Just intonation: Difference between revisions
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Similar logic may be used for instruments with timbres not aligning with the harmonic series; see [[timbral tuning]]. | Similar logic may be used for instruments with timbres not aligning with the harmonic series; see [[timbral tuning]]. | ||
==Ways of using JI== | == Ways of using JI == | ||
Here are multiple ways in which musicians and theorists have used just intonation. | Here are multiple ways in which musicians and theorists have used just intonation. | ||
[[Free style JI | ; [[Free style JI]] | ||
[[Lou Harrison]] used this term; it means that you choose just-intonation pitches from the set of all possible just intervals (not from a mode or scale) as you use them in music. | [[Lou Harrison]] used this term; it means that you choose just-intonation pitches from the set of all possible just intervals (not from a mode or scale) as you use them in music. | ||
; Harmonic limits and subgroups | |||
[[Harmonic limit | [[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 | 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 [[ | 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 | |||
These are [[Detempering|detemperings]], including [[NEJI]] systems. | These are [[Detempering|detemperings]], including [[NEJI]] systems. | ||
; Other approaches | |||
Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets|product sets]]. | Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets|product sets]]. | ||
==Approximating JI with temperaments== | |||
There are a lot of JI intervals, and it's difficult to keep track of all of them. As such, people often use simpler systems to approximate JI intervals, known as [[temperament]]s. A popular choice is [[equal temperament]]s; for example the predominant [[12edo|12et]], which is widely used to approximate [[5-limit]] JI. Other equal temperaments exist, for example [[19edo|19et]], [[22edo|22et]], and [[31edo|31et]]. Besides equal temperaments, other temperaments exist, such as [[regular temperament]]s and [[well temperament]]s. | |||
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. | |||
==Instruments== | ==Instruments== | ||
Latest revision as of 17:59, 27 December 2025
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 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 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.[1] 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 intervals and commas, two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are 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.
Consonance
LCJI intervals achieve consonance through alignment of partials if the interval has 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.
Similar logic may be used for instruments with timbres not aligning with the harmonic series; see timbral tuning.
Ways of using JI
Here are multiple ways in which musicians and theorists have used just intonation.
Lou Harrison used this term; it means that you choose just-intonation pitches from the set of all possible just intervals (not from a mode or scale) as you use them in music.
- Harmonic limits and subgroups
Harmonic limits 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. Subgroups 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"[2] (the harmonic series, isoharmonic chords, AFDOs/overtone scales, primodality, ringer scales), the "numerator to one or very few values" (the subharmonic series, IFDOs/undertone scales), or both (tonality diamonds)
- Mediants
The use of harmonic and arithmetic mediants as was common with the 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
These are detemperings, including NEJI systems.
- Other approaches
Other approaches include Meru scales, tritriadic scales, and product sets.
Approximating JI with temperaments
There are a lot of JI intervals, and it's difficult to keep track of all of them. As such, people often use simpler systems to approximate JI intervals, known as temperaments. A popular choice is equal temperaments; for example the predominant 12et, which is widely used to approximate 5-limit JI. Other equal temperaments exist, for example 19et, 22et, and 31et. Besides equal temperaments, other temperaments exist, such as regular temperaments and well temperaments.
Temperaments also create new structures not found in JI; for example, meantone temperament (which 12et supports) 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.
Instruments
- The array mbira was designed by Bill Wesley as a versatile just intonation instrument, covering a 5 octave range.
- Most of Harry Partch's instruments were designed to be for just intonation.
Music
Notation
There are various notation systems for just intonation.
See also
|
Todo: cleanup |
- List of approaches to musical tuning
- Gallery of just intervals
- Gallery of 12-tone just intonation scales
- Families of scales
- Boogie woogie scale
- Category:Just intonation
References
Further reading
- Just intonation on the Tonalsoft Encyclopedia
- Just Intonation by Mark Nowitzky
- Just Intonation Explained by Kyle Gann
- Anatomy of an Octave by Kyle Gann
- What is Just Intonation? by David B. Doty
- What is "just intonation"? by Carl Lumma
- A Response to Julia Werntz by David B. Doty
- Comma Sequences by Gene Ward Smith
- Just Intonation in Renaissance Theory & Practice by Ross W. Duffin