Equal temperament, A440 and why cents matter
How the twelve-tone equal temperament system turns notes into exact frequencies, and how cents describe the gap between a pitch and a note.
One octave, twelve equal steps
Equal temperament solves an old tuning problem by making every semitone the same ratio. Doubling a frequency raises the pitch by an octave, and equal temperament fills that octave with twelve steps whose ratios all equal the twelfth root of two. Multiply any note by that factor once and you climb a semitone, twelve times and you land exactly one octave higher. This even spacing is what lets a piano play in every key without retuning.
Why A4 is the anchor
A frequency system needs one fixed reference, and by international agreement that anchor is A4 at 440 Hz. Every other note is defined by how many semitones it sits above or below that A, using the MIDI numbering where A4 is 69. Change the anchor and the whole ladder shifts with it, which is exactly what happens when you set the tool to 432 Hz and each note drops slightly in pitch while keeping the same note names.
Reading cents and small offsets
Cents give a fine ruler for pitch, dividing each semitone into 100 equal parts and each octave into 1200. Because the scale is logarithmic, a cent is a ratio rather than a fixed number of hertz, so it means the same musical distance high or low on the keyboard. When you feed the tool a measured frequency such as 450 Hz it reports the closest note, A4, and the offset of about 39 cents, telling you the sound is noticeably sharp but still nearer to A than to the next note up.
Practical uses in tuning and audio
Turning notes into exact hertz values is handy well beyond music theory. Instrument builders set string and pipe lengths from target frequencies, synth programmers enter oscillator pitches in hertz, and audio engineers hunt resonant room modes by matching a frequency back to a note. Running the conversion both ways, note to hertz and hertz to note, lets you move freely between the language of a score and the numbers a tuner or spectrum analyser shows.