FA-81156 / Music interval and transposition theory / Open access

Written key signature for a transposing part: impossible flat concert keys accepted · case 01

A concert signature of -8 flats is accepted and transposed instead of rejected.

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

ROOT CAUSE

Only the sharp-side limit of the concert signature is checked.

VERIFIED REPAIR

Restore the range validation step so that it reads `concert < -7 or concert > 7`.

Unsuccessful approach: Checking only the flat side accepts eight or more sharps.

Case contract

Input [concert_fifths, written_semitones_above_concert] with concert_fifths -7..7 and an integer semitone shift. The key moves by (7 * shift) mod 12 fifths; the written signature is concert_fifths plus that move, then normalised into -5..6 by adding or subtracting 12 (so six flats is written as six sharps). Invalid input returns None.

Why this case matters

Part extraction must print a transposed key signature consistent with the transposed notes.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(x):
    if not isinstance(x, list) or len(x) != 2:
        return None
    concert, shift = x
    if not isinstance(concert, int) or not isinstance(shift, int) or concert > 7:
        return None
    move = (7 * shift) % 12
    r = concert + move
    while r > 6:
        r -= 12
    while r < -5:
        r += 12
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 2], 2), ([0, -2], -2), ([-3, 2], -1), ([0, 9], 3), ([0, 3], -3), ([0, 7], 1), ([8, 0], None), ([-8, 0], None)], [([0, 9], 3), ([0, 3], -3), ([0, 7], 1), ([0, 14], 2), ([1, 2], 3), ([-1, 2], 1), ([8, 0], None), ([-9, 2], None)], [([0, 14], 2), ([1, 2], 3), ([-1, 2], 1), ([4, 2], 6), ([5, 2], -5), ([-6, 0], 6), ([8, 0], None), ([-8, 0], None)], [([4, 2], 6), ([5, 2], -5), ([-6, 0], 6), ([6, 0], 6), ([7, 0], -5), ([-7, 0], 5), ([8, 0], None), ([-9, 2], None)], [([6, 0], 6), ([7, 0], -5), ([-7, 0], 5), ([0, 0], 0), ([2, -2], 0), ([0, 1], -5), ([8, 0], None), ([-8, 0], None)]]
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	`2`	`2`	Passed
oracle 1	`-2`	`-2`	Passed
oracle 2	`-1`	`-1`	Passed
oracle 3	`3`	`3`	Passed
oracle 4	`-3`	`-3`	Passed
oracle 5	`1`	`1`	Passed
oracle 6	`None`	`None`	Passed
oracle 7	`4`	`None`	Failed

SHA-256 / 5417592ca73cc1da596340c28adc2f7f6ef0f6f2da70a5a81c93f5a09fbddb86

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(x):
    if not isinstance(x, list) or len(x) != 2:
        return None
    concert, shift = x
    if not isinstance(concert, int) or not isinstance(shift, int) or concert < -7:
        return None
    move = (7 * shift) % 12
    r = concert + move
    while r > 6:
        r -= 12
    while r < -5:
        r += 12
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 2], 2), ([0, -2], -2), ([-3, 2], -1), ([0, 9], 3), ([0, 3], -3), ([0, 7], 1), ([8, 0], None), ([-8, 0], None)], [([0, 9], 3), ([0, 3], -3), ([0, 7], 1), ([0, 14], 2), ([1, 2], 3), ([-1, 2], 1), ([8, 0], None), ([-9, 2], None)], [([0, 14], 2), ([1, 2], 3), ([-1, 2], 1), ([4, 2], 6), ([5, 2], -5), ([-6, 0], 6), ([8, 0], None), ([-8, 0], None)], [([4, 2], 6), ([5, 2], -5), ([-6, 0], 6), ([6, 0], 6), ([7, 0], -5), ([-7, 0], 5), ([8, 0], None), ([-9, 2], None)], [([6, 0], 6), ([7, 0], -5), ([-7, 0], 5), ([0, 0], 0), ([2, -2], 0), ([0, 1], -5), ([8, 0], None), ([-8, 0], None)]]
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	`2`	`2`	Passed
oracle 1	`-2`	`-2`	Passed
oracle 2	`-1`	`-1`	Passed
oracle 3	`3`	`3`	Passed
oracle 4	`-3`	`-3`	Passed
oracle 5	`1`	`1`	Passed
oracle 6	`-4`	`None`	Failed
oracle 7	`None`	`None`	Passed

SHA-256 / 9db66d4d9d7406b582e7fc3a70f97c58e3e1a4b02098a04b13513c5a029a1331

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(x):
    if not isinstance(x, list) or len(x) != 2:
        return None
    concert, shift = x
    if not isinstance(concert, int) or not isinstance(shift, int) or concert < -7 or concert > 7:
        return None
    move = (7 * shift) % 12
    r = concert + move
    while r > 6:
        r -= 12
    while r < -5:
        r += 12
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 2], 2), ([0, -2], -2), ([-3, 2], -1), ([0, 9], 3), ([0, 3], -3), ([0, 7], 1), ([8, 0], None), ([-8, 0], None)], [([0, 9], 3), ([0, 3], -3), ([0, 7], 1), ([0, 14], 2), ([1, 2], 3), ([-1, 2], 1), ([8, 0], None), ([-9, 2], None)], [([0, 14], 2), ([1, 2], 3), ([-1, 2], 1), ([4, 2], 6), ([5, 2], -5), ([-6, 0], 6), ([8, 0], None), ([-8, 0], None)], [([4, 2], 6), ([5, 2], -5), ([-6, 0], 6), ([6, 0], 6), ([7, 0], -5), ([-7, 0], 5), ([8, 0], None), ([-9, 2], None)], [([6, 0], 6), ([7, 0], -5), ([-7, 0], 5), ([0, 0], 0), ([2, -2], 0), ([0, 1], -5), ([8, 0], None), ([-8, 0], None)]]
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	`2`	`2`	Passed
oracle 1	`-2`	`-2`	Passed
oracle 2	`-1`	`-1`	Passed
oracle 3	`3`	`3`	Passed
oracle 4	`-3`	`-3`	Passed
oracle 5	`1`	`1`	Passed
oracle 6	`None`	`None`	Passed
oracle 7	`None`	`None`	Passed

SHA-256 / d633811c3951ee456b8665dd3cef9755dbb3e1a69bc75d3dcf9b8ff82aaff06b

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 2026-09-29T14:50:00.529434+00:00.

Case digest / 9a45e6a23123559026d4c02a444b4e5fc26843f841ce19c725aa643e0b0343c7