• Hochschule für Musik der Stadt Basel
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This article presents selected results from a research project on cello multiphonics at the Hochschule für Musik Basel within which I am producing updated fingering charts in a smartphone application and affiliated online repository. The article details work that has informed this resource and illustrates results that reveal critical questions and point to future areas of interest. I begin by introducing cello multiphonics and contextualising my previous findings, then discuss pitch content, ‘chain’ multiphonics and the balance and intonation of multiphonic components.

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The string player controls variations in spectral content mainly via bow velocity, bow-bridge distance and bow force. Many combinations of the bowing parameters influence the pitch noticeably as well, in particular close to the upper bow-force limit in the Schelleng diagram. The influence of the bowing parameters on the spectral content and pitch were studied systematically by use of a monochord and a bowing machine. Bow force was found to be by far the most dominant parameter determining the spectral centroid. Bow-bridge distance and bow velocity serve essentially as indirect control parameters of spectral content by giving the player access to playable areas with high or low bow forces in the Schelleng diagram. Clear areas of pitch flattening could be distinguished below the upper bow-force limits in the Schelleng diagrams, confirming the role of pitch flattening as a practical bow-force limit in playing. The conditions for anomalous low frequencies (ALF), S-motion and other, higher types of string motion were analyzed, and it was shown that secondary waves might play an important role in their creation.
The history of musical instruments is nearly as old as the history of civilization itself, and the aesthetic principles upon which judgments of musical quality are based are intimately connected with the whole culture within which the instruments have evolved. An educated modem Western player or listener can make critical judgments about particular instruments or particular per­ formances but, to be valid, those judgments must be made within the appro­ priate cultural context. The compass of our book is much less sweeping than the first paragraph might imply, and indeed our discussion is primarily confined to Western musical instruments in current use, but even here we must take account of centuries of tradition. A musical instrument is designed and built for the playing of music of a particular type and, conversely, music is written to be performed on particular instruments. There is no such thing as an "ideal" instrument, even in concept, and indeed the unbounded possibilities of modem digital sound-synthesis really require the composer or performer to define a whole set of instruments if the result is to have any musical coherence. Thus, for example, the sound and response of a violin are judged against a mental image of a perfect violin built up from experience of violins playing music written for them over the centuries. A new instrument may be richer in sound quality and superior in responsiveness, but if it does not fit that image then it is not a better violin.
With a bow force greater than the Schelleng maximum and careful control, it will be demonstrated that it is possible to produce sounds on a violin of definite pitch ranging from approximately a musical third to a twelfth or more below the normal pitch. The lowered pitch is in agreement with the fundamental frequency of the observed harmonic series. The fundamental itself is very weak if the sounds are produced on the open G string. Mari Kimura has utilized the effect in performances [New York Times, 21 April 1994, p. B3, and Strings, Sept./Oct. 1994, 60?66]. These anomalous low frequencies (ALF) occur when the bow force is great enough to prevent the Helmholtz kink from triggering the normal release of the string from the bow hair. As a result of pronounced bow?nut and bow?bridge reflections there is at the bow a very complex string waveform, some portion of which regularly triggers the slipping of the string. ALF can also be produced on a bowed string mounted on a steel beam, where the motion is detected optically. Computer simulation is used to show how a string can be forced to vibrate at frequencies lower than the natural fundamental frequency of the string.
These experiments were conducted to determine the dominance of each partial in determining the residue pitch of a complex tone. Subjects were required to make pitch matches to a complex tone which had one partial slightly mistuned from its ‘‘correct’’ harmonic value. The shift in residue pitch was measured as a function of the frequency shift of the harmonic, for each harmonic in turn. For mistunings up to ±2%–3% the shift in residue pitch was approximately a linear function of the shift in the harmonic, but for greater mistunings the shift in residue pitch was reduced. The degree to which a given harmonic can influence residue pitch gives a measure of the dominance of that harmonic. The dominant harmonics were always contained within the lowest six harmonics (for fundamental frequencies of 100, 200, and 400 Hz), but there were marked individual differences in the exact distribution of dominance across harmonics. The level of a harmonic relative to adjacent harmonics can have a significant effect on its dominance. The implications of the results for theories of pitch perception are discussed.
By carefully positioning the bow and a lightly touching finger on the string, the string spectrum can be conditioned to provide narrow bands of pronounced energy. This leaves the impression of multiple complex tones with the normal (Helmholtz) fundamental as the lowest pitch. The phenomenon is seen to be caused by two additional signal loops, one on each side of the finger, which through the repeating slip pattern get phase locked to the full loop of the fundamental. Within the nominal period, however, the slip pulses will not be uniform like they are during the production of a normal "harmonic" or "flageolet" but may vary considerably in shape, size, and timing. For each string, there is a large number of bow/finger combinations that bear the potential of producing such tones. There are also two classes, depending on whether the bow or the finger is situated closest to the bridge. Touching the string with the finger closest to the bridge will somewhat emphasize the (Helmholtz) fundamental. The technique is applicable to double bass and cello, while less practical on shorter-stringed instruments. Analyses based on impulse responses and the Poisson summation formula provide an explanation to the underlying system properties.
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