Octave reduction: Difference between revisions

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'''Octave reduction''' is the process of multiplying an interval with a whole-number power of 2 ~~([[2/1]] = [[octave]])~~ until it has a real-number value greater or equal than 1 and less than 2.

'''Octave reduction''' is the process of multiplying an interval with a whole-number power of 2 until it has a real-number value greater or equal than 1 ("[[1/1]]", the [[unison]]) and less than 2 ("[[2/1]]", the [[octave]]):

~~In other words, an '''octave-reduced''' interval~~ ''r'' ~~satisfies the equation~~ 1 ~~<= r <~~ 2.

1 <= r < 2

If ''r'' does not satisfy this inequality, it has to be

* multiplied by 2 while less than 1 or

* divided by 2 while greater than or equal to 2

== Examples ==

* 3/4 is less than 1, so multiply by 2 to get [[3/2]]

* 7/2 is greater than 2, so divide by 2 to get [[7/4]]

* 4/2 is greater than 2, so divide by 2 to get 2, which is equal to 2, so divide by 2 to get 1

* Adding 4 fifths corresponds to calculating the product of 4 time ([[3/2]] the interval ratio) leading to 81/16. This interval (5.0625 in decimal representation) is greater than 2 octaves <code style="white-space: nowrap;">(2*2 = 2^2 = 4)</code>, but less than 3 octaves <code style="white-space: nowrap;">(2*2*2 = 2^3 = 8)</code>. So it gets divided by 2 (or multiplied by 1/2) two times: <code style="white-space: nowrap;">(81/16)*(1/2)*(1/2) = 81 / (16*2*2) = [[81/64]]</code>

* Subtracting a fourth ([[4/3]]) from minor third [[6/5]] corresponds to dividing 6/5 by 4/3 which is the same as <code style="white-space: nowrap;">(6/5)*(3/4) = 18/20 = 9/10</code>. The result (0.9 in decimal representation) is less than 1 but greater than 1/2 (which mean ''one octave down''). So it gets multiplied by 2 once: <code style="white-space: nowrap;">9/10*2 = 18/10 = [[9/5]]</code>.

== See also ==

* [[Octave-reduce]]

[[Category:~~method~~]]

* [[Octave]]

[[Category:~~term~~]]

* [[Octave complement]]

[[Category:Method]]

[[Category:Term]]

[[Category:Howto]]

[[Category:Octave]]

@@ Line 1: / Line 1: @@
-'''Octave reduction''' is the process of multiplying an interval with a whole-number power of 2 ([[2/1]] = [[octave]]) until it has a real-number value greater or equal than 1 and less than 2.
+'''Octave reduction''' is the process of multiplying an interval with a whole-number power of 2 until it has a real-number value greater or equal than 1 ("[[1/1]]", the [[unison]]) and less than 2 ("[[2/1]]", the [[octave]]):
-In other words, an '''octave-reduced''' interval ''r'' satisfies the equation 1 &lt;= r &lt; 2.
+<= r < 2
+If ''r'' does not satisfy this inequality, it has to be
+* multiplied by 2 while less than 1 or
+* divided by 2 while greater than or equal to 2
 == Examples ==
+* 3/4 is less than 1, so multiply by 2 to get [[3/2]]
+* 7/2 is greater than 2, so divide by 2 to get [[7/4]]
+* 4/2 is greater than 2, so divide by 2 to get 2, which is equal to 2, so divide by 2 to get 1
 * Adding 4 fifths corresponds to calculating the product of 4 time ([[3/2]] the interval ratio) leading to 81/16. This interval (5.0625 in decimal representation) is greater than 2 octaves <code style="white-space: nowrap;">(2*2 = 2^2 = 4)</code>, but less than 3 octaves <code style="white-space: nowrap;">(2*2*2 = 2^3 = 8)</code>. So it gets divided by 2 (or multiplied by 1/2) two times: <code style="white-space: nowrap;">(81/16)*(1/2)*(1/2) = 81 / (16*2*2) = [[81/64]]</code>
 * Subtracting a fourth ([[4/3]]) from minor third [[6/5]] corresponds to dividing 6/5 by 4/3 which is the same as <code style="white-space: nowrap;">(6/5)*(3/4) = 18/20 = 9/10</code>. The result (0.9 in decimal representation) is less than 1 but greater than 1/2 (which mean ''one octave down''). So it gets multiplied by 2 once: <code style="white-space: nowrap;">9/10*2 = 18/10 = [[9/5]]</code>.
 == See also ==
-* [[Octave-reduce]]
-[[Category:method]]
+* [[Octave]]
-[[Category:term]]
+* [[Octave complement]]
+[[Category:Method]]
+[[Category:Term]]
+[[Category:Howto]]
+[[Category:Octave]]