22: Inverting Large Intervals

Inverting intervals larger than an octave invites us to consider a generalization of the procedure to invert small (an octave or less) intervals. That procedure was that two intervals are inverted if their qualities are inverted (major pairs with minor; augmented pairs with diminished; and so on) and the sum of their sizes was 9 (sixths pair with thirds; sevenths pair with seconds; and so on). More abstractly, one can imagine “flipping the interval”; if there’s a C# a sixth below an A, then “flipping” the interval so that there is now an A a third below a C# does indeed correctly invert the minor sixth into a major third.

To look at intervals larger than an octave, we will use precisely that heuristic, and in so doing, come up with a theoretically infinite number of “rule of 9”-like rules.

The procedure thinking about actual notes is simple (and you already have an idea of how to do this from the smaller intervals): hold one note where it is, and bring the other note up or down a sufficient number of octaves to make their relative positions change.

When intervals are small, this number is one, so we have that 1 + 8 = 2 + 7 = 3 + 6 = 5 + 4 =  the rule of 9.

When intervals are large, we have similar rules.

Consider, for example, the inversion of the interval from A3 to C#6—a major third plus 2 octaves, or a major 17th.

Our steps to obtaining the inversion are:

·       Hold the C# constant, move the A3 to A4

o   Check: have they switched (is the A now higher than the C#?)? No? Keep going

·       Hold the C# constant, move the A4 to A5

o   Check: have they switched? No? Keep going

·       Hold the C# constant, move the A5 to A6

o   Check: have they switched? Yes—so stop.

Now here this is, in steps:


 

Or, if you prefer, the corresponding procedure works just as well.

·       Hold the A constant, move the C#6 to C#5

o   Do the same check

·       Hold the A constant, move the C#5 to C#4

o   Do the same check

·       Hold the A constant, move the C#4 to C#3

o   Do the same check, and stop, because they have now switched places

Notice here that the inversion is a minor sixth, and that 6 + 17 = 23. One octave is an octave, 2 octaves is called a “fifteenth”, 3 octaves is called a “twenty-second,” and so on.

Notice the rule we’ve discovered.

·       For the inversion of intervals within the octave, the rule of the sum is 8 + 1 = 9

·       For the inversion of intervals within the fifteenth, the rule of the sum is 15 +1 = 16

·       For the inversion of intervals within the twenty-second, the rule of the sum is 22 + 1 = 23

·       And so on

The rule of 9 we established for inverting small intervals is therefore neither unique nor a coincidence, but the first in an infinite line of applications of a single rule.

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