From: georg@renoir.nlp.physik.th-darmstadt.de (Georg Mueller) Newsgroups: rec.music.makers.synth,comp.music,alt.sci.physics.acoustics Subject: Papers on Piano Tuning (hard stuff) Date: 29 Jul 1993 15:58:44 GMT This is a short list of papers on piano tuning, have fun. Georg. ---------------------------------------------------------------------- AN 79(6):20715 PHYS TI The coupled motions of piano strings. AU Weinreich, G. SO Sci. Am. (Jan 1979). v. 240(1) p. 94-102 ISSN 0036-8733; CODEN SCAMA CY UNITED STATES DT Journal TC Popular LA English AB Most of the notes on a piano are sounded by two or three strings. The strings are not tuned to precisely the same frequency, a fact that contributes in unexpected ways to the tone of the instrument. CC *4375 CT *MUSICAL INSTRUMENTS; FREQUENCY DEPENDENCE; MUSIC AN 89(16):73286 PHYS TI The mathematics of piano tuning. AU Sanderson, Albert E. (Inventronics, 171 Lincoln Street, Lowell, MA 01852 (USA)) SO J. Acoust. Soc. Am., Suppl. (15 Apr 1989) v. 85(1) p. S65 117. Meeting of the Acoustical Society of America, Inc. (ASA), Syracuse, NY (USA), 22-26 May 1989 Published in summary form only Current Physics Microform No.: 8906A0891 ISSN 0163-0962; CODEN JACSC CY UNITED STATES DT Journal; Conference; Short Communication TC Theoretical LA English CC *4375 CT *TUNING; *MUSICAL INSTRUMENTS; MECHANICAL VIBRATIONS; SOUND WAVES; TONES; STRINGS AN 85(1):2174 PHYS TI Piano tuning mutes. AU Kovach, Bruce F. SO J. Acoust. Soc. Am. (Oct 1984) v. 76(4) p. 1283 Current Physics Microform No.: 8410C1385; Published in summary form only ISSN 0001-4966; CODEN JASMA CY UNITED STATES DT Journal; Short Communication LA English CC *4375 CT *MUSICAL INSTRUMENTS AN 84(15):84426 PHYS TI Effect of the inharmonicity of stiff strings on piano tuning. AU Suzuki, Hideo (CBS Technology Center, Stamford, CT (USA)) SO J. Acoust. Soc. Am., Suppl. (15 Apr 1984) v. 75(1) p. 10 107. Meeting of the Acoustical Society of America, Inc., Norfolk, VA (USA), 6-10 May 1984 Published in summary form only ISSN 0001-4966; CODEN JASMA CY UNITED STATES DT Journal; Conference TC Theoretical LA English AB The presently used tuning theory uses the beat rates calculated from the ideal harmonics of notes in the temperament octave. Following this theory, the note F3 is tuned sharp to C4 (middle C) with a 0.59-Hz beat rate. Then, the note F4 is tuned to the note F3, beatless. Theoretically, assuming the ideal harmonic relationship of partials, the best rate between notes C4 and F4 (test interval) is 1.18 Hz with F4 sharp. This is not true, however, for real piano strings with some amount of inharmonicity. If the inharmonicity index B, which describes the degree of inharmonicity of a string, is assumed to be 0.0004 for notes F3, C4, and F4, the beat rate between notes C4 and F4 becomes 0.696 Hz with F4 flat. This is a self-contradiction inherent in the present tuning theory. A method will be presented to calculate the beat rates when the inharmonicity indexes are known for all thirteen notes of the F3-F4 octave. CC *4375 CT *MUSICAL INSTRUMENTS; *STRINGS; ELASTICITY; TUNING; ANHARMONICITY; TONES; BEATS; CALCULATION METHODS ET F; C; B AN 79(13):42630 PHYS TI Further thoughts on piano tuning. AU Poole, L.M.G. (Department of Physics, Rhodes University, Grahamstown 6140, South Africa) SO Am. J. Phys. (Jun 1979) v. 47(6) p. 564-564 Published in summary form only ISSN 0002-9505; CODEN AJPIA CY UNITED STATES DT Journal; Short Communication LA English CC *4375 CT *MUSICAL INSTRUMENTS; MUSIC; *TUNING AN 89(4):14521 PHYS TI Fundamental theory and computer simulation concerning the decay characteristics of piano sound. AU Nakamura, Isao (Department of Computer Science and Information Mathematics, University of Electro-Communications, Chofu, 182 Japan) SO J. Acoust. Soc. Am., Suppl. (15 Oct 1988) v. 84(1) p. S135 116. Meeting of the Acoustical Society of America, Inc. (ASA) - 2. Joint Meeting with the Acoustical Society of Japan, Honolulu, HI (USA), 14-18 Nov 1988 Published in summary form only Current Physics Microform No.: 8811D0135 ISSN 0163-0962; CODEN JACSC CY UNITED STATES DT Journal; Conference TC Theoretical LA English AB It is shown by theory and computer simulation that the decay characteristics of piano sound are dependent upon the degree of coupling between the strings. String and soundboard vibration are calculated using an equivalent circuit. Each string is comprised of resonance circuits that correspond to partials, and the behaviors of individual partials are described independently of each other by these circuits. When the most simplified version of only one partial is considered, the equivalent circuit is expressed by the two resonant circuits coupled with the soundboard impedance. The degree of coupling between the two strings is dependent on the ratio of two constants: the degree of mistuning and the ratio of soundboard impedance to string impedance. If the former is smaller than the latter, the two strings are closely coupled with each other and a double decay characteristic results. If the relation between the two constants is reversed, the coupling is loose and a beat type of decay characteristic results. The decay characteristics are changed by the above two constants. The results show that the degree of tuning and the characteristics of the strings and the soundboard determine the decay characteristics of piano sound. CC *4375 CT SOUND WAVES; *COMPUTERIZED SIMULATION; *MUSICAL INSTRUMENTS; *ACOUSTICS; STRINGS; COUPLING; ACOUSTIC IMPEDANCE; EQUIVALENT CIRCUITS AN 85(1):2177 PHYS TI String inharmonicity and piano tuning. AU Heetveld, Vincent; Rasch, Rudolf A. (Utrecht Univ. (Netherlands)) SO J. Acoust. Soc. Am., Suppl. (15 Aug 1984) v. 76(1) p. 22 Current Physics Microform No.: 8408A1708; Published in summary form only ISSN 0001-4966; CODEN JASMA CY UNITED STATES DT Journal TC Experimental LA English AB Inharmonicity is a well-known property of the stiff strings as used in the modern piano. Effects on piano tuning (e.g., the stretched octave) have been suggested but never fully investigated. We have measured the inharmonicities of the strings of a medium-sized grand piano. The measured inharmonicities were in excellent correspondence with the predictions by formula from the physical properties of the strings. Six models, differing in three pairs of assumptions, were developed for describing the effects of the inharmonicity on the tuning of the piano and on the beat frequencies used when tuning. The correspondence between the model predictions and the actually applied tuning could be used to test the validity of the assumptions. One of the surprising outcomes was that the fact that higher harmonics of lower strings beat with lower harmonics of higher strings (which have, as a rule, higher inharmonicity) cancels out a part of the effects of inharmonicity on beat frequency. CC *4375