About the tuning of the Piano: Inharmonicity.
Acoustically, a note perceived to have a distinct pitch contains frequency components that are integer multiples of f0 usually known as harmonics. Each harmonic is a sine wave and since the hearing system analyses sounds in terms of their frequency components it turns out to be highly instructive in terms of understanding how to analyse and synthesise periodic sounds, as well as being central to the development of Western musical harmony to consider the musical relationship between the individual harmonics themselves. The frequency ratios of the harmonic series are known (see Table 1) and their equivalent musical intervals, frequency ratios and staff notation in the key of C are shown in the table below for the first ten harmonics. The musical intervals (apart from the octave) are only approximated on a modern keyboard due to the tuning system used.
The musical intervals of adjacent harmonics in the natural harmonic series starting with the fundamental or first harmonic, illustrated on a musical stave and as notes on a keyboard in Table 2, are: octave (2:1), perfect fifth (3:2), perfect fourth (4:3), major third (5:4), minor third (6:5), flat minor third (7:6), sharp major second (8:7), a major whole tone (9:8), and a minor whole tone 00:9). The frequency ratios for intervals between non- adjacent harmonics in the series can also be inferred from the figure. For example, the musical interval between the fourth harmonic and the fundamental is two octaves and the frequency ratio is 4:1, equivalent to a doubling for each octave. Similarly the frequency ratio for three octaves is 8:1, and for a twelfth (octave and a fifth) is 3:1.
Intervals for other commonly used musical intervals can be found from these. To demonstrate this for a known result, the frequency ratio for a perfect fourth (4:3) can be found from that for a perfect fifth (3:2) since together they make one octave (2:1): C to G (perfect fifth) and G to C (perfect fourth). The perfect fifth has a frequency ratio 3:2 and the octave a ratio of 2:1. Bearing in mind that musical intervals are ratios in terms of their frequency relationships and that any mathematical manipulation must therefore be carried out by means of division and multiplication, the ratio for a perfect fourth is that for an octave divided by that for a perfect fifth, or up one octave and down a fifth:
Table 1 The relationship between overtone series, harmonic series and fundamental frequency for the first ten components of a period waveform
Table 2 Frequency ratios and common musical intervals between the first ten harmonics of the natural harmonic series of C3 against a musical stave and keyboard.
Sound source from a struck string
When a stringed instrument is struck such as in a piano, the same relationship exists between the point at which the strike occurs and the modes that will be missing in the sound source. There is, however, an additional effect that is particularly marked in the piano to consider. Piano strings are very hard and they are under enormous tension compared with the strings on plucked instruments. When a piano string is stuck, it behaves partly like a bar because it is not completely flexible since it has some stiffness. This results in a slight raising in frequency of all the component modes with the effect being greater for the higher modes, resulting in the modes no longer being exact integer multiples of the fundamental mode. This effect, known as 'inharmonicity', varies as the square of the component mode (n2), or harmonic number, and as the fourth power of the string radius (R4). Thus for a particular string, the third mode is shifted nine times (32) as much as the first, or fundamental, mode, and a doubling in string radius increases inharmonicity by a factor of sixteen (24). The effect would therefore be considerably greater for bass strings if they were simply made thicker to give them greater mass, and in many stringed instruments, including pianos, guitars and violins, the bass strings are wrapped with wire to increase their mass without increasing their stiffness. The notes of a piano are usually tuned to equal temperament and octaves are then tuned by minimising the beats between pairs of notes an octave apart. When tuning two notes an octave apart, the components which give rise to the strongest sensation of beats are the first harmonic of the upper note and the second harmonic of the lower note. These are tuned in unison to minimise the beats between the notes. This results in the f0 of the lower note being slightly lower than half the to of the higher note due to the inharmonicity between the first and second components of the lower note.
Inharmonicity on a piano increases as the strings become shorter and therefore the octave stretching effect increases with note pitch. The stretching effect is usually related to middle C and it becomes greater the further away the note of interest is in pitch. Figure 3 illustrates the effect in terms of the average deviation from equal-tempered tuning across the keyboard of a small piano. Thus high and low notes on the piano are tuned sharp and flat respectively to what they would have been if all octaves were tuned pure with a frequency ratio of 2:1. From the Figure it can be seen that this stretching effect amounts to approximately 35 cents sharp at C8 and 35 cents flat at Cl with respect to middle C.
Figure 3 Approximate form of the average deviations from equal temperament due to inharmonicity in a smal! piano. Middle C marked with a spot.