Playing the Pythagorean C scale on violin — interval ratios, finger placement and tuning method

What is Pythagorean tuning (short)?

Pythagorean tuning builds notes by stacking pure fifths (ratio 3:2). For a diatonic C major scale the results give perfectly pure fifths and fourths, but a wide (sharp) major third. On a fretless instrument like the violin you can realize these ratios precisely by ear and by adjusting finger placement.

Pythagorean ratios for a C major scale (C = 1/1)

• C = 1/1
• D = 9/8
• E = 81/64
• F = 4/3
• G = 3/2
• A = 27/16
• B = 243/128
• C (octave) = 2/1

Two useful semitone sizes in this system: the Pythagorean limma (diatonic semitone) = 256/243 (~1.0535) and the Pythagorean apotome (chromatic semitone) = 2187/2048 (~1.0679). The Pythagorean major third (81/64 ≈ 1.2656) is noticeably sharper than the just 5/4 = 1.25 or equal-tempered 2^(4/12) ≈ 1.2599.

Practical frequency example (if C4 = 261.626 Hz)

• C4 = 261.626 Hz
• D = 9/8 × C ≈ 294.33 Hz
• E = 81/64 × C ≈ 331.13 Hz
• F = 4/3 × C ≈ 348.84 Hz
• G = 3/2 × C ≈ 392.44 Hz
• A = 27/16 × C ≈ 441.38 Hz
• B = 243/128 × C ≈ 496.17 Hz
• C5 = 523.252 Hz

How to realize these intervals on the violin — step by step

1. Pick your reference C on the fingerboard. Decide which C will be your reference (for example middle C on the G string or a C on the A or D string). You will place other notes relative to that vibrating length.
2. Use a drone. Hold the reference C as a sustained drone (or use a tuning app/synth drone). This gives you a fixed pitch to tune against while you place other notes.
3. Tune the perfect fifths first (the core of Pythagorean tuning). From C tune G as a 3:2 above C. On the violin, start by finding G and adjust until the beats between the C drone and G stop disappearing (pure fifths produce almost no slow beats). Next find D as a 3:2 above G (which will also be 9:8 above C). Then A as 3:2 above D, and E as 3:2 above A if you need those notes. Stack fifths to generate G, D, A, E as pure 3:2 intervals.
4. Derive the remaining notes by descending fifth (or by octave reduction). For F, go a fifth down from C (ratio 2/3) and then raise by an octave (multiply by 2) to get 4/3. For B you can stack fifths up and reduce an octave so that B = 243/128 relative to C (or tune the fifths chain so B falls in place).
5. Find the finger positions using the ratio method. If you have a reference vibrating length for C (call it L_C), the vibrating length for a note with ratio r (relative to C) is L_note = L_C × (1/r). Example: for D (r = 9/8) L_D = L_C × 8/9, so the stopped point for D is 8/9 of the C vibrating length from the nut. Practically, place the finger where the pitch matches the drone D (9/8 of C) and use the relative movement from C as a guide.
6. Listen to the thirds and expect a wide third. Play the C–E third: it will be sharp compared to the equal-tempered or just 5/4 third. That sharpness is normal in Pythagorean tuning (E = 81/64). Expect and learn the sound—this is characteristic of the style.
7. Practice with scales and intervals. Play C–G, G–D, D–A (perfect fifths) and then try C–E and F–A (thirds) to become comfortable with the brighter thirds and pure fifths. Use slow bowing and long drones to check beats.

How to check/tune by ear (listening tips)

• Pure fifth (3:2) -> very little or very slow beats; try to eliminate the beating between the two tones.
• Pythagorean third (81/64) -> will show faster, regular beating when played against the root (C). It won’t be beatless like a just 5/4 third.
• Use open strings as drones when possible (e.g., open G or D) to compare intervals visually and aurally.

Numeric finger-placement example (optional)

Suppose your vibrating length for the reference C (L_C) is 300 mm. To get D (9/8), L_D = 300 × 8/9 = 266.667 mm. So the finger that produced C would move 33.333 mm toward the bridge to produce D (assuming same string). This direct length- ratio method is accurate in principle; on the instrument you should fine-tune by ear.

Practical cautions and ensemble considerations

• Pythagorean tuning is excellent for music that privileges pure fifths (medieval, early music, drone-based music) and for solo or small ensemble contexts that agree on a tuning system.
• If you play with a fixed-pitch instrument tuned equal temperament (piano), many Pythagorean notes—especially thirds—will clash. Decide beforehand whether to adopt Pythagorean intervals or compromise toward equal temperament.
• Stacking many pure fifths around the circle of fifths creates the Pythagorean comma: you cannot have all fifths pure and also have perfectly matching octaves across the entire chromatic system. For diatonic C major you can keep everything sensible by placing the tuning adjustments away from the notes you use most.

Short practice routine

1. Play a sustained C drone.
2. Tune G to that drone as a pure 3:2 fifth (listen for beat reduction).
3. Find D as 3:2 above G (or 9/8 above C) and check by playing C–D–G as a triad (listen for the expected Pythagorean third when you stack).
4. Continue stacking fifths to get A and E as needed; derive F and B by descending/raising octaves.
5. Play C major scale slowly and listen carefully to the character of each interval; adjust fingers until the fifths feel pure and you are comfortable with the bright thirds.

Final notes

Because the violin is fretless, it is an ideal instrument to explore Pythagorean tuning: you can place exact ratios with small finger moves and learn to hear the difference between pure fifths and tempered thirds. Start slowly with drones and fifths, then build the scale and internalize the sound. If you want, tell me which exact C on the fingerboard you're using (which string and octave) and I can give specific stopping distances or fingerboard positions for that setup.