FA-81071 / Music interval and transposition theory / Open access

Interval-class vector of a pitch-class set: self pairs and ordered pairs counted · case 01

Every interval is counted twice and each pitch class paired with itself lands in the tritone slot.

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

ROOT CAUSE

The inner loop visits ordered pairs including i == j instead of unordered distinct pairs.

VERIFIED REPAIR

Restore the pair enumeration step so that it reads `range(i + 1, len(pcs))`.

Unsuccessful approach: Starting the inner loop at i removes the double count but still pairs each member with itself.

Case contract

Input a list of integers, reduced mod 12 and deduplicated. For each unordered pair count its interval class min(d, 12-d) in 1..6. Return the six counts [ic1..ic6]; non-integer input returns None.

Why this case matters

Interval vectors drive set-class similarity measures and Z-relation detection.

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 all(isinstance(v, int) for v in x):
        return None
    pcs = sorted(set(v % 12 for v in x))
    vec = [0] * 6
    for i in range(len(pcs)):
        for j in range(len(pcs)):
            d = (pcs[j] - pcs[i]) % 12
            ic = min(d, 12 - d)
            vec[ic - 1] += 1
    return vec
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 4, 7], [0, 0, 1, 1, 1, 0]), ([0, 1], [1, 0, 0, 0, 0, 0]), ([0, 6], [0, 0, 0, 0, 0, 1]), ([0, 7], [0, 0, 0, 0, 1, 0]), ([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([], [0, 0, 0, 0, 0, 0])], [([0, 1], [1, 0, 0, 0, 0, 0]), ([0, 6], [0, 0, 0, 0, 0, 1]), ([0, 7], [0, 0, 0, 0, 1, 0]), ([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [12, 12, 12, 12, 12, 6]), ([0, 3, 6, 9], [0, 0, 4, 0, 0, 2])], [([0, 6], [0, 0, 0, 0, 0, 1]), ([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [12, 12, 12, 12, 12, 6]), ([0, 3, 6, 9], [0, 0, 4, 0, 0, 2]), ([0, 4, 8], [0, 0, 0, 3, 0, 0]), ([11, -1], [0, 0, 0, 0, 0, 0])], [([0, 7], [0, 0, 0, 0, 1, 0]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 3, 6, 9], [0, 0, 4, 0, 0, 2]), ([0, 4, 8], [0, 0, 0, 3, 0, 0]), ([11, -1], [0, 0, 0, 0, 0, 0]), ([-1, 0, 4], [1, 0, 0, 1, 1, 0]), ([-13, 2], [0, 0, 1, 0, 0, 0])], [([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [12, 12, 12, 12, 12, 6]), ([-1, 0, 4], [1, 0, 0, 1, 1, 0]), ([-13, 2], [0, 0, 1, 0, 0, 0]), ([0, 5, 7], [0, 1, 0, 0, 2, 0]), ([1, 8], [0, 0, 0, 0, 1, 0]), ([0, 2, 7], [0, 1, 0, 0, 2, 0])]]
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, 2, 2, 2, 3]`	`[0, 0, 1, 1, 1, 0]`	Failed
oracle 1	`[2, 0, 0, 0, 0, 2]`	`[1, 0, 0, 0, 0, 0]`	Failed
oracle 2	`[0, 0, 0, 0, 0, 4]`	`[0, 0, 0, 0, 0, 1]`	Failed
oracle 3	`[0, 0, 0, 0, 2, 2]`	`[0, 0, 0, 0, 1, 0]`	Failed
oracle 4	`[2, 2, 2, 2, 2, 6]`	`[1, 1, 1, 1, 1, 1]`	Failed
oracle 5	`[2, 2, 2, 2, 2, 6]`	`[1, 1, 1, 1, 1, 1]`	Failed
oracle 6	`[0, 12, 0, 12, 0, 12]`	`[0, 6, 0, 6, 0, 3]`	Failed
oracle 7	`[0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0]`	Passed

SHA-256 / be68ae9b0ad3e6b8ae602c10984d51e036c8e518ba82733aca999f3a9e54e083

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 all(isinstance(v, int) for v in x):
        return None
    pcs = sorted(set(v % 12 for v in x))
    vec = [0] * 6
    for i in range(len(pcs)):
        for j in range(i, len(pcs)):
            d = (pcs[j] - pcs[i]) % 12
            ic = min(d, 12 - d)
            vec[ic - 1] += 1
    return vec
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 4, 7], [0, 0, 1, 1, 1, 0]), ([0, 1], [1, 0, 0, 0, 0, 0]), ([0, 6], [0, 0, 0, 0, 0, 1]), ([0, 7], [0, 0, 0, 0, 1, 0]), ([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([], [0, 0, 0, 0, 0, 0])], [([0, 1], [1, 0, 0, 0, 0, 0]), ([0, 6], [0, 0, 0, 0, 0, 1]), ([0, 7], [0, 0, 0, 0, 1, 0]), ([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [12, 12, 12, 12, 12, 6]), ([0, 3, 6, 9], [0, 0, 4, 0, 0, 2])], [([0, 6], [0, 0, 0, 0, 0, 1]), ([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [12, 12, 12, 12, 12, 6]), ([0, 3, 6, 9], [0, 0, 4, 0, 0, 2]), ([0, 4, 8], [0, 0, 0, 3, 0, 0]), ([11, -1], [0, 0, 0, 0, 0, 0])], [([0, 7], [0, 0, 0, 0, 1, 0]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 3, 6, 9], [0, 0, 4, 0, 0, 2]), ([0, 4, 8], [0, 0, 0, 3, 0, 0]), ([11, -1], [0, 0, 0, 0, 0, 0]), ([-1, 0, 4], [1, 0, 0, 1, 1, 0]), ([-13, 2], [0, 0, 1, 0, 0, 0])], [([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [12, 12, 12, 12, 12, 6]), ([-1, 0, 4], [1, 0, 0, 1, 1, 0]), ([-13, 2], [0, 0, 1, 0, 0, 0]), ([0, 5, 7], [0, 1, 0, 0, 2, 0]), ([1, 8], [0, 0, 0, 0, 1, 0]), ([0, 2, 7], [0, 1, 0, 0, 2, 0])]]
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, 1, 1, 1, 3]`	`[0, 0, 1, 1, 1, 0]`	Failed
oracle 1	`[1, 0, 0, 0, 0, 2]`	`[1, 0, 0, 0, 0, 0]`	Failed
oracle 2	`[0, 0, 0, 0, 0, 3]`	`[0, 0, 0, 0, 0, 1]`	Failed
oracle 3	`[0, 0, 0, 0, 1, 2]`	`[0, 0, 0, 0, 1, 0]`	Failed
oracle 4	`[1, 1, 1, 1, 1, 5]`	`[1, 1, 1, 1, 1, 1]`	Failed
oracle 5	`[1, 1, 1, 1, 1, 5]`	`[1, 1, 1, 1, 1, 1]`	Failed
oracle 6	`[0, 6, 0, 6, 0, 9]`	`[0, 6, 0, 6, 0, 3]`	Failed
oracle 7	`[0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0]`	Passed

SHA-256 / 3b3cd6aacefeac2c7b66ca161badda37ea7ebda7070383800c2f5ba4e39b0680

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 all(isinstance(v, int) for v in x):
        return None
    pcs = sorted(set(v % 12 for v in x))
    vec = [0] * 6
    for i in range(len(pcs)):
        for j in range(i + 1, len(pcs)):
            d = (pcs[j] - pcs[i]) % 12
            ic = min(d, 12 - d)
            vec[ic - 1] += 1
    return vec
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 4, 7], [0, 0, 1, 1, 1, 0]), ([0, 1], [1, 0, 0, 0, 0, 0]), ([0, 6], [0, 0, 0, 0, 0, 1]), ([0, 7], [0, 0, 0, 0, 1, 0]), ([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([], [0, 0, 0, 0, 0, 0])], [([0, 1], [1, 0, 0, 0, 0, 0]), ([0, 6], [0, 0, 0, 0, 0, 1]), ([0, 7], [0, 0, 0, 0, 1, 0]), ([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [12, 12, 12, 12, 12, 6]), ([0, 3, 6, 9], [0, 0, 4, 0, 0, 2])], [([0, 6], [0, 0, 0, 0, 0, 1]), ([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [12, 12, 12, 12, 12, 6]), ([0, 3, 6, 9], [0, 0, 4, 0, 0, 2]), ([0, 4, 8], [0, 0, 0, 3, 0, 0]), ([11, -1], [0, 0, 0, 0, 0, 0])], [([0, 7], [0, 0, 0, 0, 1, 0]), ([0, 1, 3, 7], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 3, 6, 9], [0, 0, 4, 0, 0, 2]), ([0, 4, 8], [0, 0, 0, 3, 0, 0]), ([11, -1], [0, 0, 0, 0, 0, 0]), ([-1, 0, 4], [1, 0, 0, 1, 1, 0]), ([-13, 2], [0, 0, 1, 0, 0, 0])], [([0, 1, 4, 6], [1, 1, 1, 1, 1, 1]), ([0, 2, 4, 6, 8, 10], [0, 6, 0, 6, 0, 3]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [12, 12, 12, 12, 12, 6]), ([-1, 0, 4], [1, 0, 0, 1, 1, 0]), ([-13, 2], [0, 0, 1, 0, 0, 0]), ([0, 5, 7], [0, 1, 0, 0, 2, 0]), ([1, 8], [0, 0, 0, 0, 1, 0]), ([0, 2, 7], [0, 1, 0, 0, 2, 0])]]
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, 1, 1, 1, 0]`	`[0, 0, 1, 1, 1, 0]`	Passed
oracle 1	`[1, 0, 0, 0, 0, 0]`	`[1, 0, 0, 0, 0, 0]`	Passed
oracle 2	`[0, 0, 0, 0, 0, 1]`	`[0, 0, 0, 0, 0, 1]`	Passed
oracle 3	`[0, 0, 0, 0, 1, 0]`	`[0, 0, 0, 0, 1, 0]`	Passed
oracle 4	`[1, 1, 1, 1, 1, 1]`	`[1, 1, 1, 1, 1, 1]`	Passed
oracle 5	`[1, 1, 1, 1, 1, 1]`	`[1, 1, 1, 1, 1, 1]`	Passed
oracle 6	`[0, 6, 0, 6, 0, 3]`	`[0, 6, 0, 6, 0, 3]`	Passed
oracle 7	`[0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0]`	Passed

SHA-256 / 0200eb258a5f344a3dfe45ae45cafb9580b7371912c6deea6c247018ee4fe39c

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 2026-09-29T14:49:59.642577+00:00.

Case digest / 18887e72e3f16e2ae24085409a9e98051c40a85afed3e12c520fd4beb9f4195a