Octave reduction: Difference between revisions

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=== 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:

* 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 \cdot 4 \\

@@ Line 53: / Line 53: @@
 === 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:
-* 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 \cdot 4 \\