Octave reduction: Difference between revisions

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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.

~~==== 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.

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* 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.

~~==== 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>: <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 ===

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.

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* 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>: <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>.

@@ Line 10: / Line 10: @@
 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.
@@ Line 24: / Line 22: @@
 * 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.
-==== 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 ===
 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.
@@ Line 51: / Line 35: @@
 * 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>.