FA-81326 / Music interval and transposition theory / Open access

Triad and seventh inversion figures: doubled triad labelled as a seventh chord · case 01

A four-note triad with a doubled root (C E G C) is labelled with seventh figures ("7") instead of root position.

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

ROOT CAUSE

Triad versus seventh figures are chosen by the number of input notes rather than distinct pitch classes.

VERIFIED REPAIR

Restore the chord size basis step so that it reads `len(rel) == 3`.

Unsuccessful approach: Counting distinct MIDI numbers still treats octave doublings as extra chord members.

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 = ['', '6', '64'] if len(x) == 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']), ([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, ''])], [([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']), ([65, 67, 71, 74], [7, '42']), ([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, ''])], [([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']), ([62, 65, 68, 71], [2, '7']), ([59, 62, 65, 69], [11, '7'])], [([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']), ([57, 60, 64], [9, '']), ([60, 62, 67], None), ([60, 64, 68], [0, ''])], [([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, '']), ([57, 60, 64], [9, '']), ([60, 62, 67], None), ([60, 64, 68], [0, '']), ([64, 60, 67], [0, '']), ([71, 62, 65], [11, '6']), ([60, 61, 62], 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, '64']`	`[0, '64']`	Passed
oracle 3	`[0, '6']`	`[0, '6']`	Passed
oracle 4	`[0, '']`	`[0, '']`	Passed
oracle 5	`[7, '7']`	`[7, '7']`	Passed
oracle 6	`[0, '7']`	`[0, '']`	Failed
oracle 7	`[0, '7']`	`[0, '']`	Failed

SHA-256 / 774378b3109df7ece77c0293766a0d06df858a5b0f88c41abd2560eaccbce557

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', '64'] if len(set(x)) == 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']), ([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, ''])], [([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']), ([65, 67, 71, 74], [7, '42']), ([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, ''])], [([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']), ([62, 65, 68, 71], [2, '7']), ([59, 62, 65, 69], [11, '7'])], [([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']), ([57, 60, 64], [9, '']), ([60, 62, 67], None), ([60, 64, 68], [0, ''])], [([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, '']), ([57, 60, 64], [9, '']), ([60, 62, 67], None), ([60, 64, 68], [0, '']), ([64, 60, 67], [0, '']), ([71, 62, 65], [11, '6']), ([60, 61, 62], 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, '64']`	`[0, '64']`	Passed
oracle 3	`[0, '6']`	`[0, '6']`	Passed
oracle 4	`[0, '']`	`[0, '']`	Passed
oracle 5	`[7, '7']`	`[7, '7']`	Passed
oracle 6	`[0, '7']`	`[0, '']`	Failed
oracle 7	`[0, '7']`	`[0, '']`	Failed

SHA-256 / f168846a4f74747d45518adc06f4d53e1071265f4da28474b9164af9ced09b73

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 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', '64'] 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']), ([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, ''])], [([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']), ([65, 67, 71, 74], [7, '42']), ([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, ''])], [([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']), ([62, 65, 68, 71], [2, '7']), ([59, 62, 65, 69], [11, '7'])], [([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']), ([57, 60, 64], [9, '']), ([60, 62, 67], None), ([60, 64, 68], [0, ''])], [([60, 64, 67, 72], [0, '']), ([48, 64, 67, 72], [0, '']), ([57, 60, 64], [9, '']), ([60, 62, 67], None), ([60, 64, 68], [0, '']), ([64, 60, 67], [0, '']), ([71, 62, 65], [11, '6']), ([60, 61, 62], 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, '64']`	`[0, '64']`	Passed
oracle 3	`[0, '6']`	`[0, '6']`	Passed
oracle 4	`[0, '']`	`[0, '']`	Passed
oracle 5	`[7, '7']`	`[7, '7']`	Passed
oracle 6	`[0, '']`	`[0, '']`	Passed
oracle 7	`[0, '']`	`[0, '']`	Passed

SHA-256 / 7fc363c9b38a67473140dd9969177d4c7294dc79ae0f705c08a75427ac430f37

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 2026-09-29T14:50:01.890491+00:00.

Case digest / 8877a1ce323ee94d4452b249efb988511c4439cae77856b7d21e4992ed3731fa