piano - virginia tech department of physicskimballton/home/pub/piano/piano.pdf · 2 piano tuning...

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1 Piano Tuning For Physicists & Engineers using your PianoTuning Laptop, Microphone, and Hammer Bruce Vogelaar 313 Robeson Hall Virginia Tech [email protected] at Room 130 Hahn North April 21, 2011 by

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Page 1: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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For Physicists & Engineersusing your

PianoTuning

Laptop, Microphone, and Hammer

Bruce Vogelaar313 Robeson Hall

Virginia [email protected]

at

Room 130 Hahn NorthApril 21, 2011

by

Page 2: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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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…

Page 3: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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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?

Page 4: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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A piano string is fixed at its two ends, and can vibrate in several harmonic modes.

02

;2

nfnLvvf

nL

n

The string vibrating at a given frequency,produces sound with the same frequency.

Depending where you pluck the string,the amplitude of different frequencies varies.

What you hear is the sum.

...)2sin()2sin()2sin()( 332211 tfatfatfatp

L

Page 5: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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5th (3/2)

4th (4/3)

3rd (5/4)

Why some notes sound ‘harmonious’

Octaves are universally pleasing; to the Western ear, the 5th is next most important.

Octave (2/1)

Page 6: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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

Page 7: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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

(this suggests perhaps12 notes per octave)

f

log2(f)

log2(f) shifted into same octave

“Wolf ” fifth

We define the number of ‘cents’ between two notes as1200 * log2(f2/f1)

Octave = 1200 cents“Wolf “ fifth off by 23 cents.

Up

by 5

ths:

(3/

2)n

“Circle of 5th s”

127 5.12

Page 8: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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log 2/1

log 3/2

log 4/3

log 5/4log 6/5log 9/8

We’ve chosen 12 EQUAL tempered steps; could have been 19 just as well…

Average deviation from ‘just’ notes

1=

0

log2 of ‘ideal’ ratios Options for equally spaced notes

Page 9: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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from: http://www.sengpielaudio.com/calculator-notenames.htm

Typically setA4 to 440 Hz

Page 10: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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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…

Page 11: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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I was hopeless,and even wrote asynthesizer to tryand train myself…

but I still couldn’t‘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.

Page 12: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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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…

Page 13: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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ning True Equal Temperament Frequencies

0 1 2 3 4 5 6 7 8C 32.70 65.41 130.81 261.63 523.25 1046.50 2093.00 4186.01C# 34.65 69.30 138.59 277.18 554.37 1108.73 2217.46D 36.71 73.42 146.83 293.66 587.33 1174.66 2349.32D# 38.89 77.78 155.56 311.13 622.25 1244.51 2489.02E 41.20 82.41 164.81 329.63 659.26 1318.51 2637.02F 43.65 87.31 174.61 349.23 698.46 1396.91 2793.83F# 46.25 92.50 185.00 369.99 739.99 1479.98 2959.96G 49.00 98.00 196.00 392.00 783.99 1567.98 3135.96G# 51.91 103.83 207.65 415.30 830.61 1661.22 3322.44A 27.50 55.00 110.00 220.00 440.00 880.00 1760.00 3520.00A# 29.14 58.27 116.54 233.08 466.16 932.33 1864.66 3729.31B 30.87 61.74 123.47 246.94 493.88 987.77 1975.53 3951.07

With a (free) “Fourier” spectrum analyzer we can set the pitches

exactly!

Page 14: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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ning 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-2

-1

0

1

2

timedomain

frequencyspectrum

Forward and Reverse Fourier Transforms

Destructive Constructive

Page 15: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Any repeating waveformcan be decomposed intoa Fourier spectrum.

But that doesn’t mean theysound good…

frequency content determines ‘timbre’

Page 16: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Given an arbitrary waveform, how does one find its spectral content (Fourier transform)?

• Collect N samples x dT = T seconds,f0 1/T.

• Multiply the input by harmonics of f0 [sin(2nf0t)], and report 2x the average for each.

eg: 200 x 1/200 = 1 sec

f0 (1 Hz in this case)

2f0

3f0

4f0

1 sec

Page 17: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Input 4Hz pure sine wave

Look for 2Hz component

Multiply & Average

2xAVG = 0

200 Samples, every 1/200 second, giving f0 = 1 Hz

1 sec

Page 18: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Input 4Hz pure sine wave

Look for 3Hz component

Multiply & Average

2xAVG = 0

1 sec

4+3 = 7 Hz

4 - 3 = 1 Hz

Page 19: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Input 4Hz pure sine wave

Look for 4Hz component

Multiply & Average

2xAVG = 1.0

1 sec

4+4 = 8 Hz

4 - 4 = 0 Hz

Page 20: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Input 4Hz pure sine wave

Look for 5Hz component

Multiply & Average

2xAVG = 0

1 sec

Page 21: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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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 = 30o; sample 4 Hz

2 x (0.432 + 0.252)1/2 = 1.0 Right On!

1 sec 1 sec

sin cos

0.43 0.25

Page 22: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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From a “math” perspective:

time average is 0, unless = R

Page 23: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Signal phase does not matter.What about input at 10.5 Hz?

Sample 2xAverage7 0.098 0.139 0.2110 0.6411 0.6412 0.2113 0.1314 0.09

Finite Resolution

Page 24: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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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 Limit Sample > 2x frequency of interest;

Page 25: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Fast Fourier Transforms• 100’s of times faster

• uses complex numbers…Free FFT Spectrum Analyzer:

http://www.sillanumsoft.org/download.htm“Visual Analyzer”

Page 26: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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But first – a critical note about ‘real’ strings (where ‘art’ can’t 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+n2

• concert grands have less inharmonicitybecause they have longer strings

Page 27: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Tuning the ‘A’ keys: Ideal strings

With 0.0001 inharmonicity

Need to “Stretch” thetuning.

Can not match all harmonics, must compromise ‘art’

sounds ‘sharp’ sounds ‘flat’

24);2(4400 nf n

32 f0

33.6 f0

Page 28: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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(how I’ve done it)

for octaves 0-2tune to the harmonics in octave 3

for octaves 6-7set ‘R’ inharmonicity to ~0.0003and use R(L) ‘Stretched’

.

Page 29: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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but some keys don’t work…

pianos were designed to come apart

(if you break a string tuning it,you’ll need to remove the ‘action’ anyway)

(remember to number the keys before removing themand mark which keys hit which strings)

“Regulation”Fixing keys, and making mechanical adjustments

so they work optimally, and ‘feel’ uniform.

Page 30: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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a pain on spinets

Page 31: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Page 32: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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ning “Voicing”

the hammersNOT for the novice

(you can easily ruin a set of hammers)

Page 33: piano - Virginia Tech Department of Physicskimballton/home/pub/piano/piano.pdf · 2 Piano Tuning What our $50 piano sounded like when delivered. So far: cleaned, fixed four keys,

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Let’s now do it for real…

pin turningunisons (‘true’ or not?)tune using FFTput it back together