# Musikalische Intervalle & Frequenzverhältnisse

Interaktive Erkundung musikalischer Intervalle, Frequenzverhältnisse und Obertöne

> Kanonische Seite: https://elysiatools.com/de/visualizations/musical-intervals

- **Kategorie:** Math

## Überblick

Interactive exploration of musical intervals, frequency ratios, and the harmonic series with real audio playback using Web Audio API. Features comprehensive music theory visualization: (1) Piano Keyboard UI - Interactive keyboard showing all notes with clickable keys, visual highlighting of root note (green) and interval note (red), 2-octave range (C3 to B5), support for selecting any root note and interval up to 2 octaves. (2) Frequency Spectrum Analyzer - Real-time FFT-style display showing fundamental frequencies and harmonics of both notes, color-coded bars (green=root fundamental, red=interval fundamental, orange=harmonics), harmonic series comparison table showing first 8 harmonics with frequency ratios, visual representation of harmonic alignment creating consonance. (3) Waveform Visualization - Three separate canvas displays: root note waveform (single frequency), interval note waveform (single frequency), combined waveform showing interference pattern and beating, real-time animation during playback, multiple wave types supported (sine, triangle, square, sawtooth). (4) Interval Calculator - Automatic frequency ratio calculation (r = f₂/f₁), cent value using formula ¢ = 1200×log₂(r), just intonation vs equal temperament comparison showing difference in cents, interval name identification (unison, octave, perfect fifth, major third, etc.). (5) Interval Presets - One-click access to common intervals: Unison (1:1, 0¢), Octave (2:1, 1200¢), Perfect Fifth (3:2, ~702¢), Perfect Fourth (4:3, ~498¢), Major Third (5:4, ~386¢), Minor Third (6:5, ~316¢), Major Sixth (5:3, ~884¢), Minor Sixth (8:5, ~814¢). Adjustable parameters: root note selection (all 12 chromatic notes), root octave (3, 4, 5), interval in semitones (0-24), wave type selection, volume control. Real-time audio playback using Web Audio API with dual oscillators (one for root, one for interval), independent gain control, master volume, support for all four standard waveforms, 60fps canvas rendering. Mathematical foundation section covering: Frequency Ratio Formula r = f₂/f₁ (ratio of higher to lower frequency), Equal Temperament Formula f = f₀×2^(n/12) (frequency n semitones above f₀, basis of modern Western music), Cents Formula ¢ = 1200×log₂(r) (logarithmic interval measure, 1200¢ = octave), Common Just Intonation Ratios: octave 2:1, perfect fifth 3:2, perfect fourth 4:3, major third 5:4, minor third 6:5, major sixth 5:3, minor sixth 8:5, Harmonic Series f, 2f, 3f, 4f, 5f... (integer multiples explaining instrument timbre and consonance). Comprehensive educational content: What are Musical Intervals? (definition as distance between two pitches, consonance vs dissonance based on frequency ratio simplicity, psychoacoustic perception of intervals), Just Intonation vs Equal Temperament (just intonation uses pure ratios from harmonic series for perfect consonance but makes modulation difficult, equal temperament divides octave into 12 equal semitones enabling key changes but slightly compromising interval purity except octave, historical context and practical trade-offs), The Harmonic Series (fundamental frequency + overtones at integer multiples, explains why certain intervals are consonant - their harmonics align, gives each instrument unique timbre, basis of timbre and sound color), Real-World Applications (Music Theory: intervals are foundation of melody, harmony, and chord construction in all musical traditions, Instrument Design: piano and fretted instrument manufacturers use equal temperament calculations for string/fret placement, Tuning Systems: different cultures/historical periods use various systems - just intonation, meantone, well temperament, Audio Engineering: harmonics understanding helps with EQ, compression, avoiding phase cancellation, Choral Singing: professional singers drift toward just intonation for purer harmonies), and Listening Guide (start with unison and octave - most consonant, waves align perfectly, perfect fifth 3:2 - most stable after octave, foundation of Western harmony, compare major third 5:4 bright/happy vs minor third 6:5 sad/dark - small ratio changes dramatically affect mood). Interactive features: Click keyboard keys to select root note, preset buttons instantly change interval with auto-play, semitone slider for custom intervals, wave type buttons change timbre, stop button terminates audio immediately, real-time updates of all calculations and visualizations. Responsive layout with keyboard section (120px height), controls panel with preset grid and note selection, three waveform displays (80px each), spectrum analyzer (250px height), harmonics comparison table, formula cards grid, and educational sections. Canvas-based rendering with smooth 60fps animations, color-coded keys (green=root, red=interval), accurate piano keyboard geometry with black key positioning, professional music theory appearance. Multi-language support (zh, en, es, fr, de, ru, pt, ja) with complete translations of musical terminology, interval names, formulas, and educational content.

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