Physics 1251 The Science and Technology of Musical Sound

1. Physics 1251The Science and Technology of Musical Sound Unit 2 Session 21 MWF Musical Scales and Strings

2. Physics 1251 Unit 2 Session 21 Scales and Strings Foolscap Quiz: If A4 is the tonic of an A-major scale (just intonation) and has a fundamental frequency of 440 Hz, what is the frequency of (1) the major third C4♯, (2) the “perfect” fifth E4, and (3) the octave A5 ? The frequency of the major third is 5/4 ftonic = 550 Hz. The frequency of the “perfect” fifth is 3/2 ftonic = 660 Hz. The frequency of the octave is 2 ftonic = 880 Hz.

3. Physics 1251 Unit 2 Session 21 Scales and Strings 1′ Lecture: • A Just Tempered Scale sets the frequencies of the notes in the scale at precise ratios of whole numbers. • Not all chords can be in tune with Just Temperament. • The Equal Tempered Scale is the compromise that sets all semitones an equal interval apart (100¢, frequency ratio:1.05946). • The frequency of vibration of a string is inversely proportional to its length.

4. ♩ Physics 1251 Unit 2 Session 21 Scales and Strings What is a scale? • “Gamut” {Note “G-Clef”} ♩ ♯♩ ♩ ♩ ♩ ♩ ♩ ♩ ♩ ♩ ♩ ♩ ♩ ♩ ♩ Do Re Mi Fa So La Ti Do Do Re Mi Fa So La Ti Do C-major G-major Guido d’Arezzo: “gamma ut→gamut” • Solfeggio G is “Do” in the G-scale

5. Physics 1251 Unit 2 Session 21 Scales and Strings What is a scale? • Solfege Do Ti La So Fa Mi Re Zoltan Kodaly Do

6. 440 950.4 633.6 A [major 3 rd] A A B G B B G G [tonic] [perfect 5th] C F C C F F 792 352 528 1188 528 792 E D E E D D Physics 1251 Unit 2 Session 21 Scales and Strings The Circle of Fifths: 4:5 5:6 2:3

7. 4 2/3 264 0 Hz 5 3/4 297 0 Hz 6 5/6 330 13Hz 7 15/16 371 -4 Hz Tonic 1 396 0 Hz 2 9/8 4466 Hz 3 5/4 495 20Hz 4 4/3 528 0 Hz Physics 1251 Unit 2 Session 21 Scales and Strings A Scale is constructed from Integer Ratios of Frequencies! Not all scales can be ‘tuned.” Tonic C 1 264 0 2 D 9/8 297 204 ¢ 3 E 5/4 317 316 ¢ 4 F 4/3 352 498 ¢ 4♯ F ♯64/45 375 5 G 3/2 396 702 ¢ 6 A 5/3 440 884 ¢ 7 B 15/8 475 1018 ¢ 8 vo C 2 528 1200 ¢

8. 4 2/3 264 0 Hz 5 3/4 297 0 Hz 6 5/6 330 13Hz 7 15/16 371 -4 Hz Tonic 1 396 0 Hz 2 9/8 4466 Hz 3 5/4 495 20Hz 4 4/3 528 0 Hz Physics 1251 Unit 2 Session 21 Scales and Strings A Scale is constructed from Integer Ratios of Frequencies! Not all scales can be ‘tuned.” Tonic C 1 264 0 2 D 9/8 297 204 ¢ 3 E 5/4 317 316 ¢ 4 F 4/3 352 498 ¢ 4♯ F ♯64/45 375 5 G 3/2 396 702 ¢ 6 A 5/3 440 884 ¢ 7 B 15/8 475 1018 ¢ 8 vo C 2 528 1200 ¢ Key of C Major Key of G Major Difference 5 th G 3/2 396 Tonic 1 396 0 Hz 7 th B 15/8 475 3 rd 5/4 495 20Hz The 3rd in the Key of G major with “howl” if the piano is tuned in C major.

9. Physics 1251 Unit 2 Session 21 Scales and Strings 80/20It is impossible to tune perfectly scales in all keys using the same frequencies and just temperament. 80/20The Equal Tempered Scale sets each semitone exactly 100 ¢ apart or at a ratio of 1.05946….

10. Physics 1251 Unit 2 Session 21 Scales and Strings A Comparison of the Just and Equal Tempered Scales: Just: Tonic C 1 264 0 2nd D 9/8 297 204 ¢ 3rd E 5/4 317 316 ¢ 4th F 4/3 352 498 ¢ 5th G 3/2 396 702 ¢ 6th A 5/3 440 884 ¢ 7th B 15/8 475 1018 ¢ 8 vo C 2 528 1200 ¢ Equal Tempered: 0¢ 1.000 261.6 -2 200¢ 1.122 293.7 -3 400¢ 1.260 329.6 13 500¢ 1.335 349.2 -3 700¢ 1.498 392.0 -4 900¢ 1.682 440 0 1100¢1.889 493.8 -1 1200¢ 2.000 523.2 -5

11. Physics 1251 Unit 2 Session 21 Scales and Strings Listen to Pachelbel’s Canon in Just and Equal Temperament: Microtonal URL

12. ♩ Physics 1251 Unit 2 Session 21 Scales and Strings Musical Notation ♯♩ ♩ ♯♩ ♯♩ ♯♩ ♩ ♯♩ G4♯ C5♯ B4 F4♯ ♯ E4 A4♯ ♭♩ ♭♩ D4♯ ♭♩ ♩ ♩♭ ♭♩ ♩ A3♭ E3♭ C4♯ D3♭ F3 G3♭ B3♭ C3

13. Physics 1251 Unit 2 Session 21 Scales and Strings On the standard Keyboard the sharps ♯ and flats ♭ are enharmonic (sound the same frequency and pitch.) In some systems this is not the case.

14. Physics 1251 Unit 2 Session 21 Scales and Strings Examine a standard Guitar: Note that each succeeding fret shortens the string by a constant fraction.

15. Physics 1251 Unit 2 Session 21 Scales and Strings What determines the fundamental frequency of oscillation of a string? • Pythagoras discovered that the frequency was inversely proportional to the length of the string. • f1 /f2 = L2 / L1 • Why?

16. Physics 1251 Unit 2 Session 21 Scales and Strings The fundamental frequency of oscillation of a string is fixed by the time required for a round trip of a string wave: L f = [1/2L] ‧ vstring

17. Physics 1251 Unit 2 Session 21 Scales and Strings Guitar frets shorten string to raise pitch. Fret Lo / Ln 1 1.06 2 1.12 3 1.20 4 1.27 5 1.33 Lo L1

18. Physics 1251 Unit 2 Session 21 Scales and Strings What determines the velocity of a wave on a string? 2 T (y/x) y = ½ a t 2 a = F/m F = 2 T y/x ∴ y = ½ (2 T y/x) t 2/m (x /t )2 = T/ (m/x) vstring = √ (T/ μ) T T y m= μx x

19. Physics 1251 Unit 2 Session 21 Scales and Strings 80/20Velocity of a wave on a string: vstring = √T/ μ T is the tension in the string [N]. μ is the mass per unit length [kg/m] “Rope Race Demonstration”

20. Physics 1251 Unit 2 Session 21 Scales and Strings 80/20f = [1/(2L)] ‧ vstring f = [1/(2L)] ‧ √(T/ μ)] Tuning pegs More Tension: Raises pitch

21. Physics 1251 Unit 2 Session 21 Scales and Strings 80/20f = [1/(2L)] ‧ √(T/ μ)] Denser wire: lower pitch Less dense wire: higher pitch

22. Physics 1251 Unit 2 Session 21 Scales and Strings Practice: 80/20f = [1/(2L)] ‧ √(T/ μ)] What is the frequency of a guitar string that is 65 cm long, has a mass density μ = 0.005 kg/m and is under a tension of T = 182 N? f = [1/(2L)] ‧ √(T/ μ)] f = [1/ (2(0.65 m)) ]√(182 N/ 0.005 kg/mμ) f = 146.8 Hz

23. Physics 1251 Unit 2 Session 21 Scales and Strings Why do you think that are the frets in this guitar oddly spaced? Microtonal music can be played on this instrument.

24. Physics 1251 Unit 2 Session 21 Scales and Strings Summary: • A Just Tempered Scale sets the frequencies of the notes in the scale at precise ratios of whole numbers.. • The Equal Tempered Scale is the compromise that sets all notes an equal interval apart (100¢, frequency ratio:1.05946). • f = [1/(2L)] ‧ √(T/ μ)]