Fretboard Leaps and Seam Crossing

Status

This page is a companion extension to Piano Triangle Notation, applying PPT’s boundary-marker principles to stringed instruments with non-uniform interval lattices.

The isomorphic lattice with one seam

The guitar fretboard is largely an isomorphic 2D lattice: across most adjacent strings, the interval is a uniform perfect fourth (5 semitones, 3-prime Pythagorean generator). However, to facilitate six-string chord voicings, standard tuning introduces a single major third seam (4 semitones, 5-prime Ptolemaic generator):

• In Standard Tuning (E2–A2–D3–G3–B3–E4), the seam sits between strings 3 and 2 (G–B).
• In D-Standard Tuning (D2–G2–C3–F3–A3–D4, the primary working tuning for PPT string research), the seam sits between strings 3 and 2 (F–A).

This asymmetry is the exact stringed-instrument analogue to the piano’s two black-key gaps (E–F and B–C). In both instruments, a predominantly uniform geometric grid contains a small, fixed number of documented exceptions where spatial shifts occur.

The general leap formula

When calculating the physical fret position of an interval leaping across multiple strings, the displacement on the destination string follows a simple linear formula:

offset = (interval_in_semitones - 5 * strings_apart) + (1 if seam crossed else 0)

Where:

• offset is the number of frets relative to the origin fret on the destination string (e.g., +1 means one fret higher, 0 means the same fret).
• interval_in_semitones is the interval size (e.g., octave = 12, major tenth = 16).
• strings_apart is the difference in string indices (destination_string - origin_string).
• The correction term +1 is added if and only if the path crosses the G–B (or F–A) seam.

Worked reference table

The following table demonstrates the formula across common melodic and harmonic leaps in both standard and D-standard tunings:

The seam-crossing flag

In notation and tabulation systems, crossing the fretboard seam can be explicitly flagged by reusing the bare-letter skip marker from Piano Triangle Notation and Rhythmic Grammar.

Rather than inventing an ad-hoc guitar symbol, the occurrence of a bare marker indicates a boundary step in an otherwise uniform lattice.

Future abstraction

The emergence of the same structural pattern across both piano and guitar — a uniform lattice + single documented seam + bare-marker flag — suggests a potential higher-level PPT primitive for instrument topography. For now, this is retained as a concrete observation across the two instrument families, awaiting further corpus and pedagogical testing before formal abstraction.

See also

• Piano Triangle Notation — core ergonomic notation and skip-marker grammar
• Rhythmic Grammar — token chaining and the axis skip-marker precedent
• Default Do (12TET Keyboard) — keyboard topographical symmetry around D
• Uniform Solfège — Overview — the isomorphic notation layer