The picture below is of a musical balance "scale" that has four "pans" represented by colored thumbtacks and distributed symmetrically around a circle, instead of the two pans of a typical balance scale. Assuming octave and enharmonic equivalence, there are only four augmented triads, which are each assigned to one of these four "pans," such that augmented triads a whole step apart are across from one another on the circle. Any group of notes can be "weighed" on this "scale." For example, the picture below evaluates an A-minor triad, the tonic triad of Paganini’s famous caprice. The single orange ring on the blue thumbtack represents the root A, and the two orange rings on the white thumbtack represents the third C and fifth E. The circle thus leans directly toward the camera, touching the table at a point along the circle’s circumference a third of the way from the C-E-G# augmented triad to the C#-F-A augmented triad, or 30° away from the C-E-G# pan.
Here's another example, using the F-major chord. Now the circle touches the table at a point 30° farther clockwise away from where the A-minor chord touched the table. Therefore, we'll say that the A-minor chord and the F-major chord differ by 30°; more precisely, we'll say that the F-major chord is 30° clockwise away from the A-minor chord.
Quinn and Cohn use the structures that "scales" like this make for different purposes:
Quinn is more interested in the magnitude by which different types of chords tip the scale, and less interested in the direction of the tip. Any augmented triad would tip this scale the most, while a chord of the fully-diminished type or four-note cluster type (among others) are perfectly balanced and do not tip this scale at all. The magnitude of the tilt can give an idea of how "major-thirdy" a certain type of chord is.
Cohn is more interested in the direction that a chord of a certain type tips the scale, and less interested in the magnitude of the tip. He has noticed that this scale can give a relatively good idea how far the notes in one chord would need to travel to move to notes in another chord. In particular, when you divide the number of degrees two three-note chords are apart by 30, this sometimes, or perhaps even often, tells you how many semitones in total the notes in one chord minimally need to move to produce the notes in the other chord. For instance, in the example above, the A-minor triad and the F-major triad differ by 30°. Divide this by 30 and you get 1, which is the sum of all of the semitones minimally needed to change the notes of an A-minor triad to the notes of an F-major triad: A-->A (0 semitones), C-->C (0 semitones), E-->F (1 semitone) (0+0+1=1). Furthermore, if the change is clockwise, the minimal voice leading goes up; if the change is counterclockwise, the minimal voice leading does down. In this case, the change is 30° clockwise; therefore, the single semitonal motion is up (E-->F).
I write "relatively good idea" and "sometimes, or perhaps often" because this method works more frequently for some chord types than others. Between augmented triads, it always works. Between major and minor triads, it almost always works. The closer the notes can be put together, irrespective of octave, the less frequently it works. Cohn focuses his attention on major and minor triads, so this approach is pretty reliable.
My Paganini variation begins, as expected, with an A-minor chord, and then follows it with eight other major or minor triads, one for each measure: EM, FM, CM, Fm, DbM, AbM, C#m, and AM. (The number 8 is useful in how it matches the number of measures in the first section of Paganini's caprice.) These triads are exactly the eight major or minor triads that are no more than 60° away from the A-minor tonic triad. Three are exactly 60° away: EM, CM, and AbM. Three are exactly 30° away: FM, DbM, and AM. Two are exactly 0° away; that is, they tilt the circular "scale" in the same direction as Am: Fm, and C#m. The order of these triads both accommodates the general shape of Paganini's tune (while, at the same time, semitonal shifts of parts of Paganini's tune in mm. 5-7 accommodate the triads) and moves consistently down by (sometimes enharmonic) major third among these three trios: Am.........Fm......C#m, ...EM...CM......AbM, and ......FM......DbM......AM.
Moreover, as one triad progresses to the next in this variation, the number of degrees the circular "scale" shifts divided by 30 matches the minimal number and direction of semitonal changes needed for the triadic progression, as the video below demonstrates. The few progressions that contradict this correlation are avoided here. (One of those progressions will be examined more closely next month.) Overall, the voice leading from one triad to the next is relatively small, because no triad in this group of nine is more than 90° away from any other.
Quinn is more interested in the magnitude by which different types of chords tip the scale, and less interested in the direction of the tip. Any augmented triad would tip this scale the most, while a chord of the fully-diminished type or four-note cluster type (among others) are perfectly balanced and do not tip this scale at all. The magnitude of the tilt can give an idea of how "major-thirdy" a certain type of chord is.
Cohn is more interested in the direction that a chord of a certain type tips the scale, and less interested in the magnitude of the tip. He has noticed that this scale can give a relatively good idea how far the notes in one chord would need to travel to move to notes in another chord. In particular, when you divide the number of degrees two three-note chords are apart by 30, this sometimes, or perhaps even often, tells you how many semitones in total the notes in one chord minimally need to move to produce the notes in the other chord. For instance, in the example above, the A-minor triad and the F-major triad differ by 30°. Divide this by 30 and you get 1, which is the sum of all of the semitones minimally needed to change the notes of an A-minor triad to the notes of an F-major triad: A-->A (0 semitones), C-->C (0 semitones), E-->F (1 semitone) (0+0+1=1). Furthermore, if the change is clockwise, the minimal voice leading goes up; if the change is counterclockwise, the minimal voice leading does down. In this case, the change is 30° clockwise; therefore, the single semitonal motion is up (E-->F).
I write "relatively good idea" and "sometimes, or perhaps often" because this method works more frequently for some chord types than others. Between augmented triads, it always works. Between major and minor triads, it almost always works. The closer the notes can be put together, irrespective of octave, the less frequently it works. Cohn focuses his attention on major and minor triads, so this approach is pretty reliable.
My Paganini variation begins, as expected, with an A-minor chord, and then follows it with eight other major or minor triads, one for each measure: EM, FM, CM, Fm, DbM, AbM, C#m, and AM. (The number 8 is useful in how it matches the number of measures in the first section of Paganini's caprice.) These triads are exactly the eight major or minor triads that are no more than 60° away from the A-minor tonic triad. Three are exactly 60° away: EM, CM, and AbM. Three are exactly 30° away: FM, DbM, and AM. Two are exactly 0° away; that is, they tilt the circular "scale" in the same direction as Am: Fm, and C#m. The order of these triads both accommodates the general shape of Paganini's tune (while, at the same time, semitonal shifts of parts of Paganini's tune in mm. 5-7 accommodate the triads) and moves consistently down by (sometimes enharmonic) major third among these three trios: Am.........Fm......C#m, ...EM...CM......AbM, and ......FM......DbM......AM.
Moreover, as one triad progresses to the next in this variation, the number of degrees the circular "scale" shifts divided by 30 matches the minimal number and direction of semitonal changes needed for the triadic progression, as the video below demonstrates. The few progressions that contradict this correlation are avoided here. (One of those progressions will be examined more closely next month.) Overall, the voice leading from one triad to the next is relatively small, because no triad in this group of nine is more than 90° away from any other.
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