Single-pitch tuning

This page presents a niche topic.

While not arbitrary, it may contain ideas whose practical applications in xenharmonic music are as yet unknown, or are narrow in scope.

Niche topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks.

Single-pitch tuning is a tuning system that contains only a single pitch. It contrasts 1edo because it does not even have octaves. It also contrasts unpitched tuning, where no recognizable pitches exist. Single-pitch tuning can be described in terms of various models of tuning.

In equal tunings

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

Single-pitch tuning can be specified as 0 equal divisions of the octave (0edo), or 0ed-p for any positive, finite number p.

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 theory

Single-pitch tuning corresponds to 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 sometimes called the Om temperment. 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 is 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.