Slide 1

For Physicists & Engineersusing your

PianoTuningLaptop, Microphone, and HammerBruce Vogelaar313 Robeson HallVirginia Techvogelaar@vt.eduat3:00 pm Room 130 Hahn NorthMarch 17, 2012by#Piano Tuning1What 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

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Bravely put your VT physics education to work on that ancient piano!

Tune: to what? why? how?Regulate: what?Fix keys: how?

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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 fundamentalPluck near edge many higher harmonics

What you hear is the sum (transferred into air pressure waves).

[v = speed of wave on string]#Piano Tuning

timedomainfrequencyspectrum

Destructive Constructive

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frequency content determines timbre

#Piano TuningGiven only the sum,what were the components? Fourier AnalysisHow much of the sum comes from individual components#Piano Tuning

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13 slides on how this is done (just cant resist)P(x):

f(x):

P(x)f(x):Consider a class gradedistribution:

P(x) is the number ofstudents versus grade

f(x) is a 1x1 blockat a certain grade

Summing the product ofP(x)f(x) gives the number ofstudents with that grade

#Piano Tuning P(x) f(x) P(x)f(x)

02310

1 2 3 4 5 1 2 3 4 5

sum components#Piano Tuning P(t) f(t) P(t)f(t)

010

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#Piano Tuningfinding am

all terms on right integrate to zero except mth !find bm using sin(mt)#Piano Tuning

typical extraction of properties from a distribution#Piano TuningInput 4Hz pure sine waveLook for 3Hz component

1 sec4+3 = 7 Hz4 - 3 = 1 HzMultiplyAverage200 Samples, every 1/200 second, giving f0 = 1 HzAVG = 0#Piano TuningInput 4Hz pure sine waveLook for 4Hz componentAVG = 1/2

1 sec4+4 = 8 Hz4 - 4 = 0 HzMultiplyAverage#Piano TuningInput 4Hz pure sine waveLook for 5Hz component

1 secMultiplyAverageAVG = 0#Piano TuningGreat, picked out the 4 Hz input. But what if the input phase is different?

0.25Use COS as well. For example: 4Hz, 0 = 30o; sample 4 Hz(0.432 + 0.252)1/2 = 1/2 Right On!1 sec1 secsincos0.430.25#Piano TuningSignal phase does not matter.What about input at 10.5 Hz?Finite Resolution

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Remember, we only had 200 samples, so there is a limitto how high a frequency we can extract. Consider 188 Hz,sampled every 1/200 seconds:Nyquist LimitSample > 2x frequency of interest;lots of multiplication & summing slow#Piano TuningFree FFT Spectrum Analyzer:http://www.sillanumsoft.org/download.htmVisual Analyzer

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40960 sample/s 32768 samples

= 1.25 Hz resolution#Piano Tuning

5th (3/2)

4th (4/3)

3rd (5/4)Why some notes sound harmoniousOctaves are universally pleasing; to the Western ear, the 5th is next most important.

Octave (2/1)#Piano Tuning

5th (3/2)

GCGC

tf#Piano Tuning

A frequency multiplied by a power of 2 is the same note in a different octave.#Piano Tuning

Going up by 5ths 12 times brings you verynear the same note(but 7 octaves up)

(this suggests perhaps12 notes per octave)

flog2(f)log2(f) shifted into same octaveWolf fifthWe define the number of cents between two notes as1200 * log2(f2/f1)

Octave = 1200 centsWolf fifth off by 23 cents.Up by 5ths: (3/2)nCircle of 5th s

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log 2/1log 3/2log 4/3log 5/4log 6/5log 9/8Weve chosen 12 EQUAL tempered steps; could have been 19 just as wellAverage deviation from just notes1=0log2 of ideal ratios Options for equally spaced notes#Piano TuningTypically set A4 to 440 Hz

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5th (3/2)

4th (4/3)

3rd (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#Piano Tuning

I was hopeless,and even wrote asynthesizer to tryand train myself but I still couldnthear itFrom C, set G above it such that an octave and a fifth above the C you hear a 0.89 Hz beatingThese beat frequencies are for the central octave.

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Is it hopeless?

not with a little help from math and a laptop

we (non-musicians) can use a spectrum analyzer#Piano TuningTrue Equal Temperament Frequencies012345678C32.7065.41130.81261.63523.251046.502093.004186.01C#34.6569.30138.59277.18554.371108.732217.46D36.7173.42146.83293.66587.331174.662349.32D#38.8977.78155.56311.13622.251244.512489.02E41.2082.41164.81329.63659.261318.512637.02F43.6587.31174.61349.23698.461396.912793.83F#46.2592.50185.00369.99739.991479.982959.96G49.0098.00196.00392.00783.991567.983135.96G#51.91103.83207.65415.30830.611661.223322.44A27.5055.00110.00220.00440.00880.001760.003520.00A#29.1458.27116.54233.08466.16932.331864.663729.31B30.8761.74123.47246.94493.88987.771975.533951.07With a (free) Fourier spectrum analyzer we can set the pitches exactly!#Piano TuningBut first a critical note about real strings (where art cant be avoided)strings have stiffnessbass strings are wound to reduce this, but not all the way to their endstreble strings are very short and stiffthus harmonics are not true multiples of fundamentals fn is increased by a factor of 1+n2concert grands have less inharmonicity because they have longer strings

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A4 (440) inharmonicity

true 8x440 pianowhich should match A7?

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Tuning the A keys: Ideal strings

With 0.0001 inharmonicityNeed to Stretch thetuning.

Can not match all harmonics, must compromise artsounds sharpsounds flat

32 f033.6 f0#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

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With Db4With Db5The effect is larger for higher harmonics,and so you simply cant match everythingat the same time.Trying to set Db7

#Piano Tuningbut 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 themand mark which keys hit which strings)

RegulationFixing keys, and making mechanical adjustmentsso they work optimally, and feel uniform.#Piano Tuning

a pain on spinets#Piano Tuning

#Piano TuningVoicingthe hammersNOT for the novice(you can easily ruin a set of hammers)#Piano TuningLets now do it for real

pin turningunisons (true or not?)tune using FFTput it back together#Piano TuningWeighted average

Typical Application (assume P and are normalized)

class grade averagecenter of massdipole moments

(same as above, but for continuous distributions)e.g.: Maxwell Boltzmann velocity distributions

Fourier component of

commuting quantum mechanical variables

non-commuting quantum mechanical variables

Fermis golden rule for transitions between two states.

IntervalEqual Temperament Frequency RatioDifferenceHarmonic Series Frequency Ratio

Octave=2.00000.00002.0000=2/1

Major Seventh=1.88770.01271.8750=15/8

Minor Seventh=1.78180.03181.7500=7/4

Major Sixth=1.68180.01511.6667=5/3

Minor Sixth=1.5874-0.01261.6000=8/5

Perfect Fifth=1.4983-0.00171.5000=3/2

Tritone=1.41420.00001.4142=

Perfect Fourth=1.33480.00151.3333=4/3

Major Third=1.25990.00991.2500=5/4

Minor Third=1.1892-0.01081.2000=6/5

Major Second=1.1225-0.00251.1250=9/8

Minor Second=1.0595-0.00721.0667=16/15

Unison=1.00000.00001.0000=1/1