FA-81261 / Music interval and transposition theory / Open access

Diatonic step transposition within a major key: notes below the tonic truncated toward zero · case 01

Notes a few steps below the tonic jump up an octave after transposition (B3 up two steps in C gives D5).

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

ROOT CAUSE

The octave index truncates toward zero instead of flooring, so notes below the tonic are placed in octave 0.

VERIFIED REPAIR

Restore the octave floor step so that it reads `(n - tonic) // 12`.

Unsuccessful approach: Using the absolute distance places every note below the tonic in a positive octave.

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:
            return None
        octave = int((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]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50])], [([60, [60], -1], [59]), ([60, [48, 50], -3], [43, 45]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [55, 54], -2], [52, 50]), ([60, [61], 1], None), ([60, [60, 65], 3], [65, 71]), ([62, [62, 66], -7], [50, 54]), ([65, [53, 58], 4], [60, 65])], [([67, [66], 1], [67]), ([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, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [55, 54], -2], [52, 50]), ([60, [], 3], []), ([60, [64], -9], [48]), ([60, [57], -2], [53]), ([69, [57, 56], 1], [59, 57]), ([60, [65, 71], 0], [65, 71])], [([60, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50]), ([65, [53, 58], 4], [60, 65]), ([60, [57], -2], [53]), ([69, [57, 56], 1], [59, 57]), ([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	`[74, 72]`	`[62, 60]`	Failed
oracle 5	`[71, 74, 78]`	`[71, 74, 78]`	Passed
oracle 6	`[79]`	`[67]`	Failed
oracle 7	`[52, 62]`	`[52, 50]`	Failed

SHA-256 / d4e72f4c3c57d686d3881087c9d5b9da3833eefeaa5d9fdd3070f26b27b895b0

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:
            return None
        octave = abs(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]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50])], [([60, [60], -1], [59]), ([60, [48, 50], -3], [43, 45]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [55, 54], -2], [52, 50]), ([60, [61], 1], None), ([60, [60, 65], 3], [65, 71]), ([62, [62, 66], -7], [50, 54]), ([65, [53, 58], 4], [60, 65])], [([67, [66], 1], [67]), ([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, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [55, 54], -2], [52, 50]), ([60, [], 3], []), ([60, [64], -9], [48]), ([60, [57], -2], [53]), ([69, [57, 56], 1], [59, 57]), ([60, [65, 71], 0], [65, 71])], [([60, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50]), ([65, [53, 58], 4], [60, 65]), ([60, [57], -2], [53]), ([69, [57, 56], 1], [59, 57]), ([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	`[74, 72]`	`[62, 60]`	Failed
oracle 5	`[71, 74, 78]`	`[71, 74, 78]`	Passed
oracle 6	`[79]`	`[67]`	Failed
oracle 7	`[76, 86]`	`[52, 50]`	Failed

SHA-256 / bae1db2311d5aea28f7d5ea417556bd5e1d284771002f3cccc73316ec8cc219d

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]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50])], [([60, [60], -1], [59]), ([60, [48, 50], -3], [43, 45]), ([67, [67, 71, 74], 2], [71, 74, 78]), ([67, [55, 54], -2], [52, 50]), ([60, [61], 1], None), ([60, [60, 65], 3], [65, 71]), ([62, [62, 66], -7], [50, 54]), ([65, [53, 58], 4], [60, 65])], [([67, [66], 1], [67]), ([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, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [55, 54], -2], [52, 50]), ([60, [], 3], []), ([60, [64], -9], [48]), ([60, [57], -2], [53]), ([69, [57, 56], 1], [59, 57]), ([60, [65, 71], 0], [65, 71])], [([60, [59, 57], 2], [62, 60]), ([60, [48, 50], -3], [43, 45]), ([67, [66], 1], [67]), ([67, [55, 54], -2], [52, 50]), ([65, [53, 58], 4], [60, 65]), ([60, [57], -2], [53]), ([69, [57, 56], 1], [59, 57]), ([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	`[71, 74, 78]`	`[71, 74, 78]`	Passed
oracle 6	`[67]`	`[67]`	Passed
oracle 7	`[52, 50]`	`[52, 50]`	Passed

SHA-256 / 2068f16fd62a5fab0eb4cf58c86414e6d12ea4d34b64f0520496db4fffb23c48

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 2026-09-29T14:50:01.272161+00:00.

Case digest / 533722dfe7241301c30882b9ce731bd6d7b86d3b2aae68ce0a0490c5c7ed4f36