Fluid just intonation: Difference between revisions

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 Each chord contains a ''common note'' shared with the previous chord. The tuning of the common note stays exactly the same, and all other notes of the new chord are tuned in pure [[JI]] ratios relative to the common note.
-The allowed ratios, and how much they differ from their [[12edo]] tempered version, are listed in the following table.
+=== Allowed ratios ===
+The sheet music for ''Microtonal Composition in "Fluid Just Intonation"''  includes a table of allowed ratios, and how much they differ from their [[12edo]] tempered version (in [[cents]]).
-(''Note: Some of the cents values given by the original table - the one included in the sheet music - do not match the actual cents values of the JI values given. So, the "original" column documents the values given by the original table, and the "corrected" column documents the accurate values for each JI interval. It is unclear which of these was used in the piece itself.'')
+However, the JI ratios and the cent values do not all agree with one another.
+So, this page will include two tables, one based on the JI ratios provided in the sheet music, and one based on the cent values.
+'''Based on the JI ratios'''
 {| class="wikitable"
-! rowspan="2" | Scale degree!! rowspan="2" | Ratio!! colspan="2" |Tuning adjustment in [[cents]]
+! Scale degree
+! Ratio
+! Tuning adjustment (cents)
 |-
-!Original
+|  1
-!Corrected
+| [[1/1]]
+|  0
 |-
-|1||1/1||0
+|  b2
-|0
+| [[16/15]]
+|  +12
 |-
-|b2||16/15||+5
+|  2
-| +12
+| [[9/8]]
+|  +4
 |-
-|2||9/8||+4
+|  b3
-| +4
+| [[6/5]]
+|  +16
 |-
-|b3||6/5||+16
+|  3
-| +16
+| [[5/4]]
+|  -14
 |-
-|3||5/4||-14
+|  4
-| -14
+| [[4/3]]
+|  -2
 |-
-|4||4/3||-2
+|  #4/b5
-| -2
+| [[11/8]]
+|  -49
 |-
-|#4/b5||11/8||-49
+|  5
-| -49
+| [[3/2]]
+|  +2
 |-
-|5||3/2||+2
+|  b6
-| +2
+| [[8/5]]
+|  +14
 |-
-|b6||8/5||+41
+|  6
-| +14
+| [[5/3]]
+|  -16
 |-
-|6||5/3||+6
+| rowspan="2" |b7
-| -16
+| [[7/4]] (''dom7'')
+|  -31 (''dom7'')
 |-
-| rowspan="2" |b7||7/4 (dom7)||-31 (dom7)
+| [[9/5]] (''m7'')
-| -31 (dom7)
+|  +18 (''m7'')
 |-
-|9/5 (m7)||+18 (m7)
+|  7
-| +18 (m7)
+| [[15/8]]
+|  -12
 |-
-|7||15/8||-12
+|  8ve
-| -12
+| [[2/1]]
+|  0
+|}
+'''Based on the cent values'''
+{| class="wikitable"
+! Scale degree!! Ratio!!Tuning adjustment (cents)
 |-
-|8ve||2/1||0
+|1||[[1/1]]||0
-|0
+|-
+|b2||[[17/16]]||+5
+|-
+|2||[[9/8]]||+4
+|-
+|b3||[[6/5]]||+16
+|-
+|3||[[5/4]]||-14
+|-
+|4||[[4/3]]||-2
+|-
+|#4/b5||[[11/8]]||-49
+|-
+|5||[[3/2]]||+2
+|-
+|b6||[[13/8]]||+41
+|-
+|6||[[27/16]]||+6
+|-
+| rowspan="2" |b7||[[7/4]] (''dom7'')||-31 (''dom7'')
+|-
+|[[9/5]] (''m7'')||+18 (''m7'')
+|-
+|7||[[15/8]]||-12
+|-
+|8ve||[[2/1]]||0
 |}
-== Subsets and supersets ==
+=== Relationship to other tunings ===
-Fluid just intonation is a subset of [[11-limit]] just intonation.
+'''According to the sheet music's stated JI ratios:'''
+Fluid just intonation is a subset of [[11-limit]] just intonation.
+[[30afdo]] is the smallest [[Overtone scale|mode of the harmonic series]] that can approximate all of fluid just intonation's allowed ratios with a median error of 0 cents and a maximum error less than 20 cents.
+[[EDOs]] with progressively better approximations of fluid just intonation's allowed ratios include: {{EDOs|'''19''', 22, 28, '''31''', '''41''', 72, '''87''', '''118'''...}}
+'''According to the sheet music's stated cent values:'''
+Fluid just intonation is a subset of [[17-limit]] just intonation.
+[[32afdo]] is the smallest mode of the harmonic series that can approximate all of fluid just intonation's allowed ratios with a median error of 0 cents and a maximum error less than 20 cents.
-It is also a subset of the [[polymicrotonal]] tuning of ([[15afdo]] and [[8afdo]]).
+[[EDOs]] with progressively better approximations of fluid just intonation's allowed ratios include: {{EDOs|'''19''', 24, 30, '''31''', 33, '''41''', 46, 66, 77, '''87''', 90, 94, '''118'''...}}
 == Influences ==