How note frequencies are calculated
Equal temperament divides the octave into twelve equal steps, and each step multiplies the frequency by the twelfth root of two, about 1.0595. Starting from A4 at 440Hz, every other note follows from one formula: f = 440 × 2^((m − 69)/12), where m is the MIDI note number.
An octave doubles the frequency, so A5 is 880Hz and A3 is 220Hz. Middle C, MIDI note 60, is 261.63Hz, and the lowest key on a piano, A0, is 27.5Hz.
Frequency to note, and cents
Going the other way, m = 69 + 12 × log2(f / 440). The whole-number part is the nearest note and the rest is how far off it is, measured in cents, where 100 cents is one semitone. A tuner showing +20 cents is reading a note a fifth of a semitone sharp.
Why your DAW calls middle C C3
MIDI note 60 is middle C everywhere. What changes is the octave number printed next to it. Scientific pitch notation, used in our guides and by most orchestration texts, calls it C4. Cubase, Logic and Kontakt call it C3 by default, and FL Studio calls it C5.
The naming only matters when you pass a note name to someone else. Say which convention you mean, or give the MIDI number, which never changes. For what the other MIDI numbers in a template do, see understanding MIDI CCs.
Changing the A4 reference
440Hz is the international standard for A4. Many orchestras tune a little higher, to 442 or 443Hz, and period-instrument ensembles often use 415Hz, close to a semitone lower. When you layer a live recording over samples, set the library's tuning to match the players, and use this chart at the same reference to check individual notes.
To see these frequencies against the instruments that play them, use the orchestral frequency chart or the instrument range chart.