Physics 1251The Science and Technology of Musical SoundUnit 2Session 21 MWFMusical Scales and Strings

Physics 1251Unit 2 Session 21Scales and StringsFoolscap 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.

Physics 1251Unit 2 Session 21Scales and Strings1 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.

Physics 1251Unit 2 Session 21Scales and StringsWhat 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 dArezzo: gamma utgamut SolfeggioG is Do in the G-scale

Physics 1251Unit 2 Session 21Scales and StringsWhat is a scale?SolfegeDoReMiFaSoLaTiDoZoltan Kodaly

Physics 1251Unit 2 Session 21Scales and StringsThe Circle of Fifths:4:52:35:6

Physics 1251Unit 2 Session 21Scales and StringsA 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 3755 G 3/2 396 702 6 A 5/3 440 884 7 B 15/8 475 1018 8 vo C 2 528 1200

Physics 1251Unit 2 Session 21Scales and StringsA 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 3755 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 MajorKey of G Major Difference5 th G 3/2 396 Tonic 1 396 0 Hz7 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.

Physics 1251Unit 2 Session 21Scales and Strings80/20It 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.

Physics 1251Unit 2 Session 21Scales and StringsA Comparison of the Just and Equal Tempered Scales:Just:Tonic C 1 264 02nd 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 -3400 1.260 329.6 13500 1.335 349.2 -3700 1.498 392.0 -4900 1.682 440 011001.889 493.8 -11200 2.000 523.2 -5

Physics 1251Unit 2 Session 21Scales and StringsListen to Pachelbels Canon in Just and Equal Temperament:Microtonal URL

Physics 1251Unit 2 Session 21Scales and StringsMusical Notation

B4C5D4E4F4G4A4C4E3D3F3A3B3G3C3

Physics 1251Unit 2 Session 21Scales 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.

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

Physics 1251Unit 2 Session 21Scales and StringsWhat determines the fundamentalfrequency of oscillation of a string?Pythagoras discovered that the frequency was inversely proportional to the length of the string.f1 /f2 = L2 / L1Why?

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

Physics 1251Unit 2 Session 21Scales and StringsGuitar frets shorten string to raise pitch.LoL1Fret Lo / Ln11.0621.1231.2041.2751.33

Physics 1251Unit 2 Session 21Scales and StringsWhat determines the velocity of a wave on a string? y = a t 2a = F/mF = 2 T y/x y = (2 T y/x) t 2/m(x /t )2 = T/ (m/x) vstring = (T/ )xyTTm= x2 T (y/x)

Physics 1251Unit 2 Session 21Scales and Strings80/20Velocity of a wave on a string:

vstring = T/ T is the tension in the string [N]. is the mass per unit length [kg/m]

Rope Race Demonstration

Physics 1251Unit 2 Session 21Scales and Strings80/20f = [1/(2L)] vstring f = [1/(2L)] (T/ )]Tuning pegsMore Tension:Raises pitch

Physics 1251Unit 2 Session 21Scales and Strings80/20f = [1/(2L)] (T/ )]Denser wire: lower pitchLess dense wire: higher pitch

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

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

Physics 1251Unit 2 Session 21Scales and StringsSummary: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/ )]