13.4 Math and Music
3 min read•june 18, 2024
Sound and music are all about waves and numbers. From the high-pitched squeal of a flute to the deep rumble of thunder, it's all just vibrations at different frequencies. We'll explore how these frequencies shape what we hear.
Musical notes and intervals are the building blocks of music. We'll look at how notes relate to each other on a keyboard and dive into the math behind octaves. Understanding these concepts helps us appreciate the science behind the tunes we love.
Sound and Music
Frequency, pitch, and decibels in sound
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- measured in (Hz) represents the number of sound wave cycles per second
- Higher sound waves result in higher perceived (flute vs. tuba)
- is the perceptual property of sound determined by the frequency of sound waves
- Higher pitch corresponds to higher frequency (whistle vs. thunder)
- (dB) measure sound intensity or volume using a
- Every 10 dB increase represents a tenfold increase in sound intensity (jet engine vs. normal conversation)
- Doubling the sound intensity results in a 3 dB increase (two identical instruments playing together)
Wave properties and sound quality
- is inversely proportional to frequency, affecting pitch perception
- determines the loudness of a sound
- is the unique quality of a sound that distinguishes different instruments or voices
- are additional frequencies above the fundamental frequency that contribute to timbre
- occurs when an object vibrates in response to sound waves of a specific frequency
Musical Notes and Intervals
Note relationships on keyboards
- () is the smallest interval between two notes on a piano keyboard
- Two adjacent keys, whether white or black (C to C# or E to F)
- () is an interval consisting of two half-steps
- Two keys with one key in between, whether white or black (C to D or F# to G#)
- is the interval between two notes with a 2:1
- 12 half-steps or 6 whole steps apart on a keyboard (C to the next higher or lower C)
- Notes an apart have the same letter name (A to A, F to F)
Frequency calculations for octaves
- Octave frequency relationship involves doubling when ascending an octave and halving when descending an octave
- If A4 (middle A) has a frequency of 440 Hz, A5 (one octave higher) has a frequency of Hz
- A3 (one octave lower) has a frequency of Hz
- Extending the concept, each subsequent octave doubles or halves the frequency
- A6 has a frequency of Hz (two octaves higher than A4)
- A2 has a frequency of Hz (two octaves lower than A4)
- are integer multiples of the fundamental frequency, contributing to the overall sound
Tuning systems
- is a tuning system that divides the octave into 12 equal parts, allowing for consistent intervals in all keys
Key Terms to Review (25)
Amplitude: Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. It is a critical aspect in understanding sound waves and musical tones, as it directly relates to the volume or loudness of the sound produced. In music, a greater amplitude means a louder sound, while a smaller amplitude results in softer sound, affecting how we perceive music and its emotional impact.
Binned frequency distribution: A binned frequency distribution is a way to organize and summarize data by grouping the data into intervals, called bins, and counting the number of observations in each bin. It helps to visualize the distribution of data points across different ranges.
Decibels: Decibels (dB) are a logarithmic unit used to measure the intensity of sound. This scale allows us to express large variations in sound intensity in a more manageable form, reflecting how humans perceive changes in loudness. In the context of sound, a difference of 10 dB typically represents a tenfold increase in intensity, while the human ear perceives this as roughly twice the loudness.
Equal Temperament: Equal temperament is a tuning system that divides the octave into 12 equal parts, creating a uniform distance between each semitone. This system allows instruments to play in any key and ensures that chords sound harmonious regardless of the key used, making it foundational in Western music. The versatility of equal temperament has made it the standard for tuning in modern Western music, influencing everything from classical compositions to contemporary genres.
Flat: In music, 'flat' refers to a pitch that is one semitone lower than its natural state. It is denoted by the symbol ♭ and alters the note downward on the musical scale.
Frequency: Frequency is the number of occurrences of a repeating event per unit of time. In music, it determines the pitch of a sound based on how many sound waves occur in one second, measured in Hertz (Hz).
Frequency: Frequency refers to the number of times a specific event or value occurs within a given data set or over a specified period. It helps in understanding the distribution of data, allowing for insights into patterns, trends, and relationships in various contexts, including statistical analysis and musical compositions.
Frequency ratio: A frequency ratio is a mathematical comparison of the frequencies of two or more sounds, expressed as a simple fraction or proportion. This concept is crucial in understanding musical intervals, where the relationship between the pitches of notes can be described using frequency ratios, allowing for the creation of harmonious sound combinations. This ratio helps musicians and composers grasp how different pitches interact within music theory.
Half-step: A half-step, also known as a semitone, is the smallest interval in Western music that represents the distance between two adjacent keys on a keyboard. This concept is crucial for understanding musical scales, chords, and overall harmony, as it forms the building block for larger intervals and contributes to the structure of melodies and compositions.
Harmonics: Harmonics refer to the integer multiples of a fundamental frequency in sound waves that create musical tones. When a musical instrument plays a note, it doesn't just produce a single frequency; it generates a series of frequencies that include the fundamental frequency and its harmonics, which are essential for the richness and color of the sound. The presence and relative strengths of these harmonics help define the timbre of the sound, making it distinct even when playing the same pitch.
Hertz: Hertz (Hz) is the unit of frequency in the International System of Units (SI), representing one cycle per second. It measures the number of times a periodic event occurs per second.
Hertz: Hertz is a unit of frequency that measures the number of cycles per second of a periodic wave or signal. It is named after Heinrich Hertz, who made significant contributions to the study of electromagnetic waves. The concept of Hertz connects deeply with music as it relates to the pitch of sounds, where different frequencies correspond to different musical notes and harmonics.
Logarithmic scale: A logarithmic scale is a nonlinear scale used for a large range of quantities, where each unit increase on the scale corresponds to a tenfold increase in the value being measured. This type of scale is particularly useful in representing data that spans several orders of magnitude, allowing for easier visualization and interpretation of exponential relationships. In music, logarithmic scales can help describe frequencies and the perceived differences in pitch, demonstrating how our ears perceive sound intensity and frequency logarithmically.
Octave: An octave is an interval between one musical pitch and another with double its frequency. In Western music, it spans eight notes in a standard scale.
Octave: An octave is a musical interval that spans eight notes in the diatonic scale, specifically from one pitch to another pitch that has double its frequency. This relationship is fundamental in music theory as it establishes the basis for harmony and melody, with the notes in an octave having a unique connection that allows for the creation of musical scales and chords.
Overtones: Overtones are the higher frequency sound waves that occur above the fundamental frequency when a musical note is played. These additional frequencies contribute to the richness and timbre of the sound, allowing us to differentiate between different instruments and voices even when they play the same note. Understanding overtones is essential for grasping how sound waves interact and how they create harmony in music.
Pitch: Pitch is the perceived frequency of a sound, determining how high or low it appears to a listener. In music, pitch is crucial for melody and harmony creation.
Pitch: Pitch refers to the perceived frequency of a sound, which determines how high or low a note sounds. It is a fundamental aspect of music, as it influences melody and harmony, allowing us to distinguish between different notes. The pitch of a sound is typically measured in Hertz (Hz), with higher frequencies corresponding to higher pitches and lower frequencies resulting in lower pitches.
Resonance: Resonance is the phenomenon that occurs when a system is driven by an external force at a frequency that matches its natural frequency, resulting in a significant increase in amplitude. This concept is crucial in understanding how musical instruments produce sound, as certain frequencies lead to enhanced vibrations and sound production. The principles of resonance also apply to various aspects of sound waves and acoustics, making it a key element in the relationship between math and music.
Semitone: A semitone is the smallest musical interval commonly used in Western music, representing the distance between two adjacent keys on a piano keyboard, whether they are white or black. It serves as a fundamental building block in music theory, allowing for the construction of scales and chords, and is crucial for understanding concepts like tuning and harmonic relationships.
Sharp: Sharp is a musical notation symbol (♯) indicating that the pitch of the note should be raised by one half step. It is a crucial element in understanding scales, chords, and harmonic functions in music theory.
Timbre: Timbre is the quality or color of a sound that distinguishes different types of sound production, such as voices and musical instruments. It encompasses the unique characteristics of a sound that allow listeners to identify its source, even when the pitch and loudness are the same. This aspect of sound is influenced by factors such as the instrument's material, shape, and how it is played, making timbre essential for understanding music and sound in depth.
Wavelength: Wavelength is the distance between consecutive crests or troughs of a wave, typically measured in meters. It is a fundamental property of waves that relates to their frequency and speed, impacting how waves behave in different mediums. In the context of sound and music, wavelength plays a crucial role in determining the pitch and tone of sounds produced by musical instruments or vocalizations.
Whole step: A whole step is a musical interval that consists of two half steps, representing the distance between two notes. This interval is crucial for understanding scales and melodies, as it helps to define the structure of various musical systems. Whole steps create a sense of movement and progression within music, often contributing to the overall harmony and rhythm.
Whole Tone: A whole tone is a musical interval that spans two half steps or semitones, making it one of the fundamental building blocks of music theory. This interval creates a sound that is often described as bright and open, contributing to various scales, including the whole tone scale. Whole tones play a crucial role in establishing the mood and structure of musical compositions across different genres.