Tools

𝑓 to MIDI frequency

MIDI pitches are expressed as integers from 0 - 127. They are calculated as a function of real-time frequency 𝑓 as: \[m = 69 + 12 \log_2({f \over 440})\]

Real time frequency, in Hz =

MIDI frequency =

MIDI frequency to 𝑓

MIDI frequency =

Real time frequency, in Hz =

Frequency Transposition

A real-time frequency 𝑓 will correspond directly to a percieved pitch, and the musical relationships betweene pitches can be expressed mathematically as so: \[f_t = 2^{h \over 12}f_i\] where \(𝑓_i\) is the initial frequency, and \(𝑓_t\) is the frequency with the halfstep relation corresponding to ℎ halfsteps.

initial frequency, in Hz =
number of halfsteps =

transposed frequency, in Hz =

A full octave corresponds to twelve halfsteps, and thus increasing by ℎ = 12 halfsteps will double the initial frequency.

Transposition Formulas for Looping Wavetables

To solve the real-time frequency 𝑓 for a desired halfstep ℎ, the formula: \[f = {2^{h \over 12}R \over N}\] Conversely, to solve the number 𝑁 of samples in a wavetable to achieve ℎ from a starting frequency 𝑓: \[N = {2^{h \over 12}R \over f}\]

The sample rate 𝑅 is typically standardized to 44100 samples/s.

Notes

A Sinusoid wave can be expressed as: \[x[n] = a cos(\omega n + \phi)\] where: 𝑛 is the sample number, an integer 𝑎 is the amplitude 𝜔 is the angular frequency 𝛷 is the initial phase, which may take values from -1 to 1 Each sample is discrete and unitless, so to produce audio samples must be played at a rate and for a duration: \[n = Rt\] where: 𝑅 is the sample rate in Hz (or \(s^{-1}\)) 𝑡 is the time in seconds (s) Real-time frequency may be related to sample rate and angular frequency as: \[f = {{\omega R} \over {2 \pi}}\] where: 𝑓 is the real-time frequency, corresponding to a pitch, in Hz (or s^(-1)) 2𝜋 is the radians conversion factor to move from angular frequency (𝜔) Examining the sinusoid function, note that the possible values for cos([anything]) run only from -1 to 1, so the amplitude 𝑎 is a multiplicative factor on our wave. An increased amplitude corresponds to increaded loudness of a sound. The peak amplitude refers to the maximum sample value within a given window, typically understood as maximum of the absolute value (so if over the window sample values range from -10 to 2, the peak amplitude would be 10.) Under ordinary conditions we can take peak amplitude A(peak) = 𝑎 and root-mean squared amplitude \[A_{RMS} = {𝑎 \over \sqrt{2}}\] Decibels are logarithmic relative units corresponding to amplitude: \[d = 20 \log_{10}{a \over a_0}\] 𝑑 = 20 log (𝑎/𝑎(0)) where: 𝑑 are decibels log is the base-10 logarithm 𝑎(0) is the reference amplitude The typical reference amplitude in digital audio assumes the hardware has a maximum amplitude of 1, and takes 𝑎(0) = 10^(-5) = 0.00001 [Return to main page](/index)