Physics 5B Lecture 12 February 8 2012 Chapter 16 Sound Musical Scales An octave is a doubling in frequency The second harmonic is an octave higher than the fundamental The fourth harmonic is two octaves above the fundamental etc In this chart an octave is divided equally into 12 intervals on a log scale scale logg 2 0 025085 12 Ratio 1 0000 1 0573 1 1221 1 1870 1 2595 1 3321 1 4122 1 4962 1 5840 1 6794 1 7786 1 8855 2 0000 Log 0 0 0242 0 0500 0 0745 0 1002 0 1245 0 1499 0 1750 0 1997 0 2252 0 2501 0 2754 0 3010 Xylophone y p B Perfect 5th 792 3 528 2 Octave This instrument has only seven intervals per octave not twelve It is missing the sharps and flats black keys on the piano Virtual Piano Unlike the xylophone the piano is tuned on an equally tempered scale equal spacing in the logarithm so that music can be played in all keys Major 3rd Fifth O Octave http www bgfl org bgfl custom resources ftp client ftp ks2 music piano Musical Intervals The next important interval after the octave 2 is the fifth 3 2 262 3 786 e g the interval from C to G 392 2 784 Perfect fifth 1 5 262 Hz to 393 Hz Equally tempered fifth 1 498 262 Hz to 392 Hz Note how with a perfect fifth the 3rd harmonic of the C will ill liline up exactly tl with ith th the 2nd harmonic h i off th the G G In the equally tempered scale they are off by a couple of Hz from liningg up p The resultingg beats are easyy to hear in the following where I generate a middle C with 3 harmonics and a G with two harmonics htt i http scipp ucsc edu johnson phys5b PerfectFifth wav d j h h 5b P f tFifth http scipp ucsc edu johnson phys5b EqualTemperedFifth wav More Musical Intervals The perfect fourth 4 3 e g approx C to F The 4th harmonic of C lines up p with the 3rd harmonic of F The major third 5 4 e g approx C to E The 5th harmonic of C lines up with the 4th harmonic of E The major sixth 5 3 e g approx C to A The 5th harmonic of C lines up with the 3rd harmonic of A And so on If we go further the intervals become less and less consonant in some sense although through history people s opinions have evolved with regard to which intervals are considered consonant and which are dissonant Minor Third An example of a minor third interval is C to D with a frequency ratio of 6 5 1 2 What is the first harmonic of C that will approximately align with ith a harmonic h i off D A Third B Fifth Fif h C Sixth O an equally On ll tempered d scale l D Eleventh El h C 262 0 6 1572 D 311 6 5 1558 Wave Interference in 2D http phet colorado edu en simulation wave interference Wave Interference 2D or 3D Speakers in phase Constructive Destructive Wave Interference http phet colorado edu en simulation sound Sound Interference The figure shows two identical speakers that are connected to the same amplifier but Speaker B is wired backwards such that it oscillates 180 out of phase with respect to