Octave reduction: Difference between revisions
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Example: | Example: | ||
* Octave-reducing 4900/243 can be done by using the formula with <math>r = 4900/243</math>:<br><math>\begin{ | * 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^{-\left\lfloor{4.33375\ldots}\right\rfloor} \\ | ||
&= 4900/243 \cdot 2^{-4} \\ | &= 4900/243 \cdot 2^{-4} \\ | ||
&= 4900/243 \cdot 1/16 \\ | &= 4900/243 \cdot 1/16 \\ | ||
&= 1225/972\end{ | &= 1225/972\end{aligned}</math> | ||
=== Logarithmic measures === | === Logarithmic measures === | ||
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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{ | * 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{ | &= 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>. | 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: | Example: | ||
* Octave-reducing 412 steps of 97edo again:<br><math>\begin{ | * 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\left\lfloor{4.24742\ldots}\right\rfloor \\ | ||
&= 412 - 97 \cdot 4 \\ | &= 412 - 97 \cdot 4 \\ | ||
&= 412 - 388 \\ | &= 412 - 388 \\ | ||
&= 24\end{ | &= 24\end{aligned}</math> | ||
== Generalizations == | == Generalizations == | ||
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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. [[ | 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 | * 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 === | ||