View source for Octave reduction

'''Octave reduction''' is the process of replacing an [[interval]] by the unique [[Equivalence|equivalent]] interval between the [[unison]] and the [[octave]]. In practice, this is done by adding or subtracting octaves from the starting interval as necessary.

== Practical methods ==
An easy way to find the reduced form of an interval is to use a specialized calculator such as [[xen-calc]]. This is especially useful when working with very complex ratios.

There are also several methods that can be followed. The choice of an appropriate method 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).

=== Linear measures ===
Stacking intervals expressed as ratios corresponds to multiplying those ratios. For instance, going up an octave means multiplying by 2, while going down an octave means dividing by 2.

# If the starting interval is greater or equal to the unison (1) and less than the octave (2), it is already in reduced form.
# If the starting interval is less than 1, multiply it by 2. Repeat until the resulting interval is greater than 1.
# If the starting interval is greater than 2, divide it by 2. Repeat until the resulting interval is less than 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.
* 4/1 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.

=== Logarithmic measures ===
Stacking intervals expressed with logarithmic measures corresponds to adding those measures. For instance, when working in cents, going up an octave means adding 1200 ¢, while going down an octave means subtracting 1200 ¢.

# Find the logarithmic measure of the octave in the same unit as the one used for your starting interval; e.g. 1200 ¢, 19 steps of 19edo, 1900 [[Relative cent|r¢]], etc.
# If the starting interval is positive and less than the octave (e.g. 1200 ¢), it is already in reduced form.
# If the starting interval is negative, add the octave. Repeat until the result is positive.
# If the starting interval is greater than the octave, subtract the octave. Repeat until the result is less than the octave.

Examples:

* 1442¢ is greater than 1200 ¢, so subtract 1200 ¢ to get 242 ¢.
* In 31edo, the octave is 31 steps and the [[patent val]] of the [[5/1|fifth harmonic]] is 72 (steps). This interval is greater than the octave, so subtract 31 to get 41, so subtract 31 again to get 10.

== General formulas ==
=== Linear measures ===
For a starting interval <math>r</math> expressed as a ratio, the reduced form <math>\text{red}(r)</math> of that interval can be found using this formula: <math>\text{red}(r) = r \cdot 2^{-\left\lfloor{\log_2 r}\right\rfloor}</math>.

Example:

* Octave-reducing 4900/243 can be done by using the formula with <math>r = 4900/243</math>:<br><math>\begin{align}\text{red}(4900/243) &= 4900/243 \cdot 2^{-\left\lfloor{\log_2 4900/243}\right\rfloor} \\
&= 4900/243 \cdot 2^{-\left\lfloor{4.33375\ldots}\right\rfloor} \\
&= 4900/243 \cdot 2^{-4} \\
&= 4900/243 \cdot 1/16 \\
&= 1225/972\end{align}</math>

=== Logarithmic measures ===
For a starting interval <math>l</math> and octave <math>e</math> expressed in the same units, the reduced form <math>\text{red}(l, e)</math> of that interval can be found using this formula: <math>\text{red}(l, e) = r \bmod e</math>, where <math>\bmod</math> is the modulo operation.

Example:

* Octave-reducing 412 steps of 97edo can be done by using the formula with <math>r = 412</math>, <math>e = 97</math>:<br><math>\begin{align}\text{red}(412, 97) &= 412 \bmod 97 \\
&= 24\end{align}</math>

This formula can also be written without the modulo operation: <math>\text{red}(l, e) = r - e\left\lfloor{l/e}\right\rfloor</math>.

Example:

* Octave-reducing 412 steps of 97edo again:<br><math>\begin{align}\text{red}(412, 97) &= 412 - 97\left\lfloor{412/97}\right\rfloor \\
&= 412 - 97\left\lfloor{4.24742\ldots}\right\rfloor \\
&= 412 - 97 \cdot 4 \\
&= 412 - 388 \\
&= 24\end{align}</math>

== Generalizations ==
=== Other equaves ===
Octave reduction is mainly used in the context of octave-equivalent tunings (eg. [[12edo]]), where equivalent notes are separated by octaves. However, this operation can be generalized to any [[Periodic scale|periodic tuning]] by replacing the octave by the interval of equivalence or  ''[[equave]]'' of that tuning.

For example, '''tritave reduction''' is the analog of octave reduction in a tritave-equivalent tuning (eg. [[Bohlen–Pierce]]), where the equave is the [[tritave]]. Therefore, a tritave-reduced interval is always obtained through transposition by tritaves, and the reduced interval lies between the unison (1/1) and the tritave (3/1).

The general formula for an interval <math>r</math> and an equave <math>e</math> is as follows: <math>\text{red}(r, e) = r \cdot e^{-\left\lfloor{\log_e r}\right\rfloor}</math>. Note that <math>e</math> is a variable and not Euler's number.

Examples:

* 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; <math>\text{red}(81/64, 3/2) = 81/64 \cdot (3/2)^{-\left\lfloor{\log_{3/2} 81/64}\right\rfloor} = 1</math>.
* In the equal-tempered Bohlen–Pierce tuning, a 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.

=== Balanced reduction ===
'''Balanced reduction''' is an alternate operation where the values are equally distributed around the unison instead of being situated between the unison and the octave (or the equave).

Examples:
* Balanced octave-reduction with ratios will lead to values greater than or equal to <math>\frac{1}{\sqrt{2}}</math>, but less than <math>\sqrt{2}</math>.
* Balanced octave-reduction with cents will lead to values greater than or equal to -600 ¢, but less than 600 ¢.
* Balanced tritave-reduction with ratios will lead to values greater than or equal to <math>\frac{1}{\sqrt{3}}</math>, but less than <math>\sqrt{3}</math>.

Here are some formulas for balanced reduction:
* Balanced octave-reduction of an interval <math>r</math> expressed as a ratio: <math>\text{reb}(r)=\frac{1}{\sqrt{2}} \text{red}(\sqrt{2} \cdot \text{red}(r))</math><ref>misotanni, [https://misotanni.github.io/fjs/en/crash.html#lesson-0 The FJS Crash Course]</ref>.
* Balanced reduction of an interval <math>r</math> and an equave <math>e</math> expressed as ratios: <math>\text{reb}(r, e)=\frac{1}{\sqrt{e}} \text{red}(\sqrt{e} \cdot \text{red}(r, e), e)</math>.
* Balanced reduction of an interval <math>l</math> and an equave <math>e</math> expressed as logarithmic measures in the same units: <math>\text{reb}(l, e)= \text{red}(\text{red}(l, e) + e/2, e) - e/2</math>.

== See also ==
* [[Octave complement]]

== References ==
<references />

== External links ==
* https://forum.sagittal.org/viewtopic.php?p=1296 (thread on mathematical functions)

[[Category:Method]]
[[Category:Terms]]
[[Category:Elementary math]]
[[Category:Octave]]