Understanding Just Intonation for Tuning a Violin: A Comprehensive Guide

Just Intonation, often misunderstood, is a tuning system in music that produces pure intervals based on simple integer ratios. Unlike the commonly used equal temperament, which divides the octave into twelve equal parts, just intonation aims to achieve perfect consonance.

Tuning a Violin in Just Intonation

The tuning of a violin in just intonation follows a specific pattern. Each string is tuned to a perfect fifth, which is a ratio of 3:2. This means that if the A string is tuned to 440 Hz, the D string will be tuned to 550 Hz (A to D ratio of 3:2), the G string to 698.78 Hz (A to G ratio of 3:2), and the E string to 659.26 Hz (D to E ratio of 3:2).

The A String and Its Variability

The A string is the most crucial string in just intonation. It is typically tuned to 440 Hz, but this can vary depending on the context. For example, if the violinists are playing with other orchestral instruments, the A string might be tuned to 441 Hz or 442 Hz to match the standard pitch for that instrument.

The Harmonic Content and Its Impact

The proper properties of the strings can subtly alter their harmonic content, which in turn can affect the exact pitch and relative distance between notes. In just intonation, the goal is to maintain these perfect intervals, even if the exact frequency may need to be adjusted slightly for performance purposes.

Examples and Practical Application

Let's walk through the tuning process of a violin in just intonation:

1. A String

The A string is tuned to 440 Hz. This is the reference frequency for many orchestral instruments.

2. D String

The D string is then tuned to a perfect fifth above the A string. In terms of frequency, this is calculated as:

440 Hz * (3/2) 660 Hz / 2 ≈ 550 Hz

3. G String

The G string is tuned to a perfect fifth above the D string. The calculation is:

550 Hz * (3/2) ≈ 698.78 Hz

4. E String

The E string is tuned to a perfect fifth above the G string. The calculation is:

698.78 Hz * (3/2) ≈ 659.26 Hz

Conclusion

Just intonation offers a different approach to tuning a violin, emphasizing pure and consonant intervals over the equally tempered system. While the exact frequencies can vary due to instrument properties, the perfect fifths remain the guiding principle for tuning in just intonation. Understanding and applying just intonation can significantly enhance the quality and purity of musical intervals, making it a valuable technique for skilled musicians.