Measuring the quality of guitar tone
Measuring the quality of guitar tone
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For a detailed description see paper: SALI Samo, Kopac Janez. Measuring the quality of guitar tone. Exp. mech., 2000, vol. 40, no. 3, p. 242-247.
Rule of consonance-dissonance:
The string exciter assured that bad and good guitars were tested under equal conditions. The sound was measured with non-directional condenser microphone, 180 mm from the front resonant board. The quality of guitar's sound was defined on the basis of three tones: - "F" 87.3 Hz (6th string) - "B" 123.5 Hz (5th string) - "g" 196.0 Hz (3rd string) Let' s see the difference between tones "F" at 0,2 s after string excitement of really bad and really good guitar (A-weighted spectrum, reference pressure = 0,00002 Pa, frequency resolution = 86,13 Hz): Both spectra have characteristically guitar sound, but what a difference between them!
A simultaneous combination of two tones, which is pleasing to the ear, is termed consonant. When a combination of tones is not pleasing to the ear, the sound is termed dissonant. The consonant combinations of tones have the ratio of fundamental frequencies of two integer numbers none of which is large; for example, 2:1, 3:2, 5:3, 4:3, etc. (H.F. Olson: Music, Physics and Engineering). The combination of two tones can be seen as an interval of the fundamental frequencies of the tones. Similarly the spectrum of any tone can be seen as a host of intervals where each interval consists of two frequency lines. This host of intervals includes both consonant and dissonant intervals. When the difference in Hz between a certain frequency line and fundamental frequency of some scale tone (scale with tone a1=440 Hz was used) is less than 1%, the frequency line can be considered as a fictitious tone and gets the name of the scale tone. For the tone "F" (recorded 0.2 s after string excitation), the fictitious tones with sample spectra of good and bad tones are shown in the following figure. The names of the fictitious tones are in parentheses under the numbers of the corresponding frequency lines (components). Note that A-weighted sound pressure level (SPL) of some frequency lines (components) differ by more than 10 dBA.
The 11th and 13th frequency lines are excluded from the analysis of intervals, because their position does not match a fundamental frequency of any scale tone. The 7th and 14th frequency lines form a consonant interval inside a tone spectrum, but they are considered as a dissonant interval, because the difference between them and the fundamental frequencies of any scale tone is bigger than 1%. Out of the 105 (15x14/2) intervals for the first 15 frequency lines, we considered 25 intervals, 19 of which are consonant. The SPL of interval Lij(k), consisting of frequency lines i and j, is:
where Li(k) and Lj(k) are the A-weighted SPL of the first and the second frequency line of the interval ij, respectively, and k indicates one of the tones "F", "B" or "g". The 25 considered combinations of ij are shown in the following table:
The difference of Lij(k) for the good and bad tones of the same pitch gives the parameter dLij(k):
dLij(k) = Lij(k: good guitar) - Lij(k: bad guitar)
The following terms can now be defined: · The consonance (CG(k)) of a good tone k relative to a bad tone k, is defined as the sum of the consonant dLij(k) that are larger than 0. · The dissonance (DG(k)) of a good tone k relative to a bad tone k, is defined as the sum of the dissonant dLij(k) that are larger than 0. · The consonance (CB(k)) of a bad tone k relative to a good tone k, is defined as the sum of the consonant dLij(k) that are smaller than 0. · The dissonance (DB(k)) of a bad tone k relative to a good tone k, is defined as the sum of the dissonant dLij(k) that are smaller than 0.
Since each tone of bad and good guitars (4 bad and 4 good guitars were tested) was recorded in three different periods after string excitation, 144 (4 guitars x 4 guitars x 3 periods x 3 tones) comparisons between good and bad tones were made. For each comparison of good and bad tone, recorded after the same time after string excitation, the following expression is significant:
(CG(k)+CB(k))>2(DG(k)+DB(k))
This expression is named the rule of consonance-dissonance and means a criterion for distinguishing the bad guitar tone from good one. When two tones of the same pitch are recorded under equal conditions, the quality of the bad tone in comparison to the good tone can be defined as:
Q(k) = DG(k) + DB(k) - CB(k) - CG(k)
Larger Q(k) indicates a better quality of the bad (i.e., tested) tone in comparison to the good tone.
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