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204 6. Analyzing Pitch and Rhythm

(b) What chord do you get from C-E-G♯ if you lower E by one half step? What chord do you get from C-E-G♯ if you raise E by one half step? (c) What chord do you get from C-E-G♯ if you lower C by one half step? What chord do you get from C-E-G♯ if you raise C by one half step? This exercise shows that there are six major and minor chords that are as close as possible—only one half step difference for just a single note—to the chord C-E-G♯.

6.5.12. Suppose you want a cyclic rhythm with 4 note onsets on a 9-hour clock. Show that the Euclidean algorithm generates the sequence: 101010100. This sequence corresponds to the Aksak rhythm from Turkey.

6.5.13. Suppose you want a cyclic rhythm with 7 note onsets on a 16-hour clock. Show that the Euclidean algorithm generates the sequence: 1001010100101010. This sequence is closely related to the Samba rhythm from Brazil. If the rhythm is started on the last onset (time-shifting by T2), then the Samba rhythm is obtained.

6.5.14. Suppose you want a cyclic rhythm with 9 note onsets on a 16-hour clock. Show that the Euclidean algorithm generates the sequence: 1011010101101010.

This sequence is closely related to several different rhythms. If it is time-shifted by T2, so that it begins on the last onset, then it is the bell pattern used in the Ngbaka-Maibo rhythm of the Central African Republic. If the rhythm is started on the fourth onset (time-shifting by T−5), then it is the cow-bell rhythm used in Brazilian Samba, and is also a rhythm played in West and Central Africa.

6.5.15. Sometimes a cyclic rhythm will correspond to a partial application of the Euclidean algorithm. Suppose, for example, that you want a cyclic rhythm with 10 note onsets on a 16-hour clock. Show that the sequence of 1’s and 0’s obtained at the step corresponding to 4=2+2 gives the note onsets for the rhythm played by the quijada in Figure 6.11 on p. 187.

6.6 A Case Study in Rhythm: Bruch’s Kol Nidrei

In this section, we analyze a portion of Max Bruch’s composition for cello and orchestra, Kol Nidrei, Op. 47. We will use a lot of the tools we have developed throughout the book to analyze a cyclic rhythm that is used frequently in the piece. Our analysis will shed some further light on the structured patterns of music referred to in the quote by Cooper and Meyer on p. 183. Much of the material in this section is adapted from work done by Emily Gullerud, in collaboration with this book’s authors. The portion of Bruch’s Kol Nidrei that we shall analyze is the melody shown in Figure 6.21. It is written as a passage for a solo cello.

Figure 6.21. Featured cello melody from Bruch’s Kol Nidrei. Played in measures 9 to 12, 13 to 16, 25 to 28, and 47 to 50.

The passage is repeated four times in the piece and is derived from a melody sung by a Cantor in Orthodox Jewish religious services, dating back several centuries prior to Bruch’s time. In Figure 6.22, we show clocks describing the rhythm of note onsets for the first two measures of the melody. These rhythm clocks are very similar, except that the onsets in the first clock do not

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Figure 6.22. Clocks showing note onsets for first two measures of solo cello melody.

have the symmetry possessed by the onsets in the second clock. If, however, we look at examples of how the first measure is typically performed, we will be able to see that the first two rhythms do have identical symmetry. For example, on the top left of Figure 6.23 we show a spectrogram of a recording10 of the first measure of the melody (as played in measure 13).

∼Cello and Orchestra ∼ Viola and Piano ∼

Figure 6.23. Top left: Spectrogram of cello performance of measure 13 of Bruch’s Kol Nidrei, with orchestral background. Top right: Spectrogram of viola performance of octave-higher version of measure 13, with piano accompaniment. For each spectrogram, vibratoed portions of second part of each quarter-note lie above time spans marked by symbol ∼. Bottom left: Modified solo passage showing quarter-note as two tied eighth-notes, with vibratoed eighth-note for its second part. Bottom right: Rhythm clock for first measure of solo melody, with onset of vibratoed eighth-note indicated.

The quarter note that begins the passage is played at first as a straight tone for about the duration of an eighth note, and then the player increases the vibrato quite noticeably. This vibratoed ending of the quarter note, lasting about the length of an eighth note, is marked by the horizontal bar at the bottom of the left-hand spectrogram in Figure 6.23. This combination of a straight tone plus vibratoed tone is reminiscent of the use of vibrato for rhythmic effect that we described in discussing Louis Armstrong’s trumpet technique in Chapter 5, p. 144. The effect in this cello passage is to mimic the Cantor’s singing of the melody. It is one of the ways that the espressivo instruction in the score

10Konzert für Violoncello und Orchester, Antonin Dvoˇrák. Kol Nidrei, Max Bruch. Deutsche Grammophon, 1990.

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is realized by the cello player. At the bottom of Figure 6.23 we have shown how the quarter note is expressed as a tied pair of eighth notes, with the second eighth note in the pair indicated as a vibratoed note (via the symbol ∼). With this expressive rhythmic effect taken into account, the rhythm clock for this first measure of the passage takes the form shown on the bottom right of Figure 6.23. The clocks for the first two measures are identical, with identical symmetry. The pattern of note onsets here is an example of a rhythm generated by the Euclidean algorithm discussed in the previous section (see Exercise 6.6.1). As further evidence for our interpretation of the performance of this passage, we show on the top right of Figure 6.23 a spectrogram of its performance on a viola (played an octave higher) with piano accompaniment.11 Here the vibratoed eighth note portion of the first quarter note is even more clearly displayed. This gives us some evidence that the first rhythm clock shown in Figure 6.23 has become an aspect of standardized performance practice for the passage. Another aspect of the passage is the rhythm of the harmoniesused. The cello solo in the passage is accompanied by various strings playing background harmony. In Figure 6.24, we show measures 13 to 16 from the full score. The solo cello melody is written on the bass clef staff at the top. The strings, playing harmonic accompaniment, are written on the bottom two staves. For the first three measures, the strings are playing stacked chords. These chords are played precisely at the rests in the cello melody. Hence the full orchestral passage follows the symmetric patterns shown in the rhythm clocks at the bottom of Figure 6.24. Each complete rhythm clock displays 180◦ rotational symmetry as well as symmetry by reflection through the hours 0 and 4, and hours 2 and 6.

Dm A7 A7 A7 Dm E◦ Dm A7 Dm

∼

Figure 6.24. Measures 13 to 16 of Bruch’s Kol Nidrei. Top staff has solo cello melody. Bottom two staves contain string notes. Chords played by strings, in root notation, are written above treble clef staff. Cyclic rhythm for first measure is shown in circle on left side, with solo cello notes indicated by solid dots and string chord onsets indicated by small circles. Second circle shows same rhythm repeated in second measure, while semicircle shows it repeated for first half of third measure (ending right before grace note).

Besides the rhythm of note onsets there is also an interesting succession of chords in the passage. Just above the treble clef staff in Figure 6.24, we have indicated the chords being played (using only root notation). These chords divide into the following two sequences:

Dm −→ A7 −→ A7 −→ A7 −→ Dm and B◦ −→ Dm −→ A7 −→ Dm (6.18)

11Kol Nidrei, Max Bruch, from the CD Bis, Encore!, Limen music & arts srl, 2014.

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The key for the piece is D-minor. Therefore, the roman numeral for the chord Dm is i. Moreover, the seventh chord A7 is the dominant seventh chord V7. For similar reasons as we discussed in Chapter 3 for the dominant V chord (see p. 60), the dominant seventh chord V7 provides a large amount of anticipation of the tonic chord i. That is why it provides a strong ending, an authentic cadence, for the following sequence of chord progressions: i −→ V7 −→ i (6.19) This sequence, which is a typical one for a minor key, occurs in both sequences in (6.18). In roman numeral notation, the first sequence in (6.18) is i −→ V7 −→ i (6.20) (long)

where (long) refers to the fact that V7 is repeated three times, and the entire sequence of chords spans almost three measures. In contrast, the second sequence in (6.18) is ii◦ −→ i −→ V7 −→ i (6.21) (short) and here (short) refers to the fact that each chord is used only once, and for significantly shorter durations, with a total span of less than one measure. In both sequences, the relatively longer durations of the ending i chords that follow the V7 chords serve to emphasize the authentic cadence V7 −→ i. The aspect of rhythm that we have been emphasizing for (6.20) and (6.21)—the rhythm of the chord durations and their changes—is known as harmonic rhythm. One final aspect of rhythm related to this passage is worth pointing out. There is a correspondence between the different tempos of the chord progressions in (6.20) and (6.21) with the melodic motion in the cello solo. To be precise, the melodic motion in the cello solo from the start of the first measure to the occurrence of the grace note can be described by this diagram:

D3 : D3 XX  XXX  Down XXXz  Up

After the grace note provides a pickup to the start of the second part of the passage at the F3 note, the tempo of up and down shifts in the melody speeds up, with several quick repetitions of the up/down pattern:

F3 D3 @  @  @  D@R U D@R U D@R U

This analysis shows a connection between the first part of the melody and the second, mirrored by a similar connection between the harmonies for each part. Our exploration of the rhythm in this iconic melody from Kol Nidrei shows how subtle the various aspects of rhythm can be. Of course, there are many more aspects to this important orchestral work. Some of them will be explored in the exercises.

Exercises

6.6.1. Show that the cyclic rhythm for each of the first two measures, as shown in the clock in Figure 6.23, is a time-shift of a rhythm obtained by using the Euclidean algorithm to 6 onsets out of 8. 6.6.2. Show that the cyclic rhythm for the chord onsets in the first two measures, as shown by the small circles in the two full clocks in Figure 6.24, is a time-shift of a rhythm obtained by using the Euclidean algorithm to 2 onsets out of 8.

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6.6.3. How does the intensity of the piano accompaniment differ from the intensity of the string accompaniment in the two versions of the Kol Nidrei passage shown in the spectrograms in Figure 6.23?

6.6.4. Explain why the dominant seventh chord V7 provides strong anticipation of the minor tonic chord i. Use the progression A7 → Dm as an example.

6.6.5. In Figure 6.25 there is a score of the violins and harp parts from a portion of Kol Nidrei. Answer the following questions about it. (a) To what chord does all of the notes belong to? (b) Find the largest palindrome in the notes for the harp. This palindrome will include notes from both staves. (c) Describe the sequencing that occurs on each side of the central note of the palindrome you found for part (b). (d) Show that the notes for Violins I are a conjoining of two palindromes intersecting at four notes. Circle these four notes, and indicate the mirror positions for the two palindromes. (e) Show that the pitches for the notes for Violins II are a conjoining of two palindromes intersecting at four notes. [NOTE: One of the notes is of a different length than the others, focus only on the pitches not their durations.] Circle these four notes, and indicate the mirror positions for the two palindromes.

Figure 6.25. Harp and Violins from Bruch’s Kol Nidrei, fourth measure after rehearsal mark E.

6.6.6. In Figure 6.25 there is a score of the violins and harp parts from a portion of Kol Nidrei. Answer the following questions about it. (a) Find the relationship between the two beamed sets of four sixteenth notes in the second beat of each measure for Violins I and II. To be specific, what transformation converts the four Violin I notes to the four Violin II notes? (b) Find the relationship between the two beamed sets of four sixteenth notes in the fourth beat of each measure for Violins I and II. To be specific, what transformation converts the four Violin I notes to the four Violin II notes?

6.6.7. In Figure 6.26 there is a score of the violins from a portion of Kol Nidrei. Answer the following questions about it. (a) Draw vertical lines that mark off that mark off the individual beats, and identify the chords that the notes belong to for each beat. (b) Draw rhythm clocks for the note onsets for Violins I.

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(c) Draw rhythm clocks for the note onsets for Violins II.

(d) Find the time-shift Tk that maps the note onsets for the rhythm clock you found in (b) to the note onsets for the rhythm clock you found in (c).

Figure 6.26. Violins from Bruch’s Kol Nidrei, seventh measure after rehearsal mark E. Violins II are played one beat ahead of Violins I, with contrary melodic motion in each beat, creating a fascinating sound.

6.6.8. In Figure 6.27 there is a score for the harp from a portion of Kol Nidrei. Answer the following questions about it. (a) What chord do the notes belong to? (b) Find the relationship between the three beamed triplets of sixteenth notes in the lower staff with the three beamed triplets of sixteenth notes in the upper staff. To be specific, what transformation converts the first set of beamed triplets to the second set?

Figure 6.27. Harp from Bruch’s Kol Nidrei, fourth measure before rehearsal mark H.

6.6.9. In Figure 6.28 there is a score for the cello solo from a portion of Kol Nidrei. What seventh chord do these notes all belong to? What is the roman numeral form for this seventh chord in the key specified by the key signature shown in the score?

Figure 6.28. Cello solo from Bruch’s Kol Nidrei, fifth measure before end.

6.7 Comparing Musical Scales and Cyclic Rhythms

In this section we look at a method for analyzing a musical scale, based on the relative frequency of harmonic intervals between the pitch classes within the scale. We relate this relative frequency to the amount of dissonance that the scale allows for. This analysis allows us to musically compare different scales. Interval frequencies will also be used to compare different cyclic rhythms, based on the mathematical equivalence of scales and cyclic rhythms.

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