Part 4 : The Chromatic Scale - A Shared Map of the Octave
The Story of Music : How twelve equal steps finally gave Western music a common language
By the early eighteenth century, Western music was full of invention - but not full of agreement. Pitch standards shifted from place to place. Temperaments - the tuning systems that decide how the steps inside the octave are spaced - varied by region and instrument. Singers carried older modal habits; keyboard and fretted‑instrument players lived inside newer harmonic ones. Everyone shared the idea of the octave, but no one divided it in quite the same way. Everyone was using a different set of steps. It wasn’t chaos, exactly — but it was close enough that musicians felt the strain every day. Sooner or later, a question had to be answered: Can we build the same number and size of steps inside the octave — a set of steps that will work for everyone?
The answer was the chromatic scale: twelve equal steps. - The idea of dividing the octave into twelve equal parts didn’t begin with mathematics. Western music already had twelve pitch‑classes - seven diatonic degrees and five chromatic alterations - long before anyone thought in terms of frequency. But tuning those twelve pitches with pure intervals never quite worked: the cycle of fifths refused to close, and every temperament left something distorted.
In 1691, the Dutch mathematician Christiaan Huygens finally expressed the physics behind the problem musicians had been wrestling with for more than a century. If the octave is a doubling of frequency, and if you want twelve equal steps inside that octave, then each step must multiply the frequency by the same constant - the twelfth root of two.
That constant sounds abstract; it sounds complicated. But it’s like standing an egg on its end: once you see it, the idea is straightforward. We want a number that, when multiplied by itself twelve times, gives exactly 2 — the 2:1 frequency ratio that defines the octave.
Why do we divide twelve times into 2, and not into 1?
Because the octave isn’t “one unit long.” It’s a doubling. We’re not dividing the length of the octave - we’re dividing the doubling of the frequency from the first pitch to the pitch an octave above. So the thing being sliced into twelve equal ratio‑steps is the 2 in the 2:1 ratio. So the mathematical expression of that looks like this:
Solving that equation gives:
In everyday terms, that number is 1.059463. It’s a small, steady rise of about 6% of whatever pitch you’re currently on - the same proportional step each time. Each new pitch is vibrating a little under 6% faster than the pitch one step below it. Multiply any frequency by 1.059463 and you move up one equal‑tempered step, the smallest interval in our equally tempered scale. Multiply by it twelve times - twelve identical steps - and the frequency has doubled. That’s all the twelfth root of two really is: the constant ratio that slices the octave into twelve identical proportional steps of frequency.
To see this in action, start with the open A string at 110 Hz. Multiply by 1.059463 and you get the pitch one fret higher:
• 110.00 Hz × 1.059463 = 116.54 Hz (A♯ / B♭)
Multiply again:
• 116.54 Hz × 1.059463 = 123.47 Hz (B)
and you get the pitch the next fret higher. Continue the process - the same multiplication every time - and after twelve steps you arrive at:
• 110 Hz × 1.059463¹² ≈ 220 Hz
The frequency has doubled - and you’ve arrived at the octave in 12 identical steps - the 12 step equal tempered chromatic scale.
Chroma: a palette of pitch
This is where the word chromatic comes from. In Greek, chroma means colour a palette. The twelve tempered steps aren’t a scale to be sung from bottom to top; they’re a set of colours you can draw from to build the diatonic scales you may already know, that you may have learned at school.
The old modal scales didn’t disappear. They simply found a new home inside a system that made their intervals consistent, portable, and shareable. For the first time, musicians across Europe could work from the same set of pitches, in the same keys, on the same instruments, without wolves, distortions, or regional traps.
From mathematics to wood and metal
The same ratio governs the layout of the guitar fretboard. Start with the full string length - say 645 mm. To find the first fret, divide the vibrating length by the twelfth root of two (1.059463). That gives the remaining vibrating length, from the saddle, to the first fret: about 608.8 mm. The first fret is placed at that point.
Divide the remaining length by the same ratio for the second fret, and keep going. After twelve such divisions, you’ve marked twelve fret positions, and the remaining vibrating length is exactly half the original. You’ve reached the twelfth fret and the octave. Equal temperament, carved into wood and metal.
Take‑away
So the chromatic scale is not a melody or a mode. It’s the underlying map:
twelve equal steps inside the octave,
twelve equally proportioned slices of frequency.
A shared palette of pitches from which every modern scale, mode, and harmony is drawn.
Twelve pitches in the palette - but we only have seven names.
And that twelve pitch palette raises a new question; one that musicians have been quietly living with for centuries: Why do we only name seven of those twelve pitches? That’s where we go in next article, next week…
Temperament
A temperament is a tuning system - a way of spacing the steps inside the octave. Different temperaments divide the octave differently, each balancing purity, practicality, and the needs of the music. Some keep certain intervals pure but distort others; some spread the compromises evenly. Equal temperament is the modern system that divides the octave into twelve identical steps - the evenly spaced framework behind the modern chromatic scale.
Pitch‑class : A pitch‑class is a group of pitches that share the same musical identity across octaves. All the Cs belong to one pitch‑class, all the Ds to another, and so on. Medieval musicians didn’t use the term, but they worked with the idea: seven modal degrees, plus their chromatic alterations, gave Western music the twelve pitch‑classes we still use today.
If today’s article leaves you curious or wanting a little more breathing room, you’re welcome to join the conversation in the Story of Music Reader’s Room →