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Logarithm
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===Music=== {{multiple image | direction = vertical | width = 350 | footer = Four different octaves shown on a linear scale, then shown on a logarithmic scale (as the ear hears them) | image1 = 4Octaves.and.Frequencies.svg | alt1 = Four different octaves shown on a linear scale. | image2 = 4Octaves.and.Frequencies.Ears.svg | alt2 = Four different octaves shown on a logarithmic scale }} Logarithms are related to musical tones and [[interval (music)|intervals]]. In [[equal temperament]] tunings, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or [[pitch (music)|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 [[semitone]]s. For example, if the [[a (musical note)|note ''A'']] has a frequency of 440 [[Hertz|Hz]] then the note [[Bβ (musical 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 (musical note)|''B'']] (frequency 493 Hz). Accordingly, the frequency ratios agree: <math display="block">\frac{466}{440} \approx \frac{493}{466} \approx 1.059 \approx \sqrt[12]2.</math> Intervals between arbitrary pitches can be measured in octaves by taking the {{Nowrap|base-{{math|2}}}} logarithm of the [[frequency]] ratio, can be measured in equally tempered semitones by taking the {{Nowrap|base-{{math|2<sup>1/12</sup>}}}} logarithm ({{math|12}} times the {{Nowrap|base-{{math|2}}}} logarithm), or can be measured in [[cent (music)|cents]], hundredths of a semitone, by taking the {{Nowrap|base-{{math|2<sup>1/1200</sup>}}}} logarithm ({{math|1200}} times the {{Nowrap|base-{{math|2}}}} logarithm). The latter is used for finer encoding, as it is needed for finer measurements or non-equal temperaments.<ref>{{Citation|last1=Wright|first1=David|title=Mathematics and music|location=Providence, RI|publisher=AMS Bookstore|isbn=978-0-8218-4873-9|year=2009}}, chapter 5</ref> {| class="wikitable" style="text-align:center;" |- ! Interval<br /> <span style="font-weight: normal">(the two tones are played<br> at the same time)</span> | [[72 tone equal temperament|1/12 tone]]<br> {{audio|1_step_in_72-et_on_C.mid|play}} | [[Semitone]]<br> {{audio|help=no|Minor_second_on_C.mid|play}} | [[Just major third]]<br> {{audio|help=no|Just_major_third_on_C.mid|play}} | [[Major third]]<br> {{audio|help=no|Major_third_on_C.mid|play}} | [[Tritone]]<br> {{audio|help=no|Tritone_on_C.mid|play}} | [[Octave]]<br> {{audio|help=no|Perfect_octave_on_C.mid|play}} |- ! '''Frequency ratio'''<br> <math>r</math> | <math>2^{\frac 1 {72}} \approx 1.0097</math> | <math>2^{\frac 1 {12}} \approx 1.0595</math> | <math>\tfrac 5 4 = 1.25</math> | <math>\begin{align} 2^{\frac 4 {12}} & = \sqrt[3] 2 \\ & \approx 1.2599 \end{align} </math> | <math>\begin{align} 2^{\frac 6 {12}} & = \sqrt 2 \\ & \approx 1.4142 \end{align} </math> | <math> 2^{\frac {12} {12}} = 2 </math> |- ! '''Number of semitones'''<br /><math>12 \log_2 r</math> | <math>\tfrac 1 6</math> | <math>1</math> | <math>\approx 3.8631</math> | <math>4</math> | <math>6</math> | <math>12</math> |- ! '''Number of cents'''<br /><math>1200 \log_2 r</math> | <math>16 \tfrac 2 3</math> | <math>100</math> | <math>\approx 386.31</math> | <math>400</math> | <math>600</math> | <math>1200</math> |}
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