1/0
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| Interval information |
- Not to be confused with 1\0, the first step of 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 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.
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 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.