~ • ~ • ~ • ~
4.2.1
THE
BASIC
INTERVALS
So far, three intervals have made an appearance:
• Octave: Pitch distance between the first note and the eighth
note of the major scale (or first note and 13th note of the
chromatic scale)
• Semitone: Pitch distance between any two adjacent notes of
the chromatic scale
• Tone:
Pitch distance of two semitones
Other intervals exist, and they all have names, but not very
interesting ones like Natasha or Engelbert. Since the semitone is the
smallest interval, you can measure the other intervals in multiples of
semitones.
Even
the tone and the semitone have their own special alternative “interval” names.
Table
11 below lists all of the intervals within an octave. Usually (but not always),
you reckon an interval—which is always two notes—as starting from the lower note
and going to the upper note, as in the “Example” column in Table 11.
TABLE 11 Names of Basic Intervals
Interval
|
Number of
Semitones
|
Example
|
Minor Second
Major Second
Minor Third
Major Third
Perfect Fourth
Augmented Fourth
Perfect Fifth
Minor Sixth
Major Sixth
Minor Seventh
Major Seventh
Octave
|
1
2
3
4
5
6
7
8
9
10
11
12
|
C – C♯
C – D
C – E♭
C – E
C – F
C – F♯
C – G
C – A♭
C – A
C – B♭
C – B
C – C
|
4.2.2
INTERVAL
NAMES
EXPLAINED
Figure 15 (below) clarifies the logic of interval names a bit:
FIGURE 15 C Major Scale with Intervals Named
For reasons that will become clearer as you get better
acquainted with intervals and scales and chords, all of the intervals
are named with reference to the first note (the tonic note) of the
major scale. For example, major second refers to the second note
of the major scale, if you start from the tonic note.
The
major scale has only eight notes. That’s why none of the intervals has a
name higher than “seventh,” even though there are 12 different intervals.
(The
intervals are not named after the notes of the chromatic scale because the
chromatic scale by itself has no use as a “musical” scale.)
Here’s
how each interval gets its name:
• Major Second and Minor Second: Both named for the
second note of the major scale. The major second is an
interval of a whole tone. The minor second is an interval of a
semitone.
• Major Third and Minor Third: Both named for the third note
of the major scale. The major third is an interval of four
semitones. The minor third is a semitone less, at three
semitones.
Minor Confusion
Yet Another Potential Point of Confusion:
The term “minor,” when referring to intervals (such as “minor third”), has a
different meaning from the term “minor” when referring to keys, (such as
“key of D minor”). Chapter 5 discusses keys.
If you confuse the meanings of “minor interval” and
“minor key,” you’re apt to get lost.
If you were to get lost, Marshal Puma would probably
conscript Deputy Fester and Doc Yada-Yadams to saddle up and fetch you back.
Deputy Fester never learned to ride so good and nobody can figure out how he got
to be a deputy. As for Doc, he’s three-fifths drunk, 80% of the time and can’t
stay on his horse unless somebody does up his seat belt for him. Neither Fester
nor Doc would be much good in a search party. So, if you steer clear of any
confusion about minor intervals and the minor keys, you’ll stay found.
|
• Perfect Fourth:
Named for the fourth note of the major scale. It’s an interval of five
semitones. It’s called “perfect” because, compared with the augmented fourth, it
sounds a lot more, um ... “perfect.” At least in the context of a chord or a
tune.
• Augmented Fourth:
A wild, unruly interval, it’s also named for the fourth note of the major scale.
However, the augmented fourth overshoots the perfect fourth by a semitone, for a
total of six semitones. This interval has several other names. It’s often called
the tritone because it spans three whole tones (six semitones). It’s also
known as the diminished fifth, because it’s a half-tone short of being a
“perfect” fifth. In the Middle Ages, they called it diabolus in
musica—the “devil in music.” Somebody had a sense of humour way back then.
Or ... maybe they believed it was the
musical devil hisself.
• Perfect Fifth:
Named for the fifth note of the major scale. It’s an interval of seven
semitones. It’s called “perfect” because, compared with the diminished fifth, it
sounds a lot more “perfect” in the context of a chord or a tune. (But, as you’ll
see, “perfect” doesn’t necessarily mean “interesting.” Just like people.)
• Major Sixth and Minor Sixth: Named for the sixth note of
the major scale. The major sixth is an interval of nine
semitones. The minor sixth, one less at eight semitones.
• Major Seventh and Minor Seventh: Named for the seventh
note of the major scale. The major seventh is an interval of
eleven semitones. The minor seventh is one less, at ten
semitones.
More Minor Confusion
Yet Another, Another Potential Point of Confusion:
Here we go again. As you’ve learned, there’s a difference between minor
intervals and minor keys.
Well, there’s also a difference between minor
intervals and minor
chords, such as, for example, the chord “D minor seventh,” which is
neither an interval nor a key.
Chapter 6 discusses chords in harrowing detail. For now, just be
aware that, if you confuse the meanings of
• minor
interval,
• minor
chord, and
• minor
key,
you could get lost.
You don’t want to get lost right now, because
Marshal Puma’s in no mood for organizing search parties. She just found out that
Ex-Marshal McDillon and Doc and Deputy Fester have all been partying on Doc’s
moonshine in a gully south of Dodge with a bunch of mid-west farmers’ daughters
from the Beach Boys song, “California Girls,” who really make them feel alright.
Looks like it’s all over between Marshal Puma and Ex-Marshal McDillon.
|
4.2.3
PERFECT
FIFTHS
AND SCALE
CONSTRUCTION:
MONTY
PYTHAGOR’S
METHOD
You may wonder who first figured out the relationship between
lovely-sounding overtones, simple frequency ratios, and their
application to scale building.
People usually credit the Greek philosopher, mathematician, and
comedian, Monty Pythagor. As you know, Mr. Pythagor also
formulated the Pythagorean Theorem about the square hide of the
hippopotamus and the sum of the other square hides, which
apparently revolutionized the footwear industry.
Mr. Pythagor (582 BC - 496 BC) may have figured out the
mathematics of overtones and scales 2,500 years ago but he
certainly was not the first to discover musically pleasing scales. As
discussed in Chapter 1, Neanderthals had bone flutes with diatonic
scale notes tens of thousands of years ago.
As for
Mr. Pythagor, it seems he realized that if you kept adding tones in consecutive
frequency ratios of 3:2 (perfect fifths), you would get a pleasing-sounding
musical scale. Next time you’re near a keyboard, try this:
• Play the note:
C
• Now play the perfect fifth (seven semitones) above, which is
G:
C G
• Next, play the perfect fifth (seven semitones) above G, which
happens to be D, like this:
G D
• Next, play the perfect fifth above D, which is A:
D A
• Then the perfect fifth above A, which is E:
A E
• Then the perfect fifth above E, which is B:
E B
So
far, you’ve played the following sequence of six notes:
C G D A E B
The highest note, B, is almost three octaves above the C you
started with.
The next step is to play all six notes in the same octave, and in
scale order. Then add another C to complete the scale. Now you
have the following seven-note scale:
There
you go. That’s almost the diatonic major scale.
You
can construct a good many of the world’s popular musical scales simply by using
notes derived from consecutive frequency jumps in the ratio of 3:2, the ratio of
the perfect fifth interval.
And, as discussed earlier, when you plunk a bunch of these
notes into the same octave, you end up with other simple frequency
ratios within the scale as well, such as 2:1 (octave), 4:3 (perfect
fourth), 5:4 (major third), and so on.
So, since Mr. Pythagor figured out the principle of creating scales
derived from simple frequency ratios, such scales are called
Pythagorean scales. The “do-re-mi” major diatonic scale is a Pythagorean
scale, even though it’s not perfectly based on
consecutive intervals in ratios of 3:2.
“Not
perfectly” means something goes awry. Here’s how:
So far, you've seen that if you use the strict Pythagorean method,
you get these six different notes (the octave note is repeated):
C D E G A B C
It’s almost
the major diatonic scale. But one note’s missing, namely F.
So, why not try to get that last note by playing the next note, a
fifth interval (seven semitones) up from B, which was the last note
you played in the series?
Try it.
What’s
the note you get?
Alas,
it’s F♯, not plain old F.
Worse,
the fifth above F♯ is C♯, not C.
Dang.
Worse still, suppose you go away from the piano and instead
decide to derive the series of notes using a calculator. You start with
the frequency 261.6 (Middle C) and use your calculator to derive the
series of fifth intervals as exact ratios of 3:2. Then you compare your
list of calculated frequencies with the actual frequencies of the corresponding
piano notes (available on Roedy Black’s Musical
Instruments Poster).
What you discover is that all the theoretical notes you calculated
are slightly but noticeably sharper than the notes on the piano!
Dang again.
In any case, the fact that you can almost get a complete major
diatonic scale simply by using notes derived from consecutive
overtone frequencies with the single simple frequency ratio 3:2 (the
perfect fifth) illustrates the central role of simple frequency ratios in
scale building.
4.2.4
THE
PYTHAGOREAN
COMMA
Suppose you were to start with the frequency for Middle C and just
keep on going, up and up in leaps of perfect fifth intervals, until you
eventually reach the note C again, in a much higher octave.
The first question is, would you ever get to C again, somewhere
over the rainbow, way up high?
Yes, indeed. Especially in Kansas.
It takes 12 leaps of perfect fifths to get to another C. You end up
seven octaves above the C that you started with.
If you
start from Middle C and use a calculator to multiply each successive frequency
by a ratio of 3:2 (the simple frequency ratio of the perfect fifth interval),
you get the data in Table 12. (It’s theoretical, because the last note is well
above the upper limit of human hearing. Way over the rainbow.)
TABLE 12 Consecutive Perfect Fifth Intervals Going
Up Seven Octaves
|
Note
|
Frequency
(Hz)
|
Middle C
G
D
A
E
B
F♯
C♯
G♯
D♯
A♯
F
C, seven
octaves up
from Middle
C
|
261.6
392.4
588.6
882.9
1,324.4
1,986.5
2,979.8
4,469.7
6,704.5
10,056.8
15,085.2
22,627.8
33,941.6
|
Now, just for fun (are you having fun?), try getting to that same
C, seven octaves above Middle C, except do your leaps in octaves,
instead of perfect fifths.
Start with Middle C at 261.6 Hz and keep doubling the frequency
to preserve the 2:1 simple frequency ratio that defines an octave
interval. Table 13 shows what you get.
TABLE 13 Consecutive Octave Intervals, Going Up
Seven Octaves
Note
|
Frequency
(Hz)
|
Middle C
C , one octave up
C, two octaves up
C, three octaves up
C, four octaves up
C, five octaves up
C, six octaves up
C, seven octaves
up from Middle C
|
261.6
523.2
1,046.4
2,092.8
4,185.6
8.371.2
16,742.4
33.484.8
|
Have a look at the last frequency in Table 12 and compare it with
the last frequency in Table 13.
They’re
both supposed to be the note C, seven octaves above Middle C, right? So the two
frequencies are supposed to be exactly the same, aren’t they?
But
they ain’t.
The ratio between them, 33,941.6 Hz : 33,484.8 Hz, boils down
to a ratio of 1.0136:1, instead of 1:1.
Dang, for the third time.
That ratio of 1.0136:1 is called the Pythagorean comma. (In
music, a tiny interval is called a comma.)
The Pythagorean comma caused all sorts of havoc with
instrument tuning for more than 2,000 years after Monty Pythagor
died of laughter, without telling anybody how to fudge the
Pythagorean comma and stay in tune.
(Chapter 5 discusses some clever jiggery-pokery, called equal
temperament, that gets around the Pythagorean comma and cures
all problems with scales forever. Well, sort of.)
4.2.5
WHY
PYTHAGOREAN
SCALES
EMERGED
INDEPENDENTLY
ON SEVERAL
CONTINENTS
As discussed in Chapter 3, the human brain has
the ability to automatically analyse a tone’s constituent harmonics and identify
the soundmaker. That means the brain has the ability to understand (and
appreciate) simple ratios of frequencies, whatever form they take—overtones of a
single tone, or scales consisting of notes in simple-frequency relationships.
So, whenever humans stumble upon a way of generating a
series of notes in simple-frequency relationships, they find the notes
pleasing and make music. Homo neanderthalensis knew how to do this, and
they weren’t even of our species, Homo sapiens.
The harmonic series is a phenomenon of nature that anybody
anywhere can generate with nothing more than a string or a piece
of catgut or sinew attached via some sort of bridge to a resonator.
Easy to make. Pleasing, You get simple-frequency-ratio discrete
notes.
It’s
no wonder, then, that Pythagorean-type scales, especially pentatonic scales
(discussed in Chapter 5), have emerged independently in the musical cultures of
all the major civilizations, from Africa to Europe to Asia. Humans everywhere
prefer music made with tones in relationships of simple frequency ratios. Even a
22-tone scale used in India shows an underlying Pythagorean structure, no doubt
derived from the harmonic series.
4.2.6
CONSONANCE
AND DISSONANCE
Some intervals sound stable, balanced, at rest,
when you play the two notes either together or successively. That’s called consonance.
Others
sound unstable, unbalanced, restless. That’s dissonance
(Table 14).
TABLE 14 Consonant and Dissonant Intervals
Interval
|
Number of
Semitones
|
Example
|
Consonant/
Dissonant
|
Minor Second
Major Second
Minor Third
Major Third
Perfect Fourth
Augmented Fourth
Perfect Fifth
Minor Sixth
Major Sixth
Minor Seventh
Major Seventh
Octave
|
1
2
3
4
5
6
7
8
9
10
11
12
|
C
– C♯
C
– D
C – E♭
C – E
C – F
C
– F♯
C – G
C – A♭
C – A
C
– B♭
C
– B
C – C
|
Dissonant
Dissonant
Consonant
Consonant
Consonant
Dissonant
Consonant
Consonant
Consonant
Dissonant
Dissonant
Consonant
|
Pick an interval, any interval. Play the two notes of the interval
simultaneously on a guitar or keyboard, the way you would play a
chord. Or successively, the way you would play a tune. Go through
the list yourself and try out all the intervals.
Consonance
vs dissonance goes straight to the heart of what helps make music exciting and
emotional (a good amount of dissonance), or predictable and dull (too much
consonance). In music, “dissonant” does not mean “grating” or “harsh.” Rather,
it refers to the sense you get of tonal unrest, the seeking of tonal
resolution which imparts motion to melody and harmony.
Later
on, you’ll find that chords, because they’re comprised of two or more intervals
(three or more notes), also have consonant or dissonant characteristics,
depending on the intervals within the chord.
The notes of a tune (melody) against the backdrop of a chord
progression produce consonant or dissonant sounds, too.
Happy Thirds and Sad Thirds: Great Country Hits of Auctioneers and
Chickadees
If you live near the sea, you may hear foghorns
every so often. What’s that interval, the descending
Dah
Dah ?
It’s a descending major third. People just love that
major third. It’s also the cheerful “ding-dong” of your doorbell.
And it’s the main interval the auctioneer uses as he
or she disposes of the family farm. In 1956, Leroy Van Dyke and Buddy Black
wrote a country classic called “The Auctioneer,” which highlights the
auctioneer’s major third sing-song patter. Gordon Lightfoot recorded a fine
version of this tune on his 1980 album
Dream Street Rose.
The minor third, on the other hand, has a
decidedly sad sound. It’s the chief interval in the children’s chant, “Ring
Around the Rosie” (the interval on the word, “ros - ie”), also known as
“Nyah-Nyah-Nyah-Nyah Nyaaaaah Nyah.”
The male chickadee uses a sliding descending minor
third during mating season. The call goes from A down to F♯, or B♭ down to G.
Women chickadees love that sad tune. The slide into the second note of the
interval is characteristic of sad country songs. Male chickadees may have been
the first true country singers.
|
4.2.7
DISSONANCE:
FREAKY
FREQUENCY
RATIOS
What causes intervals (and, by extension, chords) to sound
consonant or dissonant?
Have a look at the ratios of frequencies that correspond to
consonant vs dissonant intervals (Table 15).
TABLE 15 Frequency Ratios of the Intervals
Interval
|
Semi-
tones
|
Example
|
Freq.
Ratio
|
Consonant/
Dissonant
|
Unison
Minor Second
Major Second
Minor Third
Major Third
Perfect Fourth
Augmented Fourth
Perfect Fifth
Minor Sixth
Major Sixth
Minor Seventh
Major Seventh
Octave
|
0
1
2
3
4
5
6
7
8
9
10
11
12
|
C
C
– C♯
C
– D
C – E♭
C – E
C – F
C
– F♯
C – G
C – A♭
C – A
C
– B♭
C
– B
C – C
|
1:1
16 : 15
9 : 8
6 : 5
5 : 4
4 : 3
45 : 32
3 : 2
8 : 5
5 : 3
9 : 5
15 : 8
2 : 1
|
Consonant
Dissonant
Dissonant
Consonant
Consonant
Consonant
Dissonant
Consonant
Consonant
Consonant
Dissonant
Dissonant
Consonant
|
Some intervals have simple frequency ratios, such as the major
third (ratio of 5:4). Others have complex ratios, especially the
augmented fourth (ratio of 45:32), the freakiest of them all.
In general, you get consonant intervals from the simplest
frequency ratios, the ones with small numbers. You get dissonant
intervals from complex frequency ratios, the ones with larger
numbers.
Degree of perceived consonance vs dissonance is a function of
pitch relationships among tones. Also, as discussed a bit later (Chapter 6), consonant intervals have
overtones in common, or overlapping. Dissonant intervals tend not
to.
Infants show clear preferences for consonant intervals, based on
simple frequency ratios, such as fourths and fifths, and show a
distinct aversion to dissonant intervals, such as the tritone. This
indicates such preferences are wired in the brain at birth. It also
underscores the futility of trying to build audiences for unpalatably
dissonant music.
In an experiment comparing consonant-dissonant preferences of
humans and cottontop tamarins, the monkeys showed no clear
preference for consonant intervals over dissonant intervals. In the
same experiment, humans showed a clear preference for consonant
intervals, supporting the theory that music is a species-specific
adaptation in humans only.
4.2.8
INTERVALS
WITHIN
SCALES
So far, the discussion of intervals has focussed on intervals in which
the first of the two notes is the lowest note of the scale, the tonic.
Can an interval start on any note?
Sure.
You can start on the note A, the sixth note of the C major scale. If you then go
up three semitones to C, that’s an interval of a minor third. Any span of three consecutive semitones is a minor third
interval, no matter where it occurs in a scale. Consider, for example,
the intervals within this scale (Figure 16):
FIGURE 16 C Major Scale
Table
16 below shows intervals drawn exclusively from the C major scale—no chromatic
notes.
TABLE 16 Intervals Occurring Naturally in the Major
Scale
Interval
|
Semi-
tones
|
Example
|
Freq.
Ratio
|
Consonant/
Dissonant
|
Minor Second
Major Second
Minor Third
Major Third
Perfect Fourth
Augmented Fourth
Perfect Fifth
Minor Sixth
Major Sixth
Minor Seventh
Major Seventh
Octave
|
1
2
3
4
5
6
7
8
9
10
11
12
|
B – C
C
– D
A – C
C
– E
C
– F
F – B
C
– G
E – C
C
– A
D – C
C
– B
C
– C
|
16 : 15
9 : 8
6 : 5
5 : 4
4 : 3
45 : 32
3 : 2
8 : 5
5 : 3
9 : 5
15 : 8
2 : 1
|
Dissonant
Dissonant
Consonant
Consonant
Consonant
Dissonant
Consonant
Consonant
Consonant
Dissonant
Dissonant
Consonant
|
Of the 12 different intervals, 11 anchor naturally to the tonal
centre (the note C) at one end of the scale or the other.
And
the only one that doesn’t? It’s that diabolical diabolus in
musica, the very devil hisself, the augmented fourth. The one with
the weirdest frequency ratio, 45:32.
The same interval can occur in several places in one scale. For
example, in the C major scale ...
• The
minor second (one semitone) occurs in two places: E – F, and B – C.
• The
perfect fifth (seven semitones) occurs in four places: C – G, D – A, E – B, and
F – C.
4.2.9
COMPLEMENTARY
INTERVALS
Any two intervals that add up to an octave (which consists of 12
semitones) are called complementary intervals (Table 17).
TABLE 17 The Complementary Intervals
Minor 2nd (1 semitone)
Major 2nd (2 semitones)
Minor 3rd (3 semitones)
Major 3rd (4 semitones)
Perfect 4th (5 semitones)
|
+ Major 7th (11 semitones)
+ Minor 7th (10 semitones)
+ Major 6th (9 semitones)
+ Minor 6th (8 semitones)
+ Perfect 5th (7 semitones)
|
= Octave
= Octave
= Octave
= Octave
= Octave
|
A few
“rules” of complementary intervals:
• The complement of any minor interval is a major interval. And
vice-versa.
• The
only two “perfect” intervals—perfect fourth and perfect fifth— complement each other (wouldn’t you know it).
• There’s
no complement for the diabolical tritone (6 semitones).
Complementary intervals are important in understanding chord
changes or chord progressions, the subject of Chapter 6.
4.2.10
WHY
INTERVALS
ARE
THE REAL
MUSICAL
UNITS
OF MELODIES
AND CHORDS
A tone in isolation is just a tone. Only when two
tones are sounded, either together or in sequence, does a relationship form.
Your brain analyses that relationship. As each tone sounds in succession, your
brain tries to anticipate the new tone that might come next in the context of
the ones you’ve just heard.
If you
play a progression of chords without a tune, does your brain interpret that
chord progression as “music”?
No, it
doesn’t.
Hardly ever, anyway. All you hear is formless harmony.
To hear music, you need a tune.
Your brain demands it. You’ll see why in the discussion of harmony, chords, and
chord progressions (Chapter 6).
On the
other hand, if you play or sing a tune by itself, with no chords, does your
brain interpret that tune as “music”?
Yes, it does.
For example, most people sing national anthems without
instrumental accompaniment. Great national anthems, such as those of France, Britain, America, Russia,
and South Africa, have stood the test of time. These anthems have such powerful tunes that
they sound beautiful with or without chords.
“O’er the Land of the Fa-ree-eee- eee-eee-uh”
You’ve probably heard pop stars perform over-the-top versions of your national
anthem. Usually, such renditions ruin the anthem.
When some singer with no compositional know-how
deviates from the classic tune of a great national anthem in an effort to “make
it his (or her) own,” he or she is attempting to re-compose the tune on the
fly, incompetently improvising. It’s the musical equivalent of painting a
moustache on the Mona Lisa.
That said, occasionally a genuine musical genius
comes along and succeeds in rendering a national anthem in an awe-inspiring, yet
original way. Jimi Hendrix did it at the Woodstock music festival in 1969. But
that’s rare.
|
Most of the
time, music consists of a tune with instrumental
accompaniment. The tune seems to float or bounce along on top of
the chords, which provide depth and color. With or without
instrumental accompaniment, the tune or melody actually consists
of a succession of intervals, not a succession of notes.
The first six
notes of “The Star Spangled Banner”—“O-oh say can you see”—form five successive intervals. Here they are (Table
18):
TABLE 18
First Five Intervals of “The Star Spangled Banner”
O – oh
oh – say
say – can
can – you
you – see
|
Minor third, moving down (three semitones)
Major third, moving down (four semitones)
Major third, moving up (four semitones)
Minor third, moving up (three semitones)
Perfect fourth, moving up (five semitones)
|
Whether a tune is interesting or boring depends on its
arrangement of intervals, not individual notes. Intervals come from
scales. And scales come from overtones.
Not
only that, but, as you’ll soon see, intervals determine how
chords sound, and whether a chord progression imbues a piece of
music with purpose and feeling ... or fails to.
Only when you get to intervals does the possibility of music even
arise.
Here’s
a little flow diagram that summarizes these relationships (Figure 17). The
arrows mean “give rise to”:
