Just intonation: Difference between revisions

'''Just intonation''' ('''JI''') is an approach to [[musical tuning]] which uses tones whose frequencies are whole-number ratios of a given fundamental [[frequency]]. Just intonation includes the [[harmonic series]], which is the collection of tones found at integer multiples of a fundamental frequency; all just intervals can be found as the interval between two notes in the harmonic series. Just ratios of small numbers, called '''Low-complexity just intonation (LCJI)''' intervals, tend to be the most [[concordant|consonant]] in the sense that their sounds meld together.  

In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit|5-limit tuning]]—intervals where the numerators and denominators of any ratio used have no prime factors greater than 5. ''Extended just intonation'', a term coined by [[Ben Johnston]], refers to any tuning in the harmonic series regardless of [[prime limit]].<ref>From Ben Johnston "A Notation System for Extended Just Intonation." ''Maximum Clarity'', 2006, p. 77</ref> In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].  

The structure of just intonation has several implications on music composition. [[Wolf interval|Wolf intervals]] and [[Comma|commas]], two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[Comma pump|comma pumps]], which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as tools to use or as problems to be solved. Examples of approaches that try to solve these problems without greatly restricting the set of available ratios include pitch shifts, [[adaptive just intonation]] and [[temperament]]. Other approaches restrict the space of usable JI intervals in a way that makes these problems arise less frequently. {{todo|clarify}}

== Consonance ==

LCJI intervals achieve consonance through alignment of [[Partial|partials]] if the interval has [[Harmonic timbre|harmonic timbre]]. In fact, alignment of partials is a stronger effect with harmonic timbre: if partials align at frequency n, they will also align at every multiple of n; and in addition, two notes whose partials align with the same root note will also have partials aligning with each other. This allows for the construction of just-intonation chords of more than two notes where every comprising interval is a consonance.  

Low-complexity JI intervals and chords also achieve consonance by being the ratios between harmonics of a (possibly unplayed) fundamental even if they do not have harmonic timbre.

 

== Ways of using JI ==

; [[Free style JI]]

; Harmonic limits and subgroups

[[Harmonic limit]]s set a limit for the highest prime number in the factorization of any ratio used; for example, western music is based off the [[5-limit]]. [[Subgroup]]s name a list of allowable prime numbers used.

; Restrictions on the denominator or numerator

Some approaches restrict "the denominator to one or very few values"<ref name=":0">From Jacques Dudon, "Differential Coherence", ''1/1'' vol. 11, no. 2: p.1).</ref> (the [[harmonic series]], [[isoharmonic chord]]s, [[AFDO]]s/[[overtone scale]]s, [[primodality]], [[Ringer scale|ringer scales]]), the "numerator to one or very few values" (the [[subharmonic series]],  [[IFDO]]s/undertone scales), or both ([[Tonality diamond|tonality diamonds]])

; Mediants

The use of harmonic and arithmetic [[mediant (operation)|mediants]] as was common with the [[ancient Greek music|Ancient Greeks]]. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.{{citation needed}}

; Approximations/alterations of tempered tunings

; Other approaches

Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets|product sets]].

 

 

Temperaments also create new structures not found in JI; for example, [[meantone]] temperament (which 12et [[support]]s) tempers out 81/80, making [[5/4]] the same as the major third obtained by stacking four fifths, [[81/64]]; this structural feature is often assumed without thinking in western music.