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PAGE
INDEX
4.3.1 Musical Drama
4.3.2 Scale Degrees
4.3.3 Curvature of the Major Scale
4.3.4 Interval Dynamics: Curved Arrows Through Your Brain
4.3.5 Interval Dynamics: Musical Road Trips
4.3.6 Interval Dynamics: Curved Arrows and Context
4.3.7 Interval Dynamics: Manipulating Tonal Tension
4.3.8 The Tyrannical Octave
~ • ~ • ~ • ~
4.3.1
MUSICAL
DRAMA
Recall that an interval is a relationship
between two pitches. Why the stress on “relationship”? Because that’s where the
“music” in tunes and harmony comes from. Each note in a scale, and, ultimately,
in a tune, sounds restful or restless, relaxed or tense, depending on the note’s
position with respect to the other notes in the scale or tune. These
note-to-note relationships, the urges and forces your brain perceives when it
hears a tune are called interval dynamics.
The activity that goes on in
your brain to process these interval relationships is your
musical experience.
In
general, if music contains a large amount of unrest as the tune (melody) moves
from interval to interval and chord to chord, you have an emotionally charged
musical experience.
Intervals perform like the
characters in a novel, sit-com, movie, or play. You get interested and
emotionally involved in a dramatic story only when you perceive tension and
unrest among the characters. Similarly when you perceive tension and unrest
among the intervals as the tune and chords progress, you experience emotional
involvement.
4.3.2
SCALE
DEGREES
Figure 18 (below) shows all eight notes of the
major scale, beginning and ending with C. However, there are other scales
besides the scale of C major. So, to discuss interval dynamics in general, not
just for the C major scale, it’s necessary to assign numbers to each of
the tones of the scale.
When you number each note of
the diatonic scale, the numbered notes are called scale degrees.
FIGURE 18
The Major Scale Showing Scale Degrees (Numbers)
The first and last notes of the
scale share the same number, so “(8)” is added to the last note in the following
discussion of interval dynamics to distinguish first from last.
Each scale degree has its own
name. Only some of these names are important enough to keep in mind, the ones in
bold type (Table 19):
TABLE 19
Names of the Scale Degrees
1
2
3
4
5
6
7
1 (8)
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Tonic
Supertonic
Mediant
Subdominant
Dominant
Submediant
Leading Tone
Tonic
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4.3.3
CURVATURE
OF THE MAJOR
SCALE
How does your mind interpret what you hear when
you play a major scale? Call this scale what you like ...
do
|
re
|
mi
|
fa
|
so
|
la
|
ti
|
do
|
C
|
D
|
E
|
F
|
G
|
A
|
B
|
C
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
1 (8)
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... it’s the same scale. Figure 19 (below) gives
you a better visual representation than Figure 18 (above). Here’s how your mind
actually hears this scale:
FIGURE 19
Interval Dynamics: “Going Away, Then Coming Back”
You hear the pitch rising
higher and higher as you proceed upwards through the scale degrees, from 1 to 2
to 3 to 4, all the way up to 1 (8).
Or you hear the pitch falling
as you proceed downwards from 1 (8) to 7 to 6 to 5, all the way down to 1.
As you proceed upwards through
the scale degrees, each tone sounds like it’s not only ascending in pitch, but
also moving further away from the vertical line that runs through 1 and 1
(8).
Then, when you get to scale
degree 5, something happens. The direction of motion reverses. And,
although the pitch continues to rise, the tones sound like they’re somehow
returning home, towards 1 (8).
And yet, it’s a different
version of home, a different version of the tonal centre.
Oddly, you get this “going
away, then coming back” sensation whether you ascend the scale from one end to
the other, or descend it from one end to the other.
4.3.4
INTERVAL
DYNAMICS:
CURVED
ARROWS
THROUGH
YOUR
BRAIN
The following discussion pertains to interval
dynamics in tunes
without chords. Tunes with harmony are discussed in Chapter 6.
When you play a single note,
that’s all your brain perceives. Just a note. Not music (ignoring, for the time
being, the tiny little matter of rhythm). But when you play at least two
successive notes that are different from each other—an interval—suddenly you
have at least the possibility of music.
In Figure 20 below, the arrows
show the tensions, the unrest your brain perceives in the relationships between
the tones (that is, the intervals), as you play the scale up or down.
The term interval dynamics
refers to the fact that, once your brain understands which note is the tonic
note, it perceives the succession of tones as energized, dynamic players that
move in force fields—not as static, lifeless beads on a string. Without
interval dynamics, there’d be no music.
In Figure 20, the thicker the
arrow, the greater the dynamic tension or unrest.
FIGURE 20
Interval Dynamics: How Your
Brain Actually Hears the Major Scale
4.3.5
INTERVAL
DYNAMICS:
MUSICAL
ROAD
TRIPS
Recall that simple ratios of frequencies gave
rise to a scale in the first place. However, some frequency ratios within the
scale are simpler than others. When your brain hears two frequency ratios, one
simple, the other not-as-simple, it perceives an urging of the not-as-simple
frequency ratio to become simpler. That’s the onset of a tune.
As the tune moves from note to
note, your brain stays interested only if most of the ratios of frequencies do
not resolve to simpler ones, while holding the promise of ultimately resolving.
To get a tune started, you need
a minimum of two frequency ratios so that your brain can tell which one
is simpler than the other. This can happen only if you hear at least three
successive notes:
• A single frequency ratio is a ratio of two
different notes (one interval).
• Therefore two frequency ratios require at least
three notes (two consecutive intervals).
Every interval except the
octave creates tension or unrest. This tension creates a musical story line or
musical narrative, as musicologists call it, especially when referring to
long-form instrumental works, such as symphony movements. Here’s how Anthony
Storr describes the nature of musical adventuring:
Hero myths
typically involve the protagonist leaving home, setting out on adventures,
slaying a dragon or accomplishing other feats, winning a bride, and then
returning home in triumph... The end of the piece is usually indicated by a
return ‘home’ to the tonic; most commonly to the major triad, less commonly to
the minor. A hero myth is an archetypal pattern, deeply embedded in the psyche,
because it reflects the experience of nearly all of us. We all have to ‘leave
home’ by severing some of the ties which bind us to it ...
Musical narratives apply to any
musical form, including short songs. Here are three of many versions:
1. “The Muso of Oz”
story line. The protagonist leaves Kansas—the tonic note, the first
note of the scale—on a mysterious journey. Immediately, tension arises (the
curved arrows in Figure 20), and the tune finds itself on a yellow brick road
trip, trying find its way back home.
Will the tune find its way
back home? Will it run into more tension and unrest before it finds its way
home? Will it get hopelessly lost and have to rely on Marshal Puma to dispatch
Doc and Fester, neither of whom can even stay upright on a horse?
Usually, the tune does find
its way back to tonic Kansas. End of tune.
2. “The Escapee”
story line. The protagonist moves through various dynamic tonal
fields, hiding, disguising itself, trying to escape re-capture.
Will the fugitive, Dr.
Richard Cymbal, get caught somewhere along the road and hauled back to Tonal
Headquarters to face the music? Will everything somehow resolve in a Hollywood
ending of dramatic climax, car chases, explosions, truth, and justice?
Yes, of course. End of
story and tune.
3. The “Lord of the
Tunes” story line. The protagonist is the sovereign, the queen or
king (could it be Elvis?), the holder of authority over the tune.
The plot concerns itself with
the loss and regaining of rightful authority. The sovereign’s source of
authority, the Tonic Note, somehow passes into the possession of other notes.
The story is still a road trip—a tune would not be a tune if it didn’t move
continuously and, to a degree, restlessly. The identity of the holder of
sovereignty gets called into question.
Will the rightful sovereign
get back sovereignty? Yes, usually. End of story and tune.
Every tune’s a road trip. If
the tune’s really short, the story’s over in seconds (for example, numerous
nursery tunes). If the tune takes a lot of twists and turns, the story might go
on for 20 minutes before the tune finally finds its way back home (a symphonic
movement).
4.3.6
INTERVAL
DYNAMICS: CURVED
ARROWS
AND CONTEXT
Music arises when your brain compares frequency
ratios of a succession of notes, an order of intervals. That means your brain
needs context. If it’s a tune without chords, the first note you hear supplies
the beginning of context. The second note provides more information. The third,
still more information. And so on.
All the while, your brain is
comparing frequency ratios. If it perceives several different simple frequency
ratios (for example, 2:1, 3:2, 4:3, etc.) among the note relationships (i.e.,
the intervals), it figures out there’s an organizing principle at work that is
giving rise to the succession of simple frequency ratios it’s perceiving.
What is this organizing
principle?
A scale of some sort.
It then expects to hear more
notes from the same scale, but not necessarily in the same order.
In fact, your brain will get
bored and lose interest in the tune unless it perceives some surprises in the
relationships between the pitches (the intervals) as the tune moves on.
As soon as the tune begins
(sets the musical road trip in motion), your brain goes to work figuring out
which note is the tonic—the tonal centre. All of the frequency ratios that
define the other intervals depend on the tonal centre for context. The tonic
note acts as a kind of gravitational force on the tune as a whole, which is
why it’s called the tonal centre.
Your brain perceives a
hierarchy of stability, with scale degree 1 (the tonic note) perceived as most
stable.
The Musical Adventures of “Tritone,” the Cat
As discussed in Chapter 1, chimpanzees create
abstract paintings that sell for big bucks.
So, why couldn’t a talented cat compose music on the
piano? If people buy chimp paintings, somebody might buy cat music.
Marshal Puma inherited a piano-playing cat after
Ex-Marshal McDillon left town in a ball of feathers and humiliation. The cat,
Tritone, walks along Marshal Puma’s piano keyboard.
Yes, but is it music?
Your brain hears a succession of random notes and
can’t figure out which one is the tonal centre. Therefore, it can’t apply an
organizing principle—a scale—to the notes it hears. So it can’t make sense of
any of the intervals Tritone is playing.
Not only that, but Tritone, being a cat, has no
ability to entrain. So he can’t even walk along the keys in a recognizably
rhythmic style.
Still, people do pay thousands of dollars for
chimpanzee paintings. Who knows, Marshal Puma might want to record Tritone’s
piano playing and send a demo to a record label in some other city, such as
Wichita or even Austin. One that specializes in postmodern music.
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Here’s another way of looking
at the way the tones of the major scale gravitate towards the tonal centre
(Figure 21):
FIGURE 21
Interval Dynamics: “Gravitational Force” of the Tonic Note
Figure 21 illustrates the appropriateness of the term
“diatonic.” All the notes of this type of scale ultimately relate to each other
diatonically—“through” or “by” the “tonic” note.
When you play a simple major
scale, how does your brain automatically figure out and interpret what it’s
hearing? Table 20 below shows the basics. You need context. Your brain needs to
process all of the notes successively for you to feel these effects.
TABLE 20
Interval Dynamics, Major Scale
Note
Move-
ment
|
Interval
Name
|
Fig. 20
Graphic
|
State of Unrest/ Tension (with Respect to Tonic Note)
|
1 – 2
|
Major second (whole tone)
|
Thick arrow
|
Upper note of major second (frequency ratio of 9:8) seeks to resolve down to the
tonal centre. Motion against the natural force, 1 – 2, creates high tension.
Motion with the natural force, 2 – 1, resolves it.
|
4 – 3
|
Minor second (semi-tone)
|
Thick arrow
|
Upper note of minor second (frequency ratio of 16:15) seeks to resolve down to
the closest note, scale degree 3, with its much simpler frequency ratio of 5:4
with respect to the tonal centre. Motion against the natural force, 3 – 4,
creates high tension. Motion with the natural force, 4 – 3, resolves it.
|
7 – 1 (8)
|
Minor second (semi-tone)
|
Thick arrow
|
Lower note of minor second (frequency ratio of 16:15) seeks to resolve up to the
tonal centre. Motion against the natural force, 1 (8) – 7, creates high tension.
Motion with the natural force, 7 – 1 (8), resolves it.
|
3 – 1
|
Major third
|
Medium arrow
|
Upper note of major third (frequency ratio of 5:4) seeks to resolve down to the
closest tonal centre. Motion against the natural force, 1 – 3, creates moderate
tension. Motion with the natural force, 3 – 1, resolves it.
|
6 – 5 or
6 – 1 (8)
|
Minor second or
Minor third
|
Medium arrow
|
Scale degree 6 has a roughly equal urge to resolve either down to the simpler
frequency ratio of the nearby scale degree 5, or up to the closest tonal centre.
Motion against the natural forces, 5 – 6 or 1 (8) – 6, creates moderate tension.
Motion with the natural forces, 6 – 5 or 6 – 1 (8) resolves it.
|
5 – 1 or
5 – 1 (8)
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Perfect fifth or
Perfect fourth
|
Thin arrow
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Scale degree 5 has a only a very slight but roughly equal urge to resolve to
either tonal centre. Motion against the natural forces, 1 – 5 or 1 (8) – 5
creates slight tension. Motion with the natural forces, 5 – 1 or 5 – 1 (8)
resolves it.
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Your brain perceives all of the
notes except the tonal centres, 1 and 1 (8), in some state of unrest as you play
the scale. You can use any of several terms to characterize these interval
dynamics:
restless
unbalanced
tense
unstable
dissonant
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vs at rest
vs balanced
vs resolved
vs stable
vs consonant
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The instant these forces come
into play—the instant you hear a
series of notes played or sung (a succession of intervals)—your brain may
sense a tune (musical motion). It depends on the frequency ratios of the
intervals and whether or not your brain can sense in those intervals an
underlying organization.
Your brain automatically tries
to determine if the intervals correspond to simple ratios of frequencies. It
will also try to determine the tonal centre, the note that serves as the anchor
for purposes of identifying the simple ratios. If it identifies several familiar
simple frequency ratios, it instantly understands the organizing principle (a
diatonic scale) and perceives some sort of tune—a succession of intervals
manifesting a variety of levels of dynamic tension.
4.3.7
INTERVAL
DYNAMICS:
MANIPULATING
TONAL
TENSION
Just as a writer of a movie script or play
manipulates tension through the actions of characters, a composer or songwriter
manipulates tension through the actions of intervals. Some intervals deliver
more tonal tension than others.
Normally, a composer or
songwriter comes up with a tune without intellectualizing about it. The tune
just comes out as an effusion. However, like any good writer, a skilled
songwriter or composer will then go over the tune and recognize weak
spots—places where the tune drags (not enough high-tension intervals), or
becomes confusing (too much material for short-term memory to handle), as it
moves from note to note.
A knowledge of interval
dynamics becomes vital in revising the tune. Historically, great composers (e.
g., Beethoven) and songwriters (e. g., Leonard Cohen, Paul Simon), have sweated
over revisions until they sense the tune has its own identity and doesn’t get
tired-sounding, even after repeated listenings.
Any tune retains a distinct
identity no matter where it’s played or sung in the spectrum of pitches.
Therefore, any pitch whatsoever can serve as the tonal centre, the tonic note.
It’s the frequency ratios that matter, not the specific frequency that
serves as the foundation (tonic note) for determining the ratios.
The intervals with the simplest
frequency ratios have the lowest dynamic tension, the greatest stability. The
octave, with a frequency ratio of 2:1, is, of course, the most stable interval.
The perfect fifth, with its 3:2
frequency ratio, has very little inherent tension, and therefore serves as a
kind of counter terminus to the tonic notes at either end of the scale. The
perfect fifth has so much natural stability that many tunes end on it (instead
of the tonic, which is where most tunes end), and the listener does not feel as
though the tune has failed to come to rest.
At the other extreme, the minor
second can supply a lot of tension, especially in its role as scale degree 7
going up to 1 (8). Because scale degree 7 strongly seeks to resolve up to 1 (8),
scale degree 7 is known as the leading tone.
It’s important to reiterate
that your brain does not “learn” any of this. It’s hard-wired. You will always
sense these states of rest or unrest, tension or resolution, etc., whenever you
hear a variety of simple ratios of frequencies in succession.
4.3.8
THE
TYRANNICAL
OCTAVE
You don’t have to think of the octave as
tyrannical. But, like other natural phenomena (gravity, for instance), it is. As
Figure 20 above illustrates, all arrows curve to the octave notes. In music, you
can’t break free of the tyrannical octave.
No matter how hard your tune
may try to break the chains of 1 and 1 (8), there’s just no escaping. Your tune
merrily leaves home, lights out for the territory, and ends up ... where?
Strangely, back home. Without having turned back. Without having gone in a
circle.
The irreducible simplicity of
the 2:1 (octave) frequency ratio induces a feeling of balance or repose. All
other pitches arise from more complex frequency ratios such as 3:2, 4:3, 5:4,
and so on. Your brain distinguishes them from the octave interval notes in two
ways:
1. By associating a “different-from-octave”
qualitative sound with each note representing each “non - 2:1" frequency
ratio, within the context of the octave interval.
For example, as you play the
white keys on the piano from C up to the next C, you hear the notes D, E, F, G,
A, and B all sounding qualitatively different from the C you started the
scale with.
But when you get to the C at
the top of the scale, even though it’s a different note, it sounds
qualitatively the same as the C you started with. Yes, it’s higher in pitch,
but it still sounds to your brain like the identical note you started with, C.
2. By associating a feeling of imbalance or
unrest with each
non-octave note. This feeling of unrest or tension increases in intensity
as frequency ratios become more complex with respect to the octave interval.
You can stuff as many notes as
you want between 1 and 1 (8), but you still won’t escape the octave. You can
never pry the octave open any wider, because you can’t reduce a frequency ratio
to anything simpler than 2:1.
Paradoxically, making peace
with the smallest intervals of the octave, the semitones (through a bit of
fudging called equal temperament), provides more than ample relief, if not
escape, from the octave’s tyranny (coming up in Chapter 5).
(Tuning purists will note that
some tuning systems slightly “stretch” the octave, such as one used by
Indonesian Gamelan percussion orchestras. But such tunings are highly variable
and, in any case, unheard of in Western popular music.)
~ • ~ • ~ • ~
~ • ~ •
~ • ~
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