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</o:shapelayout></xml><![endif]--></head><body lang=FR link=blue vlink=purple><div class=WordSection1><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'>Dear colleagues,<o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'> <o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'> 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. <o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'> Major third root movements are similar since III sounds the tenth, twelfth, and fifteenth harmonics of the tonic as</span><span lang=EN-US> </span><span lang=EN-US style='font-family:"Times New Roman","serif"'>Dimitar Ninov reminds us in his May 3<sup>rd</sup> 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.<o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'> 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. <o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'> 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. <o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'> 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.<o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'> 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 1<sup>st</sup> posting. Stability is coming back home after a trip, rather than being left out on a limb, away from friends and family.<o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'><o:p> </o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'> Thank you for your attention.<o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span lang=EN-US style='font-family:"Times New Roman","serif"'><o:p> </o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span style='font-family:"Times New Roman","serif"'>Bruce Grant, DM, Indiana University<o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span style='font-family:"Times New Roman","serif"'>Chef d’orchestre et chef des chœurs,<o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span style='font-family:"Times New Roman","serif"'>Théâtre de l’opérette, Lyons, France<o:p></o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><span style='font-family:"Times New Roman","serif"'><o:p> </o:p></span></p><p class=MsoNormal style='punctuation-wrap:simple;text-autospace:none'><b><span style='font-family:"Times New Roman","serif"'> <o:p></o:p></span></b></p><p class=MsoNormal><o:p> </o:p></p></div></body></html>