Cent: Difference between revisions

(2 intermediate revisions by the same user not shown)

Line 8:

}}

The '''cent''' (symbol: '''¢''') is a [[unit of interval size]] equal to exactly 1/~~100th~~ (or 1%) of a [[12edo]] [[semitone (interval size measure)|semitone]]. In other words, cents divide the half step (semitone) of 12edo into 100 equal parts. First proposed in the late 19th century by {{w|Alexander John Ellis|Alexander J. Ellis}}, the cent may also be defined as the {{w|logarithm}} base 1200th root of 2 of a ratio.

The '''cent''' (symbol: '''¢''') is a [[unit of interval size]] equal to exactly 1/100 (or 1%) of a [[12edo]] [[semitone (interval size measure)|semitone]]. In other words, cents divide the half step (semitone) of 12edo into 100 equal parts. First proposed in the late 19th century by {{w|Alexander John Ellis|Alexander J. Ellis}}, the cent may also be defined as the {{w|logarithm}} base 1200th root of 2 of a ratio.

Cents are often used to express the size of intervals in different tuning systems, sometimes to express the accuracy of the representation of a [[just intonation]] [[interval]] in a given system.

Line 16:

== Conversion ==

=== Ratio to cents ===

To find the size ''s'' of an interval in cents from its ratio ''c'', ~~you have to~~ calculate the [[log2|binary logarithm]] (log2) of its [[frequency ratio]], and multiply ~~this~~ by 1200.

To find the size ''s'' of an interval in cents from its ratio ''r'', calculate the [[log2|binary logarithm]] (log2) of its [[frequency ratio]], and multiply it by 1200.

<math>\displaystyle s = 1200\log_2 (c)</math>

<math>\displaystyle s = 1200 \cdot \log_2 (r)</math>

~~Example (~~just perfect fifth~~): log~~2(3/2) × ~~1200 ≈~~ 0.584 ~~× 1200~~ ≈ 701.955 cents.

For example, the size in cents of a just perfect fifth is 1200⋅log2(3/2) ≈ 1200 × 0.584 ≈ 701.955 cents.

If your pocket calculator has no <code>log2</code> key, but does have a <code>log</code> (log10) or <code>ln</code> (log''e'') key, you can key it this way:

@@ Line 8: / Line 8: @@
 }}
 {{Wikipedia|Cent (music)}}
-The '''cent''' (symbol: '''¢''') is a [[unit of interval size]] equal to exactly 1/100th (or 1%) of a [[12edo]] [[semitone (interval size measure)|semitone]]. In other words, cents divide the half step (semitone) of 12edo into 100 equal parts. First proposed in the late 19th century by {{w|Alexander John Ellis|Alexander J. Ellis}}, the cent may also be defined as the {{w|logarithm}} base 1200th root of 2 of a ratio.
+The '''cent''' (symbol: '''¢''') is a [[unit of interval size]] equal to exactly 1/100 (or 1%) of a [[12edo]] [[semitone (interval size measure)|semitone]]. In other words, cents divide the half step (semitone) of 12edo into 100 equal parts. First proposed in the late 19th century by {{w|Alexander John Ellis|Alexander J. Ellis}}, the cent may also be defined as the {{w|logarithm}} base 1200th root of 2 of a ratio.
 Cents are often used to express the size of intervals in different tuning systems, sometimes to express the accuracy of the representation of a [[just intonation]] [[interval]] in a given system.
@@ Line 16: / Line 16: @@
 == Conversion ==
+{{See also| Ratio #Conversion }}
 === Ratio to cents ===
-To find the size ''s'' of an interval in cents from its ratio ''c'', you have to calculate the [[log2|binary logarithm]] (log<sub>2</sub>) of its [[frequency ratio]], and multiply this by 1200.
+To find the size ''s'' of an interval in cents from its ratio ''r'', calculate the [[log2|binary logarithm]] (log<sub>2</sub>) of its [[frequency ratio]], and multiply it by 1200.
-<math>\displaystyle s = 1200\log_2 (c)</math>
+<math>\displaystyle s = 1200 \cdot \log_2 (r)</math>
-Example (just perfect fifth): log<sub>2</sub>(3/2) × 1200 ≈ 0.584 × 1200 ≈ 701.955 cents.
+For example, the size in cents of a just perfect fifth is 1200⋅log<sub>2</sub>(3/2) ≈ 1200 × 0.584 ≈ 701.955 cents.
 If your pocket calculator has no <code>log2</code> key, but does have a <code>log</code> (log<sub>10</sub>) or <code>ln</code> (log<sub>''e''</sub>) key, you can key it this way: