inharmonicity from Acoustic Society

[Richard Moody]

Pasted from somewhere.....

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   

I don't suppose the Acoustic Society journals have been scanned and htlm
indexed?  I am 160 miles from the nearest university or I would look it up
myself. FLAT fourths because of  .0004 inharmonicity?  Perhaps a tuning
lesson in real life would reveal the true nature of this
"self-contradiction." He must mean not as sharp (a litte flatter) than
theoritical. 

Ric.

This is interesting...

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