In discussion of musical tuning there is often talk about “equal divisions” of an octave or other interval. There is potential here for confusion as there are two different ways of equally dividing an octave.

Wait, two ways? You’d think there was only one way to divide an octave equally. You make all the notes the same size! But I’m being serious. You can equally divide an interval either **arithmetically** or **logarithmically**.

Arithmetically equal = each step is the same Hz difference from the next

Logarithmically equal = each step is the same ratio difference from the next

Let’s say you have two notes, A 440Hz and A’ 880Hz. The interval between these two notes is an octave. If I want to divide this octave into arithmetically equal notes then we simply split it in a way that the difference in frequencies are equal.

For example we could split it into 8 intervals. We get the difference between the two notes by subtracting the frequencies. 880 – 440 = 440. Then divide this into 8. 440 / 8 = 55. So the base step size is 55Hz.

Constructing the scale is done by starting from 440Hz and then adding 55Hz each time until we reach the octave at 880Hz:

440Hz | 495Hz | 550Hz | 605Hz | 660Hz | 715Hz | 770Hz | 825Hz | 880Hz | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

55Hz | 55Hz | 55Hz | 55Hz | 55Hz | 55Hz | 55Hz | 55Hz | ||||||||||

9/8 | 10/9 | 11/10 | 12/11 | 13/12 | 14/13 | 15/14 | 16/15 |

Each note is 55 Hz apart from the next. You can move up and down the scale by adding/subtracting 55 Hz. The result is the harmonic series segment 8:9:10:11:12:13:14:15:16 with a fundamental frequency of 55 Hz. This scale sounds nothing at all like the equal tempered scale, in fact it sounds similar to the scales you get by playing harmonics on a guitar string.

To hear what it sounds like, you can follow this Scale Workshop link – it will open a page where you can press qwerty keys to play in this tuning.

Let’s take the octave from A 440Hz to A’ 880Hz and look at it as a ratio. 880/440. This can be simplified to 2/1.

To divide 2/1 into 12 logarithmically equal steps we need to find the step size. 2/1^{1/12} = 1.05946309436 (approx).

Constructing the scale is done by starting from 440 Hz and then multiplying this value by the step size 1.05946309436 twelve times until we reach 880 Hz:

440Hz | 466.16Hz | 493.88Hz | 523.25Hz | 554.37Hz | 587.33Hz | 622.25Hz | 659.25Hz | 698.46Hz | 739.99Hz | 783.99Hz | 830.61Hz | 880Hz | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

26.16Hz | 27.72Hz | 29.37Hz | 31.12Hz | 32.96Hz | 34.92Hz | 37Hz | 39.21Hz | 41.53Hz | 44Hz | 46.62Hz | 49.39Hz | ||||||||||||||

1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 |

These steps don’t look equal in terms of frequency – the Hz values get larger with every step. Maybe by now you have noticed one way in which the steps *are* equal… Each step is an equal ratio difference from the next.

To hear what it sounds like (I mean it’s just 12edo at A440Hz), you can follow this Scale Workshop link – it will open a page where you can press qwerty keys to play in this tuning.

In an equal temperament, you can modulate between keys and every key will sound equally in-tune. Whereas arithmetic divisions will only give you perfectly-tuned harmonics of a single fundamental with no concept of tonal modulation.

Which of these two methods *sound* equal to the ear?

It’s the logarithmic version – also known as equal temperament or EDO (equal division of the octave).

When tuning theory folks mention “equal” scales, safely assume that they’re probably talking about this method.

Equal temperaments sound equal because our perception of pitch is logarithmic itself. The hearing system isn’t listening out for equal difference in Hz, it’s listening out for equal difference in step size ratio. This isn’t immediately obvious until you’ve compared the two for yourself.

So this was just a quick and simple post to explain something that has caused a little confusion in the past. Hope someone will find it useful.