FA-81191 / Music interval and transposition theory / Open access

Roman numeral label for a chord in a key: leading-tone chord in minor given a sharp prefix · case 01

In A minor the G# diminished chord is labelled #vii° instead of vii°.

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

ROOT CAUSE

The minor-key special case for the raised leading tone is missing, so it falls into the chromatic sharp branch.

VERIFIED REPAIR

Restore the minor leading tone step so that it reads `elif mode == 'minor' and rel == 11: deg = 6`.

Unsuccessful approach: Treating the raised sixth as the leading tone too labels F# minor chords in A minor as vii instead of #vi.

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 == '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'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 6, 'min'], '#vi'), ([0, 'minor', 9, 'min'], '#vi'), ([5, 'minor', 4, 'dim'], 'vii°')], [([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 3, 'maj'], 'bIII'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 6, 'min'], '#vi'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi')], [([0, 'major', 3, 'maj'], 'bIII'), ([0, 'major', 6, 'dim'], 'bv°'), ([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 6, 'min'], '#vi'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi'), ([5, 'minor', 4, 'dim'], 'vii°')], [([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'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 9, 'min'], '#vi'), ([5, 'minor', 4, 'dim'], 'vii°')], [([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([7, 'major', 2, 'maj'], 'V')]]
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	`#vii°`	`vii°`	Failed
oracle 5	`#vi`	`#vi`	Passed
oracle 6	`#vi`	`#vi`	Passed
oracle 7	`#vii°`	`vii°`	Failed

SHA-256 / 504c08b7ea4084ef65b837b470deff0e4069c6b4f72c59adc787525ed1338f6b

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 in (11, 9):
        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'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 6, 'min'], '#vi'), ([0, 'minor', 9, 'min'], '#vi'), ([5, 'minor', 4, 'dim'], 'vii°')], [([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 3, 'maj'], 'bIII'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 6, 'min'], '#vi'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi')], [([0, 'major', 3, 'maj'], 'bIII'), ([0, 'major', 6, 'dim'], 'bv°'), ([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 6, 'min'], '#vi'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi'), ([5, 'minor', 4, 'dim'], 'vii°')], [([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'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 9, 'min'], '#vi'), ([5, 'minor', 4, 'dim'], 'vii°')], [([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([7, 'major', 2, 'maj'], 'V')]]
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	`vii°`	`vii°`	Passed
oracle 5	`vii`	`#vi`	Failed
oracle 6	`vii`	`#vi`	Failed
oracle 7	`vii°`	`vii°`	Passed

SHA-256 / 2b40cd1eaf1f415265df5eb3337335108f4d1f826f5dc63c4a226b3f08aa4372

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'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 6, 'min'], '#vi'), ([0, 'minor', 9, 'min'], '#vi'), ([5, 'minor', 4, 'dim'], 'vii°')], [([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 3, 'maj'], 'bIII'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 6, 'min'], '#vi'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi')], [([0, 'major', 3, 'maj'], 'bIII'), ([0, 'major', 6, 'dim'], 'bv°'), ([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 6, 'min'], '#vi'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi'), ([5, 'minor', 4, 'dim'], 'vii°')], [([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'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 9, 'min'], '#vi'), ([5, 'minor', 4, 'dim'], 'vii°')], [([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([7, 'major', 2, 'maj'], 'V')]]
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	`vii°`	`vii°`	Passed
oracle 5	`#vi`	`#vi`	Passed
oracle 6	`#vi`	`#vi`	Passed
oracle 7	`vii°`	`vii°`	Passed

SHA-256 / e5c149697338caaa406f66b6d7fcf662b528ce139391323bd306e9de910ba275

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 2026-09-29T14:50:00.640063+00:00.

Case digest / 52aa75981fa474fe3571718c5a62b0d0d3c5b5951fd3e499c816bda8fb3b6c37