FA-81196 / Music interval and transposition theory / Open access

Roman numeral label for a chord in a key: diminished quality marker dropped · case 01

The leading-tone chord in C major is labelled vii with no diminished marker.

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

ROOT CAUSE

The suffix table maps diminished quality to an empty marker.

VERIFIED REPAIR

Restore the diminished suffix step so that it reads `'dim': '°'`.

Unsuccessful approach: Substituting a letter o for the degree sign does not match the stipulated ° marker.

Case contract

Input [tonic_pc, mode, root_pc, quality] with mode major or minor (natural minor scale 0 2 3 5 7 8 10) and quality maj, min, dim or aug. Diatonic roots use their degree; in minor the leading tone (11 semitones) is degree VII with no prefix. Other roots in major are written b + the degree a semitone above; in minor # + the degree a semitone below. Numerals are uppercase for maj and aug, lowercase for min and dim, with suffix ° for dim and + for aug. Invalid input returns None.

Why this case matters

Harmonic analysis and ear-training tools label chords relative to the key.

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) != 4:
        return None
    tonic, mode, root, quality = x
    scales = {'major': [0, 2, 4, 5, 7, 9, 11], 'minor': [0, 2, 3, 5, 7, 8, 10]}
    if mode not in scales or quality not in ('maj', 'min', 'dim', 'aug'):
        return None
    if not all(isinstance(v, int) for v in (tonic, root)):
        return None
    rel = (root - tonic) % 12
    scale = scales[mode]
    names = ['I', 'II', 'III', 'IV', 'V', 'VI', 'VII']
    prefix = ''
    if rel in scale:
        deg = scale.index(rel)
    elif mode == 'minor' and rel == 11:
        deg = 6
    elif mode == 'major':
        prefix = 'b'
        deg = scale.index((rel + 1) % 12)
    else:
        prefix = '#'
        deg = scale.index((rel - 1) % 12)
    numeral = names[deg] if quality in ('maj', 'aug') else names[deg].lower()
    suffix = {'maj': '', 'min': '', 'dim': '', 'aug': '+'}[quality]
    return prefix + numeral + suffix
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 'major', 7, 'maj'], 'V'), ([0, 'major', 2, 'min'], 'ii'), ([0, 'major', 11, 'dim'], 'vii°'), ([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 3, 'dim'], '#iv°')], [([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 3, 'maj'], 'bIII'), ([0, 'major', 6, 'dim'], 'bv°'), ([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 11, 'dim'], 'vii°')], [([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 7, 'maj'], 'VII'), ([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 1, 'maj'], '#III'), ([0, 'minor', 11, 'dim'], 'vii°'), ([2, 'major', 1, 'dim'], 'vii°')], [([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 1, 'maj'], '#III'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 9, 'min'], '#vi'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([2, 'major', 1, 'dim'], 'vii°'), ([5, 'major', 4, 'dim'], 'vii°')], [([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([7, 'major', 2, 'maj'], 'V'), ([7, 'major', 0, 'maj'], 'IV'), ([0, 'major', 0, 'aug'], 'I+'), ([5, 'major', 4, 'dim'], 'vii°'), ([5, 'minor', 4, 'dim'], 'vii°')]]
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	`V`	`V`	Passed
oracle 1	`ii`	`ii`	Passed
oracle 2	`vii`	`vii°`	Failed
oracle 3	`bVI`	`bVI`	Passed
oracle 4	`bVII`	`bVII`	Passed
oracle 5	`bII`	`bII`	Passed
oracle 6	`vii`	`vii°`	Failed
oracle 7	`#iv`	`#iv°`	Failed

SHA-256 / dba3e10d8f75b05ddc6c067c69a61dff7d593d21636a97f9096a3617ea75a0db

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) != 4:
        return None
    tonic, mode, root, quality = x
    scales = {'major': [0, 2, 4, 5, 7, 9, 11], 'minor': [0, 2, 3, 5, 7, 8, 10]}
    if mode not in scales or quality not in ('maj', 'min', 'dim', 'aug'):
        return None
    if not all(isinstance(v, int) for v in (tonic, root)):
        return None
    rel = (root - tonic) % 12
    scale = scales[mode]
    names = ['I', 'II', 'III', 'IV', 'V', 'VI', 'VII']
    prefix = ''
    if rel in scale:
        deg = scale.index(rel)
    elif mode == 'minor' and rel == 11:
        deg = 6
    elif mode == 'major':
        prefix = 'b'
        deg = scale.index((rel + 1) % 12)
    else:
        prefix = '#'
        deg = scale.index((rel - 1) % 12)
    numeral = names[deg] if quality in ('maj', 'aug') else names[deg].lower()
    suffix = {'maj': '', 'min': '', 'dim': 'o', 'aug': '+'}[quality]
    return prefix + numeral + suffix
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 'major', 7, 'maj'], 'V'), ([0, 'major', 2, 'min'], 'ii'), ([0, 'major', 11, 'dim'], 'vii°'), ([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 3, 'dim'], '#iv°')], [([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 3, 'maj'], 'bIII'), ([0, 'major', 6, 'dim'], 'bv°'), ([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 11, 'dim'], 'vii°')], [([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 7, 'maj'], 'VII'), ([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 1, 'maj'], '#III'), ([0, 'minor', 11, 'dim'], 'vii°'), ([2, 'major', 1, 'dim'], 'vii°')], [([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 1, 'maj'], '#III'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 9, 'min'], '#vi'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([2, 'major', 1, 'dim'], 'vii°'), ([5, 'major', 4, 'dim'], 'vii°')], [([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([7, 'major', 2, 'maj'], 'V'), ([7, 'major', 0, 'maj'], 'IV'), ([0, 'major', 0, 'aug'], 'I+'), ([5, 'major', 4, 'dim'], 'vii°'), ([5, 'minor', 4, 'dim'], 'vii°')]]
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	`V`	`V`	Passed
oracle 1	`ii`	`ii`	Passed
oracle 2	`viio`	`vii°`	Failed
oracle 3	`bVI`	`bVI`	Passed
oracle 4	`bVII`	`bVII`	Passed
oracle 5	`bII`	`bII`	Passed
oracle 6	`viio`	`vii°`	Failed
oracle 7	`#ivo`	`#iv°`	Failed

SHA-256 / 8145fe4035bbdc4226ec4a42a1916d757459b951c38e03da0455509e5da82a7b

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) != 4:
        return None
    tonic, mode, root, quality = x
    scales = {'major': [0, 2, 4, 5, 7, 9, 11], 'minor': [0, 2, 3, 5, 7, 8, 10]}
    if mode not in scales or quality not in ('maj', 'min', 'dim', 'aug'):
        return None
    if not all(isinstance(v, int) for v in (tonic, root)):
        return None
    rel = (root - tonic) % 12
    scale = scales[mode]
    names = ['I', 'II', 'III', 'IV', 'V', 'VI', 'VII']
    prefix = ''
    if rel in scale:
        deg = scale.index(rel)
    elif mode == 'minor' and rel == 11:
        deg = 6
    elif mode == 'major':
        prefix = 'b'
        deg = scale.index((rel + 1) % 12)
    else:
        prefix = '#'
        deg = scale.index((rel - 1) % 12)
    numeral = names[deg] if quality in ('maj', 'aug') else names[deg].lower()
    suffix = {'maj': '', 'min': '', 'dim': '°', 'aug': '+'}[quality]
    return prefix + numeral + suffix
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 'major', 7, 'maj'], 'V'), ([0, 'major', 2, 'min'], 'ii'), ([0, 'major', 11, 'dim'], 'vii°'), ([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 3, 'dim'], '#iv°')], [([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 3, 'maj'], 'bIII'), ([0, 'major', 6, 'dim'], 'bv°'), ([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 11, 'dim'], 'vii°')], [([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 7, 'maj'], 'VII'), ([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 1, 'maj'], '#III'), ([0, 'minor', 11, 'dim'], 'vii°'), ([2, 'major', 1, 'dim'], 'vii°')], [([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 1, 'maj'], '#III'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 9, 'min'], '#vi'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([2, 'major', 1, 'dim'], 'vii°'), ([5, 'major', 4, 'dim'], 'vii°')], [([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([7, 'major', 2, 'maj'], 'V'), ([7, 'major', 0, 'maj'], 'IV'), ([0, 'major', 0, 'aug'], 'I+'), ([5, 'major', 4, 'dim'], 'vii°'), ([5, 'minor', 4, 'dim'], 'vii°')]]
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	`V`	`V`	Passed
oracle 1	`ii`	`ii`	Passed
oracle 2	`vii°`	`vii°`	Passed
oracle 3	`bVI`	`bVI`	Passed
oracle 4	`bVII`	`bVII`	Passed
oracle 5	`bII`	`bII`	Passed
oracle 6	`vii°`	`vii°`	Passed
oracle 7	`#iv°`	`#iv°`	Passed

SHA-256 / face72c9b0b32cbca9c1c93c9614455cfb5aac34799451f5cdb715192a08b089

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 2026-09-29T14:50:00.640400+00:00.

Case digest / d45bd121644a88386c9b599d479d49b4e367845e0f19c633b4d625b049fd357b