1/0: Difference between revisions

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| Name = singularitone

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~~''Not to be confused with 1\0, the first and only interval of [[0edo]], a [[1/1~~|~~perfect unison]].''~~

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).

:''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.

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.

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.

== 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]].

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 ==

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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.

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 | Name = singularitone
 }}
-''Not to be confused with 1\0, the first and only interval of [[0edo]], a [[1/1|perfect unison]].''
+{{Wikipedia| Division by zero }}
-/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).
+:''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.
+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.
+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.
 == 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]].
+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 ==
@@ Line 23: / Line 25: @@
 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 }}
-<!-- fix module: Rational or module: Infobox Interval for category -->