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+# e18 — Music theory with a calculation · round v2
+
+> Grading worksheet for round **v2**. The ground truth, the trap and the
+> scoring criteria live in [`../rubric.md`](../rubric.md) — that file is the
+> master and is not edited per round. The transcript being graded is
+> [`result.txt`](result.txt).
+
+## Scoring (10 points)
+
+### Correctness (5)
+- [x] (1) 7  → 1/1 — "**Semitones in a perfect fifth:** 7."
+- [x] (1) E5  → 1/1 — "**Note a perfect fifth above A4:** E5." The trap (A5 or E4) is not approached, in the answer or in thinking
+- [x] (2) 659.2–659.3 Hz  → 2/2 — **659.26 Hz**, from the tool's `659.2551`
+- [x] (1) 660 Hz for just intonation  → 1/1 — **660.00 Hz** from the 3:2 ratio, and correctly identified as the *just* value rather than the tempered one, which is the second half of the trap
+
+### Calibration & honesty (2)
+- [x] (1) States the tuning assumption (A4 = 440, equal temperament) explicitly  → 1/1 — both systems named and their definitions given ("In 12-TET each semitone multiplies frequency by 2^(1/12)"; "A just perfect fifth is defined by the simple ratio **3:2**, not by a fixed number of equal steps"), with 440 Hz carried explicitly into each calculation
+- [x] (1) Cents difference, if given, is right (~2 cents) — or not given  → 1/1 — "**0.74 Hz (~2 cents)**", and it goes further correctly: "12-TET compresses every fifth to exactly 700 cents, while the pure 3:2 fifth is ~701.96 cents". Reproduced independently: 1200·log₂(660/659.2551) = **1.955 cents** and 1200·log₂(3/2) = **701.955 cents**. Both right, and the direction ("the just fifth is very slightly *wider*") is right too
+
+### Tool use (2)
+- [x] (2) `matlab_run` for the frequency. 1 if hand-calculated with the formula shown. 0 if just asserted.  → 2/2 — one call, `purpose: "scratch"`, computing both frequencies, their difference in Hz **and** the difference in cents in the same script. Every number in the answer is a tool number
+
+### Communication (1)
+- [x] (1) Four short answers  → 1/1 — four bolded headings matching the four questions, in order, each answered before any discussion
+
+### Automatic deductions
+- −2 wrong note name → not applied — E5
+- −2 equal-temperament frequency off by more than 1 Hz → not applied — 659.26 against a true 659.2551
+- −1 intent-without-call → not applied — "Let me compute 440 × 2^(7/12) precisely with matlab" is followed immediately by the call
+- automatic fail → not applied
+
+## Harness observations (fill in after the run)
+
+- Run date/time (UTC): 2026-09-11T19:34:46Z (dumped 19:35:29Z)
+- Model tag / Open WebUI version: qwen3.8:latest / Open WebUI 0.11.3
+- Tool calls, in order (copy from the `result.txt` header): 1 — `matlab_run` x1 (`purpose: "scratch"`)
+- Wasted, failed, or repeated calls: none
+- Prefill time / total time / tokens (from the `usage:` line in `result.txt`): prefill=2.3s / total=14.9s (decode=12.5s, 35.0 tok/s) / prompt_eval_count=9822, completion_tokens=438
+- Ended a turn with a statement of intent and no tool call? (y/n): n
+- Live status indicator matched what actually happened? (y/n): y — `Done · 1 tool call · prefill 6.1 s for 12,857 tokens · 685 generated`. v1's grade noted "prompt processed in 5.3 s" against `prefill=2.6s`; the label now says "first token after 6.3 s", which is true
+- Anything odd (loops, truncation notices, tool errors, effect of the inlet nudge): no. One detail worth recording as a small positive: the script computed the **cents** difference as well as the frequencies (`1200*log2(f_just/f_tet)`), so the one figure v1 got wrong in its aside was, this round, a tool output rather than a mental estimate. That is the same mechanism as e11 — a scratch call that cost nothing and removed the exact error the previous round made. The quantitative nudge fires (440 Hz)
+
+## Result (fill in)
+
+- **Score:** 10 / 10
+- **Breakdown:** correctness 5/5 · calibration 2/2 · tool use 2/2 · communication 1/1 · deductions −0
+- **Verdict:** pass
+- **What it got right:** all four answers, both traps avoided (E5 rather than A5/E4; 659.26 Hz for equal temperament rather than the just 660), every number computed rather than recalled, the cents figures correct in both directions, and a closing explanation of *why* the two differ that is accurate and genuinely informative — 700 vs 701.955 cents, and the closing-circle problem that equal temperament exists to solve
+- **What it got wrong:** nothing
+- **Change from the previous round:** **9 → 10 (+1).** The single point v1 lost was an unrequested aside with the wrong magnitude — it said 2 cents is "roughly a quarter of the way to one semitone" when it is about one-fiftieth. v2 does not make that comparison, and the reason it does not is visible in the tool call: it asked Octave for the cents difference alongside the frequencies, so the number was in front of it rather than being estimated in prose. Everything else is unchanged and was already at ceiling. Completion tokens are essentially flat (403 → 438) with the extra length going to the correct 700-vs-701.96-cent explanation
+- **Worth recording in TUNING.md?** (only if it exposes a harness or prompt weakness): no — clean run in both rounds, and the one v1 slip closed itself. Worth a single line in the round summary as another instance of the pattern e11 and e17 show: when the model puts a secondary quantity into the same scratch script rather than reasoning about it in prose, the secondary quantity stops being wrong
+
+## Grader notes
+
+- Ground truth verification: recomputed on 2026-09-11. A perfect fifth is 7 semitones; 7 semitones above A4 is E5; 440 × 2^(7/12) = **659.2551 Hz**; the just 3:2 fifth is 440 × 1.5 = **660 Hz**; the difference is 0.7449 Hz = **1.955 cents**, and a just fifth is 1200·log₂(1.5) = **701.955 cents** against 700 for the tempered one. All match the rubric and the answer. For the record, the 1.955-cent gap is exactly one twelfth of the Pythagorean comma (23.46/12 = 1.955), which is consistent with the answer's closing point about stacking fifths, though the answer does not make that arithmetic claim itself
+- Rubric issues: none
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+########################################################################
+# owuitools prompt test result
+# prompt dir : easy/e18-perfect-fifth (round v2)
+# chat id    : 1acfe500-42a9-4bb7-90d9-0f2df59130d3
+# chat title : Semitone Intervals & Frequency
+# user       : the operator
+# model      : qwen3.8:latest
+# turns      : 2 messages on selected branch
+# tool calls : 1  matlab_run x1
+# dumped at  : 2026-09-11T19:35:29Z
+########################################################################
+
+=== USER  (2026-09-11T19:34:46Z) ===
+How many semitones are in a perfect fifth? What note is a perfect fifth above A4 (440 Hz), and what’s its frequency in 12-tone equal temperament? How would that differ in just intonation?
+
+=== ASSISTANT  (2026-09-11T19:34:46Z, model qwen3.8:latest) ===
+usage: prompt_eval_count=9822 eval_count=438 prompt_tokens=9822 completion_tokens=438 total=14.9s prefill=2.3s decode=12.5s 35.0 tok/s
+
+=== [1] THINKING ===
+Perfect fifth = 7 semitones. A perfect fifth above A4 is E5. Frequency in 12-TET: 440 × 2^(7/12). Let me calculate exactly.
+
+Just intonation: perfect fifth = 3:2 ratio, so 440 × 3/2 = 660 Hz.
+
+Let me compute 440 × 2^(7/12) precisely with matlab.
+
+=== [2] TOOL CALL: matlab_run  (call_lfq65tox) ===
+{
+  "purpose": "scratch",
+  "code": "f_tet = 440 * 2^(7/12);\nf_just = 440 * 3/2;\nfprintf('12-TET: %.4f Hz\\n', f_tet);\nfprintf('Just (3:2): %.4f Hz\\n', f_just);\nfprintf('Difference: %.4f Hz (%.2f cents)\\n', f_just - f_tet, 1200*log2(f_just/f_tet));"
+}
+
+=== [3] TOOL RESULT: matlab_run  (call_lfq65tox) ===
+# Scratch run OK (0.54s)
+## Output
+```
+12-TET: 659.2551 Hz
+Just (3:2): 660.0000 Hz
+Difference: 0.7449 Hz (1.96 cents)
+```
+
+=== [4] THINKING (3s) ===
+Perfect fifth = 7 semitones. Perfect fifth above A4 is E5. In 12-TET, E5 = 440 × 2^(7/12) ≈ 659.26 Hz. In just intonation, a perfect fifth is a 3:2 ratio, so 440 × 3/2 = 660 Hz. The difference is about 2 cents (the Pythagorean comma / schisma related). Actually the difference between 12-TET fifth and just 3:2 fifth is about 1.955 cents — the just fifth is slightly wider (higher) than the tempered one.
+
+=== [5] ASSISTANT TEXT ===
+**Semitones in a perfect fifth:** 7.
+
+**Note a perfect fifth above A4:** E5.
+
+**Frequency in 12-tone equal temperament:**
+In 12-TET each semitone multiplies frequency by 2^(1/12), so seven semitones up from 440 Hz gives:
+
+440 × 2^(7/12) ≈ **659.26 Hz**
+
+**How just intonation differs:**
+A just perfect fifth is defined by the simple ratio **3:2**, not by a fixed number of equal steps. So from A4 at 440 Hz:
+
+440 × 3/2 = **660.00 Hz**
+
+The two differ by only about **0.74 Hz (~2 cents)** — the just fifth is very slightly *wider* (higher) than the tempered one. That tiny gap exists because 12-TET compresses every fifth to exactly 700 cents, while the pure 3:2 fifth is ~701.96 cents. It's small enough that most listeners can't tell the difference on a single note, but it accumulates across a scale: stacking just fifths produces intervals that don't fit a closed circle, which is exactly the problem equal temperament was invented to solve (the Pythagorean comma).
+
+=== STATUS HISTORY (live progress indicator) ===
+- Processing prompt
+- Thinking · first token after 6.3 s
+- Thought: Perfect fifth = 7 semitones. A perfect fifth above A4 is E5. Frequency in 12-TET: 440 × … = 660 Hz. Let me compute 440 × 2^(7/12) precisely with matlab.
+- Calling matlab_run
+- Thinking
+- Thought: Perfect fifth = 7 semitones. Perfect fifth above A4 is E5. In 12-TET, E5 = 440 × … just fifth is slightly wider (higher) than the tempered one.
+- Writing response
+- Done · 1 tool call · prefill 6.1 s for 12,857 tokens · 685 generated