The Physics and Mathematics of Musical Tuning

1. The Standard Pitch (A4 = 440 Hz)

The baseline for modern musical tuning is A4 = 440 Hz.
This was not standardized until the 20th century (recommended in 1939, adopted by ISO in 1955).
440 Hz was chosen as a compromise: it was easy to reproduce with early electronic oscillators and prevented orchestras from continuously tuning higher to achieve a “brighter” sound.

Western music divides the octave into 12 equal logarithmic steps.
The Octave: A 2:1 frequency ratio. Doubling the frequency of a note produces the same note one octave higher (e.g., A3 = 220 Hz, A4 = 440 Hz).
The Semitone Constant: To ensure instruments can play in any key without being retuned, the “distance” between any two adjacent notes (a semitone) is a constant multiplier rather than a fixed number of Hertz.
Formula: $f_n = f_0 \times (2^{1/12})^n$
Ratio: Every adjacent key on a piano is separated by a frequency ratio of $\sqrt[12]{2}$ , or approximately 1.059463.

Human hearing is logarithmic, meaning we perceive the ratio between frequencies rather than the absolute difference in Hertz.
As you move up the keyboard, the actual gap in Hertz between notes increases, but the ear perceives the steps as musically equal because the multiplier (1.059463) remains constant.

Black Keys: Serve as the geometric mean between two white keys that are a “whole step” apart ( $f_b = \sqrt{f_1 \times f_2}$ ).
Missing Black Keys: There are no black keys between E and F or B and C. These white keys are naturally only a semitone (half step) apart.

The white keys (the C Major scale) are derived from the physics of vibrating strings and the Overtone Series, specifically the 3:2 ratio known as the Perfect Fifth.
Generating notes via Perfect Fifths yields the diatonic scale, which inherently has an uneven spacing of Whole (W) and Half (H) steps: W-W-H-W-W-W-H.
Musical Function: This uneven spacing is critical. It provides our ears with a sense of “home” (the tonic). If all keys were evenly spaced, music would lose its sense of tension, resolution, and tonal center. The black keys simply fill in the larger gaps to complete the 12-note system.