Are There Equal Frequencies Between Any Two Successive Keys in a Keyboard Instrument?

When it comes to keyboard instruments, the question of whether there is an exact same frequency multiple between any two successive keys is a fascinating one. If we consider only the white keys, the answer is often no; however, if we include the black keys and consider the tuning system, the answer is mostly yes. This is because modern keyboard instruments, particularly pianos, are typically tuned using the equal temperament system.

Equal Temperament: Tuning Each Half-Step Precisely

Equal temperament is a tuning system where the octave is divided into 12 equal parts, known as semitones. In this system, each half-step (semitone) is tuned to a frequency ratio of the twelfth root of two, mathematically represented as 21/12 1.059463094. By doing this, the frequency is divided into 12 equal parts, which is particularly useful for Western music, as it allows for a wide range of key changes without significant loss of harmony.

Piano Tuning: A Fine Balance

While the equal temperament system is widely used, high and low notes on the piano are subject to small adjustments. This is due to the inherent imperfections in the strings of the piano. Ideally, a piano string would produce overtones in a harmonic series, which would include perfect ratios such as 3:2 for fifths and 4:3 for fourths. However, the stiffness of the strings, which is a result of internal elasticity, causes the overtones to be slightly sharp.

To compensate for this, piano tuners intentionally tune the high notes slightly sharp and the low notes slightly flat. This practice is known as "stretch tuning," and it ensures that the overtones of the high notes match those of the middle strings, and vice versa for the low notes.

Harmonic Series and Keyboard Instruments

The harmonic series plays a crucial role in the sound production of keyboard instruments. In a perfectly tuned string, the overtones would follow a simple mathematical progression: the first overtone is one octave higher, the second is a fifth above the first overtone, and so on, following the ratios 2:1, 3:2, 4:3, and so forth.

However, in practical terms, the overtones of the highest and lowest notes on a keyboard instrument like a piano are not perfectly in tune. The highest notes are tuned slightly sharp, and the lowest notes are tuned slightly flat. This is a compromise made to achieve a well-rounded and pleasing sound. Other keyboard instruments, such as certain organ stops, may also benefit from similar treatment but it is not as common.

Conclusion: The Importance of Tuning Systems

The equal temperament system allows for the flexibility and versatility needed in modern Western music. By evenly distributing the octave into 12 equal parts, it facilitates a wide range of key changes and ensures that chords sound consonant across different keys. While small deviations from ideal tuning occur at the highest and lowest notes, these adjustments are made to enhance the overall listening experience.

In summary, while the exact same frequency multiple between any two successive keys is not always the case, the use of equal temperament and piano tuning practices ensures that keyboard instruments, including pianos, maintain a high degree of musical compatibility and harmony.