LESSON 1 PITCH NOTATION AND INTERVALS

FUNDAMENTALS I 1 Fundamentals I UNIT-I LESSON 1 PITCH NOTATION AND INTERVALS Sounds that we perceive as being musical have four basic elements; pitch, loudness, timbre, and duration. Pitch is the relative highness or lowness of a sound, and can be expressed as the frequency, or speed, of its vibrations. Frequency is expressed as the number of times a sound wave completes its cycle in a second. Cycles per second are expressed as Hertz, or Hz. The faster an object (string, drum head, column of air) vibrates, the higher we perceive its pitch. In musical notation, we identify pitches by letter names and octave classifications. (e.g. 440 Hz = A4). Loudness is the intensity of a sound, and is expressed as the amplitude of its vibrations. Intensity is measured in decibels, or db, which indicates the amount of displacement caused by a sound wave. Loudness is notated in music by a system of relative dynamics, ranging from pppp to ffff. Timbre is the tone quality of a sound. Timbre allows us to distinguish the sounds of the various types of instruments. We have no system of measurement for timbre, but it can be analyzed through its wave pattern or through the relative loudness of its fundamental pitch and its constituent overtones. When a note is played on a musical instrument, a series of higher frequencies is produced in addition to the fundamental frequency. In a musical sound, these higher frequencies (also known as harmonics) are in whole-number ratios to the fundamental. For example, the first overtone vibrates twice as fast (2:1) as the fundamental. The second overtone vibrates three times as fast (3:1) as the fundamental, and one and a half times as fast (3:2) as the first overtone. The presence and relative strengths of these overtones differ among various instruments. Example I.1.1: The overtone series on C2 Copy time.review Duration is the amount of time a sound lasts from its initial attack to its end in silence. Durations are expressed in musical notation through note values, which are relative to tempo, or the speed of the music. The character of a sound over time also contributes to our perception of tone quality. This can be expressed through a sound envelope, which describes the relative loudness (or intensity) of a sound through

2 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I The commonly used system of notation using a five line staff, accidentals, and various clefs to indicate specific pitches is not particularly efficient. Because we use only seven letter names (ABCDEFG) and we have twelve pitches in each octave, the lines and spaces on the printed page do not indicate the actual distance between the various notes of our chromatic scale. It can be helpful to use the piano keyboard as a visual aid in relating the lines and spaces of the various clefs to note names and intervals between notes. It is necessary not only for the performer, but for the conductor, teacher, and composer/arranger to develop fluency reading and writing in all four of the clefs in common usage. Example I.1.2 Accidentals are applied to notes to raise or lower natural, or white key pitches. A flat (@) lowers the pitch by one half step, while a sharp ( ) raises the pitch by one half step. The natural symbol ($) is used to cancel an accidental that has been previously applied. The double flat (@@) lowers the pitch by two half steps, and the double sharp (*) raises the pitch by two half steps. Notes that share the same letter name, but are an octave or more away from each other are said to be in the same pitch class. Notes that are in enharmonic relation to each other (notes that sound the same, but are spelled differently, such as A-flat and G-sharp) are also in the same pitch class. To distinguish octaves, a system of numbering is used, beginning with the lowest octave on the piano. Octaves begin on C and end on B. As a reference, remember that the note known as middle C is C4. An interval is the distance between two pitches. In tonal music theory, this distance is measured by quantity and quality. The quantity of an interval is expressed as an ordinal number, such as second, fifth, or sixth. To determine the quantity of an interval, count the number of lines and spaces, or letter names, between the pitches. The starting pitch and the ending pitch are included in the count. For example, the distance from E up to B is E-F-G-A-B, or five, so the interval between these two pitches is a fifth.

FUNDAMENTALS I 3 Example I.1.3 Interval inversion involves changing the octave of one of the pitches, so that what was the top note is now the lower note and vice versa. Intervals related by inversion sound very similar to each other, because the same two pitch classes are heard in both. When an interval is inverted, the quantity numbers of the two intervals always add up to nine (seconds invert into sevenths, fifths into fourths, etc.). Example I.1.4 Compare the two intervals in Example I.1.5. As we have already seen, the interval from E up to B is a fifth. The interval from B up to F is also a fifth, but if you play or sing these two fifths, you will notice that one is smaller than the other. Therefore, we must distinguish the difference by labeling the qualities of the two fifths. The quality of an interval gives a more precise size than quantity alone, and is expressed using the terms major, minor, perfect, diminished, and augmented to qualify the exact size of the interval. Example I.1.5 Intervals are divided into two main categories, perfect and imperfect. The perfect intervals are unisons, octaves, fourths, and fifths. Imperfect intervals are seconds, sevenths, thirds, and sixths. The quality of an interval in the perfect category can only be diminished, perfect, or augmented (i.e. there is no such thing as a major fifth ). If an interval is one half-step smaller than perfect, it is diminished. If an interval is one half-step larger than perfect it is augmented. The quality of an imperfect interval can only be diminished, minor, major, or augmented (i.e there is no such thing as a perfect third ). Minor intervals are one half-step larger than diminished, major are one half-step larger than minor, and augmented are one half-step larger than major. Just as pitches can

4 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I be enharmonic, intervals can be enharmonic as well. Two intervals that sound the same but are spelled differently are enharmonic with each other. When an interval is inverted, the quality will invert this way: perfect into perfect, diminished into augmented (and vice versa), and minor into major (and vice versa). The number of half steps contained in an interval and its inversion will always add up to 12. interval half steps inverts into half steps interval P1 0 12 P8 m2 1 11 M7 M2 2 10 m7 m3 3 9 M6 M3 4 8 m6 P4 5 7 P5 A4 6 6 d5 Counting the number of half steps in an interval to determine its quality is effective for smaller intervals, but can be very time-consuming when used for larger intervals. It is essential to develop the ability to recognize and spell intervals immediately, without pausing to think or count. The following method will help you to memorize intervals, making it easier to master the more complex musical concepts that will be studied later. All of the natural, or white key unisons and octaves are perfect. If both notes have no accidental or the same accidental, then the interval is perfect. If the accidentals do not match, the interval is either larger than perfect (augmented), or smaller than perfect (diminished). Example I.1.6 Of the seven natural fifths, all are perfect but one. The fifth from B to F is naturally smaller than perfect, and is therefore a diminished fifth. With any fifth, if the accidentals match it is perfect, unless it involves both B and F. The fifth from B to F can be made perfect by either raising its top note (F ), or lowering its lower note (B@). Because fifths invert into fourths, and perfect inverts into perfect, all of the natural fourths are perfect with the exception of the fourth from F to B. The diminished fifth B to F inverts into an augmented fourth.

FUNDAMENTALS I 5 Example I.1.7 The qualities of seconds and sevenths are imperfect, and can be major, minor, diminished, or augmented. The quality of a second is easily calculated based on the number of half steps found between the two pitches. Seconds that have only one half step between them, such as B-C, E-F, or A@-B@ are minor seconds. If two pitches are one letter name apart and have a whole step between them, such as C-D, E-F$, or A@-B@, they are major seconds. If a second is larger than major (3 half steps), it is augmented. The interval of an augmented second is enharmonic with a minor third, but is spelled differently because it has a different function in the tonal system. When heard out of context, these two intervals are identical, but they sound very different from each other when placed in the context of a scale. Example I.1.8 To determine the quality of a seventh, first invert it into a second and find the quality of the second. Then invert it back into a seventh, remembering that a major second will invert into a minor seventh and vice versa. Example I.1.9 Thirds and sixths are also imperfect, and can be major, minor, diminished, or augmented. A minor third has three half steps between its bottom note and top note, while a major third has four half steps. Diminished and augmented thirds do not occur in major or minor scales, and are relatively rare in tonal music. Because sixths are large intervals, it is easiest to invert them into thirds to determine the quality, once again remembering that minor inverts into major and major inverts into minor.

6 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I Example I.1.10 Because tonal harmony is based upon tertian chords (chords built on thirds), memorization of all major and minor thirds is an essential first step to spelling chords and becoming facile with harmony. A student who has mastery of thirds will be much better equipped to work with the harmonic material of tonal music. Intervals larger than an octave are called compound intervals, while those that are an octave or smaller are called simple intervals. The addition of an octave (or more) to the quantity of an interval results in numbers that can be misleading. The interval one step larger than an octave is a ninth, which is the compound version of a second. The compound version of a third is a tenth, a fourth becomes an eleventh, and so on. In musical math, seven is added to a simple interval to account for the octave, which is actually eight. The quality of a compound interval is identical to that of its simple equivalent (e.g. the tenth from G3-B@4 is minor, just as the third from G3-B@3 is minor).

FUNDAMENTALS I 7 LESSON 2 RHYTHM AND METER A solid understanding of how time is organized in music, and how that organization is notated and performed, is essential for musicians who wish to develop a high level of ensemble playing, where playing at the correct time is arguably more important than playing the correct pitch. Inaccurate performances are often caused by a lack of understanding of the notation used for rhythm and meter. Time in music is organized by regular pulses, or beats, which are grouped together and divided. The grouping together of these pulses results in meter. The way in which the beat is divided distinguishes the meter type. We naturally tend to group large numbers of beats into smaller units to aid in comprehension and memory. The most common groupings are into twos (duple), threes (triple), and fours (quadruple). Larger numbers of beats, such as five or six, tend to be grouped into twos (2+3, 3+3) or threes (2+2+1, 2+2+2), rather than fives or sixes. Beats are also divided and subdivided into smaller units. At the highest level, beats tend to be divided into two (simple) or three (compound) equal parts. These divisions can then be further divided into subdivisions. Listen to a performance of the theme from the third movement of Mozart s Piano Sonata No. 11 (K. 331) without looking at the score in Example I.2.2. While listening, clap or tap along and try to determine how the beats are grouped (groups of 2, 3 or 4?) and how they are divided (in half 2 parts; or thirds 3 parts). Now listen to a performance of the second movement (Rondo) from Beethoven s Piano Sonata No. 19 (Op. 49, No. 1) without looking at Example I.2.3 and do the same. How is the organization of time (groups and division of the beat) different between these two examples? How is it the same? Now look at the printed examples to see how these two pieces of music are notated, and note the difference of meter signature. In musical notation, groups of beats are marked by measures, or bars, with barlines indicating where one group ends and another begins. The number of beats in a measure will typically be two, three, or four, with exceptions possible, but not common. The way in which a beat is divided determines how a meter is indicated by its meter signature (or time signature). The music of Example I.2.2 is notated in, which is a duple simple meter. This type of meter is used when the beats are grouped in twos (duple) and divided into two equal parts (simple). In signatures for simple meters, the top number indicates the number of beats in each measure and the bottom number indicates which note value receives one beat. In, there are two beats in each measure and the quarter note represents one beat. Other common simple meters are triple simple (e.g.,, ) and quadruple simple (e.g. $, $, $ ). Note durations are indicated by various note values, which are relative to each other and to the prevailing tempo, or speed of the music. Example I.2.1 shows common note values in relation to each other. A dot immediately following a note head increases the duration of that note by one half. Example I.2.1

8 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I The Beethoven example (I.2.3) also uses groups of two beats for a duple measure, but those beats are divided into three equal parts. This music is in a duple compound meter, and is notated in ^. In contrast to notation of simple meters, the meter signature used for compound indicates the number of divisions in each measure and which note value receives the division of the beat. A duple compound meter has six divisions of the beat, which when added together make up two beats. In ^, the eighth note equals one division of the beat, and we add three eighth notes together to determine that a beat lasts for the duration of a dotted quarter note. To determine the number of beats in a measure of compound time, the top number in the meter signature is divided by three (6 3=2; 9 3=3; 12 3=4). In addition to duple compound, common compound meters include triple compound (e.g. (, y, ) and quadruple compound (e.g. W, Wy, W ). To determine which note value receives one beat, three divisions are added together. The beat in a compound meter will always be a dotted note. One way to think about the difference between beats and divisions is to determine how music in a particular meter will be conducted. A piece of music notated in ^ and performed at a moderate or fast tempo will be conducted in two, not six. Example I.2.2 Mozart Piano Sonata No. 11 in A Major K. 331, third movement mm. 1-8

FUNDAMENTALS I 9 Example I.2.3 Beethoven Piano Sonata No. 19 in g minor Op. 49 No. 1, second movement mm. 1-8 The interpretation of a meter signature is dependent upon the tempo of the music. Very fast or very slow tempos often require a conducting pattern that falls outside of the standard duple, triple, and quadruple meters. For example, when a piece in is marked at a very fast tempo, it is usually performed and conducted as a single compound meter with one beat per measure divided into three parts. The third movement of Beethoven s Symphony No. 7, marked Presto, is an example of this meter type. If the top number of a meter signature is larger than four and is not divisible by three, the meter is either irregular or complex. Irregular meters are those that have more than four beats, such as a slow or moderate % (e.g. Mars, the Bringer of War from Gustav Holt s The Planets) or a slow % or ^ (e.g. certain measures of Barber s Adagio for Strings). Complex meters have beats of unequal lengths within a measure, with some divided into three parts and others divided into two parts. An example of a complex meter is %, which is grouped into two beats (duple complex), either 2+3 or 3+2. The meter & is an example of triple complex because it has three beats that are unequal in length (3+2+2; 2+3+2; 2+2+3). * grouped into three unequal beats as a triple complex meter is often used in contemporary popular music as a contrast to the more standard $ grouping of eight eighth notes. Example I.2.4

10 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I LESSON 3 MELODY AND PHRASE STRUCTURE Example I.3.1 Mozart Piano Sonata in A Major K. 331, first movement mm. 1-8 Play or sing the melody presented in Example I.3.1. Typical of melodies from the Classical era, this exhibits a regular, balanced structure, as well as a clear sense of direction and arrival at the end. On closer examination, you will also find that the melody is built from small musical ideas to construct a larger musical statement. Similar to the way in which we use elements in spoken language, such as words and phrases, to build longer sentences and paragraphs, music is typically made up of smaller structures that are put together to produce a complete musical statement. This melody consists of two four-bar phrases, each of which ends with a cadence, or a pause that acts much as punctuation in language. The B at the end of m. 4 is the end of a phrase, but sounds as if it must continue because it is not resolved. This non-final cadence acts like either a comma or a question mark, because it does not end on the tonic pitch of the key. The sense of resolution comes with the final cadence on the tonic pitch A at the end of the second phrase in m. 8. The antecedent (first) and consequent (second) phrases together form a period, which is a group of two or more phrases that ends with a final cadence. In this example the period is parallel, because the two phrases begin with identical material but end with different cadences. Other common period types include similar, in which the musical material of the phrases is not identical, but somewhat similar; contrasting, in which the musical material is not at all similar, three-phrase, which is self-explanatory; and the double period, which consists of four phrases. Because the period is parallel, the musical material from the first phrase is re-used in the second. This efficient use of musical ideas is also reflected at a smaller level within each of the phrases. Compare the second measure to the first. How are they the same, and how are they different? The melody is the same, but it has been moved, or transposed, down by a second. This is an example of sequence, which is

FUNDAMENTALS I 11 the immediate repetition of a musical idea transposed to a different pitch. Notice that in order to stay in the key, the transposition is not exact. The first pitch C is transposed down a M2 nd to B, but the following D is transposed down a m2 nd to C. In a tonal sequence, the qualities of some intervals are adjusted to stay in the key, while in a real sequence the transposition is exact. The smallest unit of melody is the motive. A motive is a brief musical idea consisting of a distinctive melodic or rhythmic character, or sometimes both. To be considered a motive, the idea must return later, usually altered or developed in some way. The first measure contains two motives that are used to generate almost all of the other material in these eight measures. The first is the three-note figure on beat 1, which is accompanied a tenth lower in the bottom part, and then sequenced in m. 2 and repeated in mm. 5 and 6. The second motive is the repeated pitch in a quarter note/eighth note rhythmic pattern on the second beat, which is found in every measure of this theme. When analyzing the structure of a melody, the first step is to locate the cadences, which will lead to the identification of the phrases. Then a comparison of the phrases can be made by assigning a lower case letter with or without a prime symbol to indicate the exact repetition (a, a), variation (a, a ), or contrast (a, b, c, etc.) of the melodic material among the phrases. The two phrases of the Mozart example would be described as a, a because the end of the second phrase is varied from the end of the first phrase.

EXERCISES 81 EXERCISES I.1.1 I. Label each pitch by letter name and octave classification. II. Write the pitches in the indicated octave.

82 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I EXERCISES I.1.2 I. Label each pitch by letter name and octave classification. II. Write the pitches in the indicated octave.

EXERCISES 83 I. Identify the intervals by quantity only. II. EXERCISES I.1.3 Write the generic intervals above or below the given notes as indicated.

84 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I I. Identify the intervals by quantity only. II. EXERCISES I.1.4 Write the generic intervals above or below the given notes as indicated.

EXERCISES 85 EXERCISES I.1.5 I. Identify the intervals by quality AND quantity. Be sure to clearly distinguish between major (M) and minor (m). II. Write the intervals above or below the given notes as indicated.

86 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I EXERCISES I.1.6 I. Identify the intervals by quality AND quantity. Be sure to clearly distinguish between major (M) and minor (m). II. Write the intervals above or below the given notes as indicated.

EXERCISES 87 EXERCISES I.1.7 I. Identify the intervals by quality AND quantity. Be sure to clearly distinguish between major (M) and minor (m). II. Write the intervals above or below the given notes as indicated.

88 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I EXERCISES I.1.8 I. Identify the intervals by quality AND quantity. Be sure to clearly distinguish between major (M) and minor (m). II. Write the intervals above or below the given notes as indicated.

EXERCISES 89 EXERCISES I.2.1 I. List two possible meter signatures for each meter type. II. duple compound triple simple quadruple compound duple simple triple compound 1) 1) 1) 1) 1) 2) 2) 2) 2) 2) Assuming a moderate tempo, determine the meter type of each signature. Provide a full measure of beats and divisions for each, and beam divisions where necessary. Be sure to break any beams between beats.

90 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I EXERCISES I.2.2 I. List two possible meter signatures for each meter type. II. triple simple duple compound quadruple simple triple compound duple simple 1) 1) 1) 1) 1) 2) 2) 2) 2) 2) Assuming a moderate tempo, determine the meter type of each signature. Provide a full measure of beats and divisions for each, and beam divisions where necessary. Be sure to break any beams between beats.

EXERCISES 91 EXERCISES I.2.3 I. Re-write each given measure, adding beams according to the meter signature at a moderate tempo. II. Re-write the rhythms, adding beams and barlines according to the meter signature at a moderate tempo.

92 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I EXERCISES I.2.4 I. Re-write each given measure, adding beams according to the meter signature at a moderate tempo. II. Re-write the rhythms, adding beams and barlines according to the meter signature at a moderate tempo.

EXERCISES 93 EXERCISES I.3.1 Locate the cadences and phrases in each melody, and use letters to identify the relationships between or among the phrases. Identify the type of period formed by the combination of phrases. Mozart Piano Sonata in B-flat, K. 333, III Johnny Has Gone for a Soldier ( Shule Aroon ) Beethoven Piano Sonata in c, Op. 13, II

94 ESSENTIAL MATERIALS OF MUSIC THEORY: PART I EXERCISES I.3.2 Locate the cadences and phrases in each melody, and use letters to identify the relationships between or among the phrases. Identify the type of period formed by the combination of phrases. Barbara Allen Beethoven Piano Sonata in c, Op. 13, III The Devil s Questions