The Octave – Part Two: Enigma Made Easy
The octave as a physical relationship: one string, two tones
We ended Part One with the octave as something almost inevitable - a moment the ear recognises, long before the mind understands it. Part Two is about lifting the lid on that feeling. Not with equations or engineering, but with something much simpler: a string, a touch, and the sound your ear already knows.
The simplest demonstration in music
Pluck a string, and you get a note. Fret the midpoint of that string and pluck again, and you get the same note, but this time, it’s an octave higher. That’s the first ratio in Western music 2:1.
And for most of musical history, pitch was understood through length.
• Halve the string and you get the octave – ½ = a ratio of length 2:1
And if you keep going - playing the string in fractions…
• Two‑thirds of the length gives you the fifth – ⅔ = a ratio of length 3:2
• Three‑quarters gives you the fourth – 3/4 = a ratio of length 4:3
So musical notes were defined as a fraction of the length of the string you started with. (more or less). These were then expressed as ratios, and ½ becomes 2:1
A brief historical hinge: When music became multi‑voiced
For centuries, Western music was plainchant or plainsong - a single melodic line sung by every voice in a choir in unison. But around the year 1000, manuscripts started to show something new: two lines sung together. Polyphony had arrived, and with it came the first real pressure on the old tuning systems. Those simple length ratios had served plainsong well, but once music became multi‑voiced, they were no longer enough. The system had to evolve.
Eventually, around the 17th and 18th centuries, something shifted. We began to understand pitch not just as the length of a vibrating string, but as frequency - specifically how many times a second a string, or a column of air, vibrated. The ratios stayed exactly the same, but their meaning changed. Instead of 2:1 describing two lengths, it now described two speeds of vibration. The octave moved from geometry into acoustics.
Now we can say that: the octave is simply one vibration moving twice as fast as the other. And that for each and every octave, the frequency of the vibration doubles. That’s the whole secret.
This shift will matter later, when we look at how Western music ended up with twelve steps (frets on your guitar), seven note names, and the familiar bewildering pattern of sharps and flats.
What vibration really means
Every musical note is a vibration. Vibrate the air, it vibrates your ear, and you hear a sound. Now, lots of things can vibrate; the skin of a drum or a bodhrán, the metal plates of cymbals, columns of air in wind instruments; and especially the strings and the body of your guitar. What matters is what those vibrations do to the air. They push and pull it, making tiny changes in air pressure. Those tiny changes travel outward, until they reach your ear - then it becomes sound; and the vibration of strings on your guitar has become music.
We measure the frequency of those vibrations. How many times does a string vibrate in a second. That gives us a number, 110x, 220x, 440 times a second and more. We usually call them Hertz, to honour Heinrich Hertz, who helped us understand how the vibrations (waves) behave. If I tell you that your plucked open 5th string vibrates at 110 Hz, you know I’m telling you that when you pluck it, it’s vibrating 110 times a second. And you hear an A note.
The take-away
For now it’s enough to know that pitch is simply frequency - the speed of vibration. The faster the vibration, the higher the frequency - the higher the pitch. And the octave is the simplest of all musical relationships: two tones linked by a doubling of vibration. One note vibrates at a set number of times per second - say 110Hz - and the next sits exactly at twice that, 220Hz. That’s the octave.”
Before we move on next week, it’s worth defining two terms clearly…
Octave — the interval between two pitches where the higher vibrates at exactly twice the frequency of the lower.
Pitch — the perceived highness or lowness of a sound, determined by its frequency (vibrations per second, measured in Hertz).