FA-81141 / Music interval and transposition theory / Open access

Written key signature for a transposing part: transposition counted in fourths · case 01

A part written a whole step up from C major gets ten sharps-worth reduced to two flats instead of two sharps.

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

ROOT CAUSE

The key movement multiplies the semitone shift by five (fourths) instead of seven (fifths).

VERIFIED REPAIR

Restore the fifths per semitone step so that it reads `(7 * shift) % 12`.

Unsuccessful approach: Using the raw semitone count as a fifths count only works for shifts of 0 and 12.

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 or concert > 7:
        return None
    move = (5 * 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), ([-3, 2], -1), ([0, 9], 3), ([-6, 0], 6), ([6, 0], 6), ([7, 0], -5), ([-7, 0], 5), ([0, 1], -5)], [([0, -2], -2), ([0, 9], 3), ([0, 3], -3), ([-7, 0], 5), ([0, 0], 0), ([0, -1], 5), ([0, 12], 0), ([8, 0], None)], [([-3, 2], -1), ([0, 3], -3), ([0, 7], 1), ([-2, 9], 1), ([8, 0], None), ([0, '2'], None), ([-8, 0], None), ([-9, 2], None)], [([0, 2], 2), ([0, -2], -2), ([-3, 2], -1), ([0, 9], 3), ([0, 7], 1), ([0, 1], -5), ([3, -3], 6), ([-9, 2], None)], [([-3, 2], -1), ([0, 9], 3), ([0, 3], -3), ([0, 7], 1), ([0, 14], 2), ([1, 2], 3), ([0, -1], 5), ([-5, 7], -4)]]
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`	Failed
oracle 1	`-5`	`-1`	Failed
oracle 2	`-3`	`3`	Failed
oracle 3	`6`	`6`	Passed
oracle 4	`6`	`6`	Passed
oracle 5	`-5`	`-5`	Passed
oracle 6	`5`	`5`	Passed
oracle 7	`5`	`-5`	Failed

SHA-256 / aee580dbebc3728b5f74f44ff63743a07c2193836a6a33ad17212cfab3db6cdf

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 or concert > 7:
        return None
    move = 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), ([-3, 2], -1), ([0, 9], 3), ([-6, 0], 6), ([6, 0], 6), ([7, 0], -5), ([-7, 0], 5), ([0, 1], -5)], [([0, -2], -2), ([0, 9], 3), ([0, 3], -3), ([-7, 0], 5), ([0, 0], 0), ([0, -1], 5), ([0, 12], 0), ([8, 0], None)], [([-3, 2], -1), ([0, 3], -3), ([0, 7], 1), ([-2, 9], 1), ([8, 0], None), ([0, '2'], None), ([-8, 0], None), ([-9, 2], None)], [([0, 2], 2), ([0, -2], -2), ([-3, 2], -1), ([0, 9], 3), ([0, 7], 1), ([0, 1], -5), ([3, -3], 6), ([-9, 2], None)], [([-3, 2], -1), ([0, 9], 3), ([0, 3], -3), ([0, 7], 1), ([0, 14], 2), ([1, 2], 3), ([0, -1], 5), ([-5, 7], -4)]]
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	`-1`	`-1`	Passed
oracle 2	`-3`	`3`	Failed
oracle 3	`6`	`6`	Passed
oracle 4	`6`	`6`	Passed
oracle 5	`-5`	`-5`	Passed
oracle 6	`5`	`5`	Passed
oracle 7	`1`	`-5`	Failed

SHA-256 / cac457ca68952785d6220b5c41b1ca8d6cbdbbc315aec97edb966da3f4c6339c

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), ([-3, 2], -1), ([0, 9], 3), ([-6, 0], 6), ([6, 0], 6), ([7, 0], -5), ([-7, 0], 5), ([0, 1], -5)], [([0, -2], -2), ([0, 9], 3), ([0, 3], -3), ([-7, 0], 5), ([0, 0], 0), ([0, -1], 5), ([0, 12], 0), ([8, 0], None)], [([-3, 2], -1), ([0, 3], -3), ([0, 7], 1), ([-2, 9], 1), ([8, 0], None), ([0, '2'], None), ([-8, 0], None), ([-9, 2], None)], [([0, 2], 2), ([0, -2], -2), ([-3, 2], -1), ([0, 9], 3), ([0, 7], 1), ([0, 1], -5), ([3, -3], 6), ([-9, 2], None)], [([-3, 2], -1), ([0, 9], 3), ([0, 3], -3), ([0, 7], 1), ([0, 14], 2), ([1, 2], 3), ([0, -1], 5), ([-5, 7], -4)]]
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	`-1`	`-1`	Passed
oracle 2	`3`	`3`	Passed
oracle 3	`6`	`6`	Passed
oracle 4	`6`	`6`	Passed
oracle 5	`-5`	`-5`	Passed
oracle 6	`5`	`5`	Passed
oracle 7	`-5`	`-5`	Passed

SHA-256 / f956c9f6882f0b9f3bada7dd39b676102968b7a3166b8427bcf291dc9a4dee41

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 2026-09-29T14:50:00.291643+00:00.

Case digest / e054c8ff30fe193135e182c64a592144ed6cc2ef0b595806f6812f6a7d7f647e