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Spectral analysis of cello multiphonics

The following is a starting point for a physical analysis of cello multiphonics. Using the recordings made for Cellomap, Thomas Resch compares the relative amplitude of the harmonic components of a multiphonic on the four strings, and points to the inclusion of further harmonics that are not perceived in the sound and/or the involvement of which is not predicted. For a mathematical theory which predicts the component harmonics of multiphonics, the reader is referred to Caspar Johannes Walter's recent publication 'Mehrklänge auf dem Klavier. Vom Phänomen zur Theorie und Praxis mikrotonalen Komponierens' in Mikrotonalität - Praxis und Utopie ed. Cordula Pätold and Caspar Johannes Walter (Maintz: Schott 2014).





Using spectral analysis for multiphonic sounds does show frequency maxima and minima, but not necessarly the resulting audible pitches. For example, almost every multiphonic we recorded on the G-string has a maximum at the fundamental frequency and the second harmonic pitch but we only sometimes  recognized these frequencies as distinct tones. Also, even very strong maxima at higher harmonics can be heard as an overtone of a lower pitched frequency maximum.


There is not much theoretical work about multiphonic string sounds. A paper by K. Guettler and H. Thelin (J. Acoust. Soc. Am., Vol. 131, No. 1, Pt. 2, January 2012) explains the generation of multiphonics on a double bass using a Waveguide approach. Their calculations and formulas for predicting the outcome of a certain finger position combined with a certain bow position also work in some cases for the cello but matches our measurements only roughly for others. Looking at their results and their distinction between so called class 1 (bow on the nut-side of the finger) and class 2 (bow on the bridge side of the finger) multiphonics indicates that multiphonic sounds are more stable on the double bass and that the smaller cello body influences the audible outcome more strongly than on the double bass. In particular, the class 1 multiphonics with the bow position on the nut side of the finger are rather unstable on the cello. For Cellomap we focused on mutliphonics with the bow on the bridge side of the finger.


It appears, that in several cases, the recognition of harmonics as distinct notes within multiphonics is also a matter of subjective hearing. Some people recognize certain harmonics despite their very strong magnitudes merely as overtones of a lower harmonic, others can hear them very loudly and clearly.


The level of influence of the cello body has not been measured properly yet. Recordings have been done only with one instrument so far, this will be a subject of further research, as well as the subjective recognition of multiphonics and of course the influence of the string density, since the differences on the four cello strings can be also very strong.


The generation of mulitphonics is a very complex procedure combining finger-position, bow-position, string-density, bow-presure, bow-velocity and the cello body. A minor change of one of these parameters might influence the outcome very strongly. 


This analysis uses the recordings made for Cellomap which are available in the section: 'Multiphonics and Other Multiple Sounds'.


[3 4 7 11]


On the A-string, all four harmonics are visible in the spectrogram. The 3rd and 4th harmonics have only small magnitudes and are also difficult to hear. Additionally the fundamental frequency and 2nd and 8th harmonic are rather strong.


On the D-string, the 2nd harmonic is non-exisitent. Additionally, the 5th and 8th harmonics are strong. Both can be heard instead of the 4th and 7th harmonic.


Also on the G-string, [3 4 7 11] is rather unstable. Sometimes the 1st harmonic is very strong in which case the 3rd and 4th harmonics might not be recognized as distinct pitches anymore. In particular, the 4th harmonic might be recognized as an overtone of the the 1st or 2nd harmonic. As on the D-string, it seems to be a subjective matter whether the 7th or the 8th harmonic is audible.


On the C-string all 4 Harmonics are prefectly visible and audible.