Theory Piano tuning

1 theory

1.1 overtones , harmonics
1.2 temperament
1.3 stretch

theory
overtones , harmonics

schematic of vibrating string, fixed @ both ends, showing first 6 normal modes or harmonics

a stretched string vibrates in different modes, or harmonics. when piano string vibrates, harmonic modes excited simultaneously. first harmonic (or fundamental frequency) loudest, , determines pitch perceived. theoretically higher harmonics (also called overtones or partials) vibrate @ integer multiples of fundamental frequency. (e.g. string fundamental frequency of 100 hz have overtones @ 200 hz, 300 hz, 400 hz, etc.) in reality, frequencies of overtones shifted slightly, due inharmonicity caused stiffness of strings.

the relationship between 2 pitches, called interval, ratio of absolute frequencies. easiest intervals identify , tune note frequencies have simple whole-number ratio (e.g. octave 2:1 ratio, perfect fifth 3:2, etc.) because harmonics of these intervals coincide , beat when out of tune. (for perfect fifth, 3rd harmonic of lower note coincides 2nd harmonic of top note.)

temperament

the term temperament refers tuning system allows intervals beat instead of tuning pure or intervals . in equal temperament, instance, fifth tempered narrowing slightly, achieved flattening upper pitch slightly, or raising lower pitch slightly.

tempering interval causes beat. because actual tone of vibrating piano string not 1 pitch, complex of tones arranged in harmonic series, 2 strings close simple harmonic ratio such perfect fifth beat @ higher pitches (at coincident harmonics), because of difference in pitch between coincident harmonics. these frequencies can calculated, temperament may tuned aurally timing beatings of tempered intervals.

an a440 tuning fork

one practical method of tuning piano begins tuning notes in temperament octave in middle range of piano. beginning pitch tuned external reference, a440 tuning fork, , tuner successively adjusts each note s tempered intervallic relationships other notes in scale. during tuning common assess fifths, fourths, thirds (both major , minor) , sixths (also major , minor), in ascending or descending pattern hear whether progression of beat rates has been achieved.

once temperament octave tuned, tuner tunes other notes on piano using octave intervals align them, first tuned notes in temperament octave, , lastly tuned notes below , above.

the following table lists beat frequencies between notes in equal temperament octave. top row indicates absolute frequencies of pitches; a440 determined external reference. every other number indicates beat rate between 2 tones (which share row , column number) in temperament octave. slower beat rates can timed metronome, or other such device. thirds in temperament octave, difficult tune many beats per second, after setting temperament , duplicating 1 octave below, of these beat frequencies present @ half indicated rate in lower octave, excellent verification temperament correct. 1 of easiest tests of equal temperament play succession of major thirds, each 1 semitone higher last. if equal temperament has been achieved, beat rate of these thirds should increase evenly in temperament region.

the next table indicates pitch @ strongest beating should occur useful intervals. described above, when tuning perfect fifth, instance, beating can heard not @ either of fundamental pitches of keys played, rather octave , fifth (perfect twelfth) above lower of 2 keys, lowest pitch @ harmonic series overlap. once beating can heard, tuner must temper interval either wide or narrow tuning has no beatings.

stretch

the tuning described above beating plan provides approximation of equal temperament across range of temperament octave. if extended further, however, actual tuning of instrument becomes increasingly inaccurate because of inharmonicity, causes harmonics run sharp, increasingly higher tones in harmonic series reached. problem mitigated stretching octaves 1 tunes above (and extent below) temperament region. when octaves stretched, tuned, not lowest coincidental overtone (second partial) of note below, higher 1 (often 4th partial). widens intervals equally, thereby maintaining intervallic , tonal consistency.

all western music, western classical literature in particular, requires deviation theoretical equal temperament because music played within single octave. pianist plays notes spread on 3 , 4 octaves, critical mid , upper range of treble stretched conform inharmonic overtones of lower registers. since stretch of octaves perceived , not measured, tuner aware of octave needs more or less stretching. tuning requires compromise between tonal brilliance, intonation , awareness of gradation of tone through compass of instrument. amount of stretching necessary achieve function of string scaling, complex determination based on string s tension, length, , diameter. 1 tuning standard follows empirical railsback curve.

in small pianos inharmonicity extreme establishing stretch based on triple octave makes single octaves beat noticeably, , wide, fast beating intervals in upper treble—especially major 17ths—beat wildly. of necessity tuner attempt limit stretch. in large pianos concert grands, less inharmonicity allows complete string stretch without negatively affecting close octaves , other intervals. while may true smaller piano receives greater stretch relative fundamental pitch, concert grand s octaves can widened triple octaves beatless. contributes response, brilliance , singing quality concert grands offer.

a benefit of stretching octaves correction of dissonance equal temperament imparts perfect fifth. without octave stretching, slow, imperceptible beating of fifths in temperament region (ranging little more 1 beat every 2 seconds 1 per second) double each ascending octave. @ top of keyboard, then, theoretically (and ideally) pure fifth beating more 8 times per second. modern western ears tolerate fast beating in non-just intervals (seconds , sevenths, thirds , sixths), not in perfect octaves or fifths. happily pianists, string stretch accommodates inharmonicity on concert grand mitigates accumulation of dissonance in perfect fifth.

other factors, physical , psychoacoustic, affect tuner s ability achieve temperament. among physical factors inharmonic effects due soundboard resonance in bass strings, poorly manufactured strings, or peculiarities can cause false beats (false because unrelated manipulation of beats during tuning). principal psychoacoustic factor human ear tends perceive higher notes being flat when compared in midrange. stretching tuning account string inharmonicity not sufficient overcome phenomenon, piano tuners may stretch top octave or of piano more.

frequencies of audible range on twelve , 8 equal tempered scale