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Music: An Interdisciplinary Combination of Physics, Mathematics, and Biology

Music: An Interdisciplinary Combination of Physics, Mathematics, and Biology. Steven A. Jones. This Presentation Draws From. General Engineering Background One year graduate sequence in acoustics (Lecture + Lab, with thanks to Dr. Vic Anderson) Thirty years of playing guitar

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Music: An Interdisciplinary Combination of Physics, Mathematics, and Biology

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  1. Music: An Interdisciplinary Combination of Physics, Mathematics, and Biology Steven A. Jones

  2. This Presentation Draws From • General Engineering Background • One year graduate sequence in acoustics (Lecture + Lab, with thanks to Dr. Vic Anderson) • Thirty years of playing guitar • 20 Years of Research in Medical Ultrasound

  3. This Presentation Might Help • The beginning guitarist • The self-taught guitarist • The formally trained guitarist • Engineers who know little about music • Musicians who want a new perspective

  4. My Musical History • Piano Lessons (Miss Berry) • Uncles (The Banana Song) • Uncle Warren’s Guitar • Led Zeppelin • Fake Books • Classical guitar (Sor) • Lessons from • George Petch (and a new guitar) • Ron Pearl (BCGS) • Alan Goldspeil

  5. Stuff I Didn’t Know When I Started • Where the notes are on the guitar (vs piano). • Importance of rhythm • Triples • 5-tuples (hippopotamus) • Other African animals • What is interpretation? • Concept of Voices • Cerebellar Function • How keys work • How chords work

  6. More Stuff I Didn’t Know • Harmonic Analysis • What tone is • Rubato • Dynamics • Major/Minor 3rd/5th/7th (Close Encounters) • Major/Minor Scales

  7. The Guitar Thick Strings Thin Strings Bridge Nut Each string is a “vibrating string” fixed at both ends.

  8. t1 t2 y x Vibration of a String The wave equation • Behavior in time is the same as behavior in space • Wave Speed depends on tension (T) and string density per unit length ( ) (thanks to V.C. Anderson and J.W. Miles)

  9. General Solution to the Equations Meaning: The string shape can propagate along the string in the forward and/or reverse direction.

  10. Initially Stationary String If the string is not moving initially, must have: Forward and reverse waves are inverted copies (except for constant). -x +x 0

  11. Boundary Conditions Boundary Conditions Initial Conditions Plucked String Struck String

  12. Not for the Squeamish • Warning: Those who tend to pass out at the sight of math may want to leave the room before I present the next slide. I will let you know when it is safe to come back into the room again.

  13. Boundary Conditions Constrain Allowable Frequencies Assume simple harmonic motion:

  14. Harmonics of a String 1

  15. Harmonics of a String 1 Node 2

  16. Harmonics of a String 1 Node 2 3

  17. Harmonics of a String 1 Node 2 3 4

  18. String Shapes/Vibration Modes • 1st Harmonic is a Sine Wave • 2nd Harmonic is 2x the frequency of the 1st • Since the middle of the string doesn’t move for 2nd harmonic, can touch it there & still get vibration. • 3rd harmonic has two nodes (at 1/3 and 2/3rds the string length) • The “harmonics” give pure tones. • Can do harmonics with fretted strings.

  19. Color Fourier Interpretation: Tone depends on the relative loudness & phases of each harmonic. I.e. a string with 1st and 2nd harmonic excited sounds different from a string with 1st and 3rd harmonic excited. Can excite different harmonics by plucking at different locations (i.e. plucking at 1/3rd length will mute the 3rd harmonic).

  20. The Frequencies (Musician’s Terminology) C, C#, D, D#, E, F, F#, G, G#, A, A#, B Do Re Mi Fa Sol La Ti • Major 3rd (C to E) • Major 5th (C to G) • Minor 3rd (C to D#) • Major/Minor 7th (C to A# / B) • Barbershop Quartet (C, E, G, Bb)

  21. The Frequencies (Musician’s Terminology) C, C#, D, D#, E, F, F#, G, G#, A, A#, B Do Re Mi Fa Sol La Ti F D B G E Every Good Boy Deserves Favor (Moody Blues)

  22. The Circle of Fifths C C# B D A# D# A E G# F G F# Come Get Down And Eat Big Fat Cod

  23. The Circle of Fifths C C# B D A# D# A E G# F G F# Come Get Down And Eat Big Fat Cod

  24. The Circle of Fifths C C# B D A# D# A E G# F G F# Come Get Down And Eat Big Fat Cod

  25. The Circle of Fifths C C# B D A# D# A E G# F G F# Come Get Down And Eat Big Fat Cod

  26. The Circle of Fifths C C# B D A# D# A E G# F G F# Come Get Down And Eat Big Fat Cod

  27. The Circle of Fifths C C# = Db B D A# D# A E G# F G F# Come Get Down And Eat Big Fat Duck

  28. The Frequencies (Piano Keyboard) C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C Do Re Mi Fa Sol La Ti Why are there no sharps (black keys) between E&F and B&C? G C E

  29. Other Questions • Why are there 12 notes? • Why are 7 of these “in key?” • Why the “Circle of Fifths?” (Why not the “Circle of Thirds?”) • What’s all this about Major and Minor? • What do Augmented and Diminshed Mean? • Why are there different chords with the same name?

  30. Chords Major C, C#, D, D#, E, F, F#, G, G#, A, A#, B Minor C, C#, D, D#, E, F, F#, G, G#, A, A#, B Diminished C, C#, D, D#, E, F, F#, G, G#, A, A#, B Augmented C, C#, D, D#, E, F, F#, G, G#, A, A#, B

  31. Chords Diminished Seventh C, C#, D, D#, E, F, F#, G, G#, A, A#, B C7dim is the same as D#7dim, F#7dim and A7dim. This is an ambiguous chord and can resolve into many possible chords. There are only 4 diminished 7th chords.

  32. Frequencies Used in Music Frets on a Guitar • Each fret shortens the string by the same percentage (r) of it’s current length. • Frets must get closer together. • Takes 12 frets to get to ½ the length. • Must have • Thus, r = 0.943874313 • Or 1/r = 1.059463094

  33. Postulates • Tones separated by nice fractional relationships are pleasing. • Tones separated by complicated fractional relationships are less pleasing. These postulates are the basis of “Just Intonation” (Slogan: “It’s not just intonation, its Just intonation!”)

  34. Pythagorean Scale • Pythagorus proposed the scale cdefgab, based on a series of “perfect 5ths” • c = 1 • g = cx3/2 (i.e. 1 ½) • d = (gx3/2)/2 = c x (9/4)/2 (i.e. 1 1/8) • a = (dx3/2) = c x (27/16) (i.e. 1 11/16) Circulate through c-g-d-a-e-b-f-c But note that the second “c” doesn’t work. It’s

  35. “Error” of Frequencies N Ratio Note % Error Fraction 0 1.0000 C 0 1 1 1.0595 C# 0.29 1 1/16 ++/ 2 1.1225 D 0.23 1 ⅛ ++/++ 3 1.1892 D# 1.90 1 1/6 / 4 1.2599 E 0.79 1 ¼++/++ 5 1.3348 F 0.12 1 ⅓ ++/++ 6 1.4142 F# 2.77 1⅜ /++ 7 1.4983 G 0.11 1 ½ +++/+++ 8 1.5874 G# 2.37 1 ⅝ /++ 9 1.6818 A 0.90 1 ⅔ +/++ 10 1.7818 A# 1.78 1 ¾/++ 11 1.8877 B 0.68 1⅞ ++/++ 12 2.0000 C 0

  36. Harmonics of a String N Octave N/O Note 1 1 1 C 2 2 1 C 3 2 1 ½ G * 4 4 1 C 5 4 1 ¼ E * 6 4 1.5 G 7 4 1 ¾ Bb * 8 8 1 C 9 8 1 1/8 D 10 8 1 ¼ E 11 8 1 3/8 F# (ish) 12 8 1 ½ B 13 8 1 5/8 G# (ish) When you play a C, you are also playing G, E, Bb, etc. in different amounts and in Just Intonation. I.e., the combination CEG is “natural.” Higher harmonics decay rapidly.

  37. Harmonics of a String N Octave N/O Note 1 1 1 C 2 2 1 C 3 2 1 ½ G * 4 4 1 C 5 4 1 ¼ E * 6 4 1.5 G 7 4 1 ¾ Bb * 8 8 1 C 9 8 1 1/8 D 10 8 1 ¼ E 11 8 1 3/8 F# (ish) 12 8 1 ½ B 13 8 1 5/8 G# (ish) When you play a C, you are also playing G, E, Bb, etc. in different amounts and in Just Intonation. I.e., the combination CEG is “natural.” Higher harmonics decay rapidly.

  38. Harmonics • Have a “pure” tone to them. • Were not invented by Yes or Emerson, Lake and Palmer. • Can be combined with natural tones. • Granados

  39. Notes that are in key … • Are close matches to “nice” fractional values. • Match the natural harmonics of the vibrating string (C E G D B Bb) • Are early members of the circle of 5ths (C G D A E B).

  40. More Complex Relationships • Inverse Relationships • F is a Fourth to C • C is a Fifth to F • Bootstrapping • G# is not in C, but • E is the Major 3rd in C and • G# is the Major 3rd of E so • Perhaps we can get to the G# note through E

  41. Historical Notes • 12 tone system (Even Tempered Scale) relatively recent invention (ca. 1700s). • Bach used “Well Tempered Scale” • We don’t hear Toccata & Fugue the way it was written.

  42. Why 12 Notes? • Even temperament would not work as well with other spacings (besides 1/12) • Works pretty well with a spacing of 17. • Other countries use a 17 tone system. • Google: “17 tone” music

  43. 17 Tone Scale

  44. Color • Determined by the weights of the higher harmonics. • Musette …. • Bach denotes the “crisper” sound as “metallic.” • What is “metallic?”

  45. Vibrating Bar • Equation • Boundary Conditions • Harmonics • Not integer multiples • Sound speed depends on frequency • A cacaphony of sounds • Nodes • Damping

  46. Vibrating Bar • Equation • Boundary Conditions • Clamped • Displacement = 0 (y) • Slope = 0 (1st derivative wrt x) • Free • Bending Moment = 0 (2nd derivative wrt x) • Shear Force = 0 (3rd derivative wrt x)

  47. Vibrating Bar • Allowed Frequencies Are Roots of: These are not integer multiples of one another. Sound is different from Bach’s “Metallic”

  48. Vibrating Bar • A tuning fork is a bent vibrating free-free bar held at the center node. • Higher Modes Damp Out Quickly • 1st mode provides a pure tone.

  49. Tuning Fork • When you strike a tuning fork, at first the tone sounds harsh, but then it’s very very pure.1 1 With apologies to James Joyce.

  50. Breakdown of the Fourier View Notes are finite in time. Stopping Strings Damping Notes are not discrete frequencies – they are broadened.

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