quarter-tone voicings must be specified.

Measure Structure and Separators

Event Separators

Individual events (notes, rests, sustains, chords) within a measure are separated by commas:

Spaces are optional and ignored by the parser. Both 0q,2q,4q and 0q, 2q, 4q are valid.

Barlines

Barlines (|) visually separate measures but are optional. The parser strips leading and trailing barlines:

Internal barlines within a measure string are ignored as whitespace.

Time Signature Prefix

Each measure can begin with an optional time signature in brackets. If omitted, the measure inherits the time signature from the previous measure (or the global default):

Time signatures persist across measures until explicitly changed.

Chord Symbol Notation

Note: Harken Super Composer supports both chord symbols (chord track, 12-TET) and parenthetical chord notation (24-TET precision); this capability was inherited from the predecessor Harken Composer (chord symbols) and Harken Microtonal Composer (parenthetical 24-TET chords). This section documents both approaches.

Basic Structure

Chord symbols combine root pitch class, quality abbreviation, and duration:

Quality Abbreviations

Complete Chord Quality Vocabulary

Harken Super Composer (v1.1.4) implements 54 chord qualities organized by harmonic function. All qualities support alterations (b5, #5, b9, #9, #11, b13) and spectral extensions (+).

Triads: maj, min, dim, aug, 5 (power chord), sus2, sus4

Seventh Chords: dom7, maj7, min7, dim7, min7b5 (half-diminished), hdim7, minmaj7, aug7, maj7#5, dom7b5, dom7sus4, dom7sus2

Sixth Chords: maj6, min6, maj69, min69

Ninth Chords: dom9, maj9, min9, minmaj9, aug9, maj9#5, dom9b5, min9b5, dom9sus4

Eleventh Chords: dom11, maj11, min11, minmaj11, min11b5, maj7#11, maj9#11

Thirteenth Chords: dom13, maj13, min13, maj13#11

Added-Tone Chords: add2, add4, add9, add11, madd2, madd4, madd9, minadd4

Extended Added-Tone Chords: maj7add4, maj7add6, maj9sus4, minmaj7add6, minadd9b13, maj9add13, majadd9b13, maj9#5add13, aug9b13, min11b9b13 (Phrygian), min11b5b9b13 (Locrian)

Alterations: Append b5, #5, b9, #9, #11, b13 after quality (e.g., dom7b9#11w). To create spectral chords with quarter-tone intervals, append + after the quality (e.g., dom7+w yields all intervals +50¢).

24-TET Chord Symbols

Quarter-tone roots use the + modifier immediately after the root number:

These are detuned chords — all pitches shift up 50 cents from the chromatic root.

Spectral Chord Symbols: Single-Modifier Extension

A spectral chord is a chromatic chord with all intervals (except the tonic root) extended by 50 cents into quarter-tone space. The notation uses a single + modifier after the chord quality to transform the entire voicing:

The Tonic Anchor Principle: In spectral chord symbols, the root (tonic 0) remains in 12-TET as the harmonic anchor. Only the upper structure intervals—thirds, fifths, sevenths—extend into spectral space. This creates familiar bass foundation with shimmering microtonal color above.

The + Modifier: Three Usages in Chord Symbols

The quarter-tone modifier + can appear in two positions within a chord symbol, creating three distinct harmonic effects:

Detuned vs Spectral Chords

Detuned Chord (root+quality): The entire chord structure shifts uniformly by +50 cents. All intervals remain in 12-TET relationships, but the absolute pitch is a quarter-tone higher. This is equivalent to raising the root by 50 cents and maintaining standard chord construction.

Spectral Chord (root quality+): The root remains in 12-TET as the tonic anchor, while the upper structure intervals (3rd, 5th, 7th, extensions) extend +50 cents into spectral space. This creates mixed 12-TET + quarter-tone sonority with grounded bass and shimmering upper voices.

Spectral Chord Examples

Detuned Chord Examples

Carrier Wave Theory Applied to Harmony: The tonic (0) and tritone (6) act as carrier tones from 12-TET, providing harmonic stability and bass foundation. The spectral intervals (1+, 3+, 4+, 7+, 10+, 11+) form the quarter-tone halo around this chromatic skeleton. In performance, bass instruments play carrier tones while chord voicings explore the spectral halo—creating grounded microtonal harmony accessible to musicians trained in Western chord vocabulary.

Practical Advantage: One character (+) transforms an entire chord from chromatic to spectral space. Composers can write 0dom7w in the Chords Track and instantly convert to 0dom7+w for spectral color. The Melody Track auto-expands: (0, 4+, 7+, 10+)w. Immediate microtonal access with familiar chord symbols. For quarter-tone transposition of the entire progression, use detuned chords: 0+dom7w, 2+min7w, 5+dom7w.

Chord Symbol Transformation

Harken Super Composer v0.16.0+ applies geometric transformations to chord symbols alongside melodic material. When a measure containing chord symbols is reflected or rotated, the chord symbol transforms according to the CHORD_REFLECTION_MAP (54 quality pairings).

Reflection (Inversion)

Reflection across 12-TET axes (R0–R5.5) transforms both root and quality. The root reflects via the formula (2*axis - root + 12) % 12, and the quality transforms to its reflection partner:

Reflection across 24-TET-exclusive axes (R0+–R5.5+) produces spectral outputs. Quarter-tone axes yield quarter-sharp roots (e.g., 0maj7w → 0+maj7w).

Rotation (Transposition)

Rotation preserves chord quality and transposes the root by semitones. In 12-TET mode, T5 applied to 0maj7w yields 5maj7w. In 24-TET mode with quarter-tone rotations (T0+–T11+), quarter-tones propagate: T5+ applied to 3+min7w yields 8+min7w.

Multi-Measure, Multi-Track Transformation (v0.16.6)

Super Composer supports simultaneous transformation across multiple measures and all harmonic tracks. Select measures using checkboxes (shift-click for range), then apply any transformation. All harmonic tracks (piano, bass, saxophone, etc.) transform together while drum tracks (detected via /drum|perc/i pattern) copy unchanged. Result: complete harmonic pairs ready for composition.

Compositional Power: Transformations preserve harmonic function while generating variations. Reflect a progression to create its modal mirror; rotate to transpose instantly. Multi-track transformation creates complete ensemble passages with unified harmonic logic. Combined with Retrograde (time reversal), the Transform system provides 1,152 variations (24 reflections × 24 rotations × 2 time directions) of any melodic + harmonic phrase.

Complete Notation Examples

12-TET Melodic Line

Ascending C major scale in quarter and eighth notes, resolving to high tonic whole note.

24-TET Microtonal Melody

Chromatic ascent using quarter-tones: C, C+, C♯, C♯+, D, D+, E♭ (half note).

Parenthetical Chord Progression

I–IV–V–I progression using exact triad voicings.

24-TET Chord Voicing

Whole note chord: C+, E♭+, G, B♭+ — microtonal inflection of a minor 7 sonority.

Mixed Rhythm with Rests and Sustains

Tonic quarter, eighth rest, M3 eighth sustained to quarter, P5 half sustained to half.

Cross-Barline Sustain

First measure: tonic, M3, P5 (all half notes). Second measure: P5 sustained for quarter note from previous bar.

24-TET: Harmonic Perception and Aesthetic

The expansion from 12-TET to 24-TET is not merely a doubling of available pitches—it represents a fundamental shift in harmonic expressivity. Where 12-TET forces all pitch relationships into a rigid 100-cent grid, 24-TET introduces 50-cent intervals that create new zones of harmonic color.

Harmonic Breathing Room

Quarter-tone inflections produce what can be described as harmonic breathing room. In 12-TET, harmony exists in binary states: consonant or dissonant, stable or unstable. The introduction of quarter-tones creates intermediate zones where harmony possesses tension without conflict. A chord like (0+,3+,7,10+)h (C-quarter-sharp, E♭-quarter-sharp, G, B♭-quarter-sharp) is neither settled nor harsh—it occupies a middle ground of expressive color unavailable in chromatic tuning.

This quality is particularly evident in sustained harmonic contexts. Where 12-TET chords feel locked into fixed positions, 24-TET voicings create the sensation of harmony gently stretching and relaxing—a more fluid, organic sense of harmonic motion.

Natural Harmonic Series Alignment

The 24 equal divisions of the octave provide more landing points for alignment with the natural overtone series. While 12-TET averages all intervals into uniform 100-cent steps, quarter-tones can approach the pure ratios that acoustic instruments produce organically. When quarter-tone chord voicings are chosen with acoustic sensitivity, they can evoke the natural resonances of the harmonic series—resonances that 12-TET systematically flattens.

Cultural Conditioning and Release

Western ears are trained from childhood that the twelve chromatic divisions represent "correct" intonation and all other pitches are "out of tune." This is cultural conditioning, not acoustic truth. The surprise many listeners report upon hearing well-constructed 24-TET harmony—describing it as "soothing" or "strangely natural"—reveals the contingency of our tuning assumptions.

The metaphor of 12-TET "holding Western ears hostage" is apt. When quarter-tones are encountered in musical rather than academic contexts—played with conviction, integrated into meaningful harmonic progressions—many listeners experience a sensation of release, as if discovering their auditory perception has greater range than they had been allowed to believe.

Parenthetical Notation as Practical Enabler

Historically, the practical viability of microtonal harmony has been limited by notational complexity. Western staff notation handles quarter-tones awkwardly, requiring cumbersome accidentals and forcing composers to choose between readability and precision. As a result, composers have avoided quarter-tone chords not for musical reasons but for writability reasons.

Harken's parenthetical chord notation—(0+,3+,7,10+)h—makes these voicings immediate and practical. Quarter-tone harmonies become as easy to write as diatonic triads. This removes the notational barrier that has historically confined microtonality to melodic applications, enabling composers to explore quarter-tone harmony as a natural extension of chromatic practice rather than an academic exercise.

Statistical Context

From a combinatorial perspective, 12-TET provides 4,095 possible pitch class sets (2^12 − 1, excluding the empty set). 24-TET provides 16,777,215 (2^24 − 1). Most of these are musically irrelevant, but the meaningful subset—quarter-tone inflections of familiar tertian, quartal, and quintal structures—represents a genuine expansion of harmonic vocabulary with demonstrable aesthetic power.

The goal is not to replace 12-TET but to extend it. Just as chromatic harmony extends diatonic practice without invalidating it, 24-TET extends chromatic practice by providing finer gradations of harmonic color. The music decides which pitches to use.

Geometric Foundation

12-TET and the Dodecahedron

The twelve pitch classes of 12-TET map to the twelve vertices of a regular dodecahedron, a Platonic solid with twelve pentagonal faces. This mapping is not arbitrary—it emerges from the inherent symmetry of the chromatic circle and the fact that the dodecahedron is the unique regular polyhedron with twelve vertices.

The dodecahedral structure provides the geometric basis for the Harken transformation system (Reflections, Rotations, Retrograde), documented separately in the Harken Music Essay.

24-TET and the Nested-Polyhedra Lattice

The operative geometric foundation for 24-TET is a single-shape four-layer nested-polyhedra lattice — one coherent structure whose vertices sit on a shared circumsphere about a central singularity, rather than several independently-oriented solids. Its four concentric layers are:

• Octahedral layer (6 vertices): three mutually orthogonal antipodal tritone axes — the foundational core, with the tonic–tritone pair (0 and 6) on the polar axis.
• Dodecahedral layer (20 vertices): the complete 12-TET pitch-class mapping (12 pitch classes on the path vertices; the remaining 8 complete the solid).
• Decagon layer (30 vertices): additional circumsphere vertices coincident with those of an icosidodecahedron. Together with the twelve naturals of the dodecahedral layer, this layer supplies the twelve quarter-sharp positions {0+, 1+, …, 11+}, completing the 24-TET mapping.
• Icosagon layer (60 vertices): additional vertices coincident with those of a truncated icosahedron, carrying the finer 48-grid (eighth-/quarter-tone) resolution and demonstrating the framework's scalability beyond N=24.

Because all four layers share one circumsphere and one tonic–tritone polar axis, the 12-TET, 24-TET, and 48-grid resolutions remain mutually aligned (the 4⊂12⊂24 inheritance) on a single structure. This lattice is the geometry realized in Harken Super Composer's embedded Galaxy visualizer.

Historical note. An earlier model mapped the 24 pitch classes onto a subset of the 38 vertices of the pentagonal icositetrahedron (the dual of the snub cube). The discovery of a Hamiltonian path through all 38 vertices (May 15, 2026) — a 24-TET echo of William Rowan Hamilton's 1857 dodecahedral path — belongs to that conception phase. On reduction to practice the icositetrahedron carrier required independently rotated and twisted nested solids and could not bring the nested resolutions into alignment on one shared structure, so it was superseded by the single-shape lattice above. The 24-TET pitch-class cycle itself is correct in both models and is preserved unchanged; what changed is the geometric carrier, not the music. (See the Harken Master Provisional Disclosure for the full conception history.)

24-TET Interval Nomenclature

The Harken system extends Western interval nomenclature into 24-TET by adding the + modifier to indicate quarter-sharp extensions. This creates a systematic vocabulary for all twenty-four chromatic and microtonal intervals from the tonic.

Naming Convention

A quarter-tone interval is named by taking the Western interval name and appending + (plus). For example:

• 0,4 = maj3 (major third)
• 0,4+ = maj3+ (major third plus)
• 0,7 = per5 (perfect fifth)
• 0,7+ = per5+ (perfect fifth plus)

Complete Interval Set

Pedagogical Significance

This nomenclature bridges Western music theory vocabulary with microtonal practice. Musicians trained in 12-TET can immediately understand maj3+ as "major third plus a quarter-tone" without needing to learn an entirely new interval system. The + modifier scales naturally: any chromatic interval can be extended by 50 cents using the same syntax.

Application in Jazz and Blues

The interval names directly encode common microtonal performance practices:

• Blue notes: 3+ (min3+) = blues third, 7+ (per5+) = blues fifth, 10+ (min7+) = blues seventh
• Quarter-tone approach: 4+ (maj3+) resolving to 5 (per4)
• Spectral elaboration: 0,1+,3,4+,6,7+,9,10+ = 8-TET spectral diminished scale

24-TET Complete Interval Palette

In 24-TET, there are 24 pitch classes (0 through 11.5) relative to tonic, creating 24 intervals from 0. These divide into two equal sets:

12 Chromatic Intervals (Standard 12-TET)

12 Spectral Intervals (Quarter-Tone Extensions)

Total: 24 intervals (12 chromatic + 12 spectral). This is the complete 24-TET interval palette from tonic.

8-TET Spectral Subset

Within the 12 spectral intervals, four form the 8-TET quarter-tone halo — the intervals that define 8-TET when embedded in 24-TET:

• (0, 1+) = min2+ — 150 cents
• (0, 4+) = maj3+ — 450 cents
• (0, 7+) = per5+ — 750 cents
• (0, 10+) = min7+ — 1050 cents

These four quarter-tone intervals, combined with the diminished seventh skeleton (0, 3), (0, 6), (0, 9) from 12-TET, form the complete 8-TET pitch set: 0, 1+, 3, 4+, 6, 7+, 9, 10+.

Parser Architecture

The canonical implementation of Harken Numeric Notation is the parseHarkenMeasure() function in Harken Super Composer. This parser defines the grammar through its behavior, not through external specification.

Late-Binding Principle

The parser operates on late-binding tonic resolution: pitch class numbers (0–11) are parsed and stored as abstract event objects without reference to absolute pitch. The tonic key (C, D♭, F♯, etc.) is specified separately and applied only at playback or rendering time.

This architecture enables instant transposition: changing the tonic setting re-renders the entire score in a new key without rewriting a single note.

Event Object Format

The parser produces an array of event objects. Each event has a type field and type-specific properties:

The slots field represents duration in internal timing units. Slot-to-seconds conversion depends on tempo (BPM), time signature, and subdivision.

Stateful Parsing

The parser maintains state across measures to support cross-barline sustains:

• lastPc: Most recent pitch class (for sustain reference)
• lastOctShift: Most recent octave modifier (for sustain reference)

When a ~ symbol or bare duration letter is encountered, the parser references this state to determine what pitch to sustain.

Validation and Error Handling

The parser enforces strict validation:

• Pitch classes outside 0–11 range throw errors immediately
• Unclosed parenthetical chords throw errors
• Empty chords (no pitch classes) throw errors
• Invalid characters are rejected

Error messages include the invalid token and its context for debugging.

.HKN File Format

Harken compositions are stored as JSON files with the .hkn extension. The format supports both single-staff and multi-staff scores.

File Structure

Multi-Staff Architecture

The staves array defines each staff's properties. The staffOrder array controls the visual stacking order in the score (top to bottom). The measures_data.staves array contains the actual notation strings, indexed by staff number.

Example: staffOrder: [2, 0, 1] means staff 2 appears at the top, staff 0 in the middle, staff 1 at the bottom—but the notation data remains indexed as staves[0], staves[1], staves[2].

Glossary