section_path: Applications/Music
section_name: Music
paragraph_text: Music
  
  Logarithms are related to musical tones and intervals. In equal temperament tunings, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch, of the individual tones. In the 12-tone equal temperament tuning common in modern Western music, each octave (doubling of frequency) is broken into twelve equally spaced intervals called semitones. For example, if the note A has a frequency of 440 Hz then the note B-flat has a frequency of 466 Hz. The interval between A and B-flat is a semitone, as is the one between B-flat and B (frequency 493 Hz). Accordingly, the frequency ratios agree: 466 440 ≈ 493 466 ≈ 1.059 ≈ 2 12. {\displaystyle {\frac {466}{440}}\approx {\frac {493}{466}}\approx 1.059\approx {\sqrt{2}}.}
  
  Intervals between arbitrary pitches can be measured in octaves by taking the base- 2 logarithm of the frequency ratio, can be measured in equally tempered semitones by taking the base- 2 logarithm (12 times the base- 2 logarithm), or can be measured in cents, hundredths of a semitone, by taking the base- 2 logarithm (1200 times the base- 2 logarithm). The latter is used for finer encoding, as it is needed for finer measurements or non-equal temperaments.