Octave reduction: Difference between revisions

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Example:

* 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} \\

* Octave-reducing 4900/243 can be done by using the formula with <math>r = 4900/243</math>: <math>\begin{aligned}\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>

&= 1225/972\end{aligned}</math>

=== Logarithmic measures ===

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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 \\

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

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

&= 24\end{aligned}</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>.

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Example:

* Octave-reducing 412 steps of 97edo again: <math>\begin{~~align~~}\text{red}(412, 97) &= 412 - 97\left\lfloor{412/97}\right\rfloor \\

* Octave-reducing 412 steps of 97edo again: <math>\begin{aligned}\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>

&= 24\end{aligned}</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.

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.

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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 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 ===

@@ Line 40: / Line 40: @@
 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} \\
+* Octave-reducing 4900/243 can be done by using the formula with <math>r = 4900/243</math>:<br><math>\begin{aligned}\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>
+&= 1225/972\end{aligned}</math>
 === Logarithmic measures ===
@@ Line 51: / Line 51: @@
 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 \\
+* 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{aligned}\text{red}(412, 97) &= 412 \bmod 97 \\
-&= 24\end{align}</math>
+&= 24\end{aligned}</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>.
@@ Line 58: / Line 58: @@
 Example:
-* Octave-reducing 412 steps of 97edo again:<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{aligned}\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>
+&= 24\end{aligned}</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.
-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 76: / 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 ===

Octave reduction: Difference between revisions

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