Suggested Learning Resources


2) Video Lesson:

3) Online Reading: 

4) Interactive Lesson: 

Music Theory QuickThink:

Remember: Pitch Intervals means number of half steps, Ordered-Pitch-Intervals means with + or -Pitch-Class-Intervals means y-x mod 12.

You can tell if two Pitch-Class Sets are related by inversion if you (1) analyze their Pitch-Class-Intervals, and determine if the pitch-class-intervals of the two sets each add up to 12.

For Example consider the two pitch-class-sets    {8,t,0,e,1}  and {2,0,t,e,9}

Set A

Set B



Pitch-Class Intervals



Notice above that each corresponding pitch-class intervals add up to 12.  These two sets are related by inversion.

Creating an inverted set - there are several methods

Method 1 - Find the the inverse of each pitch-class in the set

- Analyze the numbers in the set, and then make a new set with corresponding numbers that add to 12 with the original set

- Example {8,t,0,e,1}   would result in {4,2,0,1,e}

Method 2 - Find numbers on the opposite side of the clock face

- Draw a clock

- The inverse of each number is on the opposite side of the clock if you make a horizontal line

- For {8,t,0,e,1}   4 is straight across from 8 on the clock... 2 is straight across from t on the clock... etc

NOTE: You can also transpose the resulting inverted set, and you will still have an inversion relation to the original set.  If you do this, it is very important INVERT FIRST and then TRANSPOSE, else it will not work.

Method 3 - ascending using the rainbow method for inversion

- For sake of analysis and discussion, it is often helpful to have both a pitch-class-set and its inversion set arrange in ascending order

- First, place your original set in ascending order {8,t,0,e,1} becomes {8,t,e,0,1}

- Now start with the last number, and find the number that adds to 12, placing it as the first number of the new set

- The second to last number corresponds with the second number of the new set and so on.

- Example   {8,t,e,0,1} ---> {e,0,1,2,4}

Objective 51.2: Examples in Music: YouTube

Objective 51.2: Calculate the inversion of a pitch-class-set