22d Discussion - PC Set Inversion

Indomitable Inversions

“How do you find the inversion of a number?”

• You “mirror” it over a given axis
• You take the distance that it is from the axis number, then go that far in the opposite direction out from the axis number
• Ex: if you invert the number 8 over 0, you get -8, which translates to 4.
• Ex: if you invert the number 7 over the number 5, you get 3.

Inversion Pairs based around 0:

• 0 <-> 0
• 1 <-> e
• 2 <-> t
• 3 <-> 9
• 4 <-> 8
• 5 <-> 7
• 6 <-> 6

Hint: you only need to memorize these pairs, then you can use this pattern over any inversion axis.

Inverting a C Major Triad in C=0

• What is the PC set for a C major triad? (0,4,7).
• What is the inversion? (0,8,5).
• (0,8,5) is an F minor triad

Practice Example

T₄I(1,5,6)

• The “T” tells you to transpose it by “4”. The “I” tells you to invert it.
• If you transpose this set before inverting it, you will have to invert it over the axis of 4.
• Conversely, if you invert this set first, you can invert it over the axis of 0, then transpose it by 4.

Step 1 - Inversion:

• (1,5,6) becomes (e,7,6)

Step 2 - Transposition:

• (e,7,6) transposed by 4 becomes (3,e,t)

Step 3 - Ascending Forms:

• (3,e,t) becomes (3,t,e) or (t,e,3) or (e,3,t)

Step 4 - Normal Form:

• the ascending form with the smallest outer interval is [t,e,3]

Further reading

From Open Music Theory

Inversion

Inversion, like transposition, is often associated with motion that connects similar objects. You need to be able to (1) invert a collection of pitches and (2) determine the inversional relationship between two collections of pitches.

This passage above from Debussy’s “Sunken Cathedral” is an example. Just as was the case in the [transpositionally-related passages][transposition], these two gestures have the same intervallic content—and so, our ears recognize them as very similar. (Debussy underscores that similarity by giving both of the gestures the same rhythmic setting.) Unlike transposition, however, the interval content of these two gestures is not arranged in the same way.

Both have the same intervals, but the {A,D,E} collection has the +5 on the bottom instead of on the top.

Inverting something is a two-step process, performed in this order: (1) Reflect the pitch classes in an object around the 0-6 axis of symmetry, and then (2) transpose it. I’ll illustrate first on a clock, and then show you an easier way:

Fortunately, there is a much quicker way to invert a pitch or collection of pitches! Given any collection of pitch classes and a TnI, simply subtract the the pitch classes from n:

Conversely, to determine the TnI that relates two collections of pitch classes, find a common value to which they all sum. That is the n in TnI: