Home of the original IBM PC emulator for browsers.
[PCjs Machine "ibm5150"]
Waiting for machine "ibm5150" to load....
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
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
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.
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.
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