Uniform Solfège — Diacritic System
The Uniform Solfège diacritic system encodes sub-semitone pitch positions through a set of geometrically distinct, prime-family-specific marks applied to the base chromatic solfège glyphs.
Diacritic System
Note: The diacritic system described here is an application of Prime Period Diacritics (PPD) to Uniform Solfège pitch space. Refer to PPD for the general specification; this document covers Uniform Solfège-specific mappings only.
Overview
The Uniform Solfège diacritic system encodes sub-semitone pitch positions by applying Prime Period Diacritics (PPD) to the base chromatic solfège glyphs. Each diacritic family corresponds to a prime number and subdivides the chromatic semitone (100¢) into exact rational intervals — no decimal approximation, no rounding.
The system is built on the PPD families: Du (Axis), DuTri, Tri, Qui, Sep, and Undec. It comprises two functionally distinct groups:
- Approximation family: Du (prime 2), including Fractal Du — a recursive binary subdivision system
- Exact families: Tri, Qui, Sep, UnDec (primes 3, 5, 7, 11) — fixed rational targets
These two families are structurally and semantically separate and should not be conflated.
Reference Interval
All diacritics operate within a single chromatic semitone. The reference interval is 100¢ (one semitone), consistent across all prime families. The base solfège syllables (Do, Di, Re, Ri, Me, Mi, Fi, Se, So, Si, La, Ti) are the shared zero-reference points — chromatic anchors common to all families.
Solfège Symbol Range
Each solfège symbol owns a 100¢ space. The boundary conditions are:
- Base (0¢): the exact chromatic anchor — undecorated glyph
- Axis (50¢): the midpoint between chromatic anchors — the terminal point of the solfège range and threshold of the Du approximation space
The full range of any solfège symbol runs from UnDecSub5 (≈ -90.91¢ from the next Base) through to Axis (50¢). The first position of any solfège symbol is UnDecSub5, whose moon diacritics open toward the previous symbol’s Axis — directionally honest across the boundary.
Glyph Forms Summary
For the full visual specification of diacritic shapes, see PPD Glyph Forms. When applied to Uniform Solfège, these forms interact with the specific geometry of the solfège characters (the rotated U with decorated arms):
| Family | Forms | Uniform Solfège Specifics |
|---|---|---|
| Du (Axis) | Horizontal stroke | Passes through the vertical arms of the base character. Extended for Fractal Du to provide legibility clearance. |
| Tri / DuTri | Triangles | Attached at the base character perimeter. |
| Qui | Triangle + T-cross | Pointing away from the base character perimeter. |
| Sep | Ticks / capped strokes | Placed on the 3 o’clock side (positive) or 9 o’clock side (negative) of the base character. |
| Undec | Moons | Placed at the cardinal points (3 o’clock or 9 o’clock). |
Pitch Position Mappings
The following tables show how the PPD positions map specifically to cents from the Base chromatic anchor.
Du (Prime 2) — Approximation Family
Fractal Du subdivides the space between Base and Axis via binary halving.
| Depth | Shorthand | Position (¢) |
|---|---|---|
| 1 | Dox | 50¢ |
| 2 | Doxo / Doxi | 25¢ / 75¢ |
| 3 | Doxoo / Doxio | 12.5¢ / 62.5¢ |
Tri (Prime 3) — Exact Family
Tri provides six evenly-spaced positions per semitone, combining to tile the 72 EDO grid.
| Position | ¢ from Base | Accidental analogy |
|---|---|---|
| Sub | −33.33¢ | 𝄫 double flat |
| HalfSub | −16.67¢ | ♭ flat |
| Base | 0¢ | ♮ natural |
| HalfSup | +16.67¢ | ♯ sharp |
| Sup | +33.33¢ | 𝄪 double sharp |
| Axis | +50¢ | threshold |
Qui (Prime 5) — Exact Family
| Position | ¢ from Base |
|---|---|
| QuiSub2 / HalfQuiSub | −40¢ |
| QuiSub1 / QuiSub | −20¢ |
| Base | 0¢ |
| QuiSup1 / QuiSup | +20¢ |
| QuiSup2 / HalfQuiSup | +40¢ |
Sep (Prime 7) — Exact Family
| Position | ¢ from Base |
|---|---|
| SepSub3 / HalfSepSub | −42.86¢ |
| SepSub2 | −28.57¢ |
| SepSub1 / SepSub | −14.29¢ |
| Base | 0¢ |
| SepSup1 / SepSup | +14.29¢ |
| SepSup2 | +28.57¢ |
| SepSup3 / HalfSepSup | +42.86¢ |
UnDec (Prime 11) — Exact Family
| Position | ¢ from Base |
|---|---|
| UnDecSub5 | −90.91¢ |
| UnDecSub4 | −81.82¢ |
| UnDecSub3 | −72.73¢ |
| UnDecSub2 | −63.64¢ |
| UnDecSub1 | −54.55¢ |
| Base | 0¢ |
| UnDecSup1 | +9.09¢ |
| UnDecSup2 | +18.18¢ |
| UnDecSup3 | +27.27¢ |
| UnDecSup4 | +36.36¢ |
| UnDecSup5 | +45.45¢ |
Universal Grid and Remainder System
LCM Grid
All family denominators taken together:
LCM(6, 4, 5, 7, 11) = 4620 units per semitone
Including Fractal Du ÷32: LCM × 32/4 = 9240 units per semitone
Every family’s positions land exactly on the 4620 grid:
| Family | Grid units per step |
|---|---|
| DuTri (÷6) | 770 |
| Fractal Du ÷4 | 1155 |
| Qui (÷5) | 924 |
| Sep (÷7) | 660 |
| UnDec (÷11) | 420 |
Cross-Family Arithmetic and Remainders
Within-family arithmetic is always closed and exact — n/p ± m/p = (n±m)/p, always the same prime family.
Cross-family arithmetic (e.g. a Qui point ± a Sep point) produces exact rationals on the 4620 grid but with denominators (e.g. 35, 55, 77) not covered by any single diacritic family. These residuals are PPT commas — irreducible gaps between prime families, exact and nameable.
Named commas identified:
- Sep/UnDec comma: 100/77¢ (≈1.30¢) — appears twice symmetrically around 50¢
- Sep/DuTri comma: 100/42¢ (≈2.38¢)
Do as Remainder Register
Any cross-family arithmetic remainder is sub-semitone by definition, so it always fits within the diacritic space. Do (Base) is the canonical remainder register — remainders are expressed as Do-anchored sub-glyphs regardless of which chromatic syllable hosts the primary diacritic.
This gives a natural canonical form for any pitch:
- Base chromatic syllable — coarse position
- Prime family diacritic — fine position within semitone
- Do-anchored remainder sub-glyph — cross-family arithmetic residual (if needed)
The remainder sub-glyph occupies the descent zone of the host glyph (see Glyph Architecture). The Do-remainder is always a U-form (Do-oriented arc) since remainders are always Do-anchored — the descent zone is semantically typed, never ambiguous.
Poly-Base Structure
The diacritic families form a parallel multi-base coordinate system on the same pitch line, unified at the chromatic anchor points. Key properties:
- Bases are parallel, not hierarchical (unlike mixed-radix systems)
- Moduli (2, 3, 5, 7, 11) are coprime — unique reconstruction from residues (cf. Chinese Remainder Theorem)
- Chromatic anchors are the common zeros across all families
- Diacritics are mutually exclusive — each pitch carries one family’s diacritic only
This is not a tensor product or direct sum — it is a partition of rational pitch space by prime family, unified at the integers. No standard algebraic name exists for this structure; it is defined here as a foundational PPT construct.
Practical Coverage
- Perceptual layer: primary diacritics to ~6¢ (Fractal Du ÷16)
- Performance layer: Fractal Du ÷32 to ~3¢ — human-articulable in rhythm, audible in sustained pitch
- Algebraic layer: 9240-grid remainders for exact cross-family arithmetic
The system is perceptually complete at the diacritic layer, algebraically complete at the remainder layer, and theoretically open via decimal extension.
Applications
Pitch: honest representation of blue notes, just intonation chords, shruti positions, spectral partials — without approximation to 12-EDO. C# and D♭ are distinct pitches (BaseTri Sup and next-symbol BaseTri Sub) rather than collapsed into one equal-tempered slot.
Rhythm: prime family subdivision applies identically to rhythmic cycles. Tuplets in 5, 7, and 11 are first-class citizens. Polyrhythm across families (5 against 7) is Qui vs Sep subdivision of the same period. The Fractal Du bitmask maps directly onto standard beam notation — the isomorphism is explicit and teachable.
Interval analysis: any two pitches have exact rational distance. Cross-family intervals produce PPT commas as algebraic residues.
Timbre: harmonic partials are a prime-ratio structure. Spectral analysis uses the same coordinate system as pitch and rhythm.
Relationship to Tuning Systems
- 72-EDO: the six DuTri positions per semitone (÷6) exactly reproduce 72-EDO within the chromatic space. 72-EDO is an emergent property of the two interlocking Tri triangles, not a design target.
- 31-EDO: BaseTri Sub/Sup at ±33.33¢ approximate the 31-EDO enharmonic distinction (~38.71¢) with a gap of ~5.38¢ — a nameable PPT comma. 72-EDO provides a good approximation grid for 31-EDO but not an exact one.
Open Questions
- Formal naming and catalogue of all PPT commas derivable from cross-family arithmetic
- Decimal-place extension convention: notation for nested prime family diacritics as successive approximation digits
- FontForge implementation: GSUB lookup structure, GPOS axis-relative mark attachment anchors
- PUA codepoint block allocation for MusiCoil
See also: Geometric Basis for glyph architecture (three-zone structure, axis-relative remainder placement, rotational identity of the four arc families).