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 nth 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: $$\frac21$$

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