Geometry / Mr. Hansen
2/13/2003

Name: _________________________

1.

Why is it especially appropriate to have Music Day at St. Albans? (Hint: Think back to the very early history of the school.) ___________________________________________

2.

All branches of mathematics involve the study of ______________ . Would you say this is true of music also? ____ Why do you think so many mathematicians are into music? ______________________________________ (But strangely, very few of the musicians I know have much interest in mathematics!)

3.

In popular music, the term ballad refers to a slow, expressive song. Is Chopin’s Ballade in F Minor a slow piece? ______ How do you know? _________________________

4.

One of the most famous movies of all time, La Dolce Vita [The Sweet Life], was released in 1960 by the Italian director Federico Fellini. Although this movie has the highest possible rating in VideoHound’s Golden Movie Retriever (4 bones), I confess that I have not seen it.

In a piece of music, what language is almost always used for instructions to the musician? _______ In what measure number is the pianist told to play "sweetly"? ______

5.

In math, we use the greater-than sign ( > ) to connect a larger quantity on the left with a smaller quantity on the right. What do you think the > symbol means in music? _________

6.

Does crescendo mean "becoming louder" or "becoming softer"? ________________ How do you know? (List a measure number if you can.) ___________________

7.

The full Italian name of the keyboard instrument we call the piano is pianoforte, which means "soft-loud" in English. What do you suppose the letter p in measure #169 means? ________ The letter f in m. 177? ________

8.

The major elements of all music, both popular and ____________ , are as follows:

i. Melody: the singing line
ii. Harmony: the extra notes that fill in the sound
iii. Rhythm: the meter or spacing of the rhythm track (often a percussion instrument)
iv. Tempo: the speed, measured in beats per minute
v. Dynamics: sound volume, and the expressiveness caused by changes in volume
vi. Timbre: waveform of the sound (for example, a violin has a different timbre from a piano, even if they both play the same note)

9.

A sine wave generates a very boring timbre. A triangle wave is a different timbre that sounds like a ____________ . The "synth lead" patch (ask Mr. Hansen to demonstrate) is a square wave that sounds ________________ . Keyboard players use the word patch as a synonym for which of the boldface terms above? ________

10.

Mr. Hansen’s Roland D-10 synthesizer has 256 patches, and the Technics digital keyboard has 16 patches. Both of these numbers are powers of _____ . Why does this come as no surprise? _____________________________________________ . However, only the Roland is called multitimbral. Why? ________________________________________ (If you can’t solve this question, ask Mr. Hansen to play a demo tune on each keyboard.)

11.

Of the 6 elements defined above, which ones are usually constant, or nearly so, from beginning to end in a popular song? _________ , _________ , and _________

In classical and romantic music, all 6 elements usually change throughout the piece of music. Do you think popular music is less artistic, or are these merely stylistic differences? _________________

12.

The melody is usually the top note and is usually not moving too fast. (The melody has to be singable, at least in theory.) In a hymnal, melodies usually have their stems pointing upward to make them easy to spot. Circle the melody in measures 170 and 171. __ [check]

13.

A countermelody is a singing line that makes a duet with the main melody. (You might say they are complementary—another math term.) In piano music, the countermelody often has stems pointing downward to avoid being confused with the melody. Circle the right-hand countermelody in m. 171. ___ [check]

14.

Harmony consists of chords (several notes struck at the same time), arpeggios (rapidly moving notes with lots of gaps), scales (runs of notes in sequence), or broken chords (a chord with a wavy vertical line). The purpose of harmony is to make music more _________________ . Circle a broken chord, a pentatonic arpeggio (hint: see m. 173), a major scale (m. 169), and a very long chromatic scale. __ __ __ __ [4 checks]

In what key is the major scale that you circled? _________

15.

Rhythm refers to the meter and spacing of the notes. The waltz that Mr. Hansen played is in 3/4 meter, meaning 3 beats per measure and a beat on each quarter note. Marches are usually in _____ (4 quarter-note beats per measure), though sometimes in 2/4 or 2/2. Polkas are always in 2/4. Nearly all ________ music has _______ beats per measure, usually with a "back beat" on beats 2 and 4. There is also a vast amount of classical music written in 4/4, although the emphasis is on beats 1 and 3 instead of on 2 and 4. More complicated meters include the following:

6/8 (i.e., ____ eighth notes per measure, played in 2 groups of 3)
______ (i.e., 9 eighth notes per measure, played in ______ groups of 3)
12/8 (i.e., 12 eighth notes per measure, played in 4 groups of 3)
5/4 (i.e., _____ quarter notes per measure)

The most famous jazz piece in 5/4 time is ____________ by The Dave Brubeck Quartet (written by saxophonist Paul Desmond).

16.

Polyrhythm, which means playing more than one ________ at the same time, is an advanced technique used in jazz and in some classical music after 1800. Also, if you have ever heard ____ Venkatraman play drums, he uses some unbelievable polyrhythms! He has performed at Encore Club and may perform again tonight at 7:30 in Hearst Hall at NCS.

Circle an example of 8 against 6, an example of 5 against 3, and an example of 7 against 4. __ __ __ [3 checks]

Why do we say that 6 against 4 (m.154) and 9 against 6 (mm. 162-163) are both really the same as 3 against 2? ____________________________________

Learn how to tap 3 against 2 (i.e., 3 beats in one hand for every 2 in the other) and get Mr. Hansen’s initials here: _____ . If you are really talented, try 4 against 3 also: _____ (initials)

17.

Mm. 175-176 have an amazing polyrhythm: 9 against 6, except that every 4th note in the right hand is also a melody note that needs to be emphasized. What is the least common multiple of 4, 9, and 6? _________ Do you see why it takes 2 full measures (i.e., 36 of the fast right-hand notes) for the pattern to come out even? ____

Ask Mr. Hansen to perform measures 175-176. Can you hear all 3 rhythms at the same time, or do they sound like a blur? __________________

18.

Mm. 166-168 are the only measures that do not contain a certain timbre-related instruction to the performer. What is the abbreviation for this instruction? _______ What does it mean? ____________________

19.

The abbreviation accel. (m. 163) means accelerando [getting faster]. In what measure does the original tempo resume? _______ During this entire time, the right hand is playing triplets against the 6 eighth-note beats per measure, which is easiest to see in m. 166. Note that the rhythm does not change, since the meter is ______ throughout the entire excerpt.

20.

How many half steps are in an octave? ______

21.

The Greeks who associated with _________________ , the mathematician for whom the right-triangle a² + b² = c² theorem is named, also studied music extensively. They believed that all musical intervals should be "pure," i.e., having frequencies that are ratios of integers. For example, a perfect fifth should be a 3:2 ratio, a major third should be a 5:4 ratio, a perfect fourth should be a 4:3 ratio, and so on. At first, this seems to work, since an octave equals a fifth plus a fourth (i.e., the fifth’s ratio times the fourth’s ratio, namely ____ times ____ , which equals _____ , exactly the ratio needed for an octave). Unfortunately, an octave also equals 3 major thirds, and if we raise _____ (the major third ratio) to the 3rd power, we get _______ , which does not equal 2.0 as required. The solution adopted by most modern piano tuners is to tweak every note a bit so that the ratio for any half step is always the 12th root of 2, approximately 1.05946. This number raised to the 7th power (since a perfect fifth is 7 half steps) gives _____ , which is quite close to the desired 3:2 ratio. Moreover, if we raise 1.05946 to the 12th power, i.e., to go an entire octave, we get something extremely close to ______ as desired.

The disadvantage of this "equal temperament" tuning used by _______ piano tuners is that the entire instrument is always slightly out of tune. However, you can play in any key without having any notes "stick out" since the errors have been spread ____________ across the keyboard. A piano tuned using a Pythagorean tuning scheme will sound excellent in some keys and dreadfully _____________ in others.

22.

Did you learn some new connections between math and music? _____

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