This ostinato has three properties. The first two are more common among melodies, particularly simple melodies, and they make the third more probable. Nonetheless, even with the first two properties in place, the third property is rather unusual, even for an ostinato with only four notes.
1) The ostinato uses exactly two different kinds of melodic intervals, irrespective of direction: seconds and thirds. I will call these pitch intervals one-steps and two-steps, respectively, since thirds are twice the size of seconds, generically speaking.
2) The ostinato uses exactly two different kinds of durations: some length of time and twice this length of time. (As I did above, these timespans are usually transcribed as an eighth note and a quarter note, respectively, when this carol is notated.) I will call these temporal durations one-spans and two-spans, respectively.
3) Every note in the ostinato that finishes an X-step commences an X-span. For example, the only instance of scale degree 2 comes after scale degree 3 in the ostinato; therefore, it finishes a descending-second interval, or, more generally, a one-step. Scale degree 2 is then held for a one-span. For this note, X = 1. This holds true for all four notes in the ostinato.
The diagram to the right visualizes this property. The circle shows the four notes of the ostinato, identified by scale degree (in green) and position in the triple-time meter (in red) and arranged so that clockwise order reflects temporal order. However, these notes are evenly spaced on the circle: their pitch and durational proportions are shown with the kite figures outside and inside the circle, respectively. The fact that the red kite’s position is rotated 90° clockwise from the green kite’s position means that each temporal interval immediately follows its corresponding pitch interval.
I can think of another repeating four-note motive from the music of Robert Schumann that has all three of these properties, but the shape is a rectangle instead of a kite. I’ll reveal it next month, unless someone beats me to it by identifying it in a comment below. Also below feel free to identify melodies that achieve some variation of these properties: a different number of notes, rotations by some amount other than 90° (including 0°), a non-repeating melody, absence or alteration of the first and second properties, etc.
3) Every note in the ostinato that finishes an X-step commences an X-span. For example, the only instance of scale degree 2 comes after scale degree 3 in the ostinato; therefore, it finishes a descending-second interval, or, more generally, a one-step. Scale degree 2 is then held for a one-span. For this note, X = 1. This holds true for all four notes in the ostinato.The diagram to the right visualizes this property. The circle shows the four notes of the ostinato, identified by scale degree (in green) and position in the triple-time meter (in red) and arranged so that clockwise order reflects temporal order. However, these notes are evenly spaced on the circle: their pitch and durational proportions are shown with the kite figures outside and inside the circle, respectively. The fact that the red kite’s position is rotated 90° clockwise from the green kite’s position means that each temporal interval immediately follows its corresponding pitch interval.
I can think of another repeating four-note motive from the music of Robert Schumann that has all three of these properties, but the shape is a rectangle instead of a kite. I’ll reveal it next month, unless someone beats me to it by identifying it in a comment below. Also below feel free to identify melodies that achieve some variation of these properties: a different number of notes, rotations by some amount other than 90° (including 0°), a non-repeating melody, absence or alteration of the first and second properties, etc.
Posted on behalf of Charles Smith:
ReplyDeleteInteresting example. I'm not sure which Schumann motive you're referring to, but the following seems to work out as two rotated rectangles:
In some kind of compound meter, say 6/8:
^3 [quarter] - ^2 [eighth] - ^7 [quarter] - ^1 [eighth]
Is this close to what you had in mind?
Charles
Charles, yes, that’s one solution. Of course, this could be transformed through inversion, rotation through the meter, relocation in the scale, and with different notated durations to make other solutions in the same solution-class.
DeleteNow, where is it in Schumann’s music? :-)
Very fun thought experiment! My answer is far below for those that don't want the **SPOILER**. For a practical generalization of the idea, I thought about how I might interpret these spans and steps while singing. When performing a 1-step or a 2-step, you might literally sing "step" or "leap" (respectively) on the second note of the interval to indicate the basic category of interval that was just performed. Alternatively, 1-spans and 2-spans may be performed with the words "short" and "long" (also respectively), but these words refer to the length of time it takes for the current note being performed. Mapping these systems together for comparison, we may rewrite your third property as: "Every note that finishes a step commences a short, and every note that finishes a leap commences a long." The mapping of step/short (or S) and leap/long (or L) helped me to quicken my search, but it was really your comment to Charles that led me to a passage in... Faschingsschwank aus Wien that fits the bill. I don't know if this is the melody you were thinking of, but it is a repeated four-note melody in E-flat major, a little over a minute into the piece: <(6, half)-(5, quarter)-(7, half)-(1, quarter)>. The melody is different than you might expect, starting its "long" on beat 3 of the 3/4 time, requiring a tie to the downbeat quarter note to finish the half-note span. The oddity of the metric placement helped it stand out in my memory when your response to Charles referred to "rotation through the meter" because the melody sounds as if it begins on the downbeat.
ReplyDeleteThat's the one I was thinking of!
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