Editing Just intonation (section)

==Terminology==

Just intonations are categorized by the notion of [[Limit (music)|limits]]. The limit refers to the highest prime number fraction included in the intervals of a scale. All the intervals of any 3&nbsp;limit just intonation will be multiples of 3. So {{sfrac| 6 | 5 }} is included in 5&nbsp;limit, because it has 5 in the denominator. If a scale uses an interval of 21:20, it is a 7&nbsp;limit just intonation, since 21 is a multiple of 7. The interval {{sfrac| 9 | 8 }} is a 3&nbsp;limit interval because the numerator and denominator are multiples of 3 and 2, respectively. It is possible to have a scale that uses 5&nbsp;limit intervals but not 2&nbsp;limit intervals, i.e. no octaves, such as [[Wendy Carlos]]'s [[Alpha scale|alpha]] and [[Beta scale|beta]] scales. It is also possible to make diatonic scales that do not use fourths or fifths (3&nbsp;limit), but use 5 and 7&nbsp;limit intervals only. Thus, the notion of limit is a helpful distinction, but certainly does not tell us everything there is to know about a particular scale.

[[Pythagorean tuning]], or 3&nbsp;limit tuning, allows ratios including the numbers&nbsp;2 and 3 and their powers, such as 3:2, a [[perfect fifth]], and 9:4, a [[ninth|major ninth]]. Although the interval from C to G is called a perfect fifth for purposes of [[Musical analysis|music analysis]] regardless of its tuning method, for purposes of discussing tuning systems musicologists may distinguish between a ''perfect fifth'' created using the 3:2&nbsp;ratio and a ''tempered fifth'' using some other system, such as [[meantone temperament|meantone]] or [[equal temperament]].

[[Five-limit tuning|5-limit tuning]] encompasses ratios additionally using the number&nbsp;5 and its powers, such as 5:4, a [[major third]], and 15:8, a [[major seventh]]. The specialized term ''perfect third'' is occasionally used to distinguish the 5:4&nbsp;ratio from major thirds created using other tuning methods. [[7-limit tuning|7&nbsp;limit]] and higher systems use higher [[prime number]] partials in the [[Harmonic series (music)|overtone series]] (e.g. 11, 13, 17, etc.)

[[Comma (music)|Commas]] are very small intervals that result from minute differences between pairs of just intervals. For example, the (5&nbsp;limit) 5:4&nbsp;ratio is different from the Pythagorean (3&nbsp;limit) major third (81:64) by a difference of 81:80, called the [[syntonic comma]]. The [[septimal comma]], the ratio of 64:63, is a 7&nbsp;limit interval which is the distance between the Pythagorean [[minor third|semi-ditone]], {{sfrac| 32 | 27 }}, and the [[septimal minor third]], 7:6&nbsp;, since  <math>\ \left( \tfrac{\ 32\ }{ 27 } \right) \div \left(\tfrac{\ 7\ }{ 6 }\right) = \tfrac{\ 64\ }{ 63 } ~.</math>

A [[cent (music)|cent]] is a measure of interval size. It is logarithmic in the musical frequency ratios. The octave is divided into 1200&nbsp;steps, 100&nbsp;cents for each semitone. Cents are often used to describe how much a just interval deviates from {{nobr|[[12 equal temperament|12 {{sc|TET}}]].}} For example, the [[major third]] is 400&nbsp;cents in 12&nbsp;TET, but the 5th&nbsp;harmonic, 5:4 is 386.314&nbsp;cents. Thus, the just major third deviates by −13.686&nbsp;cents.