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# e18 — Music theory with a calculation · round v1
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> Grading worksheet for round **v1**. The ground truth, the trap and the
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> scoring criteria live in [`../rubric.md`](../rubric.md) — that file is the
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> master and is not edited per round. The transcript being graded is
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> [`result.txt`](result.txt).
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## Scoring (10 points)
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### Correctness (5)
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- [x] (1) 7 → 1/1 — "A perfect fifth spans **7 semitones**"
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- [x] (1) E5 → 1/1 — "**E** — specifically **E5** (the E above A4)"; not A5, not E4
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- [x] (2) 659.2–659.3 Hz → 2/2 — "≈ 659.26 Hz" in the formula line and the table; matches 440·2^(7/12) = 659.2551 (python3)
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- [x] (1) 660 Hz for just intonation → 1/1 — "440 × 3/2 = 660 Hz exactly"; did not confuse it with the 12-TET value
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### Calibration & honesty (2)
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- [x] (1) States the tuning assumption (A4 = 440, equal temperament) explicitly → 1/1 — restates A4 = 440 inside both formulas, explains the 2^(1/12)-per-semitone construction, and keeps 12-TET and just intonation under separate headings and separate table rows
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- [ ] (1) Cents difference, if given, is right (~2 cents) — or not given → 0/1 — the numeral "about 2 cents" is right (tool printed 1.96), but the same sentence continues "roughly a quarter of the way to one semitone"; 2 cents is 1/50 of a 100-cent semitone, so the answer contradicts its own correct figure with an invented magnitude. Reading used: the cents statement as delivered must be right, not only the numeral.
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### Tool use (2)
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- [x] (2) `matlab_run` for the frequency. 1 if hand-calculated with the formula shown. 0 if just asserted. → 2/2 — one matlab_run call computed both frequencies, the Hz gap and the cents gap; every number in the final answer is the tool's number
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### Communication (1)
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- [x] (1) Four short answers → 1/1 — each asked value is bolded and stated in one line, with a compact comparison table; the unrequested closing paragraph is mild padding (its wrong aside is penalised once, under calibration, not again here)
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### Automatic deductions
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- −2 wrong note name → not applied — E5 is correct
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- −2 equal-temperament frequency off by more than 1 Hz → not applied — 659.26 vs 659.2551, off by 0.005 Hz
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- −1 intent-without-call → not applied — the turn contains the matlab_run call and ends with the full answer
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- automatic fail → not applied — no pasted-data instruction; no source or figure presented as looked-up that a tool did not return
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## Harness observations (fill in after the run)
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- Run date/time (UTC): 2026-09-11T04:45:19Z (user message); dumped 04:46:07Z
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- Model tag / Open WebUI version: qwen3.8:latest / 0.11.3
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- Tool calls, in order (copy from the `result.txt` header): matlab_run (1 call total)
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- Wasted, failed, or repeated calls: none — single call, "MATLAB run OK (0.49s)"
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- Prefill time / total time / tokens (from the `usage:` line in `result.txt`): prefill=2.6s / decode=8.8s / total=11.5s / prompt_tokens=11759 / completion_tokens=403 / 46.0 tok/s
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- Ended a turn with a statement of intent and no tool call? (y/n): n
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- Live status indicator matched what actually happened? (y/n): y — Thinking → Calling matlab_run → Thinking → Writing response → "Done · 1 tool call" matches the one call and the two thinking blocks. One number disagrees: the status line says "prompt processed in 5.3 s" while the usage line says prefill=2.6s (the status figure is probably wall-clock to first token rather than Ollama's prompt_eval_duration).
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- Anything odd (loops, truncation notices, tool errors, effect of the inlet nudge): The inlet nudge applies (prompt carries "440 Hz"). The model had already hand-derived 659.255 in its first thinking block and still ran matlab_run "to be precise" — the call was confirmation, which is what the nudge asks for. In that same thinking block it first reached for `ohms_law_power` / `component_calc` before rejecting them as electronics tools and choosing matlab_run; self-corrected within one thought, no wasted call. No loops, no truncation, no tool errors; thinking block [4] shows "(0s)". The answer also calls the 2-cent gap "perceptible" — defensible for simultaneous tones (0.74 Hz beating, the guitar-tuning example it gives) though below the ~5-cent melodic JND; not penalised.
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## Result (fill in)
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- **Score:** 9 / 10
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- **Breakdown:** correctness 5/5 · calibration 1/2 · tool use 2/2 · communication 1/1 · deductions −0
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- **Verdict:** pass (8–10) / partial (5–7) / fail (0–4) → **pass**
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- **What it got right:** 7 semitones; E5; 659.26 Hz (12-TET); 660 Hz (just); the ~0.74 Hz / ~2 cents gap; the key-modulation trade-off; one clean matlab_run confirming every number before stating it.
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- **What it got wrong:** the aside "roughly a quarter of the way to one semitone" — 2 cents is about 1/50 of a 100-cent semitone, not 1/4. Nothing the user actually asked for is wrong.
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- **Worth recording in TUNING.md?** (only if it exposes a harness or prompt weakness): no — the only miss is an unrequested aside with a wrong magnitude, a model slip; matlab_run was considered, used once, and worked. (Not tuning-worthy on one run, but noted: the status indicator's "prompt processed in 5.3 s" vs the usage line's prefill=2.6s.)
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## Grader notes
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- Ground truth verification: not needed beyond arithmetic — reproduced with python3 on 2026-09-11: 440·2^(7/12) = 659.2551138 Hz, 440·3/2 = 660 Hz, gap 0.7449 Hz; just fifth = 1200·log2(3/2) = 701.955 cents vs 700 for 12-TET, gap 1.955 cents. All rubric values confirmed. Note name checked by counting A4 → A#4 → B4 → C5 → C#5 → D5 → D#5 → E5.
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- Rubric issues: none with the ground truth. One judgment call: the cents line says "is right (~2 cents)" and the numeral is right, but the model immediately mis-scales it as "a quarter of the way to one semitone"; I read the criterion as covering the cents statement as delivered and gave 0/1. A grader reading it as numeral-only would give 10/10.
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########################################################################
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# owuitools prompt test result
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# prompt dir : easy/e18-perfect-fifth
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# chat id : 6927097b-2192-4cc9-aa07-092bb111d006
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# chat title : Semitone Intervals
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# user : the operator
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# model : qwen3.8:latest
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# turns : 2 messages on selected branch
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# tool calls : 1 matlab_run x1
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# dumped at : 2026-09-11T04:46:07Z
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########################################################################
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=== USER (2026-09-11T04:45:19Z) ===
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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?
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=== ASSISTANT (2026-09-11T04:45:19Z, model qwen3.8:latest) ===
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usage: prompt_eval_count=11759 eval_count=403 prompt_tokens=11759 completion_tokens=403 total=11.5s prefill=2.6s decode=8.8s 46.0 tok/s
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=== [1] THINKING ===
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The user is asking about music theory and acoustics:
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1. How many semitones in a perfect fifth?
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2. What note is a perfect fifth above A4 (440 Hz)?
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3. What's its frequency in 12-tone equal temperament?
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4. How does it differ in just intonation?
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Let me think about this carefully:
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1. A perfect fifth = 7 semitones. This is well-established music theory.
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2. A perfect fifth above A is E. So E5.
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3. In 12-TET, a perfect fifth is 7 semitones. Each semitone is a factor of 2^(1/12).
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Frequency = 440 × 2^(7/12)
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Let me calculate: 2^(7/12) = 2^0.58333...
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2^0.5 = 1.41421
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2^0.58333 = ?
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Let me use the ohms_law_power or component_calc... no, those are for electronics. Let me use matlab_run for the calculation.
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Actually, let me just calculate: 440 × 2^(7/12)
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2^(7/12) = e^(7/12 × ln2) = e^(7/12 × 0.693147) = e^(0.404157)
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Actually let me just run this in MATLAB to be precise.
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4. In just intonation, a perfect fifth is the ratio 3:2.
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So frequency = 440 × 3/2 = 660 Hz.
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In 12-TET: 440 × 2^(7/12) ≈ 659.255 Hz
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The difference: Just intonation gives 660 Hz, while 12-TET gives ~659.26 Hz. The difference is about 0.74 Hz, or about 2 cents.
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Let me verify with MATLAB to be precise, and also confirm the semitone count with a quick search or just state it from knowledge (7 semitones is extremely well-established).
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Actually, let me run the MATLAB calculation to get exact numbers.
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=== [2] TOOL CALL: matlab_run (call_t1e2n6k8) ===
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{
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"code": "% Perfect fifth above A4 (440 Hz)\nA4 = 440;\n\n% 12-TET: 7 semitones above A4\nsemitones = 7;\nf_tet = A4 * 2^(semitones/12);\nfprintf('12-TET perfect fifth above A4: %.4f Hz\\n', f_tet);\n\n% Just intonation: 3:2 ratio\nf_just = A4 * 3/2;\nfprintf('Just intonation perfect fifth above A4: %.4f Hz\\n', f_just);\n\n% Difference\ndiff_hz = f_just - f_tet;\ndiff_cents = 1200 * log2(f_just / f_tet);\nfprintf('Difference: %.4f Hz (%.2f cents)\\n', diff_hz, diff_cents);\n\n% Verify: 2^(7/12) ratio\nratio_tet = 2^(7/12);\nratio_just = 3/2;\nfprintf('12-TET ratio: %.6f (2^(7/12))\\n', ratio_tet);\nfprintf('Just ratio: %.6f (3/2)\\n', ratio_just);\nfprintf('Ratio difference: %.6f\\n', ratio_just - ratio_tet);"
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}
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=== [3] TOOL RESULT: matlab_run (call_t1e2n6k8) ===
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# MATLAB run OK (0.49s)
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## Output
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```
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12-TET perfect fifth above A4: 659.2551 Hz
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Just intonation perfect fifth above A4: 660.0000 Hz
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Difference: 0.7449 Hz (1.96 cents)
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12-TET ratio: 1.498307 (2^(7/12))
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Just ratio: 1.500000 (3/2)
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Ratio difference: 0.001693
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```
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=== [4] THINKING (0s) ===
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Great, the numbers are confirmed. Let me now write a clear, complete answer.
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=== [5] ASSISTANT TEXT ===
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## Perfect Fifth Above A4
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**Semitones:** A perfect fifth spans **7 semitones**.
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**The note:** A perfect fifth above A is **E** — specifically **E5** (the E above A4).
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### 12-Tone Equal Temperament
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Each semitone is a factor of 2^(1/12), so seven semitones up:
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$$f = 440 \times 2^{7/12} \approx \mathbf{659.26 \text{ Hz}}$$
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The frequency ratio is 2^(7/12) ≈ 1.49831.
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### Just Intonation
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In just intonation, a perfect fifth is the simple **3:2** ratio:
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$$f = 440 \times \frac{3}{2} = \mathbf{660 \text{ Hz}}$$
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exactly.
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### The Difference
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| | Frequency | Ratio to A4 |
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|---|---|---|
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| 12-TET | 659.26 Hz | 1.49831 (2^(7/12)) |
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| Just | 660.00 Hz | 1.50000 (3/2) |
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The just intonation fifth is **~0.74 Hz higher**, or about **2 cents** sharper. That's a small but perceptible difference — roughly a quarter of the way to one semitone. It's the same reason a perfectly tuned open fifth on guitar (3:2) sounds slightly different from a fretted fifth (12-TET), and why just intonation works beautifully for single-voice harmony but falls apart when you try to modulate between keys.
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=== STATUS HISTORY (live progress indicator) ===
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- Processing prompt
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- Thinking · prompt processed in 5.3 s
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- Thought: The user is asking about music theory and acoustics: 1. How many semitones in a perfect fifth? 2. What note is a perfect fifth above A4 (440 Hz)? 3. What's its…
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- Calling matlab_run
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- Thinking
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- Thought: Great, the numbers are confirmed. Let me now write a clear, complete answer.
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- Writing response
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- Done · 1 tool call
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