Tuning & Intervals
Every note is a frequency. The relationships between frequencies (octaves, fifths, thirds) follow patterns that musicians have argued about for centuries. This lesson covers the two main tuning systems and how to hear the difference between them.
What is Frequency?
When an oscillator makes sound, it creates a repeating cycle of pressure in the air, a vibration. Frequency is how many of those cycles happen in one second. We measure it in Hertz, abbreviated Hz: 1 Hz means one cycle per second.
A low bass note might vibrate at 80 Hz, meaning eighty cycles every second. A bright, piercing tone at 4000 Hz means four thousand cycles per second. The higher the number, the higher the pitch. You have been seeing Hz on sliders and displays throughout this series. Now you know what the number actually means: cycles per second.
Every Note is a Number
The note A4, the A above middle C, is defined as 440 Hz. This is the tuning standard that orchestras and synthesizers agree on. When a tuner says you are "in tune," it means your A is vibrating at 440 cycles per second.
Here is the most important rule in music: doubling the frequency raises the pitch by exactly one octave. A4 is 440 Hz. A5 is 880 Hz, double. A3 is 220 Hz, half. This works no matter where you start. 100 Hz to 200 Hz is an octave. 3000 Hz to 6000 Hz is an octave. The human ear perceives frequency ratios, not frequency differences. Doubling always sounds like "the same note, higher."
Equal Temperament
Western music divides the octave into 12 equal steps called semitones. "Equal" here means each step multiplies the frequency by the same amount: the twelfth root of 2, which is approximately 1.0595.
Why this strange number? Because if you multiply by 1.0595 twelve times, you get exactly 2, one octave. Every semitone is the same proportional distance from the last, which means you can play a melody starting on any note and it sounds the same. This system is called equal temperament, and it is how every piano, guitar, and synthesizer is tuned by default.
But equal temperament is a compromise. It was invented to solve a problem, and to understand the problem, you need to know about the older system it replaced.
Just Intonation: Pure Ratios
Before equal temperament, musicians tuned by ear using simple frequency ratios. If one note vibrates at 300 Hz and another at 200 Hz, that is a 3:2 ratio, a perfect fifth. It sounds clean and stable because the two waveforms align every 2 cycles of the lower note. A major third is 5:4. A perfect fourth is 4:3. These are called "just" intervals, and they are what your ear naturally wants to hear.
The trouble is, just intonation does not add up. If you stack twelve just-tuned fifths (each at 3:2), you do not land back on the note you started on. You overshoot by about a quarter of a semitone. This means a piano tuned with pure fifths will sound beautiful in one key and terrible in another. Equal temperament solves this by making every fifth very slightly impure (1.4983 instead of 1.5) so that all keys sound equally acceptable.
The interval buttons in the demo below use just intonation ratios, the pure ones. That is why they are labeled "3:2" and "5:4." When you play a fifth here, you are hearing the mathematically clean version that predates the modern piano.
Cents and Beating
A cent is one hundredth of a semitone. There are 100 cents in a semitone and 1200 cents in an octave. Cents are how we measure small tuning differences, the kind your ear can detect but that are too small to call "a different note."
When two oscillators play almost the same frequency (say, 440 Hz and 443 Hz) you hear a slow wobble called "beating." The beat rate is simply the difference between the two frequencies: 3 Hz means three wobbles per second. This is how piano tuners work: they listen for beats to disappear as they bring two strings into tune.
At about 5 cents apart, you hear gentle, slow beating. At 20 cents, the beating is fast and the sound feels sour. At 50 cents, half a semitone, you just hear two different notes. The detune slider below works in cents, so you can explore this entire range.