We discuss theoretical and physical models that are useful for analyzing the intonation of musical instruments such as guitars and mandolins and can be used to improve the tuning on these instruments. The placement of frets on the fingerboard is designed according to mathematical rules and the assumption of an ideal string. The analysis becomes more complicated when we include the effects of deformation of the string and inharmonicity due to other string characteristics. As a consequence, perfect intonation of all the notes on the instrument cannot be achieved, but complex compensation procedures can be introduced to minimize the problem. To test the validity of these procedures, we performed extensive measurements using standard monochord sonometers and other acoustical devices, confirming the correctness of our theoretical models. These experimental activities can be integrated into acoustics courses and laboratories and can become a more advanced version of basic experiments with monochords and sonometers.
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The saddle is the white piece of plastic or other material located near the bridge on which the strings are resting. The strings are usually attached to the bridge, which is located on the left of the saddle. On other type of guitars or other fretted instruments, the strings are attached directly to the bridge (without using any saddle). In this case the string length would be the distance between the bridge and the nut. Our analysis would not be different in this case: The bridge position would replace the saddle position.
35.
Following Eq. (4), frets number 5, 7, 12, and 19 are particularly important because they (approximately) correspond to vibrating string lengths, which are respectively 3/4, 2/3, 1/2, and 1/3 of the full length, consistent with the Pythagorean theory of monochords.
36.
38.
This statement is also an approximation because the pitch (or perceived frequency) is affected by the presence of the overtones. See, for example, the discussion of the psychological characteristics of music in Ref. 4.
39.
Our solution in Eq. (19) differs from the similar solution obtained in Ref. 24, Eq. (17). We believe that this difference is due to a minor error in their calculation, which causes only minimal changes in the numerical results. Therefore, the compensation procedure used by Byers (Ref. 24) in his guitars is practically very effective in improving the intonation of his instruments.
40.
Nevertheless some luthiers actually construct guitars where the individual frets under each string are adjustable in position by moving them slightly along the fingerboard. Each note of the guitar is then individually fine-tuned to achieve the desired intonation, requiring a very time consuming tuning procedure.
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The cent is a logarithmic unit of measure used for musical intervals. The octave is divided into 12 semitones, each of which is subdivided in 100 cents; thus the octave is divided into 1200 cents. Because an octave corresponds to a frequency ratio of 2:1, 1 cent is precisely equal to an interval of . Given two frequencies and of two different notes, the number of cents between the notes is . Alternatively, given a note and the number of cents in the interval, the second note of the interval is .
44.
We note that the frequency of the sound produced is the physical quantity we measured in our experiments. The pitch is defined as a sensory characteristic arising out of frequency but also affected by other subjective factors, which depend on the individual. It is beyond the scope of this paper to consider these subjective factors.
45.
This discrimination range was estimated for the frequencies of string 1, according to the discussion in Ref. 4, pp.
248
–252
. This range usually varies from about (±5) to (±10) cents for frequencies between 1000 and 2000 Hz to even larger values of (±40)–(±50) cents at lower frequencies between 60 and 120 Hz.© 2010 American Association of Physics Teachers.
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