Chapter 1: Sound and Music

1.1 What is music?

There are many types of music in the world: Western classical music

Hilary Hahn plays Sarabande in D Minor (1717-20) by J. S. Bach

Music notation

Actual music

What is music?

Indian classical music is one example of non-Western classical music

What is music?

Contemporary classical and experimental music

What is music?

Popular music

Michael Jackson’s “Thriller”

What is music?

Jazz

Ella Fitzgerald performs Count Basie’s “April in Paris”

What is music?

Traditional music

Folk music

What is music?

Take some time to brainstorm ideas……

Music is

• Sound
• Time
• Structure
• Art (?)
• Other…..

“Music is the art of organizing sound in time”

Challenging our expectations of music

Wolfram Tones

Are some sounds more musical than others?

Rank the following sounds from “most musical” to “least musical.”

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1.2 What is sound?

• Sound is rapid vibration in the air that is detectable by the ears of humans or other animals.
• In music, sounds have
• volume (loudness or softness)
• pitch (highness or lowness; sounds with pitch are notes or tones)
• timbre (TAM-ber - quality that distinguishes one instrument or person from another)
• Music also includes rests (silences)
• Both notes and rests have a beginning (onset) and end (offset); the difference in time between their onset and offset is called their duration.
• Found sounds are sounds that are not intended to be music, though they may be perceived as musical

Waveforms

A waveform is a graph of a sound. The horizontal axis represents time (in seconds) and the vertical axis represents displacement (essentially, vibration)

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Time in seconds

Displacement

The sound’s onset is at 0.010 seconds and its offset is at 0.025 seconds, so its duration is 0.025-0.010 = 0.015 seconds.

Waveforms for four sounds, macro and micro level

Talking

Humming

Clapping

“Shhhh”

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Talking

Humming

Clapping

“Shhhh”

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“Musical” sounds have more structure, either at the macro or micro level.

1.3 What is math?

“Math is the science of pattern.” --Michael Resnick

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Math studies the structures of numbers and space using precise definitions and logical arguments.

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Are some aspects of music more mathematical than others?

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Are some aspects of music more mathematical than others?

Are some kinds of music more mathematical than others?

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Algorithmic music Wolfram Tone

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Serialism

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Spectral

music

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Math rock

Prog rock

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Thinking about music like a mathematician

A binary code is a pattern of any length formed by two symbols. Some musical data may be represented by binary codes.

Examples of binary codes include

• 011001
• x--x-xx--xx-x
• HTHHTTTTH

Rules for binary codes:

• Each symbol may appear any number of times, including 0 times
• The order of symbols matters, so 011001 and 111000 are different codes

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Binary codes in music

𝅘𝅥 𝅘𝅥 𝄽 𝄽 𝅘𝅥 𝅘𝅥 in Western music notation �( 𝅘𝅥 = quarter note; 𝄽 = quarter rest)

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xx--x-x- in drum tab (x = drum strike; - = rest)

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− − • • − pits and lands on a CD

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Abstraction

• In asking questions about rhythm patterns, mathematicians use abstraction.
• They remove the problem from its original context and study the concept of “binary codes” or “tiling patterns.”
• An advantage of abstraction is that mathematicians are able to study all related problems at once, rather than having to redo the problem every time it appears in a different real-world situation.
• Mathematical solutions to a problem may inspire a composer.
• Writing all possible patterns could lead to the discovery of interesting patterns that are not normally used in music.
• The fact that many melodies are patterns of seven notes means that understanding codes with seven symbols (rather than two) many help us understand melodic possibilities.

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Mathematical terms

• Number
• Natural number
• Integer
• Real number
• Definition
• Divides
• Even number
• Proposition
• Axiom
• Theorem
• Proof
• Conjecture

• Universal proposition
• Example
• Counterexample
• Algorithm

Exercises

In the text: 1.10, 1.11, 1.12

Explain why “If a number is even, then it is divisible by 2” is logically equivalent (logically the same) as “If a number is not divisible by 2, then it is not even.”

Is “If a number is not divisible by 2, then it is odd” also logically equivalent?

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1.13 Explain why the universal proposition ``All music is made by musical instruments or voices'' is false.

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1.14 Explain why many examples don't prove that a universal proposition is true, but one counterexample disproves a universal proposition.

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