View source for 1/0

{{Novelty}}
{{Infobox Interval
| Name = singularitone
}}
{{Wikipedia| Division by zero }}

:''Not to be confused with 1\0, the first step of [[Single-pitch tuning|0edo]], another undefined interval.''

'''1/0''', the '''singularitone''', is a degenerate "interval" with an undefined numeric value. As a ratio, it can be taken to refer to the distance between any [[note]] and the note with a frequency of 0 Hz (equivalent to silence or a DC offset in Fourier Transform parlance), or with an infinite frequency (which does not exist).  

== Mathematics ==
As a fraction, the value of 1/0 is undefined due to the fact that 0∙''n'' = 0, causing all other intervals to vanish (as the resulting ratio can be simplified down to 1/0), which, if defined, causes absurdities.

For example, you could take a descending 1/0 from 440 Hz to 0 Hz, and then an ascending 1/0 from 0 Hz to 660 Hz, seemingly implying that a [[3/2|perfect fifth]] is the same as a [[unison]]. This problem is solved by declaring that 1/0 cannot be used to make any mathematical statements, leaving it mathematically undefined. However, it can be represented as a [[ratio]] between any number and 0. 

=== Absurdities ===

* 1/0 is [[S-expression|S1]], and as a result defines a self-referential temperament which tempers 1/0 out, yet at the same time equates it to the octave while simultaneously splitting it in two.

== In scale building ==
Building a scale out of 7 of [[21edo]]'s sharp fifths (of about 742.857 cents) gives a scale that can be interpreted as a [[diatonic]] scale with large steps of size 5 and small steps of size -2 (note that this means "ascending" small steps are actually descending). When attempting to make an [[antidiatonic|antidiatonic scale]] with the same relative step sizes, it always lands on the unison (as there are 2 large steps and 5 small steps, 5 × 2 + (-2) × 5 = 0), and as such, the sizes of the steps go to infinity – the [[generator]] for this scale is, in fact, 1/0, and the scale is represented by [[0edo]].

== Practical application ==
While 1/0 cannot be physically played, it might still be possible to imply it in a piece of music.

The list 1/1, 1/0.5, 1/0.25, 1/0.125, … gradually approaches 1/0.

This could be rewritten as 1/1, 2/1, 4/1, 8/1, …

So, if an interval of 1/1 is played, and it slides gradually wider, to 2/1, 4/1, 8/1 and so on, until it exits the human hearing range, this might be seen as implying 1/0.

{{Todo| review | explain its xenharmonic value }}