Functional harmony in rank-2 temperaments: Difference between revisions

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|+Functional ~~Roles~~ in ~~Meantone~~ and Orwell

|+ style="font-size: 105%;" | Functional roles in meantone and Orwell

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! Temperament !! Intervals of 3 !! Intervals of 5 !! Intervals of 7

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[[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.

Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (-8)–(-7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.

Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (−8)–(−7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.

[[Category:Regular temperament theory]]

[[Category:Harmony]]

[[Category:Guides]]

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 {| class="wikitable center-all"
-|+Functional Roles in Meantone and Orwell
+|+ style="font-size: 105%;" | Functional roles in meantone and Orwell
 |-
 ! Temperament !! Intervals of 3 !! Intervals of 5 !! Intervals of 7
@@ Line 88: / Line 88: @@
 [[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.
-Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (-8)–(-7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.
+Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (−8)–(−7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.
 [[Category:Regular temperament theory]]
 [[Category:Harmony]]
 [[Category:Guides]]