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Whether speaking or singing, humans automatically and effortlessly
use discrete pitches, with only the occasional slide. By contrast, our
primate cousins, such as gibbons and chimpanzees, either vocalize in pitch
glides or without distinct pitches—just grunts and pants.
Discrete pitches in speech and music serve to organize sound in
such a way that the brain can recognize patterns and make sense
of them. Once you have more than one discrete tone, you can have
a scale of some sort.
Humans undoubtedly turned discrete tones into songs long
before anybody recognized the existence of musical scales. At some
point, it must have become clear that the tunes people remembered
tended to use the same sets of notes: scales.
A tune or melody is a coherent or distinctive succession of tone
pairs called intervals. The notes of a tune (melody) move up and
down in pitch, stepping or skipping from note to note, using the
same notes time after time, like stepping up and down the same
That means the notes themselves must come from some set of
related notes of different pitches. This set of notes is called a scale.
But how does the brain recognize a set of pitches as a scale?
ON A CELLO
Imagine you have a cello. (Maybe you do have a cello.) As you
know, the fingerboard of a cello has no frets. Which makes the cello an ideal instrument for
this thought experiment.
• Take a piece of imaginary white chalk and make some
horizontal marks at random places along the fingerboard of
your imaginary cello. Say, oh, maybe eight chalk marks.
• Remove the excess imaginary chalk from your fingers by
wiping your hands on your black pants or dark skirt. Nobody
will be able to see the chalk marks on your clothing because,
even though your clothing is real, the chalk is imaginary.
pick a string, any string. Press your finger on the string over the chalk mark
nearest the narrow end of the fingerboard (the end nearest the tuning pegs).
Pluck (or bow) that string. Then move to the next chalk mark. Pluck the string.
Then the next chalk mark, and so on, until you’ve played all eight notes.
that’s a scale.
Good thing that was a thought experiment. Because the scale
you created and played on your imaginary cello sucks. Your brain
just does not recognize it as a meaningful scale. How come?
If you create a random scale, a scale comprised of notes having no
natural, physical relationship with each other (the way you did using
random chalk marks on the cello fingerboard), then try to play a tune
using that scale, your brain interprets the sound as chaos, not
Studies of both children and adults indicate your brain is
hardwired at birth to reject random scales. Infants prefer non-random
scales, as do adults. The frequencies of the notes comprising a scale have to
have some kind of internal order—ordered relationships with each other—or your brain interprets the sound as
But not just any ordered relationships.
ordered relationships that your brain recognizes: “brain-friendly” ordered
relationships of tones, as opposed to “brain-averse” chaotic non-relationships.
OF AN ORGANIZING
Recall what happens when you pluck a guitar
string that you’ve cut in halves, thirds, quarters, fifths, and so on, by
damping the string over various frets. You get a whole series of soft
overtones—overtones that sound different from the fundamental.
As the guitar-string-damping experiment reveals, each overtone
not only sounds different, it also sounds good. Brain-friendly. So it
would be a reasonable guess that a brain-friendly scale might have
something to do with the relationships of overtones to each other.
Hmmm. Maybe relationships among overtones hold the secret
that will yield a useful scale, a group of tones in a brain-friendly
Time to bring back the overtone series and have a look at
overtone frequency relationships (Table 8 below). Frequency relationships among the
first few overtones, the strongest ones, are of greatest interest. They’re the
ones you can hear by damping a guitar string at various fret positions.
Fundamental and First 15
Overtones of the “Middle C” Overtone Series
f x 2
f x 3
f x 4
f x 5
f x 6
f x 7
f x 8
f x 9
f x 10
f x 11
f x 12
f x 13
f x 14
f x 15
f x 16
• Start with the ratio of the first overtone to the fundamental
frequency, which is 523.2 Hz : 261.6 Hz, which boils down
to a simple ratio of 2:1. This simple ratio comes from the
first two numbers of the middle column.
the ratio of the second overtone to the first overtone. It’s 784.8 Hz : 523.2
Hz, a ratio of 3:2. (middle column, second and third numbers).
• Keep doing this
for the first few overtones, and you end up with a list of simple ratios
of frequencies, like this (Table 9):
Simple Ratios of Frequencies
Next step: try out simple ratios of frequencies as an organizing
principle to build a scale.
Any organizing principle worth its salt should work universally. That
is, you should be able to pick any old frequency as a starting point
for scale building.
Ratios, Ratios, Ratios—not the Overtones Themselves
Potential Point of Confusion:
Always bear in mind that it’s the
ratios of overtone frequencies that matter—not the overtone frequencies
For purposes of scale-building, it’s all about ratios of frequencies
(Table 9 above). Ratios, ratios, ratios.
If you don’t keep this distinction in mind, you
could get lost. And then the new marshal will have to organize a search party.
You heard right. Dodge City has itself a new
marshal. In a Classic Western plot twist, Ms Puma’s the new marshal now, ever
since a posse tarred and feathered Marshal McDillon and ran him out of town on a
rail. Why? For carrying on behind Ms Puma’s back. That’s why.
So, now’s not the time to cross Marshal Puma by
needlessly getting lost in a wilderness of frequencies.
• Start building the scale with the tone Middle C, the first tone
in Table 8 above, with a frequency of 261.6 Hz.
• Next, in accordance with the organizing principle of simple
frequency ratios, add a second note, derived from the simplest possible ratio,
2:1. What you get is a two-note “scale.”
scale clearly has its limitations. But you have to start somewhere (Figure 5).
Scale of “Middle C” and “C Above
• Next, add a tone derived from the next simplest ratio of
frequencies, 3:2. (The simplest possible frequency ratio that can identify a
relationship between two tones is 2:1.) For reasons that will become clear in a little
while, you can label the 3:2 tone G.
• Notice that when you add G to the scale, the relationship
between G and the C above Middle C also happens to be a
simple ratio of frequencies, 4:3.
you’ve got a scale of three notes. It sounds good, too. The organizing principle
looks promising (Figure 6).
FIGURE 6 C - G - C Scale
• Notice the big gap between Middle C and G. Fortunately, a
tone derived from the simple frequency ratio 5:4 fits
beautifully, right between Middle C and G. Call it E.
• When you add E to the scale, the relationship between E and
G turns out to be a simple ratio of frequencies, namely, 6:5.
scale grows to four notes (Figure 7 below). Sounds great, too. These four notes
correspond to the words “say, can you see,” in the American national anthem,
music composed by John Stafford Smith, a London, England, church organist.
FIGURE 7 C - E - G - C Scale
• Next, have a look at the big gap between G and C above
Middle C. It so happens that yet another tone derived from a
simple frequency ratio, 5:3, fits right in there. This tone
happens to be the lovely and talented Concert A (also
commonly called A-440). More about lovely, talented Concert
A later on.
• The scale grows to five notes (Figure 8):
FIGURE 8 C - E - G - A - C Scale
• You can fill in another big gap, the one between E and G,
using another note derived from the simple frequency ratio,
4:3. The note F relates to Middle C by the this simple ratio.
• When you insert F into the scale, it relates to Concert A by
the simple ratio of frequencies, 5:4.
• Now the scale has grown to six notes. So far, so good (Figure
FIGURE 9 C - E - F - G - A - C Scale
• Only a couple of big gaps remain, one between Middle C and
E, and another between Concert A and C above Middle C.
The simplest frequency ratio available to fit between C and E
is 9:8, which yields the note D.
• You can use the same 9:8 frequency ratio to stick a tone
between Concert A and C above Middle C. Call it B.
• When you insert these two notes (D and B), you notice a few
- The scale has no more big gaps between notes;
smallest gap between notes has a ratio of frequencies of 16:15—not exactly
- The order of the letter-names of the notes makes
some sense, though not plain, common horse sense.
The alphabet starts at C, stops at G, then starts again
you’ve got an 8-note scale. Which includes the 8th note. Which is the same as
the first note, but higher in pitch (Figure 10):
C - D - E - F - G - A - B - C Scale
This scale definitely sounds brain-friendly. Looks like the
organizing principle of tones derived from simple frequency ratios
has worked. (Whew!)
If you’re European, you’ll recognize the scale in
Figure 10 as the “do-re-mi” scale, using the solmization system, which designates
notes using syllables instead of letter names:
do re mi fa so la ti do
you’re going down the scale:
do ti la so fa mi re do
Or, as the von Trapp family sang in The Sound of Music:
Doe, a deer, a female deer
Ray, a drop of golden sun
Me, a name I call myself
Fah, a long long way to run . . .
(They could sing better than they could spell.)
get the same scale when you play the white notes on the piano starting at C. Any
old C. You don’t have to start with Middle C (Figure 11):
The “Do-Re-Mi” Scale in All Its Glory
Since the scale has eight notes (including the first and last
notes), the pitch gap between the first note and the eighth note is
called an octave.
In music, the pitch gap between any two notes is called an
interval. Think of an interval as a relationship between two
pitches. You can play the two pitches successively—usually the lower one
So, that makes the pitch relationship between the first note and
the eighth note an interval of an octave.
Melodic Intervals vs Harmonic Intervals
Another Potential Point of Confusion:
The term “interval” also has a meaning with respect to chord progressions, as
you’ll find out in Chapter 6. Harmonic intervals occupy a different musical
space than the melodic intervals discussed here.
By the time you finish Chapter 6, if you don’t
understand the distinction, you could get lost. Which might not cause you too
much trouble if you happen to meet up with Ex-Marshal McDillon, who’s still out
there, wandering around in the wilderness in his tar and feathers. He’s got
excellent survival skills, though, even without his horse, and, as a musical saw
player, he can tell you pretty much everything you need to know about the
distinction between melodic and harmonic intervals. But you have to find him,
last note of the scale sounds exactly like the first note, and yet ... well ...
“higher” in pitch. The same, but somehow different. The terminology, familiar to
everybody who plays music, goes like this: the last note of the scale is an
“octave higher” than the first note.
can see in Figure 11 above, the eight notes of the do-re-mi scale are not evenly
spaced. Still, when you play this scale, it sounds agreeable whether you play it
from bottom to top or top to bottom. It sounds as though the notes are
proceeding smoothly up and down the pitch “staircase.” As though all the notes
are the same distance apart. Even though they obviously are not.
How come? What if all the pitches were actually the same
So far, you’ve tried two different organizing
principles to construct an agreeable-sounding scale:
• A scale of random
notes—the experiment with the chalk marks on the cello fingerboard. Result? A
brain-averse scale. Chaotic and completely “unmusical.”
• A scale of notes related to each other by simple ratios of
frequencies. Result? A brain-friendly scale. Clearly “musical,” beautiful-
sounding. A scale consisting of a distinctive but uneven order of tones.
Now, just for good measure, try a third organizing principle: a
completely regular, evenly-spaced order of tones.
Start at Middle C and divide the octave into seven equal
intervals, for a total of eight notes (Figure 12 below). The lowest note
is Middle C and the highest note is C above Middle C. All eight notes
are spaced the same distance apart, frequency-wise (37.4 Hz
between each note).
No point in naming notes 2 through 7 because this scale is only
And a good thing, too. Because, like the random scale of
chalk-and-cello fame, this scale also sucks (Figure 12):
The “Eight-Note, Seven-Equal-Interval” Scale
10 below shows the frequencies for the eight notes of this scale, compared with
the “do-re-mi” scale frequencies. As you can see, they’re all different, by
roughly five to 24 Hz, except for the first and last “C” notes.
Seven-Equal-Interval” vs “Do-Re-Mi” Scale Note Frequencies
“Seven-Equal-Interval” Scale Note
1 (8) (C)
Substantial research findings show that, if you try to create music
using scales that have no tones in relationships of simple frequency
ratios, your brain stops recognizing “musical” sound and hears chaos. Like
the static you get when you move your analog radio dial between stations.
Infants respond to changes in pairs of tones only
if the tones are related by small-integer, simple frequency ratios—the tones
that emerge from the harmonic series. Tones not related by simple frequency
ratios simply do not elicit responses from babies. This strongly indicates that
the human brain has a naturally-selected specialization for simple frequency
ratios—that these preferences are not cultural constructs.
Not only that, infants remember scale tones when the intervals
of the scale are of unequal size, compared with scales having intervals
of equal size. This is consistent with the unequal-interval scales that emerge
from the harmonic series. As you’ll see in Chapter 5, most scales used commonly
worldwide have only 5 to 7 different tones (i.e., not including the second octave
note), which are
Infants also have difficulty resolving tones that are close
together. Tones spaced close together are not related by simple
summarize, your brain can make sense of, and prefers, scales of non-random but
unequally-spaced tones—pitches related to each other in simple multiples or
simple fractions of a fundamental frequency.
Look at all those (comparatively) wide intervals between some of the
notes (Figure 13 below). Between C and D. Between D and E.
Between F and G. Between G and A. Between A and B. Five
The “Do-Re-Mi” Scale
five intervals look suspiciously like they’re exactly twice as wide as the two
smaller intervals, the ones between E and F, and B and C. If the five bigger
intervals are exactly twice as wide as the two smaller ones, and if you
were to insert a tone into each of those wide gaps, you’d have:
• A 12-equal-interval scale (a total of 13 tones, including the
first and last ones, which are the same note, an octave
• A scale composed of close-together tones.
Precisely the recipe for non-musicality. So, would such a scale
actually sound chaotic?
The answer is yes, it would sound chaotic. Not at all musical.
that does not mean such a scale would have no musical value. As you’ll see
shortly, the 12-equal-interval scale serves a valuable purpose as a pool of tones you can dip into and
use in the construction of many different, truly musical scales. You
can also use the same 12-equal-interval scale as a pool you can dip
into for colourful extra notes when writing a song.
(While most equal-interval scales are inherently chaotic and
unmusical, a few are actually palatable. Chapter 5 discusses an
example of a musical-sounding equal-interval scale—an exception to the
For now, go ahead and fill in the five wide gaps in the above
scale (Figure 13) with five new notes.
before you do that, you’ll need names for the new notes. Problem is, there’s no
letter of the alphabet between C and D, or D and E, or F and G, etc. What to do?
you start at C, you’re going up to D, and you want to stick a note in between.
Since you’re going “up” in pitch, call the in-between note a sharp note
you’re going “down” in pitch, from D down to C, call the same in-between note a flat
note (symbolized ♭).
(This nomenclature will become a lot clearer shortly.)
So . .
. here’s what you get when you fill in the last five gaps of the “do-re-mi”
scale (Figure 14):
The “Do-Re-Mi” Scale With the Gaps Filled In
the piano, if you start with the note Middle C (or any other C), you’ll notice
that the in-between notes correspond to the black keys.
• The smallest interval in the above scale, the interval between
any two adjacent notes, is called a semitone or half-step
(for example, between C and C♯, or between E and F). So, an interval of an
octave is comprised of 12 semitone intervals.
• The next smallest interval, the distance covered by two
semitones, is called a tone, or a whole tone, or a step.
And the name of the above 13-note (12-semitone) scale is the
chromatic scale. The five new notes added to the do-re-mi scale are
called chromatic notes.
To play the chromatic scale, just start at any C,
then play every note ... C, C♯, D, etc., all the way up to the next C. When
you do this, you play 13 notes, but only 12 intervals of one semitone
each. (Remember, an interval is not a note. It’s the pitch distance between
La-di-da: Chromatic Solmization (You Don't Need to Know This, but It's
Kind of Interesting)
More than a thousand years ago, a nerdy Italian
friar and music theorist named Guido (Guido d'Arezzo, 995-1050), with more time
on his hands than he knew what to do with, invented solmization (“do re-mi”).
Guido also invented the basics of modern music notation.
Everyone’s familiar with “do re mi fa so la ti do”.
However, if you haven’t studied music in Europe, you may not know about the
additional syllables for the chromatic notes, syllables such as li,
te, le, fi, and so on. (Yikes!)
Not only do the chromatic notes have their own syllables, but the
syllables are different for the same note, depending on whether
you’re ascending the scale, or descending it. Here they are:
Do people actually study this stuff?
Oh yes they do. They even claim it’s useful, and so
it is, once you get into it. For instance, when you learn scales other than the
standard do-re-mi scale (scales that include some chromatic notes), you can
learn an equivalent do-re-mi syllable-based way of remembering each separate
As you'll see in later chapters, heavy metal
musicians (among others) make use of scales called modes, and each mode can be
translated into a do-re-mi type of scale using the above syllables.
the chromatic scale does sound chaotic—not naturally musical. However, you can
grab notes from the chromatic scale to craft numerous naturally musical scales.
These agreeable-sounding scales contain only eight or fewer notes, selected from
the chromatic scale. Chapter 5 discusses some of them.
now, though, a bit more about the “do-re-mi” scale.
common name is the major scale. It consists of eight notes,
spaced by seven intervals of tones and semitones in this order:
tone, tone, semitone, tone, tone, tone, semitone
This type of scale is called a diatonic
scale. “Dia” comes from the Greek word for “through” or “by.” And “tonic” refers
to the tonal anchor of the scale—the first note of the scale—called the tonic
note. So a “diatonic” scale’s notes are related to each other “through” the
first, or “tonic” note of the scale.
More on this in Chapter 5, which discusses tonal music in detail.
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