[Smt-talk] Forward and backward root movements

[Bruce Grant]

Dear colleagues,

            The discussion about descending or advancing progressions can be
resolved by observation of the fact that the notes of the dominant triad are
the third, ninth, and fifteenth harmonics of any tonic. Therefore, a I > V
root movement backs up, as does a IV > I movement, since the notes of the
tonic triad are the third, ninth, and fifteenth harmonics of the
subdominant. V > I and I > IV root movements go forward since the second
chord introduces frequencies that are not to be found in the first chord. 

            Major third root movements are similar since III sounds the
tenth, twelfth, and fifteenth harmonics of  the tonic as Dimitar Ninov
reminds us in his May 3rd posting. I sounds the tenth, twelfth, and
fifteenth harmonics of bVI if minor, and  the tenth, fifteenth, and
twenty-fifth harmonics of bVI if major. Therefore, III functions as a
dominant  and bVI as a subdominant, which is interesting as they are also
the dominant of the relative minor and subdominant of the relative major.

           The minor submediant is similar to the minor mediant because they
share a major third with the degree a minor third above, VI with I, or
below, bIII with Im. II, the subdominant of the relative minor, has long
been recognized as a substitute for IV. So it is possible to combine
Schenker’s « stufen » with the three functions of Riemann. 

            I teach my students that I, IV and V are the « primary » chords,
tonic, subdominant and dominant. In the major mode, II, III, and VI are «
secondary » chords, the subdominant, dominant and tonic of the relative
minor. VII is an incomplete dominant since it represents the fifth, sixth
and seventh harmonics of V. (If the seventh harmonic is modified in equal
temperament, so are the other harmonics!) In the minor mode, bIII, bVI, and
bVII are secondary: the tonic, subdominant and dominant of the relative
major. The diminished triad on II is an inversion of the minor triad on IV
with an added sixth, but without the fifth. 

            We can therefore chart the equivalences of the stufen and the
functions as follows: I is tonic major or minor. II is subdominant of the
relative minor or the incomplete minor subdominant with an added sixth. The
bIII is tonic of the relative major. III is dominant of the relative minor.
IV is subdominant major or minor. V is dominant major or minor. The bVI is
subdominant of the relative major. VI is tonic of the relative minor. The
bVII is dominant of the relative major. VII is an incomplete dominant
seventh.

            Dominant to tonic root movements go forward as do tonic to
subdominant movements. The subdominants IV and bVI go forward to dominants
by analogy to the II > V progression. The contrary root progressions back
up. Root movements by perfect fifths and fourths, major thirds and  major or
minor seconds change the function, but Root movements by minor thirds and
diminished fifths stay within the same function. After backing up, stability
is found by returning forwards, as in Schenker’s « ursatz »: TDT. TST is
similar if contrary. TSDT progressions are more complete since they use all
the notes of the diatonic scale and replace a back and forth movement with a
circular one as Ildar Khannanov says in his May 1st posting. Stability is
coming back home after a trip, rather than being left out on a limb, away
from friends and family.

 

            Thank you for your attention.

 

Bruce Grant, DM, Indiana University

Chef d’orchestre et chef des chœurs,

Théâtre de l’opérette, Lyons, France

 

            

 

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