FA-81101 / Music interval and transposition theory / Open access

Pythagorean ratio for a chain of fifths: flatward chains lifted by a single octave · case 01

Two fifths down returns 8/9 (below unison) instead of 16/9.

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

ROOT CAUSE

Ratios below unity are raised by one octave at most instead of repeatedly.

VERIFIED REPAIR

Restore the upward octave reduction step so that it reads `while r < 1: r *= 2`.

Unsuccessful approach: Looping on r <= 1 also doubles the unison, so zero fifths returns 2/1.

Case contract

Input an integer k with |k| <= 24 (booleans rejected). Return (3/2)**k octave-reduced into [1, 2) as [numerator, denominator]; out-of-range or non-integer input returns None.

Why this case matters

Pythagorean and meantone tuning tools stack fifths and octave-reduce the result.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    if not isinstance(x, int) or isinstance(x, bool) or abs(x) > 24:
        return None
    r = Fraction(3, 2) ** x
    while r >= 2:
        r /= 2
    if r < 1:
        r *= 2
    return [r.numerator, r.denominator]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[(0, [1, 1]), (1, [3, 2]), (2, [9, 8]), (3, [27, 16]), (4, [81, 64]), (5, [243, 128]), (-2, [16, 9]), (-5, [256, 243])], [(0, [1, 1]), (4, [81, 64]), (5, [243, 128]), (6, [729, 512]), (7, [2187, 2048]), (12, [531441, 524288]), (-3, [32, 27]), (-6, [1024, 729])], [(0, [1, 1]), (7, [2187, 2048]), (12, [531441, 524288]), (-1, [4, 3]), (-5, [256, 243]), (-12, [1048576, 531441]), (24, [282429536481, 274877906944]), (25, None)], [(0, [1, 1]), (-6, [1024, 729]), (24, [282429536481, 274877906944]), (-24, [549755813888, 282429536481]), (25, None), (-25, None), (30, None), (True, None)], [(0, [1, 1]), (-2, [16, 9]), (-3, [32, 27]), (-12, [1048576, 531441]), (30, None), (True, None), ('1', None), (1.0, 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	`[1, 1]`	`[1, 1]`	Passed
oracle 1	`[3, 2]`	`[3, 2]`	Passed
oracle 2	`[9, 8]`	`[9, 8]`	Passed
oracle 3	`[27, 16]`	`[27, 16]`	Passed
oracle 4	`[81, 64]`	`[81, 64]`	Passed
oracle 5	`[243, 128]`	`[243, 128]`	Passed
oracle 6	`[8, 9]`	`[16, 9]`	Failed
oracle 7	`[64, 243]`	`[256, 243]`	Failed

SHA-256 / 1281d2e6a2092d95c8c78553eb1f6dba0fb63609ef9fb54bde75922ee15e6d96

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    if not isinstance(x, int) or isinstance(x, bool) or abs(x) > 24:
        return None
    r = Fraction(3, 2) ** x
    while r >= 2:
        r /= 2
    while r <= 1:
        r *= 2
    return [r.numerator, r.denominator]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[(0, [1, 1]), (1, [3, 2]), (2, [9, 8]), (3, [27, 16]), (4, [81, 64]), (5, [243, 128]), (-2, [16, 9]), (-5, [256, 243])], [(0, [1, 1]), (4, [81, 64]), (5, [243, 128]), (6, [729, 512]), (7, [2187, 2048]), (12, [531441, 524288]), (-3, [32, 27]), (-6, [1024, 729])], [(0, [1, 1]), (7, [2187, 2048]), (12, [531441, 524288]), (-1, [4, 3]), (-5, [256, 243]), (-12, [1048576, 531441]), (24, [282429536481, 274877906944]), (25, None)], [(0, [1, 1]), (-6, [1024, 729]), (24, [282429536481, 274877906944]), (-24, [549755813888, 282429536481]), (25, None), (-25, None), (30, None), (True, None)], [(0, [1, 1]), (-2, [16, 9]), (-3, [32, 27]), (-12, [1048576, 531441]), (30, None), (True, None), ('1', None), (1.0, 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	`[2, 1]`	`[1, 1]`	Failed
oracle 1	`[3, 2]`	`[3, 2]`	Passed
oracle 2	`[9, 8]`	`[9, 8]`	Passed
oracle 3	`[27, 16]`	`[27, 16]`	Passed
oracle 4	`[81, 64]`	`[81, 64]`	Passed
oracle 5	`[243, 128]`	`[243, 128]`	Passed
oracle 6	`[16, 9]`	`[16, 9]`	Passed
oracle 7	`[256, 243]`	`[256, 243]`	Passed

SHA-256 / b4ce5b42443ee97aed373377ddc89da3cd058f78dacbffd9c020bb9ea022febe

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    if not isinstance(x, int) or isinstance(x, bool) or abs(x) > 24:
        return None
    r = Fraction(3, 2) ** x
    while r >= 2:
        r /= 2
    while r < 1:
        r *= 2
    return [r.numerator, r.denominator]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[(0, [1, 1]), (1, [3, 2]), (2, [9, 8]), (3, [27, 16]), (4, [81, 64]), (5, [243, 128]), (-2, [16, 9]), (-5, [256, 243])], [(0, [1, 1]), (4, [81, 64]), (5, [243, 128]), (6, [729, 512]), (7, [2187, 2048]), (12, [531441, 524288]), (-3, [32, 27]), (-6, [1024, 729])], [(0, [1, 1]), (7, [2187, 2048]), (12, [531441, 524288]), (-1, [4, 3]), (-5, [256, 243]), (-12, [1048576, 531441]), (24, [282429536481, 274877906944]), (25, None)], [(0, [1, 1]), (-6, [1024, 729]), (24, [282429536481, 274877906944]), (-24, [549755813888, 282429536481]), (25, None), (-25, None), (30, None), (True, None)], [(0, [1, 1]), (-2, [16, 9]), (-3, [32, 27]), (-12, [1048576, 531441]), (30, None), (True, None), ('1', None), (1.0, 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	`[1, 1]`	`[1, 1]`	Passed
oracle 1	`[3, 2]`	`[3, 2]`	Passed
oracle 2	`[9, 8]`	`[9, 8]`	Passed
oracle 3	`[27, 16]`	`[27, 16]`	Passed
oracle 4	`[81, 64]`	`[81, 64]`	Passed
oracle 5	`[243, 128]`	`[243, 128]`	Passed
oracle 6	`[16, 9]`	`[16, 9]`	Passed
oracle 7	`[256, 243]`	`[256, 243]`	Passed

SHA-256 / 9fec36c0bcd9309831223aa85f37a3d536cf0ab3fd724d1ebe4a4eb2e5bff6da

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 2026-09-29T14:49:59.894992+00:00.

Case digest / ecbbf73ed43823f21b106a51c59732990c6c08be5e5a83d14f726a8b2c412404