Integer Model of Pitch

In tonal music the pitches C and B# (for example) do not have the same function even though they are really the same pitch. However, in atonal music this diatonic functionality does not exist. Rahn therefor uses integers to notate pitches in atonal music. This allows the notated pitches to be ordered and equally spaced. Therefor what we hear (ie. 12 equal pitches in various octaves) is what we see:

(example 1)

This method (model) of naming pitches also has the added benefit of making sense mathematically. This becomes clear when Rahn teaches us how to calculate intervals. He does this systematically, in the fashion of a mathematician. However, sometimes his mathematical approach looses sight of its musical audience. What follows are the basic steps he uses to calculate intervals without the formal mathematical definitions. I've tried to simplify matters for the music student who is more concerned with how to use these applications in the analysis of a piece of music rather than concerned with how the math itself works (which is common sensical to begin with).

To Calculate Pitch Intervals:

Consider the following example: (example 2)

To calculate the interval between the G (7) and C (12), in the order they appear in the music, simply subtract 7 from 12 (also in this order):
12 - 7 = 5

Therefor, the pitch interval between 7 and 12 is 5. Since the 5 is positive this indicates that the interval rises in pitch. A negative interval indicates that the pitch descends. Consider this same example but where the notes G (7) and C (12) are in reverse order:

(example 3)

Here the notes within the music appear in the order: C (12), G (7). When we subtract 12 from 7 we get a negative answer:
7 - 12 = -5

Therefor, the pitch interval between 12 and 7 (in this order) is -5. The negative sign indicates that the interval is descending in pitch.

To Calculate Pitch intervals (both ascending and descending), we can use Rahn's formula: ip<x,y> = y-x

Here, the ‘ip' stands for ‘pitch interval'
x = the first note of the interval (as it appears in the music)
y = the second note of the interval
and the < > brackets indicate that the order of x and y are relevant (ie. affect the outcome)

Rahn's formula applied to our last practical example looks like this:
ip<x,y> = y-x
ip<12, 7> = 7-12
ip<12, 7> = -5

Consider the following example where the order of notes within the interval is not applicable:

(example 4)

We calculate the interval as above but drop the positive or negative sign in our final answer. Rahn uses the following formula:
ip (x, y) = | y-x |
ip (12, 7) = | 7-12 |
ip (12, 7) = | -5 |
ip (12, 7) = 5

2) Simplify Pitches to Pitch-Classes:

Rahn notes that it is handy to bring pitch numbers to their lowest common denominator, namely their pitch-class numbers. A pitch class is the lowest non-negative equivalent of a pitch between 0 and 11.

(example 5)

Pitch-class numbers ignore the importance of register the same way that letter names do. It may be useful, therefor, to have "0" correspond to a note other than "C." Rahn relates this Pitch-class system to mathematics by describing it as an example of a mod 12 system. (Bits of mathematical information like this are interesting mathematically but may be of little use to the music student interested in practical application of the theory for music analysis purposes.)

To Simplify Pitches to Pitch Classes:

add or subtract 12 form the pitch number until you reach a positive number between 0 and 11 inclusive.

Sometimes it is useful to reduce ordered pitch-classes to unordered sets (as we have already seen in our dealings with Set Theory). Sets do not take the order of pitch-classes, nor the number of times each pitch class occurs into consideration.

To Reduce Pitch-Classes to Sets:

write the pitch-classes you wish to reduce to a set in numerically increasing order including each pitch-class only once.

For example, the ordered pitch-classes: 0 4 7 4 7 0 4 7
reduces to the set {0, 4, 7}.

For practical purposes (ie. for easy comparison of different sets) it is important to further reduce this set to its normal form.

To Reduce Sets to Their Normal Form:

First, rotate the set:
0 4 7
4 7 0
7 0 4

Then, add all the intervals for each rotation together. The smallest sum of intervals indicates which rotation it the normal for of the set:
0 4 7 = 4 + 3 = 7
4 7 0 = 3 + 5 = 8 (think of the 0 as a 12)
7 0 4 = 5 + 4 = 9