FAILURE MAP
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FA-81296 / Music interval and transposition theory / Open access

Staff position and ledger lines for a clef: upper ledger count rounded up · case 01

G5 above the treble staff reports a ledger line although it sits on top of the staff.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Positions above the staff are rounded up to the next ledger line.

VERIFIED REPAIR

Restore the upper ledger count step so that it reads `(pos - 8) // 2`.

Unsuccessful approach: Offsetting from position 9 misses the ledger line through A5.

Case contract

Input [note, clef] where note is a letter, accidentals (# b x, ignored for placement) and a nonnegative octave, and clef is treble (bottom line E4), bass (G2), alto (F3) or tenor (D3). Position counts diatonic steps above the bottom line (lines are even positions 0..8). Ledger lines: below the staff (-position)//2 for negative positions, above (position-8)//2 for positions over 8, else 0. Return [position, ledgers] or None when invalid.

Why this case matters

Engraving and notation-rendering code places noteheads and ledger lines from pitch spellings.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    base = 'CDEFGAB'
    refs = {'treble': ('E', 4), 'bass': ('G', 2), 'alto': ('F', 3), 'tenor': ('D', 3)}
    if not isinstance(x, list) or len(x) != 2 or x[1] not in refs:
        return None
    note, clef = x
    if not isinstance(note, str) or len(note) < 2 or note[0] not in base:
        return None
    tail = note[1:].lstrip('#bx')
    if not tail.isdigit():
        return None
    letter, octave = refs[clef]
    pos = (int(tail) * 7 + base.index(note[0])) - (octave * 7 + base.index(letter))
    if pos < 0:
        ledgers = -pos // 2
    elif pos > 8:
        ledgers = (pos - 8 + 1) // 2
    else:
        ledgers = 0
    return [pos, ledgers]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[(['E4', 'treble'], [0, 0]), (['C4', 'treble'], [-2, 1]), (['D4', 'treble'], [-1, 0]), (['B3', 'treble'], [-3, 1]), (['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['D6', 'treble'], [13, 2]), (['E6', 'treble'], [14, 3])], [(['B3', 'treble'], [-3, 1]), (['A3', 'treble'], [-4, 2]), (['F5', 'treble'], [8, 0]), (['A5', 'treble'], [10, 1]), (['C6', 'treble'], [12, 2]), (['C#4', 'treble'], [-2, 1]), (['B5', 'treble'], [11, 1]), (['A4', 'alto'], [9, 0])], [(['G5', 'treble'], [9, 0]), (['C6', 'treble'], [12, 2]), (['C#4', 'treble'], [-2, 1]), (['Fx4', 'treble'], [1, 0]), (['Bb3', 'treble'], [-3, 1]), (['C4', 'alto'], [4, 0]), (['C4', 'bass'], [10, 1]), (['D6', 'treble'], [13, 2])], [(['C4', 'alto'], [4, 0]), (['C4', 'bass'], [10, 1]), (['E2', 'bass'], [-2, 1]), (['C4', 'tenor'], [6, 0]), (['G2', 'bass'], [0, 0]), (['B5', 'treble'], [11, 1]), (['A4', 'alto'], [9, 0]), (['E6', 'treble'], [14, 3])], [(['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['G2', 'bass'], [0, 0]), (['F3', 'alto'], [0, 0]), (['B2', 'alto'], [-4, 2]), (['H4', 'treble'], None), (['D6', 'treble'], [13, 2]), (['E6', 'treble'], [14, 3])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
oracle 0[0, 0][0, 0]Passed
oracle 1[-2, 1][-2, 1]Passed
oracle 2[-1, 0][-1, 0]Passed
oracle 3[-3, 1][-3, 1]Passed
oracle 4[9, 1][9, 0]Failed
oracle 5[10, 1][10, 1]Passed
oracle 6[13, 3][13, 2]Failed
oracle 7[14, 3][14, 3]Passed

SHA-256 / cd782171ff3d22a495f21c6c9492cdf2833e7acfc0978fcae7958b17892226c3

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    base = 'CDEFGAB'
    refs = {'treble': ('E', 4), 'bass': ('G', 2), 'alto': ('F', 3), 'tenor': ('D', 3)}
    if not isinstance(x, list) or len(x) != 2 or x[1] not in refs:
        return None
    note, clef = x
    if not isinstance(note, str) or len(note) < 2 or note[0] not in base:
        return None
    tail = note[1:].lstrip('#bx')
    if not tail.isdigit():
        return None
    letter, octave = refs[clef]
    pos = (int(tail) * 7 + base.index(note[0])) - (octave * 7 + base.index(letter))
    if pos < 0:
        ledgers = -pos // 2
    elif pos > 8:
        ledgers = (pos - 9) // 2
    else:
        ledgers = 0
    return [pos, ledgers]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[(['E4', 'treble'], [0, 0]), (['C4', 'treble'], [-2, 1]), (['D4', 'treble'], [-1, 0]), (['B3', 'treble'], [-3, 1]), (['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['D6', 'treble'], [13, 2]), (['E6', 'treble'], [14, 3])], [(['B3', 'treble'], [-3, 1]), (['A3', 'treble'], [-4, 2]), (['F5', 'treble'], [8, 0]), (['A5', 'treble'], [10, 1]), (['C6', 'treble'], [12, 2]), (['C#4', 'treble'], [-2, 1]), (['B5', 'treble'], [11, 1]), (['A4', 'alto'], [9, 0])], [(['G5', 'treble'], [9, 0]), (['C6', 'treble'], [12, 2]), (['C#4', 'treble'], [-2, 1]), (['Fx4', 'treble'], [1, 0]), (['Bb3', 'treble'], [-3, 1]), (['C4', 'alto'], [4, 0]), (['C4', 'bass'], [10, 1]), (['D6', 'treble'], [13, 2])], [(['C4', 'alto'], [4, 0]), (['C4', 'bass'], [10, 1]), (['E2', 'bass'], [-2, 1]), (['C4', 'tenor'], [6, 0]), (['G2', 'bass'], [0, 0]), (['B5', 'treble'], [11, 1]), (['A4', 'alto'], [9, 0]), (['E6', 'treble'], [14, 3])], [(['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['G2', 'bass'], [0, 0]), (['F3', 'alto'], [0, 0]), (['B2', 'alto'], [-4, 2]), (['H4', 'treble'], None), (['D6', 'treble'], [13, 2]), (['E6', 'treble'], [14, 3])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
oracle 0[0, 0][0, 0]Passed
oracle 1[-2, 1][-2, 1]Passed
oracle 2[-1, 0][-1, 0]Passed
oracle 3[-3, 1][-3, 1]Passed
oracle 4[9, 0][9, 0]Passed
oracle 5[10, 0][10, 1]Failed
oracle 6[13, 2][13, 2]Passed
oracle 7[14, 2][14, 3]Failed

SHA-256 / 494e373e413808a82ba3dc78d14d881e0c263be2c587eb36856d6736e46125a6

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    base = 'CDEFGAB'
    refs = {'treble': ('E', 4), 'bass': ('G', 2), 'alto': ('F', 3), 'tenor': ('D', 3)}
    if not isinstance(x, list) or len(x) != 2 or x[1] not in refs:
        return None
    note, clef = x
    if not isinstance(note, str) or len(note) < 2 or note[0] not in base:
        return None
    tail = note[1:].lstrip('#bx')
    if not tail.isdigit():
        return None
    letter, octave = refs[clef]
    pos = (int(tail) * 7 + base.index(note[0])) - (octave * 7 + base.index(letter))
    if pos < 0:
        ledgers = -pos // 2
    elif pos > 8:
        ledgers = (pos - 8) // 2
    else:
        ledgers = 0
    return [pos, ledgers]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[(['E4', 'treble'], [0, 0]), (['C4', 'treble'], [-2, 1]), (['D4', 'treble'], [-1, 0]), (['B3', 'treble'], [-3, 1]), (['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['D6', 'treble'], [13, 2]), (['E6', 'treble'], [14, 3])], [(['B3', 'treble'], [-3, 1]), (['A3', 'treble'], [-4, 2]), (['F5', 'treble'], [8, 0]), (['A5', 'treble'], [10, 1]), (['C6', 'treble'], [12, 2]), (['C#4', 'treble'], [-2, 1]), (['B5', 'treble'], [11, 1]), (['A4', 'alto'], [9, 0])], [(['G5', 'treble'], [9, 0]), (['C6', 'treble'], [12, 2]), (['C#4', 'treble'], [-2, 1]), (['Fx4', 'treble'], [1, 0]), (['Bb3', 'treble'], [-3, 1]), (['C4', 'alto'], [4, 0]), (['C4', 'bass'], [10, 1]), (['D6', 'treble'], [13, 2])], [(['C4', 'alto'], [4, 0]), (['C4', 'bass'], [10, 1]), (['E2', 'bass'], [-2, 1]), (['C4', 'tenor'], [6, 0]), (['G2', 'bass'], [0, 0]), (['B5', 'treble'], [11, 1]), (['A4', 'alto'], [9, 0]), (['E6', 'treble'], [14, 3])], [(['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['G2', 'bass'], [0, 0]), (['F3', 'alto'], [0, 0]), (['B2', 'alto'], [-4, 2]), (['H4', 'treble'], None), (['D6', 'treble'], [13, 2]), (['E6', 'treble'], [14, 3])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
oracle 0[0, 0][0, 0]Passed
oracle 1[-2, 1][-2, 1]Passed
oracle 2[-1, 0][-1, 0]Passed
oracle 3[-3, 1][-3, 1]Passed
oracle 4[9, 0][9, 0]Passed
oracle 5[10, 1][10, 1]Passed
oracle 6[13, 2][13, 2]Passed
oracle 7[14, 3][14, 3]Passed

SHA-256 / d92f600d5b8bc0b2494d78cf58813ef6fa2485f04087e9942e5f86417ae67d6d

Verification & scope

A deterministic bounded teaching model with a stipulated toy contract; it is not a complete music notation or theory engine. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:50:01.805987+00:00.

Case digest / 028b6f5fb7814f07f174005e0d039629c0cdd86c11fcd7d37b79dec17ebda691