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The Overtone Spectrum

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The Overtone Spectrum

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Visit https://alexisbaskind.net/teaching for a full interactive version of this course with sound and video material, as well as more courses and material.

Course series: Fundamentals of acoustics for sound engineers and music producers
Level: undergraduate (Bachelor)
Language: English
Revision: February 2020
To cite this course: Alexis Baskind, The Overtone Spectrum
course material, license: Creative Commons BY-NC-SA.


Course content:
1. What is the overtone spectrum?
Time and frequency representation of a sound, harmonic and inharmonic sounds, overtones, harmonic series, fundamental frequency, linear and logarithmic frequency scales

2. Physical generation of overtones
Tone generator, vibration modes, dependence on geometry

3. Shaping of the overtone spectrum in the instrument
resonator, resonance modes, dependence on geometry, formants, overtone singing

4. Designing the overtone spectrum by playing
harmonics glissandi, natural string harmonics, dependence of tone color on dynamics, decay, radiation patterns

5. Conclusion

Visit https://alexisbaskind.net/teaching for a full interactive version of this course with sound and video material, as well as more courses and material.

Course series: Fundamentals of acoustics for sound engineers and music producers
Level: undergraduate (Bachelor)
Language: English
Revision: February 2020
To cite this course: Alexis Baskind, The Overtone Spectrum
course material, license: Creative Commons BY-NC-SA.


Course content:
1. What is the overtone spectrum?
Time and frequency representation of a sound, harmonic and inharmonic sounds, overtones, harmonic series, fundamental frequency, linear and logarithmic frequency scales

2. Physical generation of overtones
Tone generator, vibration modes, dependence on geometry

3. Shaping of the overtone spectrum in the instrument
resonator, resonance modes, dependence on geometry, formants, overtone singing

4. Designing the overtone spectrum by playing
harmonics glissandi, natural string harmonics, dependence of tone color on dynamics, decay, radiation patterns

5. Conclusion

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The Overtone Spectrum

  1. 1. Alexis Baskind The Overtone Spectrum Alexis Baskind, https://alexisbaskind.net
  2. 2. Alexis Baskind The Overtone Spectrum Course series Fundamentals of acoustics for sound engineers and music producers Level undergraduate (Bachelor) Language English Revision January 2020 To cite this course Alexis Baskind, The Overtone Spectrum, course material, license: Creative Commons BY-NC-SA. Full interactive version of this course with sound and video material, as well as more courses and material on https://alexisbaskind.net/teaching. Except where otherwise noted, content of this course material is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License. The Overtone Spectrum
  3. 3. Alexis Baskind Outline 1. What is the overtone spectrum? 2. Physical generation of overtones 3. Shaping of the overtone spectrum in the instrument 4. Designing the overtone spectrum by playing 5. Conclusion The Overtone Spectrum
  4. 4. Alexis Baskind What is a spectrum? => Any sound is usually represented by two means: a waveform… (time representation) …or a instantaneous spectrum (frequency representation) frequency (log scale) level time The Overtone Spectrum
  5. 5. Alexis Baskind What is a spectrum? => Any sound is usually represented by two means: frequency (log scale) level time The spectrum ist the decomposition of an excerpt of the sound according to its frequency content The Overtone Spectrum
  6. 6. Alexis Baskind What are Overtones? • According to (not verifiable) legends, Pythagoras (6th century BC) discovered that all harmonic sounds are composed of several tones that share simple frequency ratios • This observation probably lead to the so-called harmonic series in music • Joseph Fourier (1768-1830) gave a modern mathematical background to this discover • The corresponding theory is called Fourier analysis • It’s been used not only for harmonic but also for inharmonic sounds The Overtone Spectrum
  7. 7. Alexis Baskind What are Overtones? original waveform (square wave) 1st harmonic 3rd harmonic 5th harmonic 7th harmonic 9th harmonic reconstructed signal until order N = sum of the first N harmonics The Fourier decomposition in the time domain The Overtone Spectrum
  8. 8. Alexis Baskind What are overtones ? Overtones are all frequency peaks in a spectrum except its fundamental frequency frequency (Hz) (linear scale) level (dB) Fundamental frequency Overtones The Overtone Spectrum
  9. 9. Alexis Baskind Frequency structure of overtones Fundamental frequency (=> most of time, provides the pitch of harmonics sounds) Overtones => also called « harmonics » for harmonic sounds F 2F 3F 4F … Harmonic sounds: overtone frequencies are multiples of the fundamental frequency frequency (Hz) (linear scale) level (dB) The Overtone Spectrum The frequency difference between neighboring overtones is always the same Partials = Fundamental Frequency + Overtones
  10. 10. Alexis Baskind Frequency structure of overtones Beware of the representation: harmonics don‘t look equally spaced on a logarithmic frequency scale, although they actually are Frequency (Hz) (linear Scale) level (dB) Frequency (Hz) (log Scale) level (dB)  F 2F 3F 4F 5F 6F 7F 8F F 2F 3F 4F ...5F The Overtone Spectrum
  11. 11. Alexis Baskind Frequency structure of overtones Harmonic sounds: overtone frequencies are multiples of the fundamental frequency Example: picked double bass, G2 frequency (log scale) level 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB The Overtone Spectrum
  12. 12. Alexis Baskind Frequency structure of overtones 98 Hz 196 Hz 294 Hz 392 Hz 490 Hz … Fundamental frequency = 98 Hz Harmonic sounds: overtone frequencies are multiples of the fundamental frequency frequency (log scale) level 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB Example: picked double bass, G2 The Overtone Spectrum
  13. 13. Alexis Baskind Frequency structure of overtones Inharmonic sounds: overtone frequencies are not multiples of the fundamental frequency Fundamental frequency Overtones => the frequency structure is not regular frequency (Hz) (linear scale) level (dB) The Overtone Spectrum Partials = Fundamental Frequency + Overtones
  14. 14. Alexis Baskind Frequency structure of overtones Example: crash cymbal hit with a stick frequency (log scale)10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB level Inharmonic sounds: overtone frequencies are not multiples of the fundamental frequency The Overtone Spectrum
  15. 15. Alexis Baskind Frequency structure of overtones Example: crash cymbal hit with a stick 105 Hz 120 Hz 154 Hz 210 Hz 276 Hz … frequency (log scale)10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB Inharmonic sounds: overtone frequencies are not multiples of the fundamental frequency level The Overtone Spectrum
  16. 16. Alexis Baskind Why is the overtone spectrum important for music? The frequencies and amplitudes over the overtones determine largely the tone color => The overtone spectrum is an important attribute of a sound, and consequently of an instrument, of a playing technique... The Overtone Spectrum
  17. 17. Alexis Baskind Outline 1. What is the overtone spectrum? 2. Physical generation of overtones 3. Shaping of the overtone spectrum in the instrument 4. Designing the overtone spectrum by playing 5. Conclusion The Overtone Spectrum
  18. 18. Alexis Baskind Physical generation of overtones Classical model of a music instrument: String instruments, Piano: strings Brass: lips Woodwinds: reed, airflow (flute) Drums: skin (drumhead) … String instruments: body and neck Piano: body, resonance board, Winds: pipe, horn Drums: body … The Radiation pattern depends among others on frequency and on the note being played Tone generator Resonator Radiation The Overtone Spectrum
  19. 19. Alexis Baskind Physical generation of overtones The Overtone Spectrum Basic vibration pattern for a bowed string Example: oscillation of bowed string for a violin, shown in slow motion Source: ViolinB0W, Creative Commons Attribution 3.0 license
  20. 20. Alexis Baskind Physical generation of overtones • The vibration of the string can be decomposed in simpler vibrations, each of them at a single frequency (see 1. part) • Each sinusoidal vibration is called Mode of vibration (or simply „Mode“) frequency (linear scale) level F 2F 3F 4F • The fundamental frequency “F” depends on the length, the diameter and the tension of the string Mode 1 Mode 2 Mode 3 Mode 4 • Each mode corresponds to the fundamental or to one overtone The Overtone Spectrum = + + + + ... Position Bow direction
  21. 21. Alexis Baskind Physical generation of overtones • The time evolution of the oscillation of the string, measured at one point, is also harmonic regarding the time axis • Therefore, the waveform on the time axis can be decomposed in fundamental frequency and overtones Position Measurement point The Overtone Spectrum time Animation generated with Matlab-Code from the University of Wyoming
  22. 22. Alexis Baskind Physical generation of overtones • In wind instruments, overtones originate from the coupled oscillation of lips (for brass) / reeds (woodwinds) and of the tube • The oscillation frequency of the lips/reed is determined by the oscillation of the air flow in the tube and depends on its length Wind Instruments (Source: IWK - Music acoustic Vienna / Matthias Bertsch) The Overtone Spectrum
  23. 23. Alexis Baskind Physical generation of overtones The Overtone Spectrum Example: vibration of a cymbal hit with a stick, shown in slow motion Inharmonic Instruments
  24. 24. Alexis Baskind • Inharmonic oscillations can also be decomposed in oscillation modes frequency level Inharmonic Instruments Physical generation of overtones • Like for harmonic oscillations, each mode corresponds to a partial (= fundamental frequency or overtone) • The frequencies of the overtones are not on an harmonic series (i.e. they are not multiple of the fundamental frequency) The Overtone Spectrum Animations from Olex Alexandrov
  25. 25. Alexis Baskind Outline 1. What is the overtone spectrum? 2. Physical generation of overtones 3. Shaping of the overtone spectrum in the instrument 4. Designing the overtone spectrum by playing 5. Conclusion The Overtone Spectrum
  26. 26. Alexis Baskind Shaping of the Overtone Spectrum String instruments, Piano: strings Brass: lips Woodwinds: tongue, airflow (flute) Drums: skin (drumhead) … String instruments: body and neck Piano: body, resonance board, Winds: pipe, horn Drums: body … The radiation pattern depends among others on frequency and on the note being played Tone generator Resonator Radiation The Overtone Spectrum • The overtones created in the generator are shaped in the resonator (=instrument body • The (body and air) vibrations in the resonator are also made of modes
  27. 27. Alexis Baskind The whole body also has vibration modes The Overtone Spectrum example: some measured vibration modes (exaggerated) for the body of a guitar Mode 1: “bending” mode (≈60 Hz) Source: Dan Russell (the red plate represents the motion of air in the soundhole)
  28. 28. Alexis Baskind The whole body also has vibration modes The Overtone Spectrum example: some measured vibration modes (exaggerated) for the body of a guitar Mode 2: “breathing” mode (≈100 Hz), synchronized with the Helmholtz resonance (see later) Source: Dan Russell (the red plate represents the motion of air in the soundhole)
  29. 29. Alexis Baskind The whole body also has vibration modes The Overtone Spectrum example: some measured vibration modes (exaggerated) for the body of a guitar Mode 3 (≈190Hz) Mode 4 (≈200Hz) Mode 5 (≈220Hz) Mode 6 (≈230Hz) Mode 7 (≈260Hz) Mode 8 (≈315Hz) Mode 9 (≈380Hz) Mode 10 (≈480Hz) Mode 11 (≈750Hz) Source: Dan Russell
  30. 30. Alexis Baskind The whole body also has vibration modes The Overtone Spectrum • The geometry of the body is carefully designed to fine- tune the modes in amplitude and frequency • For instances, guitar top plates (as well as soundboards on piano) have braces on the back Source: Neville H. Fletcher, Thomas D. Rossing, The Physics of Musical Instruments
  31. 31. Alexis Baskind The whole body also has vibration modes The Overtone Spectrum • Those vibrations entail resonances (= formants) in the overtone spectrum which are characteristic of the instrument • The formants are independent of the pitch example: some measured vibration modes (exaggerated) of the top and back of a violin 457 Hz 545 Hz 723 Hz 850 Hz Frequenz (linear) Pegel Animations by Terry Borman
  32. 32. Alexis Baskind The Human Voice In human voice, vowels are determined by a precise controle of the resonances in the mouth, the nasal cavity and the throat Source: J. Meyer, Akustics and the Performance of Music Frequency The Overtone Spectrum
  33. 33. Alexis Baskind The Human Voice The vowel („i“) remains unchanged, only the fundamental frequency changes Formants are independent of the fundamental frequency! (this is true for all instruments, not only for the voice) Formants The Overtone Spectrum
  34. 34. Alexis Baskind The Human Voice Example: Overtone singing The Overtone Spectrum 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB frequency (log scale) level Overtone singing implies a very precise control of the formants, i.e. the resonances of the mouth and the throat, independently from the vocal folds => The resonance is so strong in the 1 kHz-2,5 kHz zone that a second pitch is perceived above the fundamental
  35. 35. Alexis Baskind Outline 1. What is the overtone spectrum? 2. Physical generation of overtones 3. Shaping of the overtone spectrum in the instrument 4. Designing the overtone spectrum by playing 5. Conclusion The Overtone Spectrum
  36. 36. Alexis Baskind • Depending on the playing technique, Dynamics and radiation pattern, the overtones can be amplified or in contrary silenced • This influences two main features of the sound significantly: 1. The Pitch 2. The Tone Color Designing the overtone spectrum by playing The Overtone Spectrum
  37. 37. Alexis Baskind Selecting overtones (the slide of the trombone does not move, only the lips are more and more tightened and the blowing pressure higher and higher) 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB frequency (log scale) level Example 1: natural harmonics glissando on a bass trombone (fundamental: 59 Hz = Bb1) => The frequency of the overtones remains constant, only the amplitude is changing (in red: overtone frequencies until 1,7kHz) The Overtone Spectrum
  38. 38. Alexis Baskind Selecting overtones Example 2: natural string harmonics (“flageolets”) on a double bass based on open string D2 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB level => The frequency of the overtones remains constant, only the amplitude is changing (in red: overtone frequencies until 2,2kHz) The Overtone Spectrum The string is gently touched but not pressed on the fingerboard Frequency (log)
  39. 39. Alexis Baskind Overtones depend on dynamics In most instruments, the tone color changes drastically with dynamics Example 1: crescendo with bass trombone The volume of low-frequency overtones rises only slightly, while more and more high-frequency overtones appear during the crescendo, thus making the sound brighter and louder (= “brassy”) 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB frequency (log scale) level The Overtone Spectrum
  40. 40. Alexis Baskind Overtones depend on dynamics Example 2: crescendo roll on cymbal with wool mallets In most instruments, the tone color changes drastically with dynamics 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB frequency (log scale) level The Overtone Spectrum The volume of low-frequency overtones rises only slightly, while more and more high-frequency overtones appear during the crescendo, thus making the sound brighter and louder
  41. 41. Alexis Baskind Release: Time evolution of overtones In the resonance, high-frequency overtones decay faster than low-frequency overtones in most of acoustic instruments Example 1: picked double bass, G2 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB frequency (log scale) level The Overtone Spectrum
  42. 42. Alexis Baskind Release: Time evolution of overtones In the resonance, high-frequency overtones decay faster than low-frequency overtones Example 2: crash cymbal hit with a stick 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB frequency (log scale) level The Overtone Spectrum
  43. 43. Alexis Baskind < 450 Hz (omnidirectional) • Low frequencies radiate in all directions 650 Hz Radiation pattern of a trombone The Overtone Spectrum Example: angular width of the regions of main radiation (between -3 dB and 0 dB relative to the maximum level in the band) for a trombone for various frequency zones 500 Hz 1 kHz Between 2 and 5 kHz Between 7 and 10 kHz (highly directional) • Except for the 650 Hz zone (the first formant), directivity increases with frequency
  44. 44. Alexis Baskind Radiation pattern of a trumpet The Overtone Spectrum The trumpet has a similar radiation pattern than the trombone, but shifted up in frequency The limit for omnidirectional radiation is around 500 Hz (source: J. Meyer)
  45. 45. Alexis Baskind Radiation pattern: trumpet The Overtone Spectrum 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB frequency (log scale) level Example: trumpet miked with a TLM-103: 1 – on-axis
  46. 46. Alexis Baskind Radiation pattern: trumpet Example: trumpet miked with a TLM-103: 2 – 90° off-axis The Overtone Spectrum 10 Hz 100 Hz 1 kHz 10 kHz 0 dB -60 dB -120 dB frequency (log scale) level
  47. 47. Alexis Baskind Outline 1. What is the overtone spectrum? 2. Physical generation of overtones 3. Shaping of the overtone spectrum in the instrument 4. Designing the overtone spectrum by playing 5. Conclusion The Overtone Spectrum
  48. 48. Alexis Baskind Conclusion • The overtone spectrum is an essential component of the specific tone color of an instrument • It is related to fundamental aspects of the perception of a sound, like: - Pitch - Harmonicity - Dynamics • Its characteristics depends dramatically on the type and construction of the instrument, as on the playing technique • Those properties are used by the perception in order to recognize the instrument and the playing technique The Overtone Spectrum
  49. 49. Alexis Baskind But... • The sound of an instrument does not only consist in its overtone spectrum! • The perception of the timbre is not limited to the tone color. It‘s much more complex and involve more features, that are equally important: – The Noise component, i.e. the part of the spectrum, which cannot be modelled with sine waves – The time evolution of frequencies: for instance Vibrato – The time evolution of the amplitudes in the spectrum: for instance Tremolo, Attack, Release... => The overtone spectrum is a very important, but not unique component of the sound of a music instrument The Overtone Spectrum
  50. 50. Alexis Baskind To go further... • Jürgen Meyer, Acoustics and the Performance of Music, Springer-Verlag New York (2009) • Neville H. Fletcher, Thomas D. Rossing, The Physics of Musical Instruments, Springer-Verlag The Overtone Spectrum

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