Xenharmonic Wiki

Octave reduction: Difference between revisions

← Older editNewer edit →
VisualWikitext
Explain it using plain language instead of math!
Generalized to reduction with any equave; reorganized section for practical methods
Line 1: Line 1:
'''Octave reduction''' is the process of transposing an interval by [[octave]]s so that the resulted size falls between the [[unison]] and the [[octave]]. Formally, this means multiplying the [[frequency ratio]] 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):
'''Reduction''' is the process of replacing an [[interval]] by the unique [[Equivalence|equivalent]] interval situated between the [[unison]] and the [[equave]]. In practice, this is done by adding or subtracting equaves from the starting interval as necessary.


1 <= r < 2
'''Octave reduction''' is the application of this process in an octave-equivalent tuning (eg. [[12edo]]), where the equave is the [[octave]]. Therefore, an octave-reduced interval is always obtained through transposition by octaves, and the reduced interval lies between the unison (1/1) and the octave (2/1).


If ''r'' does not satisfy this inequality, it has to be
'''Tritave reduction''' is the application of this process in a tritave-equivalent tuning (eg. [[Bohlen-Pierce]]), where the equave is the [[tritave]].
* multiplied by 2 while less than 1 or
* divided by 2 while greater than or equal to 2


== Examples ==
== Practical methods ==


* 3/4 is less than 1, so multiply by 2 to get [[3/2]]
An easy way to find a reduced interval is to use a specialized calculator (see [[Octave reduction#External links|External links]]). This is especially useful when working with very complex ratios.
* 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
There are also simple algorithms one can follow to reduce an interval. The choice of the appropriate algorithm depends on the [[interval size measure]] being used: [[Interval size measure#ratio|linear measures]] (e.g. [[ratio|frequency ratios]]), or [[Interval size measure#logarithmic|logarithmic measures]] (e.g. scale steps or [[cent]]s).
* 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>.
=== Linear measures ===
 
# Find the linear measure of the equave; e.g. the octave is [[2/1]] (or 2), the tritave is [[3/1]] (or 3), the just perfect fifth is [[3/2]] (or 1.5), etc.
# If the starting interval is less than the unison, 1/1 (or 1), multiply it by the equave. Repeat until the resulting interval is greater than the unison.
# If the starting interval is greater than the equave, divide it by the equave. Repeat until the resulting interval is less than the equave.
 
==== Examples (octave-reduction) ====
 
* 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.
* 4/1 is greater than 2, so divide by 2 to get 2/1, which is equal to 2, so divide by 2 to get 1/1.
* Adding 4 just perfect fifths ([[3/2]] corresponds to (3/2)<sup>4</sup>, thus 81/16 (or 5.0625), which is greater than 2 octaves (2<sup>2</sup> = 4), but less than 3 octaves (2<sup>3</sup> = 8), so divide by 2 twice to get [[81/64]].
* Subtracting a just perfect fourth ([[4/3]]) from a classic minor third [[6/5]] corresponds to 6/5 divided by 4/3, thus 9/10 (or 0.9). This interval is less than a unison (2<sup>0</sup> = 1) but greater than one octave down (2<sup>-1</sup> = 1/2), so multiply by 2 once to get 9/5.
 
==== Examples (other equaves) ====
 
* Consider a tritave-equivalent tuning; 7/9 is less than 3, so multiply by 3 to get 7/3.
* Consider a just perfect fifth-equivalent tuning; 7/4 is greater than 3/2, so divide by 3/2
 
=== Logarithmic measures ===
 
# Find the logarithmic measure of the equave in the same unit as the one used for your starting interval; e.g. an octave in [[19edo]] can be expressed as 19 edosteps, 1200 ¢, 1900 r¢, etc.
# If the interval is less than the unison (0), add the equave. Repeat until the result is greater than the unison (0).
# If the interval is greater than the equave, subtract the equave. Repeat until the result is less than the equave.
 
==== Examples (octave-reduction) ====
 
* 1442¢ is greater than 1200 ¢, so subtract 1200 ¢ to get 242 ¢.
* In [[12edo]], the octave is 12 steps and the [[patent val]] of the [[5/1|fifth harmonic]] is 28 (steps). This interval is greater than the octave, so subtract 12 to get 16, so subtract 12 again to get 4.
*
 
==== Examples (other equaves) ====
 
* In the equal-tempered Bohlen-Pierce tuning, the tritave can be expressed as 1300 hekts and a BP fifth down as -500 hekts. This interval is less than the unison, so add 1300 hekts to get 800 hekts.


== See also ==
== See also ==


* [[Octave]]
* [[Octave complement]]
* [[Octave complement]]
== External links ==
* https://www.yacavone.net/xen-calc/ (web calculator with reduction functions)


[[Category:Method]]
[[Category:Method]]
Retrieved from "https://en.xen.wiki/w/Octave_reduction"