Octave reduction: Difference between revisions

(6 intermediate revisions by 3 users not shown)

Line 1:

'''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 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.

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.

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.

Line 20:

Line 18:

* 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)4, thus 81/16 (or 5.0625), which is greater than 2 octaves (22 = 4), but less than 3 octaves (23 = 8), so divide by 2 twice to get [[81/64]].

* Adding 4 just perfect fifths ([[3/2]]) corresponds to (3/2)4, thus 81/16 (or 5.0625), which is greater than 2 octaves (22 = 4), but less than 3 octaves (23 = 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 (20 = 1) but greater than one octave down (2-1 = 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 ¢.

Line 38:

Line 35:

== 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>.

Line 52:

Line 47:

=== 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>: <math>\begin{align}\text{red}(412, 97) &= 412 \bmod 97 \\

&= 24\end{align}</math>

~~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>.

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 ~~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 \\

* Octave-reducing 412 steps of 97edo again: <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 \\

Line 63:

Line 64:

&= 24\end{align}</math>

== ~~Generalization~~ ==

== 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.

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).

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.

Line 77:

Line 76:

* 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.

* 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).

Line 94:

Line 92:

== See also ==

* [[Octave complement]]

Line 101:

Line 98:

== 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]]

[[Category:Terms]]

[[Category:~~Howto~~]]

[[Category:Elementary math]]

[[Category:Octave]]

@@ Line 1: / Line 1: @@
-'''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 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.
-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.
 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.
@@ Line 20: / Line 18: @@
 * 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]].
+* 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 ¢.
@@ Line 38: / Line 35: @@
 == 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>.
@@ Line 52: / Line 47: @@
 === 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>
-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>.
+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 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 \\
+* 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 \\
@@ Line 63: / Line 64: @@
 &= 24\end{align}</math>
-== Generalization ==
+== 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.
-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).
-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.
@@ Line 77: / Line 76: @@
 * 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.
+* 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).
@@ Line 94: / Line 92: @@
 == See also ==
 * [[Octave complement]]
@@ Line 101: / Line 98: @@
 == 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]]
 [[Category:Terms]]
-[[Category:Howto]]
+[[Category:Elementary math]]
 [[Category:Octave]]