Editing Musical note (section)

=== Pitch frequency in hertz ===
{{Main|Music and mathematics|Pitch (music)}}Pitch is associated with the [[frequency]] of physical [[oscillations]] measured in [[hertz]] (Hz) representing the number of these oscillations per second. While notes can have any arbitrary frequency, notes in [[Consonance and dissonance|more consonant]] music tends to have pitches with simpler mathematical ratios to each other.

Western music defines pitches around a central reference "[[concert pitch]]" of A<sub>4</sub>, [[History of pitch standards in Western music|currently standardized]] as 440&nbsp;Hz. Notes played ''in tune'' with the [[12 equal temperament]] system will be an [[integer]] number <math>h</math> of half-steps above (positive <math>h</math>) or below (negative <math>h</math>) that reference note, and thus have a frequency of:
:<math>f = 2^\frac{h}{12} \times 440 \text{ Hz}\,</math>

Octaves automatically yield [[exponentiation|powers]] of two times the original frequency, since <math>h</math> can be expressed as <math>12v</math> when <math>h</math> is a multiple of 12 (with <math>v</math> being the number of octaves up or down). Thus the above formula reduces to yield a [[power of 2]] multiplied by 440&nbsp;Hz:

:<math>\begin{align}
f &= 2^\frac{12v}{12} \times \text{440 Hz}\\
&= 2^v \times \text{440 Hz} \,.
\end{align} </math>

==== Logarithmic scale ====
[[File:frequency vs name.svg|right|[[Logarithmic plot]] of frequency in [[hertz]] versus pitch of a [[chromatic scale]] starting on [[middle C]]. Each subsequent note has a pitch equal to the frequency of the prior note's pitch multiplied by {{radic|2|12}}.|thumb|228x228px]]The [[base-2 logarithm]] of the above frequency–pitch relation conveniently results in a linear relationship with <math>h</math> or <math>v</math>:
:<math>\begin{align}
\log_{2}(f) &= \tfrac{h}{12} + \log_{2}(\text{440 Hz})\\
&= v + \log_{2}(\text{440 Hz})
\end{align}</math>

When dealing specifically with intervals (rather than absolute frequency), the constant <math>\log_{2}(\text{440 Hz})</math> can be conveniently ignored, because the ''difference'' between any two frequencies <math>f_1</math> and <math>f_2</math> in this logarithmic scale simplifies to:
:<math>\begin{align}
\log_{2}(f_1) - \log_{2}(f_2) &= \tfrac{h_1}{12} - \tfrac{h_2}{12}\\
&= v_1 - v_2 \,.
\end{align}</math>

[[cent (music)|Cents]] are a convenient unit for humans to express finer divisions of this logarithmic scale that are {{fraction|1|100{{sup|th}}}} of an equally-[[Musical temperament|tempered]] semitone. Since one semitone equals 100&nbsp;[[cent (music)|cents]], one octave equals 12&nbsp;⋅&nbsp;100&nbsp;cents&nbsp;=&nbsp;1200&nbsp;cents. Cents correspond to a ''difference'' in this logarithmic scale, however in the regular linear scale of frequency, adding 1 cent corresponds to ''multiplying'' a frequency by {{radic|2|1200}}&nbsp;(≅&nbsp;{{val|1.000578}}).

==== MIDI ====
For use with the [[MIDI]] (Musical Instrument Digital Interface) standard, a frequency mapping is defined by:

:<math>p = 69 + 12 \times \log_2\frac{f}{440 \text{ Hz}} \, ,</math>

where <math>p</math> is the MIDI note number. 69 is the number of semitones between C<sub>−1</sub> (MIDI note 0) and A<sub>4</sub>.

Conversely, the formula to determine frequency from a MIDI note <math>p</math> is:

:<math>f=2^\frac{p-69}{12} \times 440 \text{ Hz} \, .</math>

{{anchor|History of note names}}<!-- anchor used by other linking pages -->