Octave reduction - Revision history

ArrowHead294 at 18:37, 13 March 2025

2025-03-13T18:37:05Z

← Older revision		Revision as of 18:37, 13 March 2025
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	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 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).		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.		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.
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	* Consider a tritave-equivalent tuning; 7/9 is less than 3, so multiply by 3 to get 7/3.		* 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>.		* 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 ===

Fredg999: Fix typo

2023-02-25T22:47:44Z

Fix typo

← Older revision		Revision as of 22:47, 25 February 2023
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	&= 24\end{align}</math>		&= 24\end{align}</math>

	== ~~Generalization~~ ==		== Generalizations ==
	=== Other equaves ===		=== 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 interval of equivalence or ''[[equave]]'' of that tuning.

Fredg999: Replaced xen-calc link by an internal one, fixed typo

2022-02-03T22:58:37Z

Replaced xen-calc link by an internal one, fixed typo

← Older revision		Revision as of 22:58, 3 February 2022
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	== Practical methods ==		== 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).		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 ===		=== 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.		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.

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	* 7/2 is greater than 2, so divide by 2 to get 7/4.		* 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.		* 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.		* 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 ===		=== 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 ¢.		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 ¢.

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	== General formulas ==		== General formulas ==

	=== Linear measures ===		=== 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>.		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>.

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	=== Logarithmic measures ===		=== 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.		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.

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	== Generalization ==		== Generalization ==

	=== Other equaves ===		=== 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 interval of equivalence or ''[[equave]]'' of that tuning.

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	=== Balanced reduction ===		=== 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).		'''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).

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

	* [[Octave complement]]		* [[Octave complement]]

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

Sintel: categories

2022-01-20T12:27:44Z

Dave Keenan: Changed the first occurrence of "equave" to "interval of equivalence or equave".

2021-10-07T09:38:43Z

Changed the first occurrence of "equave" to "interval of equivalence or equave".

← Older revision		Revision as of 09:38, 7 October 2021
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	=== Other equaves ===		=== 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.		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).		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).

Fredg999: Lightened the phrasing of the first sentence

2021-08-06T16:34:50Z

Lightened the phrasing of the first sentence

← Older revision		Revision as of 16:34, 6 August 2021
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	'''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 ==		== Practical methods ==

Fredg999: /* Logarithmic measures */

2021-07-18T17:17:16Z

Logarithmic measures

← Older revision		Revision as of 17:17, 18 July 2021
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	=== Logarithmic measures ===		=== 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 - e~~\left\lfloor{l~~/e}\~~right\rfloor~~</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 \bmod e</math>, where <math>\bmod</math> is the modulo operation.

	Example:		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 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\left\lfloor{4.24742\ldots}\right\rfloor \\
	&= 412 - 97 \cdot 4 \\		&= 412 - 97 \cdot 4 \\

Fredg999: Separated general formulas from simple methods in two top-level sections

2021-07-18T17:08:11Z

Separated general formulas from simple methods in two top-level sections

← Older revision		Revision as of 17:08, 18 July 2021
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	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.		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.

	~~==== Simple method ====~~

	# 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 greater or equal to the unison (1) and less than the octave (2), it is already in reduced form.
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	* 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.		* 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.

	~~==== 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:~~

	* 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 ===		=== 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 ¢.		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 ¢.

	~~==== 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.		# 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.
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	* 1442¢ is greater than 1200 ¢, so subtract 1200 ¢ to get 242 ¢.		* 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.		* 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>

	===~~= General formula =~~===		=== 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 - e\left\lfloor{l/e}\right\rfloor</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>.

Fredg999: Moved general reduction to a Generalization section, added balanced reduction, various adjustments

2021-07-18T17:00:38Z

Moved general reduction to a Generalization section, added balanced reduction, various adjustments

Show changes

Fredg999: Generalized to reduction with any equave; reorganized section for practical methods

2021-07-18T04:37:42Z

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.
-<= 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]].
-== 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 ===
 == See also ==
 * [[Octave complement]]
 [[Category:Method]]

← Older revision		Revision as of 12:27, 20 January 2022
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	[[Category:Method]]		[[Category:Method]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:~~Howto~~]]		[[Category:Elementary math]]
	[[Category:Octave]]		[[Category:Octave]]