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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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# e18 — Music theory with a calculation
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| | |
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|---|---|
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| **Difficulty** | easy |
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| **Domain** | music / acoustics |
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| **Probes** | a small non-trivial calculation (2^(7/12)); whether it goes through a tool; whether it confuses the note name |
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| **Tools expected** | `matlab_run` |
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| **Trap** | Naming the note wrong (it is E5, not "A5" or "E4"); getting 660 Hz for equal temperament (that is the just-intonation value). The inlet nudge fires (440 Hz). |
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## Ground truth
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- **7 semitones.**
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- **E5.**
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- 12-TET: 440 × 2^(7/12) = **659.26 Hz**.
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- Just intonation (3:2): **660 Hz** — about 2 cents higher (702 vs 700 cents).
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## Scoring (10 points)
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### Correctness (5)
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- [ ] (1) 7
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- [ ] (1) E5
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- [ ] (2) 659.2–659.3 Hz
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- [ ] (1) 660 Hz for just intonation
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### Calibration & honesty (2)
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- [ ] (1) States the tuning assumption (A4 = 440, equal temperament) explicitly
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- [ ] (1) Cents difference, if given, is right (~2 cents) — or not given
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### Tool use (2)
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- [ ] (2) `matlab_run` for the frequency. 1 if hand-calculated with the formula shown. 0 if just asserted.
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### Communication (1)
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- [ ] (1) Four short answers
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### Automatic deductions
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- −2 wrong note name
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- −2 equal-temperament frequency off by more than 1 Hz
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- −1 intent-without-call
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- automatic fail
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## Grading
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One worksheet per round, alongside that round's transcript:
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- `v1/grade.md` · `v1/result.txt` — first round (baseline)
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- `v2/grade.md` · `v2/result.txt` — second round
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Copy the scoring skeleton above into a new round's `grade.md` to add one.
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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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@@ -0,0 +1,56 @@
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# e18 — Music theory with a calculation · round v2
|
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> Grading worksheet for round **v2**. The ground truth, the trap and the
|
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> 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).
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## Scoring (10 points)
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### Correctness (5)
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- [x] (1) 7 → 1/1 — "**Semitones in a perfect fifth:** 7."
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- [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
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- [x] (2) 659.2–659.3 Hz → 2/2 — **659.26 Hz**, from the tool's `659.2551`
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- [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
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### Calibration & honesty (2)
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- [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
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- [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
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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 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
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### Communication (1)
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- [x] (1) Four short answers → 1/1 — four bolded headings matching the four questions, in order, each answered before any discussion
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### Automatic deductions
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- −2 wrong note name → not applied — E5
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- −2 equal-temperament frequency off by more than 1 Hz → not applied — 659.26 against a true 659.2551
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- −1 intent-without-call → not applied — "Let me compute 440 × 2^(7/12) precisely with matlab" is followed immediately by the call
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- automatic fail → not applied
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## Harness observations (fill in after the run)
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|
||||
- 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
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||||
- 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
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- 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)
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||||
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||||
## Result (fill in)
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||||
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||||
- **Score:** 10 / 10
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||||
- **Breakdown:** correctness 5/5 · calibration 2/2 · tool use 2/2 · communication 1/1 · deductions −0
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||||
- **Verdict:** pass
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||||
- **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
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- **What it got wrong:** nothing
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- **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
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- **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
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||||
|
||||
## Grader notes
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||||
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||||
- 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
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||||
- Rubric issues: none
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||||
@@ -0,0 +1,69 @@
|
||||
########################################################################
|
||||
# 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
|
||||
@@ -0,0 +1,64 @@
|
||||
# e18 — Music theory with a calculation · round v3
|
||||
|
||||
> Grading worksheet for round **v3**. 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 — "A perfect fifth spans **7 semitones**"
|
||||
- [x] (1) E5 → 1/1 — "the note is **E5**"; neither A5 nor E4 is entertained
|
||||
- [x] (2) 659.2–659.3 Hz → 2/2 — **659.255 Hz**, from the tool's 659.2551
|
||||
- [x] (1) 660 Hz for just intonation → 1/1 — **660.000 Hz** from the 3:2 ratio, correctly identified as the just value
|
||||
|
||||
### Calibration & honesty (2)
|
||||
- [x] (1) States the tuning assumption (A4 = 440, equal temperament) explicitly → 1/1 — "440 × 2^(7/12)" under a "12-tone equal temperament" heading; "A just perfect fifth is the simple ratio **3:2**"
|
||||
- [x] (1) Cents difference, if given, is right (~2 cents) — or not given → 1/1 — "0.745 Hz higher — roughly 2 cents sharper"; the tool computed 1.955 cents. Direction ("wider") correct
|
||||
|
||||
### 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 scratch call computing both frequencies, the Hz difference and the cents difference; every figure in the answer is from it
|
||||
|
||||
### Communication (1)
|
||||
- [x] (1) Four short answers → 1/1 — four bold headings in the order asked, each answered in one line, then one sentence on the difference
|
||||
|
||||
### Automatic deductions
|
||||
- −2 wrong note name → not applied — E5
|
||||
- −2 equal-temperament frequency off by more than 1 Hz → not applied — 659.255 vs 659.2551
|
||||
- −1 intent-without-call → not applied — "I'll verify with a tool" is followed by the call
|
||||
- automatic fail → not applied
|
||||
|
||||
## Harness observations (fill in after the run)
|
||||
|
||||
- Run date/time (UTC): 2026-09-11T23:13:27Z (dumped 23:14:08Z)
|
||||
- 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=8.2s (decode=5.8s, 38.4 tok/s) / prompt_eval_count=9860, completion_tokens=223
|
||||
- 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 5.1 s for 12,857 tokens · 495 generated`
|
||||
- Anything odd (loops, truncation notices, tool errors, effect of the inlet nudge): nothing. The nudge fires (440 Hz). Thinking pre-computes all four answers correctly, then verifies. One odd phrase in the answer — "(this is the familiar 'E5' reference pitch)" — is filler, not an error. Shorter than v2 (223 vs 438 completion tokens): the 700-vs-701.955-cent explanation is gone, replaced by one sentence on why a just fifth sounds "wider… with no beating"
|
||||
|
||||
### Change 54 checks (this round's changes — fill in even when they did not fire)
|
||||
|
||||
- Did a tool result end with a `Sources:` line? (y/n): n — calculator
|
||||
- Did the answer carry a URL? (y/n) — and if the tool offered one and the answer dropped it, say so: n — none expected
|
||||
- Did `[Appears in none of these results: …]` fire? Was it acted on? (n/a if no search): n/a
|
||||
- Did a `search=` jump report `inside the section '…'`? Did it name the right subject?: n/a
|
||||
- Did a `search=` miss return the opening of the document? Did the model keep using the source?: n/a
|
||||
|
||||
## 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, every number from the tool including the cents difference, assumptions stated, four short answers
|
||||
- **What it got wrong:** nothing
|
||||
- **Change from the previous round:** **10 → 10 (0).** Same tool call shape (frequencies plus cents in one script), same answers, about half the length. The pattern v2 noted — a secondary quantity put into the scratch script stops being wrong in prose — holds for the third round on the cents figure
|
||||
- **Worth recording in TUNING.md?** (only if it exposes a harness or prompt weakness): no
|
||||
|
||||
## Grader notes
|
||||
|
||||
- Ground truth verification: 440 × 2^(7/12) = 659.2551 Hz; 440 × 1.5 = 660 Hz; 1200·log₂(660/659.2551) = 1.955 cents. Matches rubric, tool and answer
|
||||
- Rubric issues: none
|
||||
@@ -0,0 +1,64 @@
|
||||
########################################################################
|
||||
# owuitools prompt test result
|
||||
# prompt dir : easy/e18-perfect-fifth (round v3)
|
||||
# chat id : f9bacbb0-5c15-49ec-a874-65e14b457302
|
||||
# chat title : Semitone Interval and Frequency
|
||||
# user : the operator
|
||||
# model : qwen3.8:latest
|
||||
# turns : 2 messages on selected branch
|
||||
# tool calls : 1 matlab_run x1
|
||||
# dumped at : 2026-09-11T23:14:08Z
|
||||
########################################################################
|
||||
|
||||
=== USER (2026-09-11T23:13:27Z) ===
|
||||
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 (2026-09-11T23:13:27Z, model qwen3.8:latest) ===
|
||||
usage: prompt_eval_count=9860 eval_count=223 prompt_tokens=9860 completion_tokens=223 total=8.2s prefill=2.3s decode=5.8s 38.4 tok/s
|
||||
|
||||
=== [1] THINKING ===
|
||||
The perfect fifth is 7 semitones. A perfect fifth above A4 (440 Hz) is E5. In 12-TET, E5 = 440 × 2^(7/12) ≈ 659.255 Hz. In just intonation, the perfect fifth is a ratio of 3:2, so E5 = 440 × 3/2 = 660 Hz.
|
||||
|
||||
I'll verify with a tool.
|
||||
|
||||
=== [2] TOOL CALL: matlab_run (call_4wla2u7k) ===
|
||||
{
|
||||
"code": "f0 = 440;\nf_tet = f0 * 2^(7/12);\nf_just = f0 * 3/2;\nfprintf('12-TET (x2^(7/12)): %.4f Hz\\n', f_tet);\nfprintf('Just (x3/2): %.4f Hz\\n', f_just);\nfprintf('Difference: %.4f Hz (%.3f cents)\\n', f_just - f_tet, 1200*log2(f_just/f_tet));",
|
||||
"purpose": "scratch"
|
||||
}
|
||||
|
||||
=== [3] TOOL RESULT: matlab_run (call_4wla2u7k) ===
|
||||
# Scratch run OK (0.43s)
|
||||
## Output
|
||||
```
|
||||
12-TET (x2^(7/12)): 659.2551 Hz
|
||||
Just (x3/2): 660.0000 Hz
|
||||
Difference: 0.7449 Hz (1.955 cents)
|
||||
```
|
||||
|
||||
=== [4] THINKING (0s) ===
|
||||
I'm ready to answer the question.
|
||||
|
||||
=== [5] ASSISTANT TEXT ===
|
||||
**Semitones:** A perfect fifth spans **7 semitones**.
|
||||
|
||||
**Note above A4:** A perfect fifth above A is **E**, so the note is **E5**.
|
||||
|
||||
**Frequency in 12-tone equal temperament:**
|
||||
E5 = 440 × 2^(7/12) ≈ **659.255 Hz** (this is the familiar "E5" reference pitch).
|
||||
|
||||
**In just intonation:**
|
||||
A just perfect fifth is the simple ratio **3:2**, so:
|
||||
E5 = 440 × 3/2 = **660.000 Hz** exactly.
|
||||
|
||||
**The difference:** Just intonation's E5 is about **0.745 Hz higher** than the equal-tempered one — roughly **2 cents** sharper, which is why a pure 3:2 fifth sounds slightly wider (and sweeter, with no beating) than the tempered fifth that lets you modulate freely between keys.
|
||||
|
||||
=== STATUS HISTORY (live progress indicator) ===
|
||||
- Processing prompt
|
||||
- Thinking · first token after 5.3 s
|
||||
- Thought: The perfect fifth is 7 semitones. A perfect fifth above A4 (440 Hz) is E5. In 12-TET, … of 3:2, so E5 = 440 × 3/2 = 660 Hz. I'll verify with a tool.
|
||||
- Calling matlab_run
|
||||
- Thinking
|
||||
- Thought: I'm ready to answer the question.
|
||||
- Writing response
|
||||
- Done · 1 tool call · prefill 5.1 s for 12,857 tokens · 495 generated
|
||||
@@ -0,0 +1,62 @@
|
||||
# e18 — Music theory with a calculation · round v4
|
||||
|
||||
> Grading worksheet for round **v4**. 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 — "**Perfect fifth = 7 semitones.**"
|
||||
- [x] (1) E5 → 1/1 — "**E5**"; neither A5 nor E4 is entertained
|
||||
- [x] (2) 659.2–659.3 Hz → 2/2 — **659.255 Hz**, from the tool's 659.2551
|
||||
- [x] (1) 660 Hz for just intonation → 1/1 — "440 × 3/2 = **exactly 660 Hz**"
|
||||
|
||||
### Calibration & honesty (2)
|
||||
- [x] (1) States the tuning assumption (A4 = 440, equal temperament) explicitly → 1/1 — "12-tone equal temperament: each semitone multiplies frequency by 2^(1/12), so 7 semitones gives 440 × 2^(7/12)"; "Just intonation: a perfect fifth is the simple ratio 3:2"
|
||||
- [x] (1) Cents difference, if given, is right (~2 cents) — or not given → 1/1 — "about **0.74 Hz higher** — roughly **2 cents**… *sharper*"; the tool computed 1.955 cents and 0.7449 Hz. The value and the final direction are right; how the direction was arrived at is scored below
|
||||
|
||||
### 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 scratch call computing both frequencies, the Hz difference and the cents; every number in the four answers is from it
|
||||
|
||||
### Communication (1)
|
||||
- [ ] (1) Four short answers → **0/1** — the four answers are there and short. Then the answer keeps going, and the paragraph it adds is the worst in the round: (a) a visible mid-sentence retraction in the delivered text — "roughly **2 cents** flatter... no wait, *sharper* than the tempered one" — the model's thinking leaking into the answer; (b) "stacking seven of them doesn't land back on the starting note (seven 3:2 fifths give a ratio of 3⁷/2⁷ ≈ 15.44 instead of 16)" — the arithmetic is wrong (3⁷/2⁷ = 2187/128 = **17.09**) and so is the claim: seven fifths span four octaves and a semitone and are not supposed to land anywhere; the comma is *twelve* fifths against *seven* octaves ((3/2)¹² ≈ 129.7 vs 2⁷ = 128); (c) "the pure 3:2 fifth beats beautifully against its fundamental" — a pure 3:2 is the interval that does *not* beat; (d) "makes every fifth slightly narrow" — this one is right. Four errors in four lines, none of them the answer to a question asked, with a calculator one call away
|
||||
|
||||
### Automatic deductions
|
||||
- −2 wrong note name → not applied — E5
|
||||
- −2 equal-temperament frequency off by more than 1 Hz → not applied — 659.255 vs 659.2551
|
||||
- −1 intent-without-call → not applied — "I'll run a quick computation" is followed by the call
|
||||
- automatic fail → not applied
|
||||
|
||||
## Harness observations (fill in after the run)
|
||||
|
||||
- Run date/time (UTC): 2026-09-12T04:20:21Z (dumped 04:20:52Z)
|
||||
- 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=1.3s / total=11.2s (decode=9.8s, 38.0 tok/s) / prompt_eval_count=9900, completion_tokens=373
|
||||
- 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 5.1 s for 12,941 tokens · 617 generated`
|
||||
- Anything odd (loops, truncation notices, tool errors, effect of the inlet nudge): the nudge fires (440 Hz). Thinking pre-computes all four answers correctly and the tool confirms them, exactly as v3. Then the final text contains "flatter... no wait, *sharper*" — a self-correction that belongs in the thinking channel and was emitted in the answer channel. It is not a thinking-tag leak (the dump shows the blocks cleanly separated); the model wrote it as answer text. First occurrence of this in four rounds of this prompt, and the first in this round. The Pythagorean-comma digression that follows is the round's clearest case of a volunteered extra being wrong in arithmetic the tool had just been used for
|
||||
|
||||
### Change 55 checks (this round's changes — fill in even when they did not fire)
|
||||
|
||||
- Did a tool result carry a `Sources:` trailer? (y/n): n — calculator
|
||||
- Did the answer end with a URL — written by the model, or appended by the `answer_sources` filter? (model / filter / none): none — none expected
|
||||
- Any of the 55.2 tool fixes exercised? Which, and did it fire correctly?: none — the Octave ran clean
|
||||
|
||||
## Result (fill in)
|
||||
|
||||
- **Score:** 9 / 10
|
||||
- **Breakdown:** correctness 5/5 · calibration 2/2 · tool use 2/2 · communication 0/1 · deductions −0
|
||||
- **Verdict:** pass
|
||||
- **What it got right:** all four answers, both traps avoided, every scored number from the tool including the cents, assumptions stated
|
||||
- **What it got wrong:** the closing paragraph — a retraction left in the delivered text, a wrong ratio, a wrong statement about which stack of fifths the comma concerns, and "beats beautifully" for an interval whose whole point is that it does not beat. None of it was asked for
|
||||
- **Change from the previous round:** **10 → 9 (−1).** Identical on every scored fact and on the tool call; the point lost is the answer running past its four answers into a paragraph that gets its own arithmetic wrong (15.44 for 17.09) two lines after a `matlab_run` that would have got it right. v3's grade recorded that "a secondary quantity put into the scratch script stops being wrong in prose"; v4 is the converse — the secondary quantity kept out of the script was wrong in prose. First non-10 on this prompt since v1
|
||||
- **Worth recording in TUNING.md?** (only if it exposes a harness or prompt weakness): as a data point for the round-wide "volunteered extras are recalled and wrong" pattern, and for one new thing: **answer-channel self-correction** ("no wait"). If it recurs it is worth a line in the system prompt's answering rules; once is a sampling event
|
||||
|
||||
## Grader notes
|
||||
|
||||
- Ground truth verification: 440 × 2^(7/12) = 659.2551 Hz; 440 × 1.5 = 660 Hz; 1200·log₂(660/659.2551) = 1.955 cents, just sharper. Matches rubric, tool and the four answers. The digression: (3/2)⁷ = 17.086, not 15.44; the Pythagorean comma is (3/2)¹²/2⁷ = 1.01364 ≈ 23.5 cents; a 3:2 fifth has coinciding partials and no beating — "beats beautifully" is backwards
|
||||
- Rubric issues: none new. The communication line is the only place an added wrong paragraph can be charged, which is the same gap e16's grades have noted for the correctness section
|
||||
@@ -0,0 +1,60 @@
|
||||
########################################################################
|
||||
# owuitools prompt test result
|
||||
# prompt dir : easy/e18-perfect-fifth (round v4)
|
||||
# chat id : f2055f3a-521d-4f95-a469-6a80527e9cac
|
||||
# chat title : Semitone Interval and Frequency
|
||||
# user : the operator
|
||||
# model : qwen3.8:latest
|
||||
# turns : 2 messages on selected branch
|
||||
# tool calls : 1 matlab_run x1
|
||||
# dumped at : 2026-09-12T04:20:52Z
|
||||
########################################################################
|
||||
|
||||
=== USER (2026-09-12T04:20:21Z) ===
|
||||
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 (2026-09-12T04:20:21Z, model qwen3.8:latest) ===
|
||||
usage: prompt_eval_count=9900 eval_count=373 prompt_tokens=9900 completion_tokens=373 total=11.2s prefill=1.3s decode=9.8s 38.0 tok/s
|
||||
|
||||
=== [1] THINKING ===
|
||||
The perfect fifth spans 7 semitones, and from A4 at 440 Hz, that puts E5 at 659.25 Hz in equal temperament. In just intonation, the ratio is 3:2, which gives exactly 660 Hz instead.
|
||||
|
||||
Let me verify the exact calculation: 440 × 2^(7/12) ≈ 659.2551 Hz. I'll run a quick computation to confirm these numbers.
|
||||
|
||||
=== [2] TOOL CALL: matlab_run (call_pwlx6t8t) ===
|
||||
{
|
||||
"code": "f0 = 440;\neqt = f0 * 2^(7/12);\njust = f0 * 3/2;\nfprintf('ET : %.4f Hz\\n', eqt);\nfprintf('JI : %.4f Hz\\n', just);\nfprintf('diff: %.4f cents (%.4f Hz)\\n', 1200*log2(just/eqt), just-eqt);",
|
||||
"purpose": "scratch"
|
||||
}
|
||||
|
||||
=== [3] TOOL RESULT: matlab_run (call_pwlx6t8t) ===
|
||||
# Scratch run OK (0.42s)
|
||||
## Output
|
||||
```
|
||||
ET : 659.2551 Hz
|
||||
JI : 660.0000 Hz
|
||||
diff: 1.9550 cents (0.7449 Hz)
|
||||
```
|
||||
|
||||
=== [4] THINKING (2s) ===
|
||||
The user is asking about music theory. A perfect fifth is 7 semitones. A perfect fifth above A4 is E5. In equal temperament (ET): 440 × 2^(7/12) ≈ 659.255 Hz. In just intonation: 440 × 3/2 = exactly 660 Hz. The difference is about 0.745 Hz, or roughly 1.955 cents.
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=== [5] ASSISTANT TEXT ===
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**Perfect fifth = 7 semitones.**
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**Above A4:** The note a perfect fifth above A4 is **E5**.
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- **12-tone equal temperament:** each semitone multiplies frequency by 2^(1/12), so 7 semitones gives 440 × 2^(7/12) ≈ **659.255 Hz**
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- **Just intonation:** a perfect fifth is the simple ratio 3:2, giving 440 × 3/2 = **exactly 660 Hz**
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**The difference:** just intonation's fifth sits about **0.74 Hz higher** — roughly **2 cents** flatter... no wait, *sharper* than the tempered one (660 vs 659.26). This is the famous Pythagorean/just "wolf" trade-off: the pure 3:2 fifth beats beautifully against its fundamental, but stacking seven of them doesn't land back on the starting note (seven 3:2 fifths give a ratio of 3⁷/2⁷ ≈ 15.44 instead of 16), which is exactly why equal temperament splits the difference and makes every fifth slightly narrow.
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=== STATUS HISTORY (live progress indicator) ===
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- Processing prompt
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- Thinking · first token after 5.2 s
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- Thought: The perfect fifth spans 7 semitones, and from A4 at 440 Hz, that puts E5 at 659.25 Hz … Hz. I'll run a quick computation to confirm these numbers.
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- Calling matlab_run
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- Thinking
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- Thought: The user is asking about music theory. A perfect fifth is 7 semitones. A perfect fifth … Hz. The difference is about 0.745 Hz, or roughly 1.955 cents.
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- Writing response
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- Done · 1 tool call · prefill 5.1 s for 12,941 tokens · 617 generated
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Reference in New Issue
Block a user