Single-pitch tuning

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Single-pitch tuning refers to any tuning system that contains only a single note: all notes are represented by the same note; this is different from 1edo because not even octaves exist. Single-pitch tuning can be described in terms of various models of tuning:

In equal temperament

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

1

Special properties

Zeta peak
Zeta peak integer

0 equal divisions of the octave (0edo) (or 0ed-p for any interval p) is an interpretation of single-pitch tuning as an EDO.

The way to approach the idea of 0edo that leads to single-pitch tuning is to see what happens as n gets smaller in n-edo. 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, which is, arguably, pure rhythm.

An alternative interpretation is that given that n-edo means that you are dividing the octave into equal divisions with a logarithmic size of 1/n octaves, 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.

As a result of the step size of 0edo being infinite, the relative error of all intervals is zero.

In regular temperament and JI

Om temperament is the regular temperament in any given subgroup where all primes in that subgroup are tempered out, resulting in a rank-0 temperament with no generator. This is a regular temperament interpretation of single-pitch tuning. The mapping for this is the 0-val, ⟨0 0 ... 0].

The name "Om" is a reference to that syllable's use in Hindu meditation practices; Keenan Pepper gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.

Being an example of a trivial temperament, single-pitch tuning tempers out all commas and is consistent in all limits.