1/0: Difference between revisions

(15 intermediate revisions by 6 users not shown)

Line 1:

{{Infobox Interval

| Name = singularitone

}}

~~{{Infobox Interval|ratio=~~1/0~~|cents=undefined}}~~

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

~~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)~~.

'''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]].

== 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.

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]].

@@ Line 1: / Line 1: @@
+{{Novelty}}
+{{Infobox Interval
+| Name = singularitone
+}}
+{{Wikipedia| Division by zero }}
-{{Infobox Interval|ratio=1/0|cents=undefined}}
+:''Not to be confused with 1\0, the first step of [[Single-pitch tuning|0edo]], another undefined interval.''
-/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).
+'''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]].
+== 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.
-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]].
+{{Todo| review | explain its xenharmonic value }}