IN THE COURT OF PRACTICE ROOM NO. 1

Re: Pythagorean Scale and Interval Ratio Units — Student Progress Report

Presiding Teacher: (me, with mild giddiness)

Respondent (the student): Beginning violinist who began this year and completed the Pythagorean units.

Statement of Facts

• The student independently constructed a Pythagorean C scale, identifying concrete interval ratios and the octave bounds.
• She demonstrated decimal-to-fraction proficiency after light scaffolding, including guidance on truncating versus rounding.
• She produced a mathematically sound interval-ratio table with written justification and clear process work.
• Practice habits tied to ear training, tuning experiments, and repertoire study were established and sustained across the unit.

Analysis (Step-by-Step Findings)

1) Construction of the Pythagorean C scale

Method used: stacking pure fifths (ratio 3:2) and reducing by octaves (2:1) to place notes in the C octave. Resulting ratios (relative to C = 1:1):

• C — 1:1 (unison)
• D — 9:8 (major second)
• E — 81:64 (Pythagorean major third)
• F — 4:3 (perfect fourth)
• G — 3:2 (perfect fifth)
• A — 27:16 (major sixth)
• B — 243:128 (Pythagorean major seventh)
• C — 2:1 (octave)

Justification: stacking fifths (3:2) and moving notes into the desired octave by multiplying or dividing by powers of two yields the listed ratios. The student showed this process explicitly, demonstrating the generative nature of Pythagorean tuning.

2) Decimal-to-fraction work and notation

Observed skillset:

• Converted decimal approximations of ratios into reduced fractions (and vice versa) with clear steps: multiply to eliminate decimals, reduce common factors, and present final simplified ratio.
• Responded well to nudges about truncation vs rounding: she adopted rounding to sensible precision when reporting decimal approximations, and used exact fractions for theoretical work.

3) Interval-ratio table quality

The student presented a rationale-based table with:

• Fraction form (exact rational ratios),
• Decimal approximations to a sensible number of places (with rounding rather than truncation), and
• Brief notes linking each ratio to how it was derived (e.g., 'E = 3/2 × 3/2 ÷ 2 = 81/64').

4) Ear training and tuning precision

Practical outcomes:

• Developed the ear for pure fifths by tuning strings in fifths and listening for slow/fast beats.
• Conducted tuning experiments using a drone (C) to place other notes precisely according to the Pythagorean ratios; noted how Pythagorean thirds and other intervals differ from equal temperament (e.g., the Pythagorean major 3rd ~407.8 cents vs equal 400 cents).
• Linked tuning experiments to repertoire: identifying when a piece benefits from pure fifths versus tempered intervals.

Conclusions

Verdict: The student meets the intended learning objectives for this unit. She has:

1. Built a correct Pythagorean C scale with justified ratios,
2. Shown reliable decimal-to-fraction technique with proper notation and rounding awareness, and
3. Established disciplined practice habits that connect theory, ear training, and tuning experiments.

Recommendations — Next Steps (practical and measurable)

1. Continue tuning practice with drones: pick a drone (C or A), tune by stacked 3:2 fifths, and check for beats at each step. Keep short logs: target ratio, measured cents (if using a tuner), and audible beat behavior.
2. Practice singing or humming each ratio against a drone to internalize the relative size of intervals (major second 9:8 is smaller than two semitones in ET; the Pythagorean third is noticeably wider than the equal-tempered third).
3. In repertoire lessons, deliberately choose passages to play with Pythagorean tuning (open-string fifths, drone-based folk tunes) versus equal temperament (virtuosic passages needing compromise). Document musical and intonational effects in brief reflection notes.
4. Strengthen the math connection: use the student’s interval table to compute cents (1200·log2(ratio)) for a handful of intervals to see the log relationship between ratios and perceived pitch spacing. This builds math-musical fluency (logarithms, precision, and context for why tuning systems differ).
5. Introduce controlled comparisons: play an interval in Pythagorean tuning, then in equal temperament, and describe the difference by ear and in cents. Record short audio examples for later review.

Closing Remarks (a brief Ally McBeal aside)

She picked up the violin this year. She did the math. She listened until the beats said, 'okay, that’s right.' There were gentle nudges (round, don’t chop), a few triumphant smiles, and a table of ratios that reads like a little argument in favor of curiosity. (She’s ready — quietly, confidently — for the next act.)

Signed,
Teacher and counsel for good intonation