Introduction

Pitch and frequency are two interrelated but distinct terms used in the fields of music and electronics. Frequency, being a physical attribute, is defined as the number of oscillations per second, while pitch is the psychoacoustic perception of tone height. This article aims to elucidate the relationship between these two concepts in detail, providing both theoretical explanations and practical examples.

What is Frequency and Pitch?

Frequency is a fundamental physical parameter that refers to the number of complete cycles an oscillating wave completes in a given time period, most commonly measured in Hertz (Hz). In contrast, pitch is the subjective perception of frequency, processed by the human auditory system. The relationship between pitch and frequency is not direct but rather follows an exponential relationship, highlighting the complexity of the auditory perception.

The Direct Relationship Between Pitch and Frequency

At a superficial level, pitch and frequency are directly related. As the frequency increases, so does the pitch, and vice versa. However, this relationship is slightly different in the context of music and electronic sound. Musicians and sound engineers often use this relationship to create and manipulate sounds, but they are aware that this relationship can be more nuanced than a simple linear correlation.

Exponential Relationship Between Pitch and Frequency

The actual relationship between pitch and frequency is not a direct one but an exponential one. This means that when the pitch increases, the frequency does not merely increase but rather increases in an exponential manner. Specifically, each octave (a doubling of the frequency) corresponds to a multiplication of the original frequency by a factor of 2, rather than an additive increase.

To illustrate, consider an example: If a note has a frequency of 440 Hz (concert A), the frequency of the note one octave higher (A5) will be 880 Hz (2*440 Hz). Similarly, the frequency of the note one octave lower (A3) will be 220 Hz (440 / 2). This relationship holds true for all twelve notes in a musical scale due to the equal temperament system used in music tuning.

The Equal Tempered Scale and Its Mathematical Formulation

The equal tempered scale, first popularized by Bach, divides the octave into 12 equal parts. This means that each half-step represents a frequency ratio of the twelfth root of 2 (approximately 1.059463). The formula used to calculate the frequency of any note in the equal tempered scale is given by:

Formula:

f_n f_0 times 2^{frac{n}{12}}

Where:

f_n is the frequency of the note that is n half-steps away from the reference frequency. f_0 is the reference frequency, typically set at 440 Hz for A4 in Western music. n is the number of half-steps away from the reference note. A positive n denotes a note that is higher, while a negative n denotes a note that is lower. 2^{frac{1}{12}} is the twelfth root of 2, which represents the frequency increase for each half-step in the equal temperament system.

Practical Examples

Using the formula, we can calculate the frequency of specific notes in the equal tempered scale. For example, to find the frequency of a C (C5), which is 3 half-steps above A4 (440 Hz), we would perform the following calculation:

C5 440 times 2^{frac{3}{12}} approx 440 times 1.1892 approx 523.25 Hz

Similarly, to find the frequency of a middle C (C4), which is 9 half-steps below A4 (440 Hz), we calculate:

C4 440 times 2^{frac{-9}{12}} approx 440 times 0.5 approx 220 Hz

Both of these calculations demonstrate how the exponential relationship between pitch and frequency can be used to generate the exact frequencies of musical notes within the equal tempered scale.

Conclusion

Pitch and frequency share a profound relationship, but this relationship is not simply linear. The exponential nature of this relationship, as described by the equal tempered scale, provides a robust framework for understanding and manipulating musical sounds. Understanding these concepts is crucial for composers, musicians, sound engineers, and anyone involved in the creation or analysis of musical sounds.

Keywords: frequency, pitch, harmonic relationship