Orders of magnitude: use semitones, not decibels

# Orders of magnitude: use semitones, not decibels
By Oliver Sourbut
Published: 2026-04-01
I'm going to teach you a secret. It's a secret known to few, a secret way of using parts of your brain _not meant for mathematics_... for mathematics. It's part of how I (sort of) do logarithms in my head. This is a nearly purposeless skill.

What's the growth rate? What's the doubling time? How many orders of magnitude bigger is it? How many years at this rate until it's quintupled?

All questions of ratios and scale.

Scale... hmm.

![chromatic scale](https://assets.tonegym.co/posts/BB4YD2.png)

'Wait', you're thinking, 'let me check the date...'. Indeed. But please, stay with me for the logarithms.

## Musical intervals as ratios, and God's joke

If you're a music nerd like me, you'll know that an octave (abbreviated 8ve), the fundamental musical interval, represents a doubling of vibration frequency. So if [A440](https://en.wikipedia.org/wiki/A440_(pitch_standard)) is at 440Hz, then 220Hz and 880Hz are also 'A'. Our ears tend to hear this as 'the same note, only higher'.

That means the 'same' interval, an octave, corresponds to successively greater gaps in frequency. First a doubling, then a quadrupling, an octupling, and so on. Our perception, and musical notation, maps the space of frequencies logarithmically.

You'll also know that a '[perfect fifth](https://en.wikipedia.org/wiki/Perfect_fifth)' is a ratio of $3:2$. A to the E above it, C# to the G# above it, etc. Consonance is _all about nice ratios_! (Ask [Pythagoras](https://en.wikipedia.org/wiki/Pythagorean_tuning).)

At least, the really sweet, in tune fifths are this ratio. Because God is an absolute wheeze, you can keep moving in fifths ($3:2$) and octaves and get 'new notes' eleven times. That's where we get our Western scale from, originally (except it's _originally_ originally Mesopotamian probably). The twelfth time ($(3:2)^{12}$) gets you to a ratio of roughly $129.7:1$. That's _almost exactly_ seven doublings, seven octaves (7 * 8ve)! That'd be $128:1$. God's joke is in [that roughly 1% margin](https://youtu.be/1DUZsQ2by2s?si=4fGwAq6dub-eyV2T&t=96), and musicians have been arguing about what to do about it for centuries. [It's a whole thing](https://en.wikipedia.org/wiki/Musical_temperament).[^equal]

[^equal]: [Usually nowadays](https://en.wikipedia.org/wiki/Equal_temperament) we squish all the fifths a tiny bit so that when stacked up they get to that delicious 128:1.

Cutting a long story short, that leaves us with twelve different notes dividing up the octave. They 'repeat', with 'the same' note again and again at either higher or lower octaves (a full doubling of frequency).

![chromatic scale](https://musiccrashcourses.com/scores/ChromScale.png)

In between octaves, those twelve divisions need to 'add up to' a doubling. For reasons, two steps (a sixth of the overall scale) is referred to as a 'tone', and a single step (a twelfth of the scale) is thus a 'semitone'. That means each 

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