Physics in the Arts Lecture S7 Superposition Beats and Harmony Kali Malone The Sacri cal Code Superposition of Waves Reminder Waves can interfere constructively or destructively 1 951 0 982 0 986 0 434 0 0 434 0 720 0 854 Constructive Interference Destructive Interference 1 574 Constructive Interference 2 0 1 5 1 0 0 5 0 5 1 0 1 5 Constructive Interference Destructive Interference Superposition of Waves You have seen these many times already Standing waves Plucking a string https physics bu edu duffy HTML5 transverse standing wave html This work by Andrew Duffy is licensed under a Creative Commons Attribution NonCommercial ShareAlike 4 0 International License Superposition of Waves You have seen these many times already Standing waves Plucking a string https physics bu edu duffy HTML5 transverse standing wave html This work by Andrew Duffy is licensed under a Creative Commons Attribution NonCommercial ShareAlike 4 0 International License Beat frequencies Superposition of two simple harmonic waves f1 1 Hz 1 2 3 time s 4 T1 1s Beat frequencies Superposition of two simple harmonic waves f1 1 Hz f2 0 75 Hz T2 1 33s T1 1s 1 1 2 2 3 3 time s 4 4 Beat frequencies Superposition of two simple harmonic waves f1 1 Hz f2 0 75 Hz T2 1 33s T1 1s 1 1 1 2 2 2 T 4 s 3 3 3 time s 4 4 4 Beat frequencies Superposition of two simple harmonic waves f1 1 Hz f2 0 967 Hz The waves are slightly out of phase with respect to each other time s Constructive Interference Constructive Interference a bit less Beat frequencies Superposition of two simple harmonic waves f1 1 Hz f2 0 967 Hz 15s After have opposite phases the waves are time s Constructive Interference 100 Destructive Interference Beat frequencies Superposition of two simple harmonic waves f1 1 Hz f2 0 967 Hz 15s After have opposite phases the waves are After 30s the wave form repeats time s Constructive Interference 100 Destructive Interference Constructive Interference Beat frequencies Beat frequency fB f1 f2 time s Beat frequencies Beat frequency fB f1 f2 Fast beats when time s frequencies are quite di erent inaudible Very slow beats when frequencies are almost identical audible Beat frequencies Beat frequency fB f1 f2 Fast beats when time s frequencies are quite di erent inaudible Very slow beats when frequencies are almost identical audible fB 1 0 967 Hz 0 033Hz TB 1 fB 30s our example f1 1 Hz f2 0 967 Hz Another example 100Hz 100Hz with together 150Hz 100Hz together with what other frequency Beat frequency fB f1 f2 Discuss with your neighbor Hint there are two possible answers Another example 100Hz 100Hz with together 150Hz 100Hz together with what other frequency Beat frequency fB f1 f2 Discuss with your neighbor Hint there are two possible answers Another example 100Hz 100Hz with together 150Hz 100Hz together with what other frequency Beat frequency fB f1 f2 Discuss with your neighbor Hint there are two possible answers Another example 100Hz 100Hz with together 150Hz 100Hz together with what other frequency Beat frequency fB f1 f2 Discuss with your neighbor Hint there are two possible answers Beat frequencies Dissonance Roughness Dissonance audible beats when two notes frequencies are played simultaneously wikipedia Beat frequencies Dissonance Roughness Dissonance audible beats when two notes frequencies are played simultaneously Sympathetic strings on Indian Sarangi But slow enough beats create a pleasant chorus e ect and beat frequencies are useful for tuning instruments wikipedia Beat frequencies Dissonance Roughness Dissonance audible beats when two notes frequencies are played simultaneously Sympathetic strings on Indian Sarangi But slow enough beats create a pleasant chorus e ect and beat frequencies are useful for tuning instruments Example C major chord C major chord doubled detuned by 8 cents wikipedia Beat frequencies Dissonance Roughness Dissonance audible beats when two notes frequencies are played simultaneously Sympathetic strings on Indian Sarangi But slow enough beats create a pleasant chorus e ect and beat frequencies are useful for tuning instruments Example C major chord C major chord doubled detuned by 8 cents wikipedia Beat frequencies Dissonance Roughness Dissonance audible beats when two notes frequencies are played simultaneously Sympathetic strings on Indian Sarangi But slow enough beats create a pleasant chorus e ect and beat frequencies are useful for tuning instruments Example C major chord C major chord doubled detuned by 8 cents wikipedia Beat frequencies Dissonance Roughness Dissonance audible beats when two notes frequencies are played simultaneously Sympathetic strings on Indian Sarangi But slow enough beats create a pleasant chorus e ect and beat frequencies are useful for tuning instruments Example C major chord C major chord doubled detuned by 8 cents demo guitar wikipedia Harmonious Intervals Unison both frequencies are the same Unison both frequencies are the same The octave Harmonious Intervals f2 2 f1 100 Hz E g f1 and f2 200 Hz time s Unison both frequencies are the same The octave Harmonious Intervals f2 2 E g f1 and Overtones time s f1 100 Hz 200 Hz 300 Hz 400 Hz 500 Hz 600 Hz 700 Hz 800 Hz 900 Hz 1000 Hz f2 200 Hz 400 Hz 600 Hz 800 Hz 1000 Hz 1200 Hz 1400 Hz 1600 Hz 1800 Hz 2000 Hz Unison both frequencies are the same The octave Harmonious Intervals f2 2 E g f1 and Overtones time s f1 100 Hz 200 Hz 300 Hz 400 Hz 500 Hz 600 Hz 700 Hz 800 Hz 900 Hz 1000 Hz f2 200 Hz 400 Hz 600 Hz 800 Hz 1000 Hz 1200 Hz 1400 Hz 1600 Hz 1800 Hz 2000 Hz Higher note provides no new overtones Two notes an octave apart are perceived almost as if they were the same Harmonious Intervals f2 3 2 f1 200 Hz E g f1 and f2 300 Hz The Fifth time s Harmonious Intervals f2 3 2 E g f1 and The Fifth Overtones time s f1 200 Hz 400 Hz 600 Hz 800 Hz 1000 Hz 1200 Hz 1400 Hz 1600 Hz 1800 Hz 2000 Hz f2 300 Hz 600 Hz 900 Hz 1200 Hz 1500 Hz 1800 Hz 2100 Hz 2400 Hz 2700 Hz 3000 Hz Harmonious Intervals f2 3 2 E g f1 and The Fifth Overtones time s f1 200 Hz 400 Hz 600 Hz 800 Hz 1000 Hz 1200 Hz 1400 Hz 1600 Hz 1800 Hz 2000 Hz f2 300 Hz 600 Hz 900 Hz 1200 Hz 1500 Hz 1800 Hz 2100 Hz 2400 Hz 2700 Hz 3000 Hz Every other overtone of second note is already present in overtone spectrum of rst Tonal Fusion Two notes sharing many partials are perceived consonant creating new tone Harmonious Intervals f2 4 3 E g f1 and The Fourth Overtones time s f1 300 Hz 600 Hz 900 Hz 1200 Hz 1500 Hz 1800 Hz 2100 Hz 2400 Hz 2700 Hz 3000 Hz f2 400 Hz 800 Hz 1200 Hz 1600