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ANALYSIS.TXT
PRINCIPLES OF MUSICAL ANALYSIS
We have all along been employing the chief principle of musical
analysis, which is simply to make observations about constituent
elements of a piece of music. Of course, this may not always be
such a simple matter, since prior assumptions, misleading appear-
ances, overlooking some detail, or a whole host of other subtle
factors may impede the accuracy of our observations.
Nonetheless, analysis, at its most straightforward level, is a matter
of making observations, attempting to be as accurate as possible in
simply and clearly stating what is presented to the senses. At this
level, one is collecting data. At the same time, we begin to compare
one datum with another, and in this process analysis-as-observation
attains a new level. Observation is not only made of constituent parts
within a whole. We then attempt to comprehend how those parts are
organized into patterns which constitute the whole.
Musical analysis is fundamentally an act of pattern recognition. And
this is seemingly a natural activity of the human brain. The primary
goal of such activity is to achieve a higher degree of understanding
of the subject under analysis.
The chief means by which this greater understanding is achieved
is through the paying of attention to details. The more detail
we can take in, while not losing sight of the whole, the closer
we can be to the inner workings of music. Then as we "map" out
the patterns which make up the whole, we may arrive at insights
about how a given piece of music works. While the map is never
the territory, the process of mapping, of laying down a template,
does provide a means for better making our way around the terrain.
DIVISION OF THE WHOLE INTO PARTS
The initial step in analysis is to experience an entire entity. This
may mean a whole work, as in a multi-movement symphony, a one-movement
piece, as in a folk or pop song, or even a segment of a larger work
which has the sense of being more or less complete. Listen to the
"Ode to Joy" segment of Beethoven's Ninth Symphony, which can be
extracted from that larger work as an entity onto itself.
I recommend a simple procedure of diagramming music which begins
first of all with presenting an image of the whole structure under
investigation:
The next step is to "map" the major subdivisions of the whole struc-
ture. This involves perceiving where the primary inception points are
in relation to the primary goal areas. The goal areas may be easist to
perceive in that that are cadences (literally, "to fall"), instants of
relative repose. There is a sense here of arrival. Quite obviously, the
strongest cadence is almost inevitably going to be the final notes, the
goal of the whole piece. In tonal music, this is most certainly going to
be an arrival on the tonic (I) note or chord. Interior cadences will be
less strong, though themselves having a distinct sense of arrival. Any
material following a cadence (and definitely the very first notes), will
sound like some form of beginning, an inception. Depending on the length
of the music under investigation, observation of the next level "down"
from the whole will reveal either sectional division or phrase division.
In the Beethoven there are 4 distinct cadences (including the final)
which constitute the largest divisional segments of the whole. <RET>
Once the areas from inception to goal have been mapped, comparison
of one section to another will reveal relatedness of material. The
Beethoven is clearly divided into 4 segments, with the 1st, 2nd, and
4th segments containing nearly identical material and the 3rd section
containing material which contrasts with these. A reasonable labelling
procedure entails associating letters of the alphabet with sections.
The "Ode to Joy" could thus be described as AA'BA', with the 2nd and
4th sections indicated with the "prime" mark to represent the slight
change from the original "A." And since the 2nd cadence is stronger
then the first (because of its arrival on the tonic note), the AA'
section could be considered as a section grouping and labelled as
a larger "A." The overall form may thus be ABA:
There is another way of viewing this structure, however. Because
of the clear balance of measures, the second 8 measures (BA') might
be grouped as an entity, thus making a two-part structure which
could conceivably be labelled AA'. This kind of ambiguity is
common in making an attempt to pin down the details of patterning.
The interplay between a sense of 3-part and 2-part organization
produces an overall asymmetry which may be a factor contributing
to the effectiveness of this very simple and straightforward piece.
This kind of structural analysis reveals groupings of events which
have been traditionally designated as phrases. A musical phrase,
to quote Douglas Green in his "Form in Tonal Music" may be defined
as "the shortest passage of music which, having reached a point of
relative repose, has expressed a more or less complete musical thought."
The qualifying terms here are very important, for the strength of
cadences is relative, and the sense of wholeness is relative. This
is certainly made clear by the Beethoven example. The first cadence
in measure 4 is not as strong as that in measure 8, and the final
cadence is the strongest of all.
DETAILS OF MELODIC SHAPING
Mapping the overall structure of a piece of music provides insights
into its sense of direction. A proper understanding of phrase struc-
ture can aid the performer to think ahead to goal areas and to give
them their proper emphasis. Composers can likewise gain understanding
of how to shape their own creations to best effect.
Just as an entire structure can be divided into sub-entities, so may
those parts be divided. Examination of the details of the very first
phrase or two may generate further insights into the music's workings.
There are at least four significant aspects of organization here. The
first is the coherency of pitch relationship. All adjacent pitches
are no more than an interval of a second. In melodies, this deploy-
ment of pitches is called "conjunct." Conjunct motion is a natural
outgrowth of vocal music because it is the easiest to remember and
sing. It contributes to creating unity. Melodic intervals larger than
a second are called "disjunct," and their use contributes to creating
variety. We might question the balance of unity versus variety in
this passage, since there are no disjunct intervals, but further in-
vestigation uncovers other factors at work which may establish a
better balance. <RET>
The second significant aspect of organization here is choice of pitches
occurring on strong beats versus those falling on weak beats. Those on
strong beats (beats 1 and 3 in a 4-beat measure) outline the 1st, 3rd
and 5th notes of a D-major scale. As we'll discuss in the greater detail
in the chapter on harmony, this is the tonic chord, and it must in some
way be given emphasis within the first few measures in order to thor-
oughly establish the key center. That the first 4 measures cadence on
a pitch not contained within this chord is the chief reason we sense
a need to hear more. Any pitch other than the tonic is unstable, un-
resolved, in relation to the tonic. Only the tonic note and the tonic
chord are wholly stable. <RET>
The third significant detail here is the specific contour formed by
the choice of pitches and the overall pitch range formed by this con-
tour. Most traditional melodies flow upwards and downwards in more or
less standard patterns. Many begin with the tonic note, ascend some
distance (often to the 5th or 8va), then return back downwards to
tonic. Some start with tonic, leap upwards and descend. Some start
high and move downward. In this instance, the initial pitch is above
the tonic, there is a movement upwards followed by descent to the
tonic. Combining elements of contour with intervallic movement from
pitch to pitch results in a melody's overall range. Here it is an
interval of a 5th, from the tonic to the 5th scale degree, which
further reinforces the sense of key orientation. <RET>
The subtle rhythmic deployment is the fourth important detail. In
what would otherwise be a dull, plodding rhythm, Beethoven has intro-
duced two chief factors of variety. The most obvious is the simple shift
from the straight quarter notes to the dotted-quarter/eighth at the end
of the phrase. Listen to what a slight but significant difference it would
make if only quarter notes were used:
The other factor is the offsetting of the repeated notes, which con-
tributes to extending the length of time the 1st, 3rd, and 5th are heard
This also creates a very gentle syncopated effect "across" the bar-line,
highlights the 5th and the 3rd and thus strengthens the movement to the
tonic note at the conclusion of the second phrase.
CELLULAR ORGANIZATION
While we examine these details, we may also observe relationships
between one little idea and another. Quite clearly, the opening upward
gesture (F#-G-A) is a unit which in some ways could be considered the
"cell" from which the rest of the tune grows. The second measure pre-
sents this cell in reverse order and "straightens out" the repeat,
thus extending it one note downwards. We could consider this relation-
ship either a mirror image or a retrograde (backwards). This kind of
correspondance contributes to unity at the same as as to variety. The
second measure is a form of varied repetition of the first.
The 3rd measure re-initiates the same contour at a lower pitch level.
This is likewise a form of varied repetition, specificically called
"sequence." Of course, it doesn't last very long here, otherwise
we would be carried beyond the tonic note.
It appears that cellular ideas, 3- or 4-note gestures, grow into
parts of phrases, and these parts of phrases grow into larger en-
ties, the phrases themselves. Phrases combine to make sections,
and sections are joined to form entire works. There is, in fact,
and entire branch of traditional music theory, form and analysis,
devoted to studying these relationships and to labelling one part
in relation to another. And we could continue examining this simple
tune, making ever new discoveries regarding relationship of parts.
What, for instance, is the specific relationship of the 3rd phrase
to the first? We know it is contrasting. Of the most obvious items,
are the fact that this section contains the only instants or disjunct
motion, and that it reaches the lowest pitch in the whole of the tune.
Also it contains the only occurrances of consecutive 8th-notes. But
are there factors linking it to the "A" section? Go back to page 3
and listen again to the whole tune and make your own observations
about this!
MUSICAL ANALYSIS
Princlples of Analysis......................................p. 1
Divisions of the Whole into Parts...........................p. 3
Details of Melodic Sharping.................................p. 7
Cellular Organization.......................................p. 13
INTERVAL.TXT
REVIEW
We have previously learned that a musical interval is the difference
between two frequencies. This difference is expressed in terms of
the numerical count from one pitch to, another together with a qual-
itative designation which enables describing the difference between
two intervals which have the same count. Thus the interval from C3
to G3 is a Perfect 5th. The interval C3 to E3 is a Major 3rd. And
the interval C3 to Eb3 is a Minor Third. Each of these intervals is
one which we have discussed. We will now learn about others.
INTERVALS IN THE MAJOR SCALE
One approach to learning intervals is to think of them in relation
to a known pattern. The major scale is one such pattern. Its famil-
iarity makes it useful for learning intervals. Because scale degrees
are numbered, 1 through 8, the numerical designation of an interval
from "prime" (or "unison") to "octave" is immediately at hand. In
relation to the key note, any other scale degree may be labelled
according to its position in the scale. The second scale degree is
an interval of a 2nd in relation to 1; the 3rd scale degree is a
3rd, and so on. In the major scale pattern these intervals have
the following qualitative designations:
EXERCISE 1
"H" = to hear key tone
"I" = to hear note within key (no larger than P8)
"R" = ready to identify
"N" = next
"Alt/F" = to save score to disk
"+,-" = to advance or reverse pages
Listen to key tone. Listen to 2nd pitch. Identify with number and
quality. M for Major (M2,M3,M6,M7). P for Perfect (P1,P4,P5,P8).
INTERVAL TRANSFORMATIONS
Once the quality of a given interval is known, altering the actual
number of half-steps contained in the interval while retaining the
numerical designation transforms the quality. We have previously
learned, for instance, that a Major 3rd made smaller by a half step
(lowering the upper note, or raising the lower note) becomes Minor.
The intervals just learned are either Major or Perfect. Here are
the "rules of transformation" involving these intervals:
-Major intervals made larger by a half-step become Augmented
-Major intervals made smaller by a half-step become Minor
-Perfect intervals made larger by a half-step become Augmented
-Perfect intervals made smaller by a half-step become Diminished
-Minor intervals made smaller by a half-step become Diminished
-Diminished intervals made smaller by a half-step become Doubly-
Diminsihed
-Augmented intervals made larger by a half-step become Doubly-
Augmented
I like to put these formulas into a simple code, in which ">" (greater
than) means "made larger" and "<" (less then) means "made smaller."
M > 1/2 = A
M < 1/2 = m
P > 1/2 = A
P < 1/2 = d
m < 1/2 = d
d < 1/2 = dd
A > 1/2 = AA
Examining each of the intervals of the major scale with their half-
step transformations leads to the following qualitative changes:
This procedure reveals a difficulty we've encountered before:
enharmonic equivalents. The interval of an augmented second sounds
the same as a minor third. How is one to know when to use which,
or, if identifying intervals aurally, which is which? As with most
everything else within the realm of an art which is based upon the
manipulation of patterns, the answer is CONTEXT.
Understanding a given context requires knowledge of all the partic-
ulars constituting that context. This is often a life-long pursuit.
We can, fortunately, learn enough about the particulars early on
to at least begin understanding the larger context and thus make
decisions about specific usages, e.g. when an interval should be
an augmented 2nd versus a minor 3rd.
We have, in fact, seen this specific choice previously. Let's look
at it again as an illustration.
The largest context here is TONALITY. Within the system of tonality
there exists a pattern of pitches known as MINOR. There are three
essential configurations of minor. To maintain the integrity of the
pattern, the interval between the 1st and 3rd scale degrees in each
of the three configurations is minor. In one of the configurations,
namely HARMONIC MINOR, the 7th scale degree is raised from its
"natural" pitch according to the key signature. This results in
the interval of an augmented 2nd from the 6th to the 7th scale
degrees. In the context of tonality and within the context of the
minor scale, these two intervals are readily distinguished both in
spelling and in sound. The minor 3rd sounds like a minor 3rd; the
augmented 2nds sounds like an augmented 2nd, even though if heard
in isolation each would sound identical to the other.
INTERVALS BY HALF-STEPS
The more ways we have of approaching a subject the more prospect
there may be for understanding it. Another way of learning inter-
vals is to count half-steps. Taking the intervals examined through
the process of transformation above, we can construct a chart:
Half-Steps: 0 1 2 3 4 5 6 7 8 9 10 11 12
Interval: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8
A1 A2 d5 d7
The top row of intervals are the most common. The second row are
likely enharmonic equivalents. Other possibilities, like d2 as an
enharmonic equivalent for P1 are rare. We have also exluded from
this chart any spellings involving doubly-diminished or doubly-
augmented intervals, which are also uncommon. You should under-
stand how these may be derived, however.
INTERVALS IN THE MINOR SCALE
Another means of identifying intervals is to see and hear them in
the context of the minor scale. The intervals which are different
here from the major scale pattern are the minor 3rd, the minor 6th,
and the minor 7th.
EXERCISE 2
"H" = to hear key tone
"I" = to hear note within key (no larger than P8)
"R" = ready to identify
"N" = next
"Alt/F" = to save score to disk
"+,-" = to advance or reverse pages
Listen to key tone. Listen to 2nd pitch. Identify with number and
quality. M for Major (M2). P for Perfect (P1,P4,P5,P8).
m for minor (m3,m6,m7).
IDENTIFYING INTERVALS BY EAR
We have already had some practice in identifying intervals by ear,
in the context of either the major or minor scale. Another way to
hear isolated intervals is in association with the interval with
which a familiar tune begins. Here are some examples:
INTERVAL IDENTIFICATION
"H" = hear interval "S" = see 2nd note "R" = ready to identify
At 1st prompt, enter QUALITY (P,M,m,d,A) and NUMERAL (e.g. P5).
At 2nd prompt enter 2nd pitch (use bb for Double Flat, ## for Double Sharp)
("H" to hear again "S" to see 2nd pitch on staff or to see name of note)
"N" = next Alt/F = save score F =return to menu
RHYTHM.TXT
MUSICAL TIME
The word "rhythm" is derived from a Greek word meaning "measure,
measured motion." In a general sense, rhythm is the organization
of events into patterns which are "measurable." In music, rhythm
is the organization of time into patterns of relative loudness
versus relative quite.
Just as space is made perceivable and measurable by objects differ-
entiating one location from another, so time is perceived by
relationships of events: the heart contracting and expanding; the
sun appearing overhead or setting; the sound of a drop of water
falling, then another...and another... Of course, all these events
are spatial as well as temporal, and in our associating the two we
comprehend the elementary continuum which they form.
In less formal language, rhythm is the "swing" of things, the flow
of events as they relate one to another. Rhythm is felt to be
generative. It propels events, gives life to them, stimulates our
sense that the elements of life create patterns. Rhythm occurs at
all levels of our experience: the cycles of our lives, our heart's
pulse, the hours, days, weeks, and months into which we make the
divisions of our time; changing seasons; design of flowers, magne-
tic fields, snowflakes, galaxies... This may be why the rhythmic
patterns of sound in music have such a fundamental effect upon us.
Musical rhythm is the articulation of time, which resonates with
all the other rhythms we instinctively understand.
Rhythm, pulse, cycles of the seasons, the days,
our blood, life and death, questions of why and
how and what shaped into communal expressions
danced and sung and pounded out on drums; rhythms
and sounds and silences and gestures groping for
...meaning...
PRELIMINARY EXERICISE
The relatively longer versus shorter sounds of words have been
traditionally represented by = long and = short.
<1> = Enter text. At prompt, write a text no longer than one line
with a maximum of 20 syllables. <RET>
<2> = Enter number of syllables in text. <RET>
<3> = Enter "S" for Short, "L" for long, trying to produce an
approximation of the rhythm of your text.
<P> = Play rhythm.
[You may continue above sequence as many times as you like.]
PULSE
In music, time is measured by rhythm, an articulation of time
created by contrasts in relative loudness or intensity levels.
These contrasts cause pulses to occur, which may be either regular
(periodic) or irregular (nonperiodic).
Pulse is distinguished by difference between emphasis (oomph!) and
de-emphasis (pah) or any combination of these contrasting events:
<RETURN>
We know that this computer is not capable of varying loudnesses,
so the sense of emphasis and de-emphasis just heard was created by
varying pitch level. This reinforces the notion that we perceive
rhythmic patterns through differences.
We are most familiar in traditional music with regular pulse, the
articulation of time into periodic recurrence of stress versus
nonstress. But rhythm may also be nonperiodic--stress and
nonstress occuring irregularly--and much of the beautiful complex-
ity of primal music, as well as the exciting tensions of twentieth-
century Western music, is created by diverging from regularity of
pulse. <RETURN>
Pulse is just what we understand it to be in relation to our heart
beat: contraction and expansion (called systole and diastole).
Pulse is a basic aspect of music which occurs on several levels: a
piece of music as a whole is a kind of pulse in relation to the
relative silence which surrounds it; there is usually sectional
division of structure within the music which creates large-scale
pulses of contrasting intensity; and there is a basic pulsation of
sound which we commonly refer to as the beat, which is the
ongoing, second-to-second, propelling rhythmic force of music,
providing the basic sense of motion through time essential to
Western music. The beat is what we clap our hands or stamp our
feet to, and it is this level of rhythm for which we need to
formulate principles of notation in order to read and compose music.
PERIODIC RHYTHM
1. Beat Groupings
As has been stated, rhythm in music requires the alternation of
sound and silence, or relative change in loudness levels. A
metronome produces such an alternation. <RET> [any key to stop]
You have just heard a regular and constantly intense pulse juxta-
posed with silence. These pulses are undifferentiated. But if we
listen for any length of time, chances are that we begin to hear
them in groups, most likely in 2's or 3's. Listen again to a
series of beats and hear how easy it is to change from hearing
them in groups of 2 to groups of three. <RET> [any key to stop]
It is possible to hear these beats in larger groupings, of course,
but one must consciously add the beats up to 4 or 5 or 6. These
larger groupings are more likely heard as combinations of 2 or 3.
Undifferentiated beats are essentially as meaningless to us as
eternal silence or featureless space. Things seem to come alive
only when we perceive change. So it is with rhythm. Just as the
ear imposes patterns on consistantly intense beats, so when music
is composed, beats are traditionally organized into patterns by
emphasising certain beats and underplaying others. And just as
the ear most readily organizes undifferentiated beats into groups
of 2 and 3, so does composed music most readily deploy these beat
patterns. Longer groupings are possible, but just as with the
imagined groupings, them are most often heard as combinations of 2-
and/or 3-beat groupings.
A beat grouping, in the terminology of music, is called a measure.
In traditional notation, the measure symbol is a vertical line--the
measure or bar line. <RET>
A 2-beat measure is duple, a 3-beat measure triple, and so on. In
the heirarchy of rhythmic levels, the measure is the next duration-
al level up from the beat (the next longest). In other words, if
the first beat of a measure is listened to as such, it can be
heard as establishing a new pulse at a longer duration than the
beat: <RET>
If, on every 4th measure, say, the elements of a piece of music
were to combine to emphasize the first beat of that measure above
any other beat, yet another level would be established--a pulse
every 4 measures: <RET>
This latter grouping of four measures could be called a musical
phrase. How phrases are grouped determines the larger musical
structure. The sense of the different levels on which rhythm is
operating is important for understanding how patters of rhythms
may be built up to shape an entire piece of music, contributing to
a sense of forward motion, fullfilling a listener's expectations
or thwarting them.
EXERCISE 1
1. "H"= (H)ear tune which has periodic rhythm. You may listen to
tune as many times as you wish, using "H".
2. "R"= ready to enter the number of beats you perceive per grouping
(measure). You get only 1 chance here, since your options are so
limited!
3. At the next prompt, enter the number measures in a phrase.
4. "N"=(N)ext tune - up to 10.
5. "P"=(P)rior tune
Beat Divisions
Careful listening to the tunes in the above exercise will have re-
vealed a couple additional pieces of information. One is that most
phrases in common tunes are either 2 or 4 measures long. Another
is that what is perceived to be the main beat is sometimes divided
into shorter sub-pulses. In fact, the sub-pulse may occasionally
be confused with the beat, depending upon the tempo (beat rate).
In slow tempos, the sub-pulse may assume the role of the beat, in
the same way that at very fast tempos, the first beat of each meas-
ure may be heard as the beat. <RET>
Just as a measure may have, at its simplest ordering, either 2 or
3 beats, so a beat may be divided most simply into either 2 or 3
sub-pulses. If the beat is divided into 2 equal sub-pulses, the
division is simple; if into 3 equal sub-pulses, the division is
compound. <RET>
By varying combinations of divisions of the beat, beats, measure
lengths, and phrase lengths, a great variety of rhythms is possible.
In most traditional music of the Western European heritage, the
phrase length, measure length, and beat length remain fairly con-
stant, but even with the permutations possible at the sub-pulse
level, tremendous rhythmic interest can be generated.
EXERCISE 2
1. "H"= (H)ear tune which has periodic rhythm. You may listen to
tune as many times as you wish, using "H".
2. "R"= ready to enter the perceived divisional value of the beat:
SIMPLE, or COMPOUND>
3. "N"=(N)ext tune - up to 10.
4. "P"=(P)rior tune.
Non-Periodic Rhythm
Beats, measures, and beat divisions, as just discussed, are
categories of periodic rhythm, all relating to a steady rate of
pulsation. Non-periodic rhythm has no steady underlying pulse rate
and therefore does not require symbols for measures, beats, and
beat division. Or if these symbols are used, some factor must be
added which allows for a changing rate. If a composition is con-
ceived in which great flexibility of pulse is desired, it is easier
to accomodate this with newly designed symbols and instructions.
For instance, an accellerating series of articulations might be
represented by a "time" line, read from left to right, with dots
standing for the articulations. Thus:
. . . . . . . . . . . . ........
visually suggests a slow rate gradually getting faster. The inter-
pretation of this could not be precise, and it is this very issue
which underlies non-periodicity. To precisely control non-periodi-
city requires extraordinary notational as well as performing skill
and the results, for the most part, are no better than approximat-
ing the same effect with a less precise system. Hit <RET> to hear
interpretation of the above notation.
Levels of Rhythmic Activity
The beat is the most basic level rhythmic activity. It is at this
level that we tap our foot, snap our fingers...dance. Measures (beat
groupings) are a "higher" or "macro" level of rhythmic organization,
and beat divisions are "lower" or "micro" level.
Our perception of the level of rhythmic activity is entirely
dependent upon our perception of tempo (the rate of the beat,
measure in beats per minute). <RET>
This tune is in 3 beats per measure with simple division. Each note
heard is a beat length. At this slow tempo, we are likely to start
hearing the divisional value (half the beat length) as the beat. At
tempos slower than around 50 bpm, we would not tap our feet to the
designated beat, but to the division of the beat. <RET>
The measure at this tempo is clearly heard. At rates anywhere from
about 60 to 140 bpm, we have no difficulty responding by matching
toe-tapping to the beat. <RET>
Now the tempo is fast enough that we hear each beat as a division,
and we group these into "the beat". Since there were 3 per measure,
we now hear 3 units per beat and thus have compound division. Note
that you may now hear 2 of these units per measure.
THAT MOMENT IS ALWAYS CHANGING...AND WHILE WE ARE THINKING I AM
TALKING AND CONTEMPORARY MUSIC IS CHANGING.
LIKE LIFE IT CHANGES. IF IT WERE NOT CHANGING IT WOULD
BE DEAD, AND, OF COURSE, FOR SOME OF US, SOMETIMES IT IS DEAD,
BUT AT ANY MOMENT IT CHANGES AND IS LIVING AGAIN.
TALKING FOR A MOMENT ABOUT CONTEMPORARY MILK: AT
ROOM TEMPERATURE IT IS CHANGING, GOES SOUR ETC., AND THEN A
NEW BOTTLE ETC., UNLESS BY SEPARATING IT FROM ITS
CHANGING BY POWDERING IT OR REFRIGERATION
(WHICH IS A WAY OF SLOWING DOWN ITS LIVELINESS) (THAT IS
TO SAY MUSEUMS AND ACADEMIES ARE WAYS OF PRESERVING)
WE TEMPORARILY SEPARATE THINGS FROM LIFE (FROM CHANGING)
BUT AT ANY MOMENT DESTRUCTION MAY COME SUDDENLY
AND THEN WHAT HAPPENS IS FRESHER
(John Cage)
RHYTHMIC NOTATION
Brief Review - Periodic rhythm is organized into measures of 2, 3,
or more beats. Beats are divided into 2 or 3 sub-pulses. If a beat
is divided into 2 equal parts, the division is simple. If the beat
is divided into 3 equal parts, the division is compound.
Labelling Measures
Combinations of particular numbers of beats in a measure and how
each beat is divided determines how measures are labelled. A 2-
beat measure is called duple; 3-beat measure = triple; 4-beat meas-
ure = quadruple; 5-beat measure = quintuple, etc.
REMEMBER: If the basic beat is divided into 2, the beat is simple;
likewise the measure having simple division of the beat. If the
basic beat is divided into 3, the beat is compound; likewise the
measure having compound division of the beat.
Thus, a measure containing 2 beats (duple), which are divided into
2 sub-pulses (simple) is labelled duple simple. Duple and quadruple
measures, either simple or compound, are often difficult to distin-
guish with the phrase structure of the piece providing the only
basis upon which to make the distinction.
A measure containing a given number of beats (2,3,4, etc), each of
which is divided into 3 sub-pulses would be a duple, triple, quad-
ruple, etc., compound measure. Compound meter will be discussed
at greater length later.
SIMPLE DIVISION
Notation of the Beat
Its Division and Subdivision
The Time Signature
A system of noteheads, stems, flags and beams, has been standard-
ized to notate rhythm. These symbols are used to indicate relative
duration of tones. Any one of these may represent the basic pulse
or beat, depending upon the time signature, which is discussed
below. The notehead most commonly associated with the beat in
simple-division measures is called a quarter note:
"H" to hear above tune. Observe, by listening, that the quarter
note does indeed represent each basic pulse (at the given tempo).
There is also a regularly occuring accent every 2 quarter notes,
and thus a measure (or bar) line is drawn to distinguish one
grouping from another. The true meaning of the measure line is to
indicate where a point of natural stress occurs. The numbers at
the beginning of the tune comprise the time signature. In simple
division measures, the upper number indicates the number of beats
per measure, while the lower number represents the kind of note
designated as the symbol for the beat, in this case a quarter note.
(Please note that the time signature is not a fraction; there is
no divider line which would indicate that the upper number is to
be divided by the lower number, although the lower number does
represent a fractional durational value).
EXERCISE 3
H=Hear (to hear example) P=Prior (prior example)
N=Next (next example R=Ready (to input information)
+ = Next page, as usual - = Previous page, as usual
Use above keystrokes to hear examples (5) and answer questions.
Note that some of the tunes do not begin on the first beat of a
measure, but with a pickup beat (anacrusis, or upbeat); the pickup
is unstressed and has the effect of leading into a stressed beat,
as in the sentence "Oh I've been workin' on the Rail Road..."
pickup
"down" beat - 1st of measure
The quarter note is normally divided into two equal sub-pulses:
The note with the single flag is an eighth-note. The quarter note
thus has a simple division, and so the above tunes are either
duple simple, triple simple, etc., depending on how many quarter
notes (beats) there are in the measure. The further sub-division
of the quarter note is indicated below.
(Note: the beam joins two or more flagged notes together into a
unit; as a general rule, it is a good idea to beam all possible
single-beat units, so that they are perceived as a unit; sometimes
4 eighth-notes are beamed together as a 2-beat unit, and sometimes
8 sixteenth-notes).
Most tunes, including ones above, do not have articulations on
every beat. There are inevitably times when one wishes to sustain
a tone for longer than a quarter note (or a single beat). The quar-
ter note bears the same relations to longer-duration notes that
division of the quarter note bears to it. Two quarter notes equal
a half-note; two half-notes equal a whole-note, etc., as shown below.
You've probably noticed by now that in several of the tunes cited
as examples, not all of the notes bear the standard durational
relationship of 1:2 (quarter to eighth, eighth to sixteenth, etc),
but rather have a 1:1.5 ratio, and that these notes have dots
after them. The dot following a notehead lengthens the duration of
the note being dotted by half again as much. Put more simply, a
dot lengthens a note by half the length of the note being dotted.
A new symbol, the tie, is employed in the following examples. It
joins notes together as single durations.
RESTS
For every notehead value that may exist, there is a corresponding
rest value (no sound). These are exhibited below.
Dots are added to rests in the same way as they are to noteheads,
to lengthen the duration of the rest by half the time of the rest
dotted.
Writing and Reading Rhythms
Whatever the durational values of notes or rests may be, they must
add up to the number of beats designated by the time signature.
Among the very first things that needs to be dealt with when
deploying rhythm is thus to make sure there are the proper number
of durations in each measure. A second "rule of thumb" regarding
rhythmic notation is to try, in the notation, to show where each
beat is. In 4-beat measures, another level of organization may be
perceived: 2-beat units. These issues will become clearer as we
proceed.
As with any aspect of music, intellectual comprehension of funda-
mentals is not the same as performance, "doing" the music, either
composing or performing. Performance necessitates a body/mind co-
ordination that demands practice. While some people may have an
inate sense of rhythm, anyone, with practice, can improve her or
his rhythmic skills. Also, the reciprocal processes of writing and
reading rhythms within the traditional system demands experience.
The main issue in both the notating and reading of rhythms is to
determine where the beat is, and to feel this beat in an internal-
ized physical and consistent way. The biggest problem I have seen
in working with students as composers is that few have approached
understanding rhythm in a disciplined enough way to achieve a
sense of feeling accurately the rhythms they are imagining. So
when it comes to writing their pieces, rhythmic notation is often
sloppy. This applies also to beginning music students who perhaps
respond instinctively to rhythm, but who have not had to think
much about what they are actually responding to. Because of this,
I generally recommend the use of a metronome when learning and
practicing rhythms, to assure consistency. Flexibility, that marv-
elous ability to bend and shape music spontaneously and individu-
ally, can come after one has acquired skill in maintaining rhythms
precisely.
The first premise in writing and reading rhythms is: feel the beat
consistently. Then: sense the beat groupings, the number of beats
that may comprise a measure. Next: sense the divisions and subdi-
visions of the beat. And always remember: keep everything regular
(a consistent, periodic beat). It is much better to start very
slowly, maintaining accuracy and relative proportion of note
values, than to attempt to do something too quickly and thereby
continually alter the duration of the given beat.
In notating rhythms, to state it from a slightly different angle,
the prinicple is: make absolutely clear where each beat is. This
is done by grouping rhythms by beats, and with a few exceptions,
this means in such a way that each single beat and its division
and subdivision units are readily seen.
COMPOUND DIVISION
Listen to the following examples (5), with particular attention
directed to the number of beats per measure and the beat division.
You will observe above standard 2 and 4-beat measures with more or
less obvious tripartate beat division. These tunes are thus in com-
pound meters. Because the notehead value designated as the beat
needs to be divided into 3, compound meters use dotted values for
the beat. And because a dotted note is the beat, the time
signature itself cannot state directly the true number of beats in
the measure.
The most common notehead value for a beat in compound time is the
dotted quarter. A two-beat measure would contain two of these
units; a four-beat measure four of them, and so on. But convention
doesn't allow us to notate this thusly:
While this might be logical, the given system rather calls for us
to notate compound division meters as if they have three times as
many beats, with the upper number thus representing the number of
divisions in a measure and the lower number indicating the kind of
durational value which represents the division rather than the
beat. A two-beat compound-division meter, using the dotted quarter
as the beat, is thus, as represented in the common folk tune "The
Irish Washerwoman:"
Because the tempo is sufficiently fast here, rather than hearing
each eighth-note as a beat, we hear the eighth-notes grouped into
threes. All compound division meters have an upper number
divisible by 3 (except 3 itself, which is usually grouped with the
simple-division meters). Dividing the upper number by 3 reveals
the number of beats in the measure, while adding 3 of the lower
number values together reveals the notehead value of the beat.
The relationship of the dotted quarter to its division and subdi-
vision is as follows:
Note that the subdivision of a compound beat traditionally divides
the division into 2, not 3.
The relation of the dotted quarter to larger duration notes in
compound meters is:
EXERCISE
H=Hear N=Next P=Prior R=Ready A=Answer +,- per usual
H.= dotted half Q.= dotted quarter Q= quarter
E.= dotted eighth E= eighth S = sixteenth
Listen to example. When ready, hit "R" and enter durational values.
(4-beat pattern is preceded by 4 beats to establish tempo).
BORROWED DIVISION
If rhythmic structures contain sub-pulses that are half the length
of the beat, the meter is simple; if the sub-pulses are a third
the length of the beat, the meter is compound. For variety, in
simple-division meters one may occasionally employ a compound
division unit, and vice-versa. This is technically considered to
be a "borrowing" of the common division from the "opposite" config-
uration, simple/compound. Such borrowings must be specially marked.
When in a simple-division meter, the compound "borrowing" is
called a triplet. The comparable unit in a compound-division meter
is a duplet.
SCALES.TXT
Half-Steps/Whole-Steps
With sharps and flats (accidentals), it is possible to notate all
twelve distinct half-step increments within the octave. To do this
however, it is necessary to know that basic notes on the 5-line
staff are not all equi-distant from one to the next. If there are
only 7 basic notes, it is obvious that the distance between some
of these adjacent pitches must be more than a half-step. The dis-
tance from one frequency (pitch) to another is known as an inter-
val. The half-step is the smallest interval in twelve-tone equal
temperament. By custom, only two adjacent pairs of pitches are
separated by an interval of a half-step. These are the intervals E
to F, and B to C, in whatever octave range they appear:
Two half-steps comprise a whole step. All of the basic-note inter-
vals of adjacent pitches, except those mentioned above (E to F and
B to C) are whole-steps.
The way a note appears on the staff ("basic," with sharp, flat,
natural, etc) is its spelling. Given twelve pitches within the
octave, and the flexibility of applying accidentals to basic notes,
it's readily apparent that the same pitch can be spelled in differ-
ent ways. For instance, D-sharp and E-flat are the same frequency,
but are spelled differently. The reason for these alternative
spellings will become clear in later discussion of pitch patterns.
Tones which sound the same but which are spelled differently are
enharmonic equivalents.
The Chromatic Scale
When all 12 notes within an octave are sounded as a scale, the
distance between each adjacent note is a half-step. This scale has
the name chromatic, in analogy to the word chroma, meaning color.
In spelling an ascending chromatic scale, it is customary to spell
the appropriate half-steps with sharps:
Conversely, a descending chromatic scale is spelled with flats:
Half-steps which result from accidentalizing a basic note are
chromatic half-steps. Do not confuse this term with chromatic
scale. Chromatic half-steps may occur within a melody without
being part of a chromatic scale. <RET>
Half-steps that are chromatic involve no change of line or space.
If adjacent half-steps are from a line to a space,or from a space
to a line, they are diatonic: <RET>
EXERCISE 1
The screen below displays the basic note pattern for the major scale.
The pattern is WWHWWWH. Your assignment is to construct this same
pattern beginning at each successively higher basic note.
N = next pattern to enter. Beginning pitch is shown.
P = prior pattern.
R = ready to enter. At prompt, enter each note name, with necessary
# or b (lower case B). Leave no spaces and enter in either
lower or upper case. 3 chances allowed on each pattern.
H = hear notes.
S = see answer.
PITCH PATTERNS: SCALES
The previous exercise introduced you to a pattern of pitches
referred to as a scale (from scala, "ladder"). We've all heard
scales practiced by budding musicians and know how tedious, boring
and unmusical they are. While practicing scales on an instrument
is a useful means of acquiring technical proficiency, we rarely
hear scales as componants of actual pieces of music. From the point
of view of analysing music, scales are abstracted "pitch sets"
which help us to better understand pitch organization.
For the past 400 or so years, composers working within traditions
developed in Western Europe have focused upon a means of organizing
pitch materials known as "tonality." The fundamental notion of
tonality is that within any given piece, there is a single tone
which functions as a kind of center of gravity. This single tone
is called the TONIC. The other pitches (within the framework of
the 12 pitches of equal temperament) all have a tendency to sound
unstable (up in the air?) in relation to this tone.
Tonality is an issue we will continue to examine. For the moment,
however, we'll look at it in the context of some particular scalar
patterns. Look at and listen to the following tune, the first
segment of the familiar "classic," Somewhere Over the Rainbow.
Given the conditioning we've experienced by way of hearing tunes
in the tonal system, you should have no difficulty hearing the
beginning pitch, C, and especially the same pitch used at the end,
as the "center of gravity" among all the other pitches. This is
the tonic note, one which I sometimes refer to as the generating
tone. If we write this pitch down on a staff and then deploy the
other pitches in ascending order 'til we reach the octave, we have
the pitch set from which this piece is made. This is, as you know,
a scale.
This particular pitch set should sound (and look) familiar. It is
one of the most common pitch sets of tonal music, known as a MAJOR
SCALE. The whole-step, half-step pattern, in ascending order, is:
WWHWWWH.
EXERCISE 2
In the first window, you can cycle through 5 examples of sections
from common tunes which use major pitch sets. You can hear these
tunes by hitting "H", cycle through them by hitting "N" (for next)
and "P" (for prior). Use the Up/Dn Arrows to set the cursor on the
line or space (in the second window) of the pitch you deduce is
the tonic note in the given example. Hit <ENTER> when ready...
If you are right, the rest of the pitches will appear in
ascending order to form a scale. Examine this scale to determine
indeed, consists of the correct pattern for MAJOR. If you're not
right, you get one more chance before being given the answer.
PREVIEW: INTERVALS
Understanding scale patterns can be enhanced by a preliminary
expansion of knowledge of intervals. A musical interval is the
frequency difference between one pitch and another. In traditional
terms, this difference is expressed as the numerical count from
one letter-name to another, including the starting and ending
letter-names in the count. Thus, the interval from C to D is a
2nd, C to E a third, and so on. And since the same frequency may
be spelled in more than one way (enharmonic equivalents), there re-
sults different varieties of the same numerically-named interval.
The interval C to E is, numerically, a 3rd. So is the interval C
to E-flat. The former contains 4 half-steps, while the latter
consists of only 3 half-steps. C to E is a MAJOR 3RD. C to E-flat
is a MINOR 3RD. The interval C to D is a 2nd. So is the interval C
to D-sharp. C to D contains 2 half-steps (thus a whole-step); C to
D-sharp is 3 half-steps. C to D is a MAJOR 2ND. C to D-sharp is
an AUGMENTED 2ND. This latter interval sounds exactly the same as
a minor 3rd, but is spelled differently. Thus these two intervals
are enharmonically equivalent.
A musical interval has two designations. One is numerical. The
other is qualitative. We will discuss these designations in
greater detail later. For now, there is one interval in particular
which is most helpful in further understanding scales. It is the
PERFECT 5TH. The interval from the first scale degree in the major
scale we have been considering (the TONIC, or generating tone) and
the 5th scale degree (called the DOMINANT), is, numerically, a 5th.
This can be readily deduced in the C-Major scale, for instance, by
counting C-D-E-F-G, a count of 5. IN ANY MAJOR SCALE, THE QUALITY
OF THE INTERVAL OF A 5TH FROM THE TONIC TO THE DOMINANT IS
PERFECT. The perfect 5th contains 7 half-steps.
The perfect 5th is a useful interval for examining certain relation-
ships among scale patterns. These relationships are the fundamental
building-blocks of tonality: KEYS. Any tonal piece of music is
said to be in a particular key. The key is the name of the
generating tone. For example, if a tune's tonic note is C, and the
pitch pattern is Major, the key is C-Major. As we have seen, the
key of C-Major needs neither sharps nor flats. It is the "basic-note"
pattern for the major scale.
In EXERCISE 1 you "built" a major scale beginning on the basic
note G. You would have determined that this scale necessitated
raising the F a half-step (F#) to replicate the major-scale
pattern. You also made a major scale beginning on D. The D-Major
scale uses both an F# and a C#. Now you can begin seeing the
pattern in which sharps are added. If you "project" upwards by the
interval of a perfect 5th you arrive at the next "sharped" key.
Each successively arrived at key adds a sharp, and that sharp is
added to the 7th scale degree. This pattern forms the basis for
what is known as the CIRCLE OF 5THS, a commonly-used device for
learning key relationships.
We do run into a problem in this circle of 5ths. Any keys beyond 7
sharps need double-sharps. This is very cumbersome. The remedy is
to respell a key using flats. This usually is done when the use of
6 sharps is reached (F#-Major) because the respelling produces a
key with 6 flats (Gb-Major):
Now, if we continue on with the circle of 5ths, spelling scales
with flats, we note that adding a sharp to the 7th scale degree is
equivalent to elliminating a flat. The circle is completed when
all flats are gone, and the key of C-Major is reached.
Another way to approach the keys which employ flats is to project
downwards by perfect 5ths. From the key of C-Major, the next key,
with one flat, is F-Major. You spelled this as the only scale in
EXERCISE 2 with a flat. To make the correct whole-step, half-step
pattern for major beginning on F, the 4th scale degree has to be
flatted. F-Major therefore has one flat, Bb. A perfect 5th down
from F is Bb. The new flat added is Eb. The counterclockwise
series in the circle of 5ths adds flats:
MAJOR KEY SIGNATURES
If a piece of music is tonal and therefore cast within any one
given key, it will consistently use a given number of sharps or
flats. Rather than employing accidentals, having to write the
sharp or flat every time it is called for, the sharps or flats are
written at the beginning of each staff line as a key signature.
The placement of the sharps or flats follows a consistent pattern:
EXERCISE 3
Enter signature for given major key.
ARROW KEYS = move cursor to proper location
# (or 3)=SHARP b (or B)=FLAT E=ERASE (reverse order)
ENTER = when you think you have proper signature
M=more (continue as long as you wish)
Alt/F = to store score to disk +,- = next or prior page
MINOR SCALES
Examine the following tune to determine its key tone:
All of the clues discussed with regard to the major scale lead to
the conclusion that this tune's tonic note is A. Deployment of the
other pitches to generate a scale produces a basic-note pattern
with a different configuration from major: WHWWHWW.
This pattern is, in the system of tonality, the complement to major:
MINOR. The most crucial difference between the two patterns is the
quality of the 3rd scale degree. The interval between the tonic
note and the 3rd in major contains 4 half-steps and is referred to
as a Major 3rd. The distance from the tonic to the 3rd scale
degree in minor consists of only 3 half-steps and its quality is
labelled minor.
There are traditional learned associations with the difference in
quality between these two patterns. Major scales, keys, and the
major 3rd are "bright," "cheerful," "open," "up-lifting." Minor
scales, keys, and the minor 3rd are "sad," "poignant," "doleful,"
"closed." Whatever associations you may have, the difference in
sound is distinct. If you perceive sounds in terms of colors (syn-
esthesia: one type of stimulus producing a secondary sensation),
you may even "see" different colors in association with major
versus minor.
Transposition of the minor scale pattern via the circle of 5ths
reveals the respective key signatures for minor. At the same time,
another method for deriving minor signatures is exhibited. If C
major has no sharps or flats in its signature and A minor is
similarly disposed, C major and A minor have the same signature.
Since A is the 6th scale degree of C major, you can deduce that
every major key has a related minor which shares its signature,
found on its 6th scale degree. This is a common way for learning
minor keys, in association with major. Cycling through the circle
of 5ths corroborates this relationship. Note that it is now the
2nd scale degree which is raised in each successive scale, rather
than the 7th.
Again, we encounter the problem of needing to use double sharps if we
were to continue with the above process. Conversion to flats when
6 sharps are reached will result in the enharmonically equivalent
scale of Eb minor.
MINOR KEY SIGNATURES
Minor key signatures have the same patern as major. Hit "B" (b) to
see the flat keys, "#" to toggle to sharp keys.
EXERCISE 4
Enter signature for given minor key.
ARROW KEYS = move cursor to proper location
#(or 3)=SHARP b(or B)=FLAT E=ERASE (reverse order)
ENTER = when you think you have proper signature
M = more (continue as long as you wish)
Alt/F = to store score to disk +,- = next or prior page
MINOR VARIATIONS
Here's an English folk song which is clearly in minor. Note,
however, the use of accidentals.
For reasons having to do with the sense of direction of a melodic
line and underlying harmonic implications, strict adherence to the
notes called for by the signature of a minor key is rare. There
are two common variants of the minor scale pattern. The example
above refers to both.
If a tune is in minor and the melodic contour contains a segment
which ascends up through the 6th and 7th scale degrees to the
tonic note, it is common for these pitches to be raised a half
step. If the melodic contour, however, is descending through the
7th to the 6th and on downwards, it is standard practice for these
pitches to be in their "key signature" positions. "The Three
Ravens" has a measure demonstrating the latter, and the last
measure exhibits the trait of the 7th raised a half-step to lead
to the tonic.
The scale abstracted from this pattern is called the melodic
minor. As a scale pattern, it raises the 6th and 7th a half-step
from "key signature" position when ascending, while lowering them
back down to "correct" spelling when descending.
Very few melodies actually appear having the exact pattern of
ascending 6th-7th-tonic, then tonic-7th-6th, but a variety of uses
could be construed as fitting the basic framework of melodic
minor. Even without this specific pattern, however, unless the
melody is a direct linear descent through the 7th to the 6th, the
7th scale degree is nearly always raised, especially if this note
is going to be harmonized by the Dominant triad (the chord
constructed from the 5th scale degree of the given key) because
this chord sounds best (in the system of tonality) if it is Major.
In a minor key, the dominant triad would "naturally" be minor, so
its "third" needs to be raised a half-step to make the major chord.
The "third" of the Dominant triad is the 7th scale degree of the
given key.
Abstracting from a minor key the pitch set which uses a raised 7th
results in the scale pattern called harmonic minor. This pattern
is the same ascending as descending, with a 7th scale degree
raised a half-step. As you can hear, this scale has a distinctly
Eastern European "flavor," which results from the augmented 2nd
interval between the 6th and 7th scale degrees. As a strict
melodic contour, this is not common to most traditional minor
tunes. In this instance, one can understand directly how it is
that a scale pattern is an abstraction.
There are, then, three common variants of minor pitch sets. The
one based upon the key signature, without accidentals, is called
natural minor. It is the "model" scale from which the melodic and
harmonic minor are derived.
EXERCISE 5
"P"= play scale. "M"= more. Alt/F= save score.
"R"= ready to identify quality. At prompt, enter MAJOR, NATURAL, MELODIC,
or HARMONIC (upper or lower case). At 2nd prompt, spell scale in
ascending and descending order, USING CAPS with no spaces
(e.g. ABCDEF#G#A-AGnFnEDCBA for "A Melodic Minor" - note hyphen
between up/down and lower-case N for "natural"). "##" for double sharp.
"M"= more. Alt/F= save score.
Directory of PC-SIG Library Disk #3471
Volume in drive A has no label
Directory of A:\
RHYTHM MNU 1312 7-07-89 8:07a
RHYTHM TBC 89148 2-17-90 9:11p
RHYTHM TXT 25490 1-22-90 3:54p
SCALES MNU 973 8-17-89 11:37a
SCALES TBC 94799 3-31-90 4:08p
SCALES TXT 18193 10-03-89 4:04p
INTERVAL MNU 679 7-10-89 10:37a
INTERVAL TBC 51253 1-22-90 1:40a
INTERVAL TXT 7531 7-10-89 10:13a
ANALYSIS TBC 30451 1-22-90 7:38p
ANALYSIS TXT 12013 7-10-89 5:24p
11 file(s) 331842 bytes
25600 bytes free