2.2 Frequency Spectrum and Pitch

Frequency spectrum and pitch are crucial elements in sound design. They determine how we perceive different tones and musical notes. Understanding these concepts helps us manipulate and create sounds with specific characteristics.

Frequency is measured in Hertz, with humans hearing between 20 Hz and 20 kHz. The fundamental frequency sets a sound's pitch, while harmonics and overtones shape its timbre. These elements combine to create the unique qualities of different instruments and voices.

Frequency and Pitch

Measuring Frequency in Hertz (Hz)

• Frequency measures the number of cycles or oscillations per second
• The unit of frequency is Hertz (Hz), named after Heinrich Hertz
• 1 Hz equals one cycle per second
• Human hearing range spans approximately 20 Hz to 20,000 Hz (20 kHz)
• Higher frequencies correspond to higher pitched sounds, while lower frequencies correspond to lower pitched sounds

Fundamental Frequency and Harmonics

• The fundamental frequency (f0) is the lowest frequency in a periodic waveform
• Determines the pitch of a sound
• Musical notes have specific fundamental frequencies (A4 = 440 Hz)
• Harmonics are integer multiples of the fundamental frequency
  • 1st harmonic = fundamental frequency (f0)
  • 2nd harmonic = 2 * f0
  • 3rd harmonic = 3 * f0, and so on

Octaves and Frequency Doubling

• An octave is the interval between two frequencies with a 2:1 ratio
• Doubling the frequency of a sound results in a pitch one octave higher
• Halving the frequency of a sound results in a pitch one octave lower
• Octaves are a fundamental concept in music theory and acoustics
• The human ear perceives octaves as having a similar quality or character

Harmonic Content

Harmonics and Overtones

• Harmonics are integer multiples of the fundamental frequency present in a sound
• Overtones refer to any frequency above the fundamental in a sound, including both harmonics and inharmonic partials
• The relative amplitudes of harmonics contribute to a sound's timbre or tone color
• Musical instruments produce different sets of harmonics, giving them distinct timbres (violin vs. flute)

Timbre and Tone Color

• Timbre is the quality that distinguishes sounds with the same pitch and loudness
• Determined by the relative amplitudes and phases of harmonics and overtones
• Allows us to differentiate between different voices and musical instruments
• Timbre is influenced by factors such as materials, shape, and size of the sound source (wooden vs. metal flute)
• Synthesizers can create unique timbres by manipulating harmonic content

Inharmonicity and Complex Tones

• Inharmonicity occurs when overtones are not exact integer multiples of the fundamental frequency
• Inharmonic partials contribute to the complex tones of certain instruments (piano, bells)
• The degree of inharmonicity affects the perceived warmth or harshness of a sound
• Inharmonicity is an important consideration in tuning and synthesizing realistic instrument sounds

Frequency Analysis

Fourier Analysis and Spectral Decomposition

• Fourier analysis is a mathematical technique that decomposes a complex waveform into its constituent frequencies
• Allows for the examination of the frequency content of a sound
• The Fourier transform converts a time-domain signal into a frequency-domain representation
• Reveals the amplitudes and phases of individual frequency components
• Fourier analysis is the foundation for many audio processing techniques (equalization, filtering)

Spectrum Analyzers and Frequency Domain Visualization

• A spectrum analyzer is a device or software that displays the frequency content of a signal
• Provides a visual representation of the amplitudes of frequency components over time
• Horizontal axis represents frequency, vertical axis represents amplitude
• Spectrum analyzers are used for analyzing and troubleshooting audio systems (identifying feedback frequencies, room modes)
• Real-time analyzers (RTAs) display the frequency spectrum in real-time, useful for live sound applications
• Waterfall plots show the evolution of the frequency spectrum over time, helpful for analyzing transient sounds (percussion, speech)