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]].

==Ways of using JI==

== Ways of using JI ==

Here are multiple ways in which musicians and theorists have used just intonation.

[[Free style JI~~|'''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.

~~'''~~Harmonic limits and subgroups~~''' ~~

; Harmonic limits and subgroups

[[Harmonic limit~~|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]]. [[Subgroup]]s name a list of allowable prime numbers used.

[[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~~''' ~~

; 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~~''' ~~

; Mediants

The use of harmonic and arithmetic [[~~Mediant~~ (operation)|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.

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~~''' ~~

; Approximations/alterations of tempered tunings

These are [[Detempering|detemperings]], including [[NEJI]] systems.

~~'''~~Other approaches~~''' ~~

; Other approaches

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

@@ Line 22: / Line 22: @@
 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.
-[[Free style JI|'''Free style JI''']] <br />
+; [[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.
-'''Harmonic limits and subgroups'''<br />
+; Harmonic limits and subgroups
-[[Harmonic limit|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]]. [[Subgroup]]s name a list of allowable prime numbers used.
+[[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'''<br />
+; 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'''<br />
+; Mediants
-The use of harmonic and arithmetic [[Mediant (operation)|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.
+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''' <br />
+; Approximations/alterations of tempered tunings
 These are [[Detempering|detemperings]], including [[NEJI]] systems.
-'''Other approaches'''<br />
+; Other approaches
 Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets|product sets]].