0edo

1edo →

Prime factorization

n/a

Step size

0¢

Fifth

0\0 (0¢)

Semitones (A1:m2)

0:0 (0¢ : 0¢)

Consistency limit

∞

Distinct consistency limit

∞

Special properties

Zeta peak
Zeta peak integer

0 equal divisions of the octave (0edo) is the tuning system that contains a single note.

Theory

There are two ways to approach this idea.

Given that n-edo means that you are dividing the octave into 1/n equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus one could not use it as a tuning system.

The other way of looking at it is to see what happens as n gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As n gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves.

Being an example of a trivial temperament, 0edo tempers out all commas and is consistent in all limits. As a result of the step size of 0edo being infinite, the relative error of all intervals is zero.

0edo is equivalent to 0ed-p of any positive, finite number p.