Physics-first music theory
Physics-first music theory
The canonical vocabulary reference for the PhizMusic framework. Every novel term used in this wiki is defined here. Definitions are grounded in physics and neuroscience. Western equivalents are noted in parentheses where applicable.
12-TET — Twelve-tone equal temperament. Divides the octave into 12 equal chromatic steps of 100 cents each (frequency ratio 2^(1/12) ≈ 1.05946 per step). The dominant tuning system in modern Western music. See twelve-tet.md.
ADC (analog-to-digital conversion) — The process of sampling a continuous pressure waveform at discrete time intervals (sampling period Ts = 1/Fs) and quantizing each sample to a finite number of amplitude levels. Converts physical sound into PCM digital audio. See digital-audio.md.
ADSR — Attack, Decay, Sustain, Release. The four phases of a sound’s amplitude envelope over time. Attack is onset to peak; decay is peak to sustain level; sustain is the held level; release is sustain to silence. See adsr-envelope.md.
Additive synthesis — Building complex timbres by summing individual sine waves (partials) at specified frequencies and amplitudes. The inverse of Fourier analysis: if analysis decomposes, synthesis recomposes. See fourier-analysis.md.
Air column — A vibrating column of air inside a tube (pipe) that supports standing waves. Open-open pipes produce all integer harmonics; open-closed pipes produce only odd harmonics. See instrument-physics.md.
Aliasing — The misrepresentation of frequencies above the Nyquist limit as lower-frequency components in a sampled signal. Occurs when the sampling rate is less than twice the signal’s highest frequency. Prevented by anti-alias filtering before sampling. See digital-audio.md.
Amplitude — Peak displacement of a pressure wave from equilibrium. Measured in pascals (Pa) or decibels (dB SPL). Determines perceived loudness — though the relationship is frequency-dependent (see equal-loudness.md).
Antinode — A point on a standing wave where displacement is maximal. Located between nodes. The number of antinodes equals the mode number. See standing-waves.md.
Auditory masking — The phenomenon where one sound renders another inaudible. Includes simultaneous masking (frequency domain) and temporal masking (time domain). Exploited by lossy audio compression (MP3, AAC). See auditory-masking.md.
Auditory nerve — The neural pathway that transmits electrical signals generated by inner hair cells from the cochlea to the brain for pitch, loudness, and source identification processing. See ear-cochlea.md.
Auditory scene analysis — The brain’s process of parsing a complex mixture of sound into separate perceptual streams (sources), using cues like harmonic template matching, onset synchrony, and spatial location. See harmonic-series.md.
Auditory streaming — The perceptual grouping of successive sounds into coherent sequences (streams) based on spectral proximity, timing, and spatial cues. When streams split, listeners hear two independent melodies from interleaved notes. See timbre.md.
Backward masking — A temporal masking effect where a loud sound reduces the audibility of a softer sound that occurred just before it (within ~5–10 ms). A post-stimulus neural suppression phenomenon. See auditory-masking.md.
Basilar membrane — The tapered membrane inside the cochlea whose varying stiffness mechanically separates frequency components: high frequencies resonate near the base, low frequencies near the apex. The ear’s physical Fourier analyzer. See ear-cochlea.md.
Beat rate — The rate of amplitude fluctuation when two nearby frequencies sound simultaneously, equal to |f₁ − f₂| Hz. At slow rates (<15 Hz) perceived as distinct beats; at faster rates transitions into roughness. See beating-roughness.md.
Beating — Periodic amplitude fluctuation produced by the interference of two sinusoidal frequencies close in pitch. The envelope oscillates at the difference frequency |f₁ − f₂|. See beating-roughness.md.
Bessel function — Mathematical functions whose zeros determine the modal frequencies of circular vibrating membranes (drums, cymbals). Unlike strings and pipes, membrane modes are not integer multiples of a fundamental, producing inharmonic spectra. See instrument-physics.md.
Bit depth — The number of bits used to represent each audio sample. Determines the number of quantization levels (2^N) and theoretical dynamic range (≈ 6.02N + 1.76 dB). CD audio uses 16-bit; professional audio uses 24-bit. See digital-audio.md.
Blue note — A pitch deliberately inflected away from the 12-TET grid toward a harmonic-series target. Common examples include the blue third (~310–330 cents, between steps 3 and 4) and blue seventh (~969 cents, near the 7:4 ratio). See pentatonic-cross-cultural.md.
Boundary condition — A physical constraint at the endpoint of a vibrating system (fixed or open) that determines which standing-wave modes are allowed. Fixed ends force nodes; open ends force antinodes. See standing-waves.md.
Cadence — A harmonic arrival point where a chord progression reaches a point of relative resolution or pause. Defined by the spectral fusion pattern of the final step-combo relative to the key’s tonal center. See chord-progressions.md.
Categorical perception — The tendency to perceive continuously varying stimuli (such as tuning) in discrete categories. Listeners often hear intervals as “in-tune” or “out-of-tune” rather than perceiving a smooth continuum. See consonance-dissonance.md.
Cent — Logarithmic unit for measuring frequency ratios. 1 octave = 1200 cents. 1 chromatic step in 12-TET = 100 cents. Formula: c = 1200 × log₂(f₂/f₁). Essential for comparing intervals across tuning systems.
Chord inversion — A rearrangement of the same step-combo that places a different chord member as the lowest pitch. Changes the bass frequency ratios and thus the perceived spectral weight, without changing the set of pitch classes. See chords.md.
Chroma — The cyclical aspect of pitch that repeats every octave. Two tones an octave apart share the same chroma but differ in height. Chroma is what makes step 0 in octave 3 and step 0 in octave 5 sound “the same.” See harmonic-series.md.
Chromatic grid notation — A notation system where pitch is mapped to equally-spaced rows (one per chromatic step) and time flows proportionally along the horizontal axis. Transposition becomes geometric translation. See notation-layer.md.
Chromatic step — One of the 12 equal divisions of an octave in 12-TET. The smallest standard interval, with frequency ratio 2^(1/12) ≈ 1.05946. (Western equivalent: “half step” or “semitone.”)
Clipping — Distortion that occurs when a signal exceeds the maximum representable amplitude in a digital system. Truncates waveform peaks, introducing spurious high-frequency harmonics. See digital-audio.md.
Close voicing — A chord arrangement where all notes fall within one octave span. Produces a compact spectral cluster. Contrasts with open voicing, where notes are spread across multiple octaves. See chords.md.
Cochlea — The fluid-filled, spiral-shaped organ in the inner ear that mechanically decomposes incoming pressure waves into frequency-specific channels via basilar membrane resonance. The biological Fourier analyzer. See ear-cochlea.md.
Cochlear amplifier — Active mechanical feedback by outer hair cells that sharpens frequency tuning and increases sensitivity on the basilar membrane. Responsible for the ear’s extraordinary dynamic range and frequency selectivity. See ear-cochlea.md.
Combination tone — A phantom tone generated by nonlinear cochlear processes when two tones sound simultaneously. The most audible is the difference tone at frequency f₂ − f₁ (also called a Tartini tone). See consonance-dissonance.md.
Consonance — The perceptual quality of tones that fuse or blend. Arises from aligned harmonics, absence of roughness within critical bandwidth, and harmonic template matching. Whether consonance is aesthetically preferred is cultural, but the perceptual fusion itself is biological. See consonance-dissonance.md.
Continuous spectrum — A spectrum where energy is spread across a continuous range of frequencies rather than concentrated at discrete peaks. Characteristic of non-periodic sounds such as noise, transients, and percussive attacks. See fourier-analysis.md.
Convergent engineering — The observation that unrelated musical cultures independently arrived at similar pitch structures (especially the pentatonic scale) because they are solving the same physics and psychoacoustics constraints. See pentatonic-cross-cultural.md.
Coprime — Two integers whose greatest common divisor is 1. A step-interval g generates all 12 chromatic steps under modular arithmetic if and only if gcd(g, 12) = 1. The generators of ℤ₁₂ are {1, 5, 7, 11}. See step-7-cycle.md.
Critical bandwidth — The frequency range within which two simultaneous tones interact perceptually, producing roughness or beating. Approximately 10–20% of center frequency, determined by the physical properties of the basilar membrane. See ear-cochlea.md.
Cycle — A repeating temporal pattern of defined length, measured in multiples of pulse period T. A cycle of 4T contains 4 pulse beats. (Western equivalent: “measure” or “bar.”) See rhythm.md.
dB SPL (decibel sound pressure level) — A logarithmic measure of sound pressure relative to the reference threshold of hearing (20 µPa). Formula: L = 20 × log₁₀(p / p_ref). A 6 dB increase ≈ doubling of pressure. See sound-waves.md.
Diatonic — A 7-of-12 step-subset formed by selecting 7 contiguous positions on the step-7 cycle. Produces the familiar major and minor scale families with their characteristic mix of 1-step and 2-step gaps. See scales.md.
Dichotic presentation — An experimental technique where different frequency components are delivered to different ears. Used to demonstrate that the missing fundamental is computed centrally in the brain, not generated by cochlear distortion. See missing-fundamental.md.
Dissonance — The perceptual quality of tones that clash or create tension. Caused by beating of nearby harmonics within critical bandwidth (roughness) or failure to match harmonic series templates. See consonance-dissonance.md.
Dissonance curve — A plot of total sensory dissonance as a function of frequency ratio for a given timbre. Minima correspond to the most consonant intervals for that specific spectrum. Computed by summing pairwise Plomp–Levelt contributions across all partials. See dissonance-curves.md.
Dodeka syllable system — Twelve unique syllables — Do (0), Ka (1), Re (2), Xo (3), Mi (4), Fa (5), Hu (6), So (7), Bi (8), La (9), Ve (10), Si (11) — assigning one name per chromatic step with no sharps or flats. Adapted from Jacques-Daniel Rochat’s Dodeka system. See naming-system.md.
Dominant function — The harmonic role of step-combos built on scale degree 5 (or other positions with strong leading-tone content). Characterized by high spectral tension relative to the tonic, creating a pull toward resolution. See functional-analysis.md.
Duration — How long a sound event lasts, expressed as a fraction or multiple of pulse period T. A sound lasting one pulse = 1T; half a pulse = 0.5T. (Western equivalent: “note value” — e.g., quarter note = 1T at standard tempo.) See rhythm.md.
Equal-loudness contour — A curve showing combinations of frequency and SPL that are perceived as equally loud. Defined by ISO 226 (historically Fletcher–Munson curves). Reveals that human hearing is most sensitive around 2–4 kHz and least sensitive at low frequencies. See equal-loudness.md.
Equal temperament — Any tuning system that divides an interval (usually the octave) into equal logarithmic steps. 12-TET is the most common form, trading pure harmonic ratios for uniform transposability across all keys. See twelve-tet.md.
Formant — A resonant frequency peak in a sound’s spectral envelope, particularly prominent in voice and acoustic instruments. Formants are determined by the resonating body’s geometry and remain relatively fixed as pitch changes, contributing to source identity. See timbre.md.
Forward masking — A temporal masking effect where a loud sound raises the detection threshold for quieter sounds that follow it (for ~50–200 ms). Used by lossy audio codecs to identify inaudible post-masker content. See auditory-masking.md.
Fourier analysis — The mathematical decomposition of any complex waveform into a sum of sine waves at different frequencies and amplitudes. The cochlea physically performs this decomposition via basilar membrane resonance. See fourier-analysis.md.
Fourier transform — The mathematical operation that converts a time-domain waveform into its frequency-domain spectrum (and vice versa). Waveform and spectrum are two representations of the same information. See fourier-analysis.md.
Frequency — The number of complete oscillations per second, measured in hertz (Hz). Determines perceived pitch. Human hearing range: approximately 20 Hz to 20,000 Hz. See sound-waves.md.
Frequency domain — A representation of sound showing amplitude (or energy) as a function of frequency. The spectrum is the frequency-domain view; the waveform is the time-domain view. Both contain the same information. See fourier-analysis.md.
Frequency ratio — The ratio of two frequencies, f₂/f₁. Determines the perceived interval between them. Simple integer ratios (2:1, 3:2, 5:4) correspond to perceptually consonant intervals. See frequency-ratios.md.
Fundamental — The lowest frequency component of a complex tone. Determines the perceived pitch. All other frequency components (harmonics) are integer multiples of the fundamental. See harmonic-series.md.
Gap pattern — The sequence of step-interval sizes between consecutive notes of a scale when arranged in ascending order. Different rotations of the same gap pattern produce different modes of the same scale. See scales.md.
Gaussian amplitude envelope — A bell-shaped weighting function in log-frequency space used in Shepard tone construction to set each octave-spaced component’s amplitude. Keeps the spectral centroid fixed as components shift. See shepard-tones.md.
Generator (of ℤ₁₂) — A step-interval g that, when repeatedly added modulo 12, visits all 12 chromatic positions. Only step-intervals coprime to 12 (g ∈ {1, 5, 7, 11}) are generators. See step-7-cycle.md.
Groove — The perceptual quality arising from systematic micro-timing deviations (ahead of or behind the pulse grid) that make rhythm feel propulsive or relaxed. A temporal texture layered on the pulse-fraction framework. See rhythm.md.
Hair cells (inner / outer) — Sensory cells in the cochlea. Inner hair cells transduce basilar membrane motion into neural signals encoding frequency content. Outer hair cells actively amplify and sharpen basilar membrane response (the cochlear amplifier). See ear-cochlea.md.
Harmonic — A frequency component that is an integer multiple of the fundamental. The 2nd harmonic = 2×f₁, the 3rd = 3×f₁, etc. Instruments with harmonic spectra (strings, pipes) support harmonic-series-based tuning systems. See harmonic-series.md.
Harmonic rhythm — The rate at which step-combos (chords) change within a progression. Faster harmonic rhythm increases information density; slower rhythm emphasizes stability and weight. See chord-progressions.md.
Harmonic series — The sequence of all integer multiples of a fundamental frequency: f₁, 2f₁, 3f₁, 4f₁, … . Naturally produced by vibrating strings and air columns. The foundation of tonal music across cultures. See harmonic-series.md.
Harmonic template matching — The auditory system’s process of fitting observed spectral components to an integer-harmonic template to infer a single source and fundamental pitch. The mechanism behind missing fundamental perception and auditory scene analysis. See harmonic-series.md.
Impedance matching — The mechanical amplification performed by the middle ear’s ossicles (malleus, incus, stapes) to efficiently transfer acoustic energy from air to the fluid-filled cochlea. Compensates for the ~30 dB loss that would otherwise occur at the air–fluid boundary. See ear-cochlea.md.
Inharmonic — Describing a spectrum whose overtone frequencies are NOT integer multiples of the fundamental. Bells, drums, and metallophones produce inharmonic spectra. Inharmonic instruments favor different tuning systems than harmonic ones (Sethares’ timbre-tuning coupling). See instrument-physics.md.
Isomorphic notation — A notation system where identical musical intervals always appear as identical geometric shapes, regardless of key. Transposition is pure translation. The chromatic grid achieves this by eliminating enharmonic distinctions. See notation-layer.md.
Just intonation — A tuning system using exact small-integer frequency ratios (e.g., 3:2, 5:4, 7:4) for intervals. Produces pure harmonic alignment at the cost of unequal step sizes and limited transposability. See frequency-ratios.md.
Key — A step-subset (scale) combined with a designated tonal center (root). Defines which pitches are “home” and establishes the tension-resolution framework for functional analysis. See functional-analysis.md.
Leading tone — A pitch one chromatic step below the tonal center that creates strong directional pull toward resolution. Its proximity (small step-interval) produces spectral tension that resolves when the tonal center arrives. See functional-analysis.md.
Line spectrum — A spectrum consisting of discrete peaks at specific frequencies (harmonics), characteristic of periodic sounds. Each peak represents a sinusoidal component at an integer multiple of the fundamental. Contrasts with continuous spectrum. See fourier-analysis.md.
Loudness — The perceptual magnitude of a sound, determined by both amplitude and frequency. Not linearly related to physical pressure — heavily shaped by equal-loudness contours. Measured in phons (loudness level) or sones (perceived loudness). See equal-loudness.md.
Membrane — A vibrating two-dimensional surface (e.g., drumhead) whose modes are governed by Bessel functions rather than the simple integer harmonics of strings and pipes. Produces inharmonic spectra with less definite pitch. See instrument-physics.md.
Missing fundamental — The perception of a pitch not physically present in a sound, inferred by the auditory system from the harmonic pattern of the upper partials. Demonstrates that pitch is a neural construct, not a direct physical measurement. See missing-fundamental.md.
Mode — A rotation of a step-subset that treats a different member as the starting point, producing a different gap pattern from the same pitch collection. Each mode has a distinct intervallic character. See scales.md.
Modular arithmetic (mod 12) — The number system where values wrap around every 12, making 13 ≡ 1 and −1 ≡ 11. The mathematical foundation of step-number transposition, cycle generation, and interval calculation in 12-TET. See step-7-cycle.md.
Modulation — Shifting the tonal center from one key to another during a piece. Can be achieved by introducing step-combos that belong to the new key but not the old, pivoting the listener’s harmonic expectations. See functional-analysis.md.
Natural triad (4:5:6) — The frequency ratio pattern formed by harmonics 4, 5, and 6 of a single fundamental. Produces maximum spectral overlap and perceptual fusion. The physics behind what Western theory calls a “major triad.” See harmonic-series.md.
Neural entrainment — The tendency of neural oscillations to synchronize with a periodic external stimulus (the pulse). Underlies the perception of beat and groove, and explains why rhythm feels physically compelling. See rhythm.md.
Node — A point on a standing wave where displacement is always zero. Nodes occur at fixed boundaries and between antinodes. The number of nodes on a string equals the mode number plus one. See standing-waves.md.
Nyquist frequency — Half the sampling rate (Fs/2). The highest frequency that can be accurately represented in a sampled signal. Frequencies above the Nyquist limit alias into lower frequencies. See digital-audio.md.
Octave — The interval between two frequencies in a 2:1 ratio. Pitches separated by an octave are perceived as the same “chroma” (pitch class) at different heights. The octave is the most universal musical interval across cultures. See frequency-ratios.md.
Octave equivalence — The perceptual principle that tones separated by exact octaves (2:1 ratio) are perceived as the same pitch class. Enables Shepard tone recycling and underlies the 12-step chromatic system. See shepard-tones.md.
Octave.step notation — PhizMusic’s formal pitch coordinate system: octave number followed by chromatic step position. Example: 4.7 = octave 4, step 7 = So4. Makes transposition arithmetic: add a constant to the step (mod 12). See naming-system.md.
Ombak — A deliberate beating texture in Balinese gamelan created by tuning paired instruments slightly apart. The controlled roughness is an aesthetic goal, not an error — demonstrating that consonance preferences are culturally shaped. See pentatonic-cross-cultural.md.
Onset — The moment a sound event begins, measured as a time position within a cycle. Expressed as a fraction of the cycle length. (Western equivalent: beat position within a measure.) See rhythm.md.
Open voicing — A chord arrangement where notes are spread across more than one octave. Creates a wider spectral distribution with more space between partials, often perceived as richer or more spacious than close voicing. See chords.md.
Overtone — Any frequency component above the fundamental. May be harmonic (integer multiple of fundamental) or inharmonic (non-integer multiple). “Overtone” is a general term; “harmonic” specifically means integer-multiple overtones.
Partial — Any individual frequency component of a complex sound. Includes the fundamental and all overtones, whether harmonic or inharmonic. The most general term for a spectral component. See fourier-analysis.md.
Pentatonic scale — A 5-of-12 step-subset, most commonly {0, 2, 4, 7, 9} (major pentatonic). Maximizes consonant 3:2 and 4:3 intervals while avoiding roughness-producing small step-intervals. Appears independently in cultures worldwide. See pentatonic-cross-cultural.md.
Perceptual compression — Audio encoding (MP3, AAC, Opus) that exploits auditory masking to remove frequency components the listener cannot hear, achieving significant data reduction with minimal perceived quality loss. See auditory-masking.md.
Period — The time for one complete oscillation of a periodic waveform, measured in seconds. Inversely related to frequency: T = 1/f. A 440 Hz tone has a period of ≈ 2.27 ms. See sound-waves.md.
Phase — The position of a sinusoidal component within its cycle at a given moment, measured in degrees or radians. Different phase relationships between components affect waveform shape but have minimal effect on perceived timbre. See fourier-analysis.md.
Phase locking — The tendency of auditory nerve fibers to fire at a consistent phase of the stimulus waveform. Provides temporal pitch information for frequencies below ~4 kHz, complementing place coding on the basilar membrane. See ear-cochlea.md.
Phon — Unit of perceived loudness that accounts for frequency-dependent hearing sensitivity. Defined relative to a 1 kHz reference: N phon = the loudness of a 1 kHz tone at N dB SPL. A 100 Hz tone needs more dB SPL to achieve the same phon level. See equal-loudness.md.
Piano roll — A time-proportional, chromatic-grid notation used in digital audio workstations (DAWs) where pitch rows are equally spaced and time flows horizontally. Transposition is geometric translation. See notation-layer.md.
Pitch — The perceptual attribute of sound corresponding to frequency (or inferred fundamental). Experienced as “high” or “low.” Pitch is a neural construct — the missing fundamental demonstrates it can exist without a matching physical frequency. See sound-waves.md.
Pitch circularity — The auditory illusion where pitch appears to rise (or fall) endlessly without actually getting higher (or lower). Achieved in Shepard tones by recycling octave-spaced components under a fixed spectral envelope. See shepard-tones.md.
Place coding — The cochlear encoding mechanism where which position along the basilar membrane responds most strongly indicates the frequency content of the input sound. Dominant for frequencies above ~4 kHz. See ear-cochlea.md.
Plomp–Levelt model — A psychoacoustic function describing the roughness (sensory dissonance) between two pure tones as a function of their frequency separation relative to critical bandwidth. The building block of Sethares’ dissonance curves. See dissonance-curves.md.
Polyrhythm — Two or more independent pulse streams with different periods sounding simultaneously. Creates a composite rhythmic texture where the streams periodically align and diverge. See rhythm.md.
Position (functional) — The role of a step-combo within a key, determined by which scale degree serves as its root and how its spectral content relates to the tonal center. Positions create the tension-resolution framework of harmonic function. See functional-analysis.md.
Prime limit — The largest prime factor appearing in the frequency ratios of a tuning system. 3-limit (Pythagorean) uses only ratios with factors of 2 and 3. 5-limit adds factor 5 (major/minor thirds). 7-limit and 11-limit incorporate progressively more complex ratios. Unifies world tuning systems: 3-limit (Chinese), 5-limit (Indian/Western), 11-limit (Arabic). See frequency-ratios.md.
Progression — A sequence of step-combos (chords) ordered in time. Characterized by voice-leading distances between successive combos and the tension-resolution arc created by their functional positions. See chord-progressions.md.
Pulse — A regular temporal reference beat. Measured by period T (seconds between pulses) or rate (BPM = 60/T). The pulse is the temporal anchor for all rhythm. (Western equivalent: “beat.”) See rhythm.md.
Pythagorean comma — The small interval (~23.46 cents) by which 12 stacked 3:2 ratios overshoot 7 octaves. Demonstrates that pure-ratio tuning cannot close the octave exactly — the fundamental motivation for equal temperament. See twelve-tet.md.
Quantization — The process of rounding each audio sample’s amplitude to the nearest available discrete level. Introduces quantization noise proportional to step size. Higher bit depth = more levels = less noise. See digital-audio.md.
Railsback curve — The empirical observation that well-tuned pianos have stretched octaves (wider than 2:1) to accommodate the inharmonicity of stiff piano strings. The ear prefers aligned partials over mathematically exact octaves. See twelve-tet.md.
Ratio-set — A chord described by the frequency ratios of its components. Example: 4:5:6 describes the ratio relationship of a chord whose components are in the 4th, 5th, and 6th harmonic positions. Ratio-sets explain why a chord sounds the way it does (physics). See chords.md.
Resolution — The perceptual arrival at a stable, high-fusion spectral state after a period of tension. Occurs when a progression moves from a dissonant or unstable step-combo to one with strong harmonic template alignment. See functional-analysis.md.
Resultant bass — A low pitch perceived when organ pipes sound upper harmonics without the fundamental itself. An application of the missing fundamental phenomenon — the brain infers the bass pitch from the harmonic pattern. See missing-fundamental.md.
Room modes — Three-dimensional standing-wave resonances in an enclosed space that create frequency-specific peaks and nulls at different positions. Depend on room dimensions and the speed of sound. See digital-audio.md.
Root — The reference pitch of a step-combo (chord), typically the lowest note in root position. The root’s frequency serves as the denominator when expressing the chord as a ratio-set. See chords.md.
Roughness — The perceptual quality caused by two frequencies within critical bandwidth beating against each other. Creates a “buzzy” or “grating” sensation. Maximum roughness occurs when two tones are separated by approximately 1/4 of the critical bandwidth. See consonance-dissonance.md.
RT60 (reverberation time) — The time for sound energy in a room to decay by 60 dB. Estimated by the Sabine equation: RT60 = 0.161 × V / A, where V is room volume and A is total absorption area. See digital-audio.md.
Sampling rate — The number of discrete samples taken per second during analog-to-digital conversion, measured in Hz. Must be at least twice the highest frequency of interest (Nyquist criterion). CD audio uses 44,100 Hz. See digital-audio.md.
Sanfen sunyi (三分損益) — The ancient Chinese method of “subtract and add one-third” for deriving pitch sets by repeatedly applying the 3:2 ratio. Produces the same pentatonic and heptatonic structures as Western Pythagorean tuning, demonstrating convergent engineering. See pentatonic-cross-cultural.md.
Sensory dissonance — Roughness quantified: the total perceptual clash computed by summing pairwise Plomp–Levelt contributions across all interacting partials of two tones. Depends on both frequency separation and timbre. See dissonance-curves.md.
Shepard tone — An auditory illusion constructed from octave-spaced sinusoidal components weighted by a fixed Gaussian spectral envelope. When shifted by semitone steps, produces the perception of endlessly ascending or descending pitch. See shepard-tones.md.
Simultaneous masking — Masking that occurs when two sounds are present at the same time. A louder sound raises the hearing threshold for nearby frequencies, with an asymmetric spread toward higher frequencies (upward spread of masking). See auditory-masking.md.
Spectral centroid — The amplitude-weighted average frequency of a spectrum; a measure of spectral “brightness.” In Shepard tones, the centroid is held approximately constant by the Gaussian envelope while component frequencies shift. See shepard-tones.md.
Spectral envelope — The smooth curve describing the overall amplitude distribution across frequencies in a sound’s spectrum. Determined by the resonating body (formants) and excitation method. Shapes timbre independently of fundamental pitch. See timbre.md.
Spectrum — The distribution of energy across frequencies in a sound. A pure sine wave has a single-frequency spectrum. A complex tone has energy at multiple frequencies (harmonics). The spectrum determines timbre. See fourier-analysis.md.
Standing wave — A stationary vibration pattern formed by the interference of two waves traveling in opposite directions within a bounded system. Characterized by fixed nodes (zero displacement) and antinodes (maximum displacement). See standing-waves.md.
Step-5 cycle — The modular arithmetic cycle generated by repeatedly adding 5 (mod 12). Since gcd(5, 12) = 1, it visits all 12 chromatic positions — traversing the step-7 cycle in reverse, approximating the 4:3 frequency ratio. See step-7-cycle.md.
Step-7 cycle — The modular arithmetic cycle generated by repeatedly adding 7 (mod 12). Since gcd(7, 12) = 1, it visits all 12 chromatic positions, approximating the acoustically salient 3:2 frequency ratio at each step. (Western equivalent: “circle of fifths.”) See step-7-cycle.md.
Step-combo — A chord described by the chromatic step positions of its components relative to the lowest note. Example: {0,4,7} describes a chord with root, a note 4 steps above, and a note 7 steps above. Step-combos specify which notes to play (practical). See chords.md.
Step-interval — The number of chromatic steps between two pitches, ranging from 0 to 11 within one octave. This is the PhizMusic measure of interval size. The 7-step-interval = 3:2 frequency ratio ≈ “perfect fifth” in Western terms. See intervals.md.
Step-number — A pitch identified by its octave number plus chromatic position (0–11). Formal PhizMusic pitch notation. Example: 4.7 = octave 4, step 7 = So4 = G4. Step-numbers make transposition trivial: transpose up by 3 = add 3 (mod 12). See naming-system.md.
Step-subset — A scale defined as a selection of step-numbers from {0, 1, 2, …, 11}. Example: pentatonic major = {0, 2, 4, 7, 9} — five notes selected from the 12 available. Different step-subsets produce different scale “flavors.” See scales.md.
Stretched partials — Overtones whose frequencies are slightly higher than exact integer multiples of the fundamental, caused by physical stiffness in vibrating strings or bars. Shifts dissonance curve minima and motivates octave stretching in piano tuning. See dissonance-curves.md.
String — A vibrating one-dimensional object (fixed at both ends) whose modal frequencies are integer multiples of the fundamental: fₙ = n × (1/2L) × √(T/μ). Produces a harmonic spectrum. See instrument-physics.md.
Subdominant function — The harmonic role of step-combos built on scale degree 4 (or related positions). Creates moderate tension — more motion than tonic but less directional pull than dominant. See functional-analysis.md.
Subtractive synthesis — Building timbres by starting with a spectrally rich waveform (e.g., sawtooth) and removing frequencies with filters. The complement of additive synthesis. See fourier-analysis.md.
Swing — A rhythmic feel produced by systematically lengthening certain subdivisions and shortening others (e.g., long-short alternation on eighth-note pairs). A specific type of groove pattern. See rhythm.md.
Syllable — The Dodeka-derived short name for each chromatic step: Do (0), Ka (1), Re (2), Xo (3), Mi (4), Fa (5), Hu (6), So (7), Bi (8), La (9), Ve (10), Si (11). Casual PhizMusic pitch notation, designed for speaking and singing. See naming-system.md.
Syncopation — Rhythmic emphasis placed on normally weak positions within the cycle, creating tension against the expected pulse pattern. Delays or advances onsets relative to the regular pulse grid. See rhythm.md.
Tempo — The rate of the pulse, measured in BPM (beats per minute) or equivalently by pulse period T (seconds). BPM = 60/T. Determines the absolute time scale of all rhythmic events. See rhythm.md.
Tension — The perceptual state created by step-combos with poor harmonic template alignment or strong roughness relative to the established tonal center. Tension drives the desire for resolution. See functional-analysis.md.
Tetrad — A four-note step-combo (chord). Adds spectral complexity beyond triads by introducing a fourth pitch that modifies the ratio relationships. See chords.md.
Timbre — The perceptual quality of a sound determined by its spectrum — which harmonics are present and at what amplitudes. Same pitch + different spectrum = different timbre. Timbre is why a piano and violin playing the same note sound different. See timbre.md.
Timbre-tuning coupling — Sethares’ principle that the most consonant intervals for a given instrument depend on its spectral content (timbre). Harmonic timbres favor integer-ratio tunings; inharmonic timbres (gamelan, bells) favor non-standard tunings where dissonance curve minima align with the partials. See dissonance-curves.md.
Time domain — A representation of sound showing pressure as a function of time (the waveform/oscillogram). Related to the frequency domain by the Fourier transform. See fourier-analysis.md.
Tonal center — The pitch that serves as the perceptual “home” of a key — the point of maximum stability and resolution. All other pitches’ functional roles are defined by their relationship to this center. See functional-analysis.md.
Tonic function — The harmonic role of step-combos that provide maximum stability and resolution within a key. Typically built on the tonal center itself. Represents the resting state of a harmonic progression. See functional-analysis.md.
Tonotopic organization — The systematic spatial mapping of frequency along the basilar membrane (and preserved through the auditory nerve to the auditory cortex). High frequencies map to the base of the cochlea; low frequencies to the apex. See ear-cochlea.md.
Transposition — Shifting all pitches in a musical passage by the same step-interval. In step-number arithmetic: add a constant (mod 12) to every step-number. In chromatic grid notation, transposition is geometric translation. See functional-analysis.md.
Triad — A three-note step-combo (chord). The most common building block of harmony, with spectral character determined by the frequency ratios between its three components. See chords.md.
Tritone paradox — A perceptual ambiguity where a pair of Shepard tones separated by 6 steps (half an octave) can be heard as ascending or descending, depending on the listener’s internal pitch-class template. See shepard-tones.md.
Tympanic membrane — The eardrum: a thin membrane separating the outer and middle ear that vibrates in response to incoming pressure waves and transmits those vibrations to the ossicles. See ear-cochlea.md.
Unison — The interval of 0 chromatic steps (frequency ratio 1:1). Two tones at the same frequency. The point of maximum fusion and zero roughness. See intervals.md.
Virtual pitch — Another term for the missing fundamental. The perceived pitch that the brain infers from the harmonic pattern of upper partials when the fundamental frequency is physically absent. See missing-fundamental.md.
Voice-leading — The movement between successive chords, measured by the total step-interval traveled by individual notes. Smooth voice-leading minimizes total movement. (Western equivalent: “voice leading” or “part writing.”) See chord-progressions.md.
Voicing — The specific vertical arrangement (register distribution) of a step-combo’s notes. Same step-combo, different voicing = same pitch classes in different octave positions, producing different spectral weight and perceived character. See chords.md.
Waveform — The shape of a pressure wave plotted over time (oscillogram). Related to spectrum by Fourier transform: waveform and spectrum contain the same information in different representations. See sound-waves.md.
Wavelength — The spatial distance between successive compressions (or rarefactions) of a pressure wave. Related to frequency by λ = v/f, where v is the speed of sound. See sound-waves.md.
Weber–Fechner law — The psychophysical principle that perceived intensity is proportional to the logarithm of physical stimulus intensity. Explains why pitch perception maps logarithmically to frequency and loudness maps logarithmically to pressure. See sound-waves.md.
Whole-tone scale — A 6-of-12 step-subset with uniform 2-step gaps: {0, 2, 4, 6, 8, 10}. Has no leading tones and no 3:2 intervals, giving it a floating, directionless quality. See scales.md.
The syllable system (Do, Ka, Re, Xo, Mi, Fa, Hu, So, Bi, La, Ve, Si) is adapted from the Dodeka Music system by Jacques-Daniel Rochat. The 12th syllable is “Si” (not “Ti”). See Naming System for the full naming convention.