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

3. Physics 1251Unit 2 Session 21 Scales and Strings Foolscap Quiz: If A 4 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 C 4 ♯, (2) the “perfect” fifth E 4, and (3) the octave A 5 ? The frequency of the major third is 5/4 f tonic = 550 Hz. The frequency of the “perfect” fifth is 3/2 f tonic = 660 Hz. The frequency of the octave is 2 f tonic = 880 Hz.

4. Physics 1251Unit 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. 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. 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 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. The frequency of vibration of a string is inversely proportional to its length.

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

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

7. A B C D E F G A B C D E F G A B C D E F G Physics 1251Unit 2 Session 21 Scales and Strings The Circle of Fifths: [tonic] [major 3 rd ] [perfect 5 th ] 4:5 352440528 2:3 5:6 528633.6792792950.41188

8. Physics 1251Unit 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 ¢ 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 9/8 446 6 Hz 3 5/4 495 20Hz 4 4/3 528 0 Hz

9. Physics 1251Unit 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 ¢ 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 9/8 446 6 Hz 3 5/4 495 20Hz 4 4/3 528 0 Hz Key of C MajorKey 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 3 rd in the Key of G major with “howl” if the piano is tuned in C major.

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

11. Physics 1251Unit 2 Session 21 Scales and Strings A Comparison of the Just and Equal Tempered Scales: Just: Tonic C 1 264 0 2 nd D 9/8 297 204 ¢ 3 rd E 5/4 317 316 ¢ 4 th F 4/3 352 498 ¢ 5 th G 3/2 396 702 ¢ 6 th A 5/3 440 884 ¢ 7 th 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

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

13. Musical Notation Physics 1251Unit 2 Session 21 Scales and Strings ♩ ♯♩♯♩♯♩♯♩ ♯♩♯♩♯♩♯♩ ♯♩♯♩♯♩♯♩ ♩ ♯♩♯♩♯♩♯♩♩ ♯♩♯♩♯♩♯♩ ♭♩♭♩♭♩♭♩ ♭♩♭♩♭♩♭♩ ♭♩♭♩♭♩♭♩ ♩ ♩♭♩♭♩♭♩♭ ♭♩♭♩♭♩♭♩ ♩ B4B4B4B4 C5♯C5♯C5♯C5♯ D4♯D4♯D4♯D4♯ E4E4E4E4 F4♯F4♯F4♯F4♯ G4♯G4♯G4♯G4♯ A4♯A4♯A4♯A4♯ C4♯C4♯C4♯C4♯ ♯ E3♭E3♭E3♭E3♭ D3♭D3♭D3♭D3♭ F3F3F3F3 A3♭A3♭A3♭A3♭ B3♭B3♭B3♭B3♭ G3♭G3♭G3♭G3♭ C3C3C3C3

14. On the standard Keyboard the sharps ♯ and flats ♭ are enharmonic (sound the same frequency and pitch.) On the standard Keyboard the sharps ♯ and flats ♭ are enharmonic (sound the same frequency and pitch.) In some systems this is not the case.

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

16. Physics 1251Unit 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. Pythagoras discovered that the frequency was inversely proportional to the length of the string. f 1 /f 2 = L 2 / L 1 f 1 /f 2 = L 2 / L 1 Why? Why?

17. Physics 1251Unit 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: f = [1/2L] ‧ v string L

18. Physics 1251Unit 2 Session 21 Scales and Strings Guitar frets shorten string to raise pitch. LoLoLoLo L1L1L1L1 Fret L o / L n 11.06 21.12 31.20 41.27 51.33

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

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

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

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

23. Physics 1251Unit 2 Session 21 Scales and Strings Practice: 80/20 f = [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

24. Physics 1251Unit 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.

25. Physics 1251Unit 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.. 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). 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)] f = [1/(2L)] ‧ √(T/ μ)]