Converting a Pitch Class to Score Representation

Problems always emerge in converting a pitch class to an accidental name/note name representation. Pitch classes are discrete, insofar as each has only one possible sound in any octave. Conversely, any sound in an octave can only be a member of one pitch class, provided that acceptable frequency limits for the pitch class have been specified. Pitch classes are context independent. When, however, they are captured in a tonal score, they are written within the context of a particular key and mode. In notating student performance in sight singing or in keyboard harmony exercises, the correct spelling of a pitch class can be determined in one of two ways:

• 1. By comparing it with the note that was expected to be sung or played, and by notating it as a chromatic alteration of that note, if it is necessary or possible to do so
• 2. By comparing it with all degrees of the current key and mode, and by notating it as the scale degree that requires either no chromatic alteration or only a one sharp or one flat change in spelling.

Attempting to discover the key and mode within which a note appears in addition to determining its function in that context is not a task that can be carried out in real time. To discover the key of a set of pitches entails listening to a number of them and then postulating the key and mode from the position of half steps within the set of notes that have been heard. Once the key has been determined, the pitches can be notated in that key context.

In CBI lessons that notate student singing or keyboard playing, key should never be determined from context. It should always be established in advance. In fact, since notes can only be sung one at a time or played one chord at a time, a lesson can interrogate the student for the particular note that it wishes to evaluate and can then compare the student's performance against the expected note. This procedure involves several steps for each note:

• 1. Displaying the note that is to be sung or played
• 2. Listening to the pitch sung or determining which keys have been depressed on a keyboard
• 3. Converting the pitch sung or played to a pitch class, as described in the last section above
• 4. Using the pitch class that has been detected to reference the data structure presented in Figure 5.9
• 5. Drawing the note and accidental that best represent what the student performed.

The pairs of base 7 numbers that represent accidental name and note name are aligned in Figure 5.9 so that all pairs representing a single pitch class are in the same column. Note that the sums of the digits in any pair always have some fixed relationship with the pitch class they represent. In the three diagonals with odd-numbered t's that run from lower left to upper right, including < 0,1> to < 4,1> through < 0,5> to < 4,5> , the sums of the digits in the pair are always three greater (mod 12) than the pitch class they represent. For example, if < 2,1> represents PC 0, 2 + 1 = 3, which is 3 greater than 0. If < 0,1> represents PC 10, 0 + 1 = 1, and 1 - 10 = -9 + 12 = 3, with the mod 12 adjustment. In the four diagonals with even-numbered t's, which include < 0,0> to < 4,0> through < 0,6> to < 4,6> , the sums of the digits are always three less (mod 12) than the pitch class they represent. For example, if < 2,2> represents PC 7, 2 + 2 = 4, and 7 - 4 = 3. If < 3,6> represents PC 0, 3 + 6 = 9, and 0 - 9 = -9 + 12 = 3, with the mod 12 adjustment.

Because the note name of the expected note would already have been calculated in the process of displaying it for the student, the accidental name of the note sung or played can be determined directly from the pitch class of the note. To eliminate the problem of having certain sums of digits be three greater and some three less than the pitch class, add 6 to all note names represented by even numbers. The association between note names and their numbers

 C	D	E	F	G	A	B
1	3	5	0	2	4	6

thus becomes

C	D	E	F	G	A	B
1	3	5	6	8	10	12

The accidental name of the note corresponding with a pitch class can be computed simply by subtracting the adjusted name from the value (PC + 3). If the pitch class can be represented by the desired note name, the result of this subtraction will be a value between 0 and 4. The accidentals represented by these values are

bb  b   n   #   x
0   1   2   3   4

If, for example, PC 7 is to be notated as a G, the value for G (8) is subtracted from (7 + 3) or 10. The result is, of course, 2, the value that represents a natural. If PC 3 is to be notated as an E, the value for E (5) is subtracted from (3 + 3) or 6. The result, 1, represents a flat. If PC 0 is to be notated as a B, the value for B (6) is subtracted from (0 + 3) or 3. The result is -9 + 12 = 3, with the mod 12 adjustment. This PC 0 is spelled as a B#. If PC 10 is played and if a C was to have been played, the operation (10 + 3) - 1 is performed. The result is 12 - 12 = 0, with the mod 12 adjustment. The note played was a Cbb.

If the calculated accidental is not in the range 0 to 4, the student undoubtedly misinterpreted the note to be sung or played instead of mistuning it. In such cases, it probably is better pedagogically to notate the pitch class with respect to whatever note name requires the smallest accidental change within a key. A CBI lesson that can notate what a student actually sings or plays can inform him or her of tuning errors or improper placement of half steps within a scale or mode. It can also provide feedback that enables him or her to correct conceptual errors, such as singing a fifth when a fourth is expected or singing an octave when a sixth is expected.

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