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About tuning


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About tuning

  1. 1. What is tuning? How is it done?“No tuning is ever finished - just left behind....” Ron Nossaman, piano builder.How is a piano tuned? What is piano tuning? Here’s my own definition:Part arithmetic and part flower arranging.In this article we’ll look first at the physical/mechanical aspects of tuning a piano, and then consider sometheory about the musical scale and what it means to be “in tune”.MECHANICAL ASPECTSA piano’s sound is produced when little wooden hammers covered in highly compressed wool felt strikesteel wire “strings” and set them in motion. The strings are secured at each end, and they pass over araised bridge that is attached to the soundboard. The soundboard is a large slightly domed board made(usually) of spruce. When the string moves, its motion is transferred through the bridge, and sets thesoundboard in motion. This arrangement might be called an acoustic transformer. Loudness is gained, asthe soundboard sets a much larger volume of air in motion than the string could on its own. Taking energyfrom the string to do this, means that the string vibrates for a shorter time than it would without theconnection to the bridge and soundboard.At the bottom of an upright piano or the far end of a grand the string is secured round a “hitch pin”integral to the cast iron plate. The other end of the string, at the top of an upright, or near the pianist in agrand piano, is wrapped (coiled) around a metal tuning pin like this one: 1
  2. 2. You can see the little hole in the tuning pin, where the end of the string gets inserted.More than two hundred strings are needed for the eighty-eight notes of the piano and the end of eachstring is coiled around its own tuning pin. The tuning pins fit very tightly into a laminated wood plank whichin UK piano terminology is called the Wrest Plank and in USA parlance, the Pinblock.In the photo above, the top grey and gold part represents the cast iron plate. The steel tuning pins are avery tight fit in the wooden Pinblock. Tuning consists of adjusting the tension of each string by using aspecial wrench called a Tuning Lever to very slightly move the tuning pins. The movements are extremelysubtle and much of the skill in tuning lies in ability to manipulate the tuning lever appropriately.Shown here are are some different tuning levers. In the first photo the top one is of a traditional design,and the lower one can take different heads so as to accommodate some older styles of tuning pins. Ingeneral tuning pins have square ends, but some older ones are occasionally found that are oblong. Inrecent years some excellent engineering has gone into a re-think of the traditional tuning lever, notably bySteve Fujan, and the lower picture on the next page shows my Fujan carbon fibre tuning lever, which islightweight and extremely rigid, and makes the job easier. 2
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  4. 4. The Fujan lever can take different tips for different tuning pin sizes. Tuning pin tips, or heads, have a “star”profile so as to fit square section pins in various positions.The technique of manipulating the pin and string is subtle. The strings pass over and under variouscomponents, including the bridge, and these create pressure points. The strings have to be moved withsome vigour so that the tension in the segments of the string evens out. In this photo you can see thatwhen the strings come from the coils around the pins, they quickly pass under a pressure bar, and thenover a V-bar. These help to position the strings. You can readily imagine that there is friction at thesepoints, and the string has to be coaxed to move such that the tension is evened out along the full length.The tuner has to know and use the right techniques to “set the pin” and “set the string”. In other words, toensure that things stay put when he has finished with one note and moves to the next. Poor techniquewith these aspects can result in a tuning that is not very stable. It is difficult to describe just how thetuning lever is manipulated, and we won’t attempt it here. It’s not especially necessary for understandingwhat’s to follow. Could an amateur musician tune a piano using a mechanic’s socket set, with a smallenough socket? The short answer is no. It is just about possible to correct a single string which for somereason has gone badly “out”. But to achieve an entire piano tuning is such a manner is an impossibility.We are going to move on now from the physical, mechanical aspects, to the other part: 4
  5. 5. THEORY OF THE SCALE AND TUNINGBooks have been written about how our Western musical scale is derived, and we can’t go too deeply intothat here. Pythagoras in ancient Greece is noted for carrying out experiments and making discoveriesabout musical sounds. We like a scale of musical notes in which we go from one pitch, or frequency, toexactly double that pitch, or frequency, in a series of twelve steps. All of western music is based on suchscales, or arrangements of notes.An item which moves four hundred and forty times per second, shoves the air around it that many timesper second, and the air shoves our eardrums that many times per second, and the movement of theeardrum is transmitted by three tiny bones to the inner ear and in a mechanism of great beauty andcomplexity, converted into electrical nerve impulses for interpretation by the brain. The sensation weexperience, is what we call the note A above Middle C. If we go down twelve steps from there, we getanother A, which moves two hundred and twenty times per second. Going up instead of down, we get an Aat eight hundred and eighty movements per second. These vibrations, or “cycles per second” are namedafter the physicist Hertz, with the abbreviation Hz. Thus we say that the note A above Middle C has afrequency of 440 Hz.The first note on the piano is an A, and its frequency is 27.5Hz. If we go up the piano in seven “doublings”or seven octaves, calling the first note A0, we get:A0 27.5 HzA1 55 HzA2 110 HzA3 220 HzA4 440 HzA5 880 HzA6 1760 HzA7 3520 HzThis is a geometric progression, but let’s not get too far into maths!What about the other notes in the scale, between the A notes? Pythagroras by experiment found pleasingmathematical ratios that worked well to give pleasant sounding notes. He found that if you multiply thefrequency of a note – say A3, 220Hz, by 1.5, you get the musical interval we call a Fifth; the note E abovemiddle C. Using Pythagoras’ multiplier of 1.5, the frequency of that E would be 330Hz.If we start at the first note on the piano keyboard, A0, and go up in a series of “Fifths”, we find that on the12th jump, we are at the same place as on the seventh Octave jump, back to an A. The notes would be(follow it on the piano keyboard if it’s easier): 5
  6. 6. Starting point A0,First jump E1Second jump B1Third jump F#2Fourth jump C#3Fifth jump Ab3Sixth jump Eb4Seventh jump Bb4Eighth jump F5Ninth jump C6Tenth jump G6Eleventh jmp D7Twelfth jump A7However, if we use Pythagoras’ multiplier of 1.5 to get the frequency of each fifth from the fifth below,something odd happens! Here are the frequencies starting at A0 and multiplying by 1.5 each time:Starting point A0, 27.50First jump E1 41.25Second jump B1 61.88Third jump F#2 92.82Fourth jump C#3 139.23Fifth jump Ab3 208.85Sixth jump Eb4 313.28Seventh jump Bb4 469.92Eighth jump F5 704.88Ninth jump C6 1057.32Tenth jump G6 1585.98Eleventh jump D7 2378.97Twelfth jump A7 3568.46Oh dear! When we get to the A seven octaves up from the starting point, the frequency, calculated using12 jumps of 1.5 (the “right” ratio for an interval of a Fifth), that A is 48.46Hz different!A perhaps simpler example, would be to consider the musical interval of a Major Third. Three Major Thirdsmake up an octave. Starting at A 220Hz, the first Major Third up is C# then F then A. Pythagorasdetermined that to go up by a Major third, you should multiply the starting note by 1.25. So, we wouldhave A 220Hz, C# 275, F 343.75, A 429.69.Erk! The next A up is supposed to be an Octave, double the frequency, 440Hz. Not 429.69 Hz! 6
  7. 7. These differences are sometimes called the Pythagorean Comma. This problem happens with all thedifferent ratios Pythagoras worked out for the series of notes in a scale. Pythagoras wasn’t wrong. It’s aproblem in physics. If you tune each of the thirds above to the frequencies given, you will get threebeautiful thirds when played individually. But they will sound awful played together, and won’t add up toan actave!In order to get a scale on the piano to “fit” properly, it is necessary to “temper” the intervals – to alterthem to fit, as it were. Taking the example of the three Thirds, you can see that there is a “shortfall” of10.31 Hz; they fail to add up to an octave by that amount. The Thirds therefore need to be “widened” a bit,so that they add up to an octave.The generally accepted system of “tempering” the scale, or adjusting the spaces between the notes awayfrom Pythagoras’ theoretical intervals, such that they will all “fit” into a scale, is called Equal Temperament.(Again, books have been written on the theory of temperament, and we can’t go too deeply into it here).The idea in Equal Temperament is to go from one note to the same note an Octave above, is a series of 12steps of equal proportion. To do that, you multiply each time, by the 12th root of 2. (The number which,multiplied by itself 12 times, gives 2). That number (to seven decimal places) is 1.0594631.If we multiply A 220 by 1.0594631, we get the frequency of the next note in the scale, A# (B flat). Doing thistwelve times brings us to A440 (almost – for all practical purposes anyway!)This is all complicated by the fact that musical strings in a piano, like any other vibrating object, do notvibrate in a simple manner. A string vibrates in sections as well as along its full length, and each of these“sections” contributes to the sound. The string for A220 also moves as two halves, three thirds, fourquarters, etc. Thus it produces sounds, “harmonics”, at 440Hz, 660Hz, 880Hz etc.Every string in the piano behaves in this way. Each string produces a “harmonic series”, a whole set offrequencies. The frequency of a particular harmonic of one note, can be close to the frequency of aharmonics of another note and this gives rise to “beats”.When two sources of sound have a frequency that is near, but not identical, an effect called “beating” isproduced. You might have observed this with two propeller engines in a plane; the drone has a kind of“pulse” in it, when they are not at identical revs. Or two motor lawnmowers, as local authority workmencut the grass in summer!For a very clear demonstration of this effect of “beating” or “beats”, go to the bottom of the page, are buttons to switch on sounds for 440Hz, 441Hz and 442Hz. Try 440 with 441,then 440 with 442. You will clearly hear the “beats”, a pulse in the sound, once per second, and twice persecond, respectively. 7
  8. 8. This phenomenon of “beats”, is the basis for tuning a piano. Most notes have three strings per note (twoin the lower tenor section and one in the low bass notes). Each of the three strings should be at absolutelyidentical frequency, therefore “beatless”.Where does the “flower arranging” come in, that we referred to at the start? No piano conforms to aperfect mathematical model. There are compromises with scale design, so that strings are not impossiblylong. And within an individual string, as the string vibrates in fractions, the fractions have a higher relativestiffness, which can make the harmonic have a higher frequency that the theoretical (this is calledinharmonicity).In addition, there is a tendency, which varies from person to person, for the ear to prefer a gradualincreasing of the frequencies of the notes beyond their theoretical positions, in the top register of thepiano. And in the very low notes, the fundamental, “whole string” frequency is not very dominant; our earextrapolates it down from the higher harmonics.All of this (it’s a lot more complex than this brief explanation!) adds up to the fact that the tuner has toarrange the tuning, to place the notes, as it were, in a way that is best for the particular instrument (andclient). It is never a matter of conforming to some mathematical formula. That is why, while the tuner hasto understand the arithmetic of the theoretical scale, he or she must also be a “flower arranger”.(With thanks to Chuck Behm for the use of his Pinblock photo).© David Boyce 2012 8