(musth212) Transposing and Inverting Pitch-Class Sets

Transposition

Transposition of pitch-class sets is by ordered pitch-class interval. Before going any further, consider the implications of that first statement.

• Transposition is by pitch class, which means that a transposition could contain octave displacements and still be a transposition.
• Transposition is by ordered pitch-class interval, which means that we always count the transposition distance in a positive direction and the distance will be 11 or less.

Given a pitch-class set in normal form, the transposition number is the pitch-class interval that must be added to all the pitch classes to obtain the second set. If [5,7,8,11] is our starting set, then T8* is [5+8, 7+8, 8+8, 11+8] giving [13, 15, 16, 19], which reduces to [1,2,4,7].

*8 should be in sub-script, as should all transposition numbers.

Make sure to keep in mind that the transposition number is not the same as an integer representing a pitch class. The transposition number represents the relationship between two sets.

Inversion

Inverting a pitch-class set simply involves reversing the succession of ordered pitch intervals in a set. If you have the set [4,5,8], the order of pitch intervals is +1, +3. If you start of pitch class 8, the resulting inversion is [8, 11, 0], with the order of intervals +3, +1. You could also start the inversion of pitch class 4, resulting in [4,7,8].

You might have already noticed that [8,11,0] is T4 of [4,7,8]. Properly labeling inversions means that you have to also label the transposition number. T0I is obtained by inverting pitch classes around 12 (0). Subtract each pitch class in the first set by 12 to get the pitch class of the inversion.

[4,5,8] = [12-4, 12-5, 12-8] = T0I=[8, 7, 4]

The pitch classes need to be reversed after this procedure to obtain normal form. [4,7,8]

If you are given two pitch class sets that are related by inversion you can calculate the transposition number by adding their corresponding elements. Remember that corresponding elements of inverted pitch-class sets are in reverse order. Given [4,5,8] and [8,11,0], you obtain the transposition number by adding 4 and 0, 5 and 11, and 8 and 8. Each pair adds up to interval 4, so the two sets are related by T4I.