FA-81301 / Music interval and transposition theory / Open access

Staff position and ledger lines for a clef: alto clef anchored at its middle line · case 01

Middle C in alto clef is placed on the bottom line instead of the middle line.

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

ROOT CAUSE

The alto reference uses the clef middle-line pitch C4 instead of the bottom-line pitch F3.

VERIFIED REPAIR

Restore the alto clef reference step so that it reads `'alto': ('F', 3)`.

Unsuccessful approach: Using F4 keeps the letter but puts the reference an octave too high.

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': ('C', 4), '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]), (['A3', 'treble'], [-4, 2]), (['C4', 'alto'], [4, 0]), (['B2', 'alto'], [-4, 2]), (['A4', 'alto'], [9, 0])], [(['B3', 'treble'], [-3, 1]), (['A3', 'treble'], [-4, 2]), (['F5', 'treble'], [8, 0]), (['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['C4', 'alto'], [4, 0]), (['F3', 'alto'], [0, 0]), (['A4', 'alto'], [9, 0])], [(['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['C6', 'treble'], [12, 2]), (['C#4', 'treble'], [-2, 1]), (['Fx4', 'treble'], [1, 0]), (['C4', 'alto'], [4, 0]), (['F3', 'alto'], [0, 0]), (['B2', 'alto'], [-4, 2])], [(['C#4', 'treble'], [-2, 1]), (['Fx4', 'treble'], [1, 0]), (['Bb3', 'treble'], [-3, 1]), (['C4', 'bass'], [10, 1]), (['E2', 'bass'], [-2, 1]), (['F3', 'alto'], [0, 0]), (['B2', 'alto'], [-4, 2]), (['A4', 'alto'], [9, 0])], [(['C4', 'alto'], [4, 0]), (['C4', 'bass'], [10, 1]), (['E2', 'bass'], [-2, 1]), (['C4', 'tenor'], [6, 0]), (['G2', 'bass'], [0, 0]), (['B2', 'alto'], [-4, 2]), (['H4', 'treble'], None), (['A4', 'alto'], [9, 0])]]
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 fixture	Actual	Expected	Outcome
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	`[-4, 2]`	`[-4, 2]`	Passed
oracle 5	`[0, 0]`	`[4, 0]`	Failed
oracle 6	`[-8, 4]`	`[-4, 2]`	Failed
oracle 7	`[5, 0]`	`[9, 0]`	Failed

SHA-256 / c71cbf5c4e94010b7e1bf7cf40f483bd149b77441d747d76d3d61a390075a66e

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', 4), '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]), (['A3', 'treble'], [-4, 2]), (['C4', 'alto'], [4, 0]), (['B2', 'alto'], [-4, 2]), (['A4', 'alto'], [9, 0])], [(['B3', 'treble'], [-3, 1]), (['A3', 'treble'], [-4, 2]), (['F5', 'treble'], [8, 0]), (['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['C4', 'alto'], [4, 0]), (['F3', 'alto'], [0, 0]), (['A4', 'alto'], [9, 0])], [(['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['C6', 'treble'], [12, 2]), (['C#4', 'treble'], [-2, 1]), (['Fx4', 'treble'], [1, 0]), (['C4', 'alto'], [4, 0]), (['F3', 'alto'], [0, 0]), (['B2', 'alto'], [-4, 2])], [(['C#4', 'treble'], [-2, 1]), (['Fx4', 'treble'], [1, 0]), (['Bb3', 'treble'], [-3, 1]), (['C4', 'bass'], [10, 1]), (['E2', 'bass'], [-2, 1]), (['F3', 'alto'], [0, 0]), (['B2', 'alto'], [-4, 2]), (['A4', 'alto'], [9, 0])], [(['C4', 'alto'], [4, 0]), (['C4', 'bass'], [10, 1]), (['E2', 'bass'], [-2, 1]), (['C4', 'tenor'], [6, 0]), (['G2', 'bass'], [0, 0]), (['B2', 'alto'], [-4, 2]), (['H4', 'treble'], None), (['A4', 'alto'], [9, 0])]]
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 fixture	Actual	Expected	Outcome
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	`[-4, 2]`	`[-4, 2]`	Passed
oracle 5	`[-3, 1]`	`[4, 0]`	Failed
oracle 6	`[-11, 5]`	`[-4, 2]`	Failed
oracle 7	`[2, 0]`	`[9, 0]`	Failed

SHA-256 / 4d12f0f5ffa28fc28aedccfa41390b875649059322f93756ea6573e0d4271e29

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]), (['A3', 'treble'], [-4, 2]), (['C4', 'alto'], [4, 0]), (['B2', 'alto'], [-4, 2]), (['A4', 'alto'], [9, 0])], [(['B3', 'treble'], [-3, 1]), (['A3', 'treble'], [-4, 2]), (['F5', 'treble'], [8, 0]), (['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['C4', 'alto'], [4, 0]), (['F3', 'alto'], [0, 0]), (['A4', 'alto'], [9, 0])], [(['G5', 'treble'], [9, 0]), (['A5', 'treble'], [10, 1]), (['C6', 'treble'], [12, 2]), (['C#4', 'treble'], [-2, 1]), (['Fx4', 'treble'], [1, 0]), (['C4', 'alto'], [4, 0]), (['F3', 'alto'], [0, 0]), (['B2', 'alto'], [-4, 2])], [(['C#4', 'treble'], [-2, 1]), (['Fx4', 'treble'], [1, 0]), (['Bb3', 'treble'], [-3, 1]), (['C4', 'bass'], [10, 1]), (['E2', 'bass'], [-2, 1]), (['F3', 'alto'], [0, 0]), (['B2', 'alto'], [-4, 2]), (['A4', 'alto'], [9, 0])], [(['C4', 'alto'], [4, 0]), (['C4', 'bass'], [10, 1]), (['E2', 'bass'], [-2, 1]), (['C4', 'tenor'], [6, 0]), (['G2', 'bass'], [0, 0]), (['B2', 'alto'], [-4, 2]), (['H4', 'treble'], None), (['A4', 'alto'], [9, 0])]]
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 fixture	Actual	Expected	Outcome
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	`[-4, 2]`	`[-4, 2]`	Passed
oracle 5	`[4, 0]`	`[4, 0]`	Passed
oracle 6	`[-4, 2]`	`[-4, 2]`	Passed
oracle 7	`[9, 0]`	`[9, 0]`	Passed

SHA-256 / bcbdbc54d48281f628097f298917064012fbb82eb2e6321d4ab9c266671e2831

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 2026-09-29T14:50:01.820268+00:00.

Case digest / 83bfa68929badb989419bc83ff5efa1da38c8822d822fbaab4ab9df4fe1676b3