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|~~2/1]] = [[~~Octave|~~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 ([[2/1]] = [[octave]]) until it has a real-number value greater or equal than 1 and less than 2.

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

[[~~Category~~:~~interval~~]]

== Examples ==

* 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 forth ([[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>.

[[Category:method]]

[[Category:term]]

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-'''Octave reduction''' is the process of multiplying an interval with a whole-number power of 2 ([[2/1|2/1]] = [[Octave|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 ([[2/1]] = [[octave]]) until it has a real-number value greater or equal than 1 and less than 2.
 In other words, an '''octave-reduced''' interval ''r'' satisfies the equation 1 &lt;= r &lt; 2.
-[[Category:interval]]
+== Examples ==
+* 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 forth ([[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>.
 [[Category:method]]
 [[Category:term]]