# Pitch and Frequency

Sketching with Math and Quasi Physics

Sound is vibration transmitted through a medium, i.e. a solid, liquid or gas. A high pitch sound corresponds to a high frequency sound wave and a low pitch sound corresponds to a low frequency sound wave. The frequency is most often measured in **hertz(Hz)**. One hertz means that an event repeats once per second, in case of wave, one cycle per second.

## Twelve Tone Equal Temperament

Twelve-tone equal temperament divides the octave into 12 equal parts(semitones). The frequency ratio of two adjacent notes is the twelfth root of two:
\(12\sqrt2=2^{1⁄12}\approx1.059463\).
The frequency \(f\) of the n^{th} key on a piano is

$$f(n)=2^\frac{n-49}{12}\times(A=440Hz)$$

* While A=440Hz is widely used and accepted as standard, slightly different frequencies are used in some cases. See Concert pitch(Wikipedia) for reference.

## Just Intonation

In just intonation, the frequencies of notes are defined by ratios of small whole numbers, e.g.

Major 2nd: \(\frac98\) (= \(\frac32\times\frac32\times\frac12\) = an octave below the perfect 5th of the perfect 5th)

Major 3rd: \(\frac54\)

Perfect 4th: \(\frac43\) (= \(\frac21/\frac32\) = perfect 5th below the octave)

Perfect 5th: \(\frac32\)

Major 6th: \(\frac53\) (= \(\frac54\times\frac43\) = the perfect 4th of the major 3rd)

Major 7th: \(\frac{15}8\) (= \(\frac54\times\frac32\) = the perfect 5th of the major 3rd)

Octave: \(\frac21\)