Editing Equal temperament (section)

== General properties ==
{{Unreferenced section|date=June 2011}}

In an equal temperament, the distance between two adjacent steps of the scale is the same [[Interval (music)|interval]]. Because the perceived identity of an interval depends on its [[ratio]], this scale in even steps is a [[geometric sequence]] of multiplications. (An [[arithmetic sequence]] of intervals would not sound evenly spaced and would not permit [[Transposition (music)|transposition]] to different [[Key (music)|keys]].) Specifically, the smallest [[Interval (music)|interval]] in an equal-tempered scale is the ratio:

:<math>\ r^n = p\ </math>
:<math>\ r = \sqrt[n]{p\ }\ </math>

where the ratio {{mvar|r}} divides the ratio {{mvar|p}} (typically the octave, which is 2:1) into {{mvar|n}} equal parts. (''See [[#Twelve-tone equal temperament|Twelve-tone equal temperament]] below.'')

Scales are often measured in [[cent (music)|cents]], which divide the octave into 1200&nbsp;equal intervals (each called a cent). This [[logarithm]]ic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in [[ethnomusicology]]. The basic step in cents for any equal temperament can be found by taking the width of {{mvar|p}} above in cents (usually the octave, which is 1200&nbsp;cents wide), called below {{mvar|w}}, and dividing it into {{mvar|n}} parts:
:<math>\ c = \frac{\ w\ }{ n }\ </math>

In musical analysis, material belonging to an equal temperament is often given an [[Musical notation#Integer notation|integer notation]], meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the [[logarithm]] of a multiplication reduces it to addition. Furthermore, by applying the [[modular arithmetic]] where the modulus is the number of divisions of the octave (usually 12), these integers can be reduced to [[pitch class]]es, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g., {{mvar|c}} is 0 regardless of octave register. The [[MIDI]] encoding standard uses integer note designations.

=== General formulas for the equal-tempered interval ===
{{Missing information|section|the general formulas for the equal-tempered interval|date=February 2019}}