1/0: Difference between revisions

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| Name = singularitone

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1/0 is an "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 with an infinite frequency (which does not exist).

= Mathematics =

== 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 440Hz to 0Hz, and then an ascending 1/0 from 0Hz to 660Hz, 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.

== In scale building ==

= 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.

~~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]]~~.

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.

Another way to imply 1/0 might be for a piece of music to be written in an antidiatonic scale with an L:s (Large step size:small step size) ratio of 6:-2 or 5:-3, and to gradually slide that ratio towards 5:-2 throughout the course of the piece.

The listener, however, may hear such a piece as a wall of incomprehensible dissonance, so it might be advisable to use forgiving timbres such as sine waves for such a piece and to keep the slide very gradual to allow their ears time to adjust.

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 | Name = singularitone
 }}
 /0 is an "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 with an infinite frequency (which does not exist).
+= Mathematics =
-== 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 440Hz to 0Hz, and then an ascending 1/0 from 0Hz to 660Hz, 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.
-== In scale building ==
+= 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.
-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]].
+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.
+Another way to imply 1/0 might be for a piece of music to be written in an antidiatonic scale with an L:s (Large step size:small step size) ratio of 6:-2 or 5:-3, and to gradually slide that ratio towards 5:-2 throughout the course of the piece.
+The listener, however, may hear such a piece as a wall of incomprehensible dissonance, so it might be advisable to use forgiving timbres such as sine waves for such a piece and to keep the slide very gradual to allow their ears time to adjust.
 {{Todo| review | explain its xenharmonic value }}
 <!-- fix module: Rational or module: Infobox Interval for category -->