1 Piano Tuning For Physicists & Engineers using your Piano Tuning Laptop, Microphone, and Hammer Bruce Vogelaar 313 Robeson Hall Virginia Tech email@example.com at 3:00 pm Room 130 Hahn North March 17, 2012 by
2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys, raised pitch a half- step to set A4 at 440 Hz, and did a rough tuning…
3 Piano Tuning Bravely put your VT physics education to work on that ancient piano! Tune: to what? why? how? Regulate: what? Fix keys: how?
4 Piano Tuning A piano string is fixed at its two ends, and can vibrate in several harmonic modes. frequency of string = frequency of sound ( of string of sound) Pluck center mostly fundamental Pluck near edge many higher harmonics What you hear is the sum (transferred into air pressure waves). [v = speed of wave on string]
5 Piano Tuning time domain frequency spectrum Destructive Constructive
6 Piano Tuning frequency content determines timbre
7 Piano Tuning Given only the sum, what were the components? Fourier Analysis How much of the sum comes from individual components
9 13 slides on how this is done (just cant resist) P(x): f(x): P(x)f(x): Consider a class grade distribution: P(x) is the number of students versus grade f(x) is a 1x1 block at a certain grade Summing the product of P(x)f(x) gives the number of students with that grade
10 Piano Tuning P(x) f(x) P(x)f(x) 0 2 3 1 0 1 2 3 4 5 sum components
13 Piano Tuning finding a m all terms on right integrate to zero except m th ! find b m using sin( m t)
14 Piano Tuning typical extraction of properties from a distribution
15 Piano Tuning Input 4Hz pure sine wave Look for 3Hz component 1 sec 4+3 = 7 Hz 4 - 3 = 1 Hz Multiply Average 200 Samples, every 1/200 second, giving f 0 = 1 Hz AVG = 0
16 Piano Tuning Input 4Hz pure sine wave Look for 4Hz component AVG = 1/2 1 sec 4+4 = 8 Hz 4 - 4 = 0 Hz Multiply Average
17 Piano Tuning Input 4Hz pure sine wave Look for 5Hz component 1 sec Multiply Average AVG = 0
18 Piano Tuning Great, picked out the 4 Hz input. But what if the input phase is different? 0.25 Use COS as well. For example: 4Hz, 0 = 30 o ; sample 4 Hz (0.43 2 + 0.25 2 ) 1/2 = 1/2 Right On! 1 sec sin cos 0.43 0.25
19 Piano Tuning Signal phase does not matter. What about input at 10.5 Hz? Finite Resolution
20 Piano Tuning Remember, we only had 200 samples, so there is a limit to how high a frequency we can extract. Consider 188 Hz, sampled every 1/200 seconds: Nyquist Limit Sample > 2x frequency of interest; lots of multiplication & summing slow…
25 Piano Tuning A frequency multiplied by a power of 2 is the same note in a different octave.
26 Piano Tuning Going up by 5 th s 12 times brings you very near the same note (but 7 octaves up) (this suggests perhaps 12 notes per octave) f log 2 (f) log 2 (f) shifted into same octave Wolf fifth We define the number of cents between two notes as 1200 * log 2 (f 2 /f 1 ) Octave = 1200 cents Wolf fifth off by 23 cents. Up by 5ths: (3/2) n Circle of 5 th s
27 Piano Tuning log 2/1 log 3/2 log 4/3 log 5/4 log 6/5 log 9/8 Weve chosen 12 EQUAL tempered steps; could have been 19 just as well… Average deviation from just notes 1= 0 log 2 of ideal ratios Options for equally spaced notes
29 Piano Tuning 5 th (3/2) 4 th (4/3) 3 rd (5/4) for equal temperament: tune so that desired harmonics are at the same frequency; then, set them the required amount off by counting beats. Octave (2/1) What an aural tuner does…
30 Piano Tuning I was hopeless, and even wrote a synthesizer to try and train myself… but I still couldnt hear it… From C, set G above it such that an octave and a fifth above the C you hear a 0.89 Hz beating These beat frequencies are for the central octave.
31 Piano Tuning Is it hopeless? not with a little help from math and a laptop… we (non-musicians) can use a spectrum analyzer…
32 Piano Tuning True Equal Temperament Frequencies 012345678 C32.7065.41130.81261.63523.251046.502093.004186.01 C#34.6569.30138.59277.18554.371108.732217.46 D36.7173.42146.83293.66587.331174.662349.32 D#38.8977.78155.56311.13622.251244.512489.02 E41.2082.41164.81329.63659.261318.512637.02 F43.6587.31174.61349.23698.461396.912793.83 F#46.2592.50185.00369.99739.991479.982959.96 G49.0098.00196.00392.00783.991567.983135.96 G#51.91103.83207.65415.30830.611661.223322.44 A27.5055.00110.00220.00440.00880.001760.003520.00 A#29.1458.27116.54233.08466.16932.331864.663729.31 B30.8761.74123.47246.94493.88987.771975.533951.07 With a (free) Fourier spectrum analyzer we can set the pitches exactly!
33 Piano Tuning But first – a critical note about real strings (where art cant be avoided) strings have stiffness bass strings are wound to reduce this, but not all the way to their ends treble strings are very short and stiff thus harmonics are not true multiples of fundamentals – their frequencies are increased by 1+ n 2 concert grands have less inharmonicity because they have longer strings
34 Piano Tuning A4 (440) inharmonicity true 8x440 piano which should match A7?
35 Piano Tuning Tuning the A keys: Ideal strings With 0.0001 inharmonicity Need to Stretch the tuning. Can not match all harmonics, must compromise art sounds sharpsounds flat 32 f 0 33.6 f 0
36 Piano Tuning (how Ive done it) octaves 3-5: no stretch (laziness on my part) octaves 0-2: tune harmonics to notes in octave 3 octaves 6-7: set R inharmonicity to ~0.0003 load note into L and use R(L) Stretched
37 Piano Tuning With D b 4 With D b 5 The effect is larger for higher harmonics, and so you simply cant match everything at the same time. Trying to set D b 7
38 Piano Tuning but some keys dont work… pianos were designed to come apart (if you break a string tuning it, youll need to remove the action anyway) (remember to number the keys before removing them and mark which keys hit which strings) Regulation Fixing keys, and making mechanical adjustments so they work optimally, and feel uniform.