Octave reduction: Difference between revisions

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'''~~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.

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

'''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).

== Practical methods ==

~~'''Tritave reduction''' is~~ the ~~application~~ of ~~this process in~~ a ~~tritave-equivalent tuning~~ (~~eg.~~ [[~~Bohlen-Pierce~~]])~~, where the equave~~ is ~~the [[tritave]]~~.

An easy way to find the reduced form of an interval is to use a specialized calculator (see [[Octave reduction#External links|§ External links]]). This is especially useful when working with very complex ratios.

~~== Practical~~ methods ==

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).

~~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.~~

=== Linear measures ===

~~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)~~.

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.

=== ~~Linear measures~~ ===

==== Simple method ====

# ~~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 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 ~~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 less than 1, multiply it by 2. Repeat until the resulting interval is greater than 1.

# If the starting interval is greater than ~~the equave~~, divide it by ~~the equave~~. Repeat until the resulting interval is less than ~~the equave~~.

# If the starting interval is greater than 2, divide it by 2. Repeat until the resulting interval is less than 2.

~~====~~ Examples ~~(octave-reduction) ====~~

Examples:

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

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* 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 (20 = 1) but greater than one octave down (2-1 = 1/2), so multiply by 2 once to get 9/5.

==== ~~Examples (other equaves)~~ ====

==== General formula ====

For a starting interval <math>r</math>, 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:

* ~~Consider a tritave~~-~~equivalent tuning; 7~~/~~9 is less than 3, so multiply~~ by ~~3 to get 7~~/3.

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

* Consider a just perfect fifth-~~equivalent tuning; 7~~/~~4 is greater than 3~~/~~2, so divide by 3~~/2

&= 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 ===

# 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.

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 ¢.

# 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~~.

==== Simple method ====

# 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 ~~(octave-reduction) ====~~

Examples:

* 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.

* 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 formula ====

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 - e\left\lfloor{l/e}\right\rfloor</math>.

Example:

* Octave-reducing 412 steps of 97edo can be done by using the formula with <math>r = 412</math>, <math>e = 97</math>: <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>

== Generalization ==

=== 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 [[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 ===

~~==== Examples~~ (~~other equaves~~) ~~====~~

'''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).

* ~~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~~.

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 ==

== External links ==

* https://www.yacavone.net/xen-calc/ (web calculator with reduction functions)

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

[[Category:Method]]

@@ Line 1: / Line 1: @@
-'''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.
+'''Octave reduction''' is the process of replacing an [[interval]] by the unique [[Equivalence|equivalent]] interval situated between the [[unison]] and the [[octave]]. In practice, this is done by adding or subtracting octaves from the starting interval as necessary.
-'''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).
+== Practical methods ==
-'''Tritave reduction''' is the application of this process in a tritave-equivalent tuning (eg. [[Bohlen-Pierce]]), where the equave is the [[tritave]].
+An easy way to find the reduced form of an interval is to use a specialized calculator (see [[Octave reduction#External links|§ External links]]). This is especially useful when working with very complex ratios.
-== Practical methods ==
+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).
-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.
+=== Linear measures ===
-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).
+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.
-=== Linear measures ===
+==== Simple method ====
-# 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 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 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 less than 1, multiply it by 2. Repeat until the resulting interval is greater than 1.
-# If the starting interval is greater than the equave, divide it by the equave. Repeat until the resulting interval is less than the equave.
+# If the starting interval is greater than 2, divide it by 2. Repeat until the resulting interval is less than 2.
-==== Examples (octave-reduction) ====
+Examples:
 * 3/4 is less than 1, so multiply by 2 to get 3/2.
@@ Line 25: / Line 25: @@
 * 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) ====
+==== General formula ====
+For a starting interval <math>r</math>, 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:
-* Consider a tritave-equivalent tuning; 7/9 is less than 3, so multiply by 3 to get 7/3.
+* 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} \\
-* Consider a just perfect fifth-equivalent tuning; 7/4 is greater than 3/2, so divide by 3/2
+&= 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 ===
-# 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.
+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 ¢.
-# 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.
+==== Simple method ====
+# 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 (octave-reduction) ====
+Examples:
 * 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.
+* 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 formula ====
+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 - e\left\lfloor{l/e}\right\rfloor</math>.
+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 - 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>
+== Generalization ==
+=== 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 [[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 ===
-==== Examples (other equaves) ====
+'''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).
-* 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.
+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://www.yacavone.net/xen-calc/ (web calculator with reduction functions)
+* https://forum.sagittal.org/viewtopic.php?p=1296 (thread on mathematical functions)
 [[Category:Method]]