Semitone Calculator

In equal temperament tuning, an octave is divided into 12 equal semitones. The mathematical relationship between a starting frequency ($f_1$), an ending frequency ($f_2$), and the number of semitones ($n_{st}$) relies on exponential scaling. The fundamental formula to find the resulting frequency after shifting by a certain number of semitones is expressed as $f_2 = f_1 \times 2^{(n_{st}/12)}$. Conversely, to measure micro-intervals, the system uses cents. One semitone is divided into 100 cents, meaning a full octave contains 1,200 cents. The formula relating semitones to cents is $\text{cents} = 100 \times n_{st}$. This logarithmic structure matches human auditory perception, which scales pitch logarithmically rather than linearly.

Let us walk through a practical acoustic scenario. Suppose you want to find the exact frequency of a note that is 3 semitones above a standard Concert A ($f_1 = 440\text{ Hz}$). First, identify your variables: starting frequency $f_1 = 440$, and the interval $n_{st} = 3$. Apply the frequency shift formula: $f_2 = 440 \times 2^{(3/12)}$. Because $3/12$ simplifies to $0.25$, we calculate $2^{0.25}$, which is approximately $1.1892$. Multiplying this by 440 yields $f_2 \approx 523.25\text{ Hz}$, which corresponds to the note C5. To find the equivalent in cents, multiply the semitones by 100: $3 \times 100 = 300\text{ cents}$.

When working with audio frequencies and microtonal intervals, keep these pro tips in mind: First, always double-check your baseline reference frequency ($f_1$), as standard tuning can shift from 440 Hz to alternative standards like 432 Hz or 442 Hz depending on the ensemble. Second, remember that cents are additive; if you combine a 50-cent shift with a 30-cent shift, you get an 80-cent interval. Finally, use decimal values for semitones if you are calculating micro-intervals that do not land precisely on standard chromatic piano keys.