FA-81271 / Music interval and transposition theory / Open access

Diatonic step transposition within a major key: chromatic notes snapped to the scale · case 01

A C# in C major is silently lowered to C and transposed instead of rejecting the input.

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

ROOT CAUSE

Non-scale notes are snapped down a semitone rather than rejected.

VERIFIED REPAIR

Restore the chromatic rejection step so that it reads `if rel not in scale: return None`.

Unsuccessful approach: Skipping chromatic notes drops them from the output instead of rejecting the melody.

Case contract

Input [tonic, notes, steps] where tonic is a MIDI reference note. Each note must belong to the major scale on the tonic (else return None). Its scale position is its degree plus 7 per octave counted from the tonic note with floor division (notes below the tonic note are in negative octaves). Move by the signed number of steps and return the resulting MIDI notes.

Why this case matters

Harmonisers and sequence generators transpose melodies diatonically rather than chromatically.

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) != 3:
        return None
    tonic, notes, steps = x
    scale = [0, 2, 4, 5, 7, 9, 11]
    out = []
    for n in notes:
        rel = (n - tonic) % 12
        if rel not in scale:
            rel -= 1
        octave = (n - tonic) // 12
        pos = octave * 7 + scale.index(rel) + steps
        out.append(tonic + 12 * (pos // 7) + scale[pos % 7])
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([60, [60, 62, 64], 2], [64, 65, 67]), ([60, [60, 64, 67], 1], [62, 65, 69]), ([60, [71], 1], [72]), ([60, [60], -1], [59]), ([60, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([60, [61], 1], None)], [([60, [60], -1], [59]), ([60, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50]), ([60, [61], 1], None), ([60, [60, 65], 3], [65, 71])], [([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50]), ([60, [61], 1], None), ([60, [60, 65], 3], [65, 71]), ([62, [62, 66], -7], [50, 54]), ([60, [72, 74], 7], [84, 86]), ([65, [53, 58], 4], [60, 65])], [([60, [61], 1], None), ([60, [60, 65], 3], [65, 71]), ([62, [62, 66], -7], [50, 54]), ([60, [72, 74], 7], [84, 86]), ([65, [53, 58], 4], [60, 65]), ([60, [], 3], []), ([60, [64], -9], [48]), ([60, [57], -2], [53])], [([60, [61], 1], None), ([65, [53, 58], 4], [60, 65]), ([60, [], 3], []), ([60, [64], -9], [48]), ([60, [57], -2], [53]), ([69, [57, 56], 1], [59, 57]), ([60, [65, 71], 0], [65, 71]), ([60, [40, 41], 10], [57, 59])]]
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	`[64, 65, 67]`	`[64, 65, 67]`	Passed
oracle 1	`[62, 65, 69]`	`[62, 65, 69]`	Passed
oracle 2	`[72]`	`[72]`	Passed
oracle 3	`[59]`	`[59]`	Passed
oracle 4	`[62, 60]`	`[62, 60]`	Passed
oracle 5	`[43, 45]`	`[43, 45]`	Passed
oracle 6	`[71, 74, 78]`	`[71, 74, 78]`	Passed
oracle 7	`[62]`	`None`	Failed

SHA-256 / 24173a8e6758ceae9083835bb250e7cd8de15eee3e74138cec659cf23d026b12

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) != 3:
        return None
    tonic, notes, steps = x
    scale = [0, 2, 4, 5, 7, 9, 11]
    out = []
    for n in notes:
        rel = (n - tonic) % 12
        if rel not in scale:
            continue
        octave = (n - tonic) // 12
        pos = octave * 7 + scale.index(rel) + steps
        out.append(tonic + 12 * (pos // 7) + scale[pos % 7])
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([60, [60, 62, 64], 2], [64, 65, 67]), ([60, [60, 64, 67], 1], [62, 65, 69]), ([60, [71], 1], [72]), ([60, [60], -1], [59]), ([60, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([60, [61], 1], None)], [([60, [60], -1], [59]), ([60, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50]), ([60, [61], 1], None), ([60, [60, 65], 3], [65, 71])], [([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50]), ([60, [61], 1], None), ([60, [60, 65], 3], [65, 71]), ([62, [62, 66], -7], [50, 54]), ([60, [72, 74], 7], [84, 86]), ([65, [53, 58], 4], [60, 65])], [([60, [61], 1], None), ([60, [60, 65], 3], [65, 71]), ([62, [62, 66], -7], [50, 54]), ([60, [72, 74], 7], [84, 86]), ([65, [53, 58], 4], [60, 65]), ([60, [], 3], []), ([60, [64], -9], [48]), ([60, [57], -2], [53])], [([60, [61], 1], None), ([65, [53, 58], 4], [60, 65]), ([60, [], 3], []), ([60, [64], -9], [48]), ([60, [57], -2], [53]), ([69, [57, 56], 1], [59, 57]), ([60, [65, 71], 0], [65, 71]), ([60, [40, 41], 10], [57, 59])]]
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	`[64, 65, 67]`	`[64, 65, 67]`	Passed
oracle 1	`[62, 65, 69]`	`[62, 65, 69]`	Passed
oracle 2	`[72]`	`[72]`	Passed
oracle 3	`[59]`	`[59]`	Passed
oracle 4	`[62, 60]`	`[62, 60]`	Passed
oracle 5	`[43, 45]`	`[43, 45]`	Passed
oracle 6	`[71, 74, 78]`	`[71, 74, 78]`	Passed
oracle 7	`[]`	`None`	Failed

SHA-256 / 67ea2b9ed47cd40f7ecdaa9235d5f587426b0cf78ff51b25eb88773e23c7ed9f

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) != 3:
        return None
    tonic, notes, steps = x
    scale = [0, 2, 4, 5, 7, 9, 11]
    out = []
    for n in notes:
        rel = (n - tonic) % 12
        if rel not in scale:
            return None
        octave = (n - tonic) // 12
        pos = octave * 7 + scale.index(rel) + steps
        out.append(tonic + 12 * (pos // 7) + scale[pos % 7])
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([60, [60, 62, 64], 2], [64, 65, 67]), ([60, [60, 64, 67], 1], [62, 65, 69]), ([60, [71], 1], [72]), ([60, [60], -1], [59]), ([60, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([60, [61], 1], None)], [([60, [60], -1], [59]), ([60, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50]), ([60, [61], 1], None), ([60, [60, 65], 3], [65, 71])], [([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50]), ([60, [61], 1], None), ([60, [60, 65], 3], [65, 71]), ([62, [62, 66], -7], [50, 54]), ([60, [72, 74], 7], [84, 86]), ([65, [53, 58], 4], [60, 65])], [([60, [61], 1], None), ([60, [60, 65], 3], [65, 71]), ([62, [62, 66], -7], [50, 54]), ([60, [72, 74], 7], [84, 86]), ([65, [53, 58], 4], [60, 65]), ([60, [], 3], []), ([60, [64], -9], [48]), ([60, [57], -2], [53])], [([60, [61], 1], None), ([65, [53, 58], 4], [60, 65]), ([60, [], 3], []), ([60, [64], -9], [48]), ([60, [57], -2], [53]), ([69, [57, 56], 1], [59, 57]), ([60, [65, 71], 0], [65, 71]), ([60, [40, 41], 10], [57, 59])]]
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	`[64, 65, 67]`	`[64, 65, 67]`	Passed
oracle 1	`[62, 65, 69]`	`[62, 65, 69]`	Passed
oracle 2	`[72]`	`[72]`	Passed
oracle 3	`[59]`	`[59]`	Passed
oracle 4	`[62, 60]`	`[62, 60]`	Passed
oracle 5	`[43, 45]`	`[43, 45]`	Passed
oracle 6	`[71, 74, 78]`	`[71, 74, 78]`	Passed
oracle 7	`None`	`None`	Passed

SHA-256 / 987978e0b0224ed3ed01a585d1891e7c979061fc1a7e6516d3c5b5d67bafa6f1

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 2026-09-29T14:50:01.395318+00:00.

Case digest / 2ee33e89835e6c882db3743267d6cb0b32c6c3d128716fdb61646c7b0310ffa4