5.4
The major scale
A scale is a set of pitches from which melodies are selected. The white keys on the
piano form a scale, as do the black keys. Many other scales are commonly used, some
even containing notes that aren’t on the piano. The notes of a scale, written in
ascending order, typically form a pattern of intervals that repeats every octave.
The pitches in a major scale form a repeating pattern of seven intervals, whose sizes,
measured in semitones, are 2, 2, 1, 2, 2, 2, 1, after which the same list of intervals
repeats. Notice that 2 + 2 + 1 + 2 + 2 + 2 + 1 = 12, so the pattern starts over again an
octave higher. If you play the white notes on a piano, starting at middle C and going
up to your right, you play a major scale; however, a major scale can start at any pitch,
even one not on the piano, as long as it has the 2, 2, 1, 2, 2, 2, 1 pattern of intervals.
Because the scale is determined by its intervals, not its starting pitch, there are many
major scales on the piano. Therefore, we don’t give letter names (like “A” or “C”) to
the pitches in a generic major scale. Instead, musicians use the solfege system
do
re
mi
fa
sol
ray
me
fah sew
la
ti
pronounced
doe
la
tea
which you might recognize from the song “Doe, a deer” (Rodgers & Hammerstein,
1959):
Doe, a deer, a female deer
Ray, a drop of golden sun
Me, a name I call myself
Far, a long long way to run
Sew, a needle pulling thread
La, a note to follow sew
Tea, I drink with jam and bread
That will bring us back to doe.
The starting note of the major scale is always called do, then re is two semitones
above do, mi is two semitones above re, etc. Because the pattern of intervals repeats
after an octave, the names are recycled—the last line of the song, “that will bring us
back to doe” actually ends on the pitch that is one octave higher than the original
starting pitch. The fact that solfege names are recycled means that there is no single
meaning for “the interval between do and sol.” Rather, we must specify a direction.
The interval from do up to sol is 7 semitones, while the interval from do down to sol is
5 semitones, which is the shortest interval between any do and any sol.
fa
sol
la
ti do
5 semitones
re
2
mi fa
2
1
7 semitones
sol
2
la
2
ti do
2
re
mi fa
1
Exercise 9. Find the number of semitones from do up to mi and from do down to mi.
Exercise 10. Find all the pairs of solfege syllables that are separated by a ifth (seven
semitones). Hint: there are six pairs, and (do, sol) is one of them.
5.5
Pitch classes and interval classes
As we have seen both with the letter names of keys on the piano and with the solfege
syllables, pitch names may be reused after an octave. This is because our brains are
hard-wired to hear pitches that are an octave apart as similar.
A pitch class is the set of all pitches that are separated from a given pitch by a whole
number of octaves. Every pitch class has in initely many members, because you could
theoretically keep going up or down by octaves forever, though you would eventually
exceed the limits of human hearing. For example, pitches 57, 69, 81, 93, all belong to
the same pitch class. This pitch class is labelled by the letter “A” and all its members
are called “A”s. Using set notation,
A = {. . . 45, 57, 69, 81, 93, . . .}.
We need an ellipsis (“. . .”) on each end of the list because there are in initely many
pitches in each pitch class.
Pitch classes corresponding to black keys on the piano are labelled with accidental
signs: the lat symbol ♭ means “minus one semitone” and the sharp symbol ♯ means
“plus one semitone.” In normal piano tuning, the black keys on the piano can be
labeled in two different ways, one using a sharp sign and the other using a lat sign.
For example, C♯ and D♭ are the same pitch class.
Because the system of pitch classes repeats after
C
C#/Db
B
an octave, they are often depicted on a circle,
0
11
1
called the pitch class circle. Normally, C is at the
A#/Bb
top and numbered 0. You can think of pitch
C# D#
F# G# A#
2
10
Db Eb
Gb Ab Bb
classes as the analog of beat classes: two pitches
are in the same pitch class if they are in the same
A 9
position in an octave, just as two beats are in the
C D E F G A B
same beat class if they are in the same position in
8
a measure. Although there are twelve pitch
4
G#/Ab
classes represented on a piano, there are actually
5
7
an in inite number of pitch classes, because there
6
F
G
are microtonal pitches “between the cracks”
F#/Gb
whose MIDI numbers are not whole numbers.
D
3
E
Exercise 11. Write the pitch class C in set notation. What is the relationship between pitch
classes and modular equivalence?
Exercise 12. Name the pitch classes in a C major scale (use do = C). Name the pitch classes in
a D major scale (use do = D). When naming the accidentals, every letter must be used once. So
the third pitch class is called F♯, not G♭.
D#/Eb
Converting pitch to pitch class. Two pitches belong to the same pitch class if and
only if their MIDI numbers are equivalent mod 12. First, note that since middle C is
pitch 60 and 60 is divisible by 12, every MIDI number that is divisible by 12 belongs
to pitch class C. In the pitch class circle, C is labelled “0” and the other pitch classes
are labelled by the remainders when their MIDI numbers are divided by 12. To
convert a pitch to a pitch class, divide by 12 and locate the remainder on the pitch
class circle. For example, to ind the pitch class of pitch 50, compute 50 ÷ 12 = 4 r.2.
Since the remainder is 2, pitch 50 belongs to pitch class 2, or D.
Exercise 13. Compute the pitch class number and letter of pitches (a) 80, (b) 84, (c) 100.
Interval classes. Measuring intervals between pitch classes presents a problem,
because there are two ways to measure distance on a circle—clockwise (ascending in
pitch) and counterclockwise (descending in pitch). Although the interval between
pitches 60 (C4) and 69 (A4) is 9 semitones, the shortest distance between any C and
any A is 3 semitones. In general, an interval class is the shortest distance between
two pitch classes on the pitch class circle, in semitones. Interval classes are numbers
between 0 and 6. For example, the interval class of the pair {C, A} is 3, because that is
the shortest distance between any C and any A.
Exercise 14. Find the interval classes of the following pairs of pitch classes: (a) E♭ and C, (b)
E and F♯, (c) F and B
Exercise 15. If the interval between two pitches is 19 semitones, what is the interval class
de ined by their corresponding pitch classes?
Answer of exercise 9. 4 and 8
Answer of exercise 10. (do, sol); (re, la); (mi, ti); (fa, do); (sol, re); (la, mi)
Answer of exercise 11. C = {. . . , 48, 60, 72, 84, 96, . . .} Two pitches are in the same pitch
class if and only if their MIDI numbers are equivalent modulo 12.
Answer of exercise 12. C = { C, D, E, F, G, A, B }; D = { D, E, F♯, G, A, B, C♯ }
Answer of exercise 13. (a) 8 = G♯/A♭; (b) 0 = C; (c) 4 = E
Answer of exercise 14. (a) 3, (b) 2, (c) 6
Answer of exercise 15. Their pitch classes are separated by 7 semitones on the pitch class
circle, because 19 ≡ 7 (mod 12). However, those pitch classes are 5 semitones apart
measured the opposite direction on the circle, so the answer is 5.