FA-81331 / Music interval and transposition theory / Open access

Triad and seventh inversion figures: inversion figures swapped · case 01

First-inversion triads are labelled 64 and second-inversion triads 6.

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

ROOT CAUSE

The triad figure list is ordered with the second-inversion figure before the first.

THE FAILURE

The triad figure list is ordered with the second-inversion figure before the first.

Unsuccessful approach: Writing 46 instead of 64 corrupts the second-inversion figure.

Case contract

Input 3 or 4 MIDI integers in any order. The pitch-class set must match a template relative to some root (triads 047 037 036 048; sevenths 047t 037t 047e 036t 0369, t=10 e=11); roots are tried in ascending pitch-class order. The bass is the lowest MIDI note. Return [root_pc, figure] with figures "", "6", "64" for triads and "7", "65", "43", "42" for sevenths by the bass position in the template; otherwise None.

Why this case matters

Figured-bass realisation and harmonic analysis depend on identifying chord root and inversion.

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 not 3 <= len(x) <= 4 or not all(isinstance(v, int) for v in x):
        return None
    bass = min(x)
    pcs = sorted(set(v % 12 for v in x))
    for r in pcs:
        rel = sorted((p - r) % 12 for p in pcs)
        if rel in ([0, 4, 7], [0, 3, 7], [0, 3, 6], [0, 4, 8], [0, 4, 7, 10], [0, 3, 7, 10], [0, 4, 7, 11], [0, 3, 6, 10], [0, 3, 6, 9]):
            pos = rel.index((bass - r) % 12)
            figs = ['', '64', '6'] if len(rel) == 3 else ['7', '65', '43', '42']
            return [r, figs[pos]]
    return None
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([60, 64, 67], [0, '']), ([64, 67, 72], [0, '6']), ([67, 72, 76], [0, '64']), ([72, 64, 67], [0, '6']), ([76, 60, 67], [0, '']), ([55, 59, 62, 65], [7, '7']), ([59, 62, 65, 67], [7, '65']), ([62, 65, 67, 71], [7, '43'])], [([67, 72, 76], [0, '64']), ([59, 62, 65, 67], [7, '65']), ([62, 65, 67, 71], [7, '43']), ([65, 67, 71, 74], [7, '42']), ([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, '']), ([60, 63, 66, 69], [0, '7']), ([71, 62, 65], [11, '6'])], [([64, 67, 72], [0, '6']), ([67, 72, 76], [0, '64']), ([72, 64, 67], [0, '6']), ([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, '']), ([60, 63, 66, 69], [0, '7']), ([62, 65, 68, 71], [2, '7']), ([59, 62, 65, 69], [11, '7'])], [([67, 72, 76], [0, '64']), ([62, 65, 68, 71], [2, '7']), ([59, 62, 65, 69], [11, '7']), ([57, 60, 64], [9, '']), ([60, 62, 67], None), ([60, 64, 68], [0, '']), ([64, 60, 67], [0, '']), ([71, 62, 65], [11, '6'])], [([64, 67, 72], [0, '6']), ([67, 72, 76], [0, '64']), ([72, 64, 67], [0, '6']), ([60, 62, 67], None), ([60, 64, 68], [0, '']), ([64, 60, 67], [0, '']), ([60, 61, 62], None), ([60, 64], None)]]
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	`[0, '']`	`[0, '']`	Passed
oracle 1	`[0, '64']`	`[0, '6']`	Failed
oracle 2	`[0, '6']`	`[0, '64']`	Failed
oracle 3	`[0, '64']`	`[0, '6']`	Failed
oracle 4	`[0, '']`	`[0, '']`	Passed
oracle 5	`[7, '7']`	`[7, '7']`	Passed
oracle 6	`[7, '65']`	`[7, '65']`	Passed
oracle 7	`[7, '43']`	`[7, '43']`	Passed

SHA-256 / 1c25d0cf63b69e8f54fc40a6ea07e00edbc2eab6479bbe744845f9ed2c81a7e1

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 not 3 <= len(x) <= 4 or not all(isinstance(v, int) for v in x):
        return None
    bass = min(x)
    pcs = sorted(set(v % 12 for v in x))
    for r in pcs:
        rel = sorted((p - r) % 12 for p in pcs)
        if rel in ([0, 4, 7], [0, 3, 7], [0, 3, 6], [0, 4, 8], [0, 4, 7, 10], [0, 3, 7, 10], [0, 4, 7, 11], [0, 3, 6, 10], [0, 3, 6, 9]):
            pos = rel.index((bass - r) % 12)
            figs = ['', '6', '46'] if len(rel) == 3 else ['7', '65', '43', '42']
            return [r, figs[pos]]
    return None
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([60, 64, 67], [0, '']), ([64, 67, 72], [0, '6']), ([67, 72, 76], [0, '64']), ([72, 64, 67], [0, '6']), ([76, 60, 67], [0, '']), ([55, 59, 62, 65], [7, '7']), ([59, 62, 65, 67], [7, '65']), ([62, 65, 67, 71], [7, '43'])], [([67, 72, 76], [0, '64']), ([59, 62, 65, 67], [7, '65']), ([62, 65, 67, 71], [7, '43']), ([65, 67, 71, 74], [7, '42']), ([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, '']), ([60, 63, 66, 69], [0, '7']), ([71, 62, 65], [11, '6'])], [([64, 67, 72], [0, '6']), ([67, 72, 76], [0, '64']), ([72, 64, 67], [0, '6']), ([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, '']), ([60, 63, 66, 69], [0, '7']), ([62, 65, 68, 71], [2, '7']), ([59, 62, 65, 69], [11, '7'])], [([67, 72, 76], [0, '64']), ([62, 65, 68, 71], [2, '7']), ([59, 62, 65, 69], [11, '7']), ([57, 60, 64], [9, '']), ([60, 62, 67], None), ([60, 64, 68], [0, '']), ([64, 60, 67], [0, '']), ([71, 62, 65], [11, '6'])], [([64, 67, 72], [0, '6']), ([67, 72, 76], [0, '64']), ([72, 64, 67], [0, '6']), ([60, 62, 67], None), ([60, 64, 68], [0, '']), ([64, 60, 67], [0, '']), ([60, 61, 62], None), ([60, 64], None)]]
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	`[0, '']`	`[0, '']`	Passed
oracle 1	`[0, '6']`	`[0, '6']`	Passed
oracle 2	`[0, '46']`	`[0, '64']`	Failed
oracle 3	`[0, '6']`	`[0, '6']`	Passed
oracle 4	`[0, '']`	`[0, '']`	Passed
oracle 5	`[7, '7']`	`[7, '7']`	Passed
oracle 6	`[7, '65']`	`[7, '65']`	Passed
oracle 7	`[7, '43']`	`[7, '43']`	Passed

SHA-256 / 963f10c23be12540dcdd21ca9f191cd128a1aa1ce295b318d4e3afe686f3f6de

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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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 2026-09-29T14:50:01.904376+00:00.

Case digest / fea22ad6c66c9f59450af955971d54f307edee3c6bac6fe7b5b380d7937e8543