HISTORY OF MUSICAL SCALES. 441 



APPENDIX. 



The laws briefly stated on pajie 487 for the neveral kinds of instnnncnts discussed 

 in the paper may be expressed more accurately by the following formuke: 



Let N = number of complete vibrations per second. 

 I = length of string or column of air or bar. 



(I = diameter of mouth-hole of resonator, corrected for tliickness of wall. 

 ^ 1) = diameter of finger-holes of resonator, corrected for thickness of wall. 

 ;( = number of tinger-holes opened on resonator. 

 f =z thickness of bar. , 



K = constant, depending on material and units of measurement. 

 Assuming centimeter-gram-second units and ordinary temperatures, 



K' = s/Xension in dynes -^- mass in grams per cm. = velocity ; e. g., in piano 



strings 17,000 to 40,000 cm. -sec; in violin strings from lo,000 for the 



covered string to 4.3,000 for the gut E-string; in weak primitive insti-u- 



ments probably much less. 



K" = 34,000 cm. -sec, the velocity of sound in air. 



K'" = 520,000 cm.-sec for iron bars; 340,000 to 520,000 for wood bars supported 



as usual in a xylophone. 

 K"' = 5,500. 



Then, corresponding with the brief laws, 



K^ 1 



(In) For strings: N =—- = ——^ v^tension -:- linear density- 



, K" 17,000 

 (lb) For columns of air: ^ = WT= — /4^" 



(2) For bars: N = K'" ~= 340,000 to 520,000 4 



(3) For resonators: N = K' 



\^a + sum ol b _ 5,500 



\/ volume \/volume 



V<'+-0- 



These constants are sufficiently accurate for the general purposes of the anthro- 

 pologi«t and musician. But the results should be expressed in musical terms. The 

 French standard pitch, now adopted by the Piano Makers' Association, gives 

 A = 435 d. v., or C = 258.7 d. v., and the ratio for any interval of j) piano semitones 

 is 2r!!. In most cases it is much more convenient to have intervals than ratios; and 

 incomparal)ly the most convenient unit of intervals is the piano semitone, of which 

 12 by definition make an octave; these can readily be grouped by anyone with slight 

 musical knowledge into larger intervals. Thirds, etc., and the musical value of any 

 whole number of them can instantly be found on a well-tuned piano. 



Since the reduction of ratios to intervals can not ordinarily be done without 

 logarithms, a short table has been calculated and is appended by the use of which 

 the reduction may be done by inspection in most practical cases. This table gives 

 the logarithm of every whole numl)er from 1 to 40, and the product of these by 40, 

 less one three-hundredth, together with the successive differences; these are in 

 semitones; for the factor is so chosen that when the logarithm of the ratio 2:1 is 

 multiplied by it the product will l)e 12, which is the number of semitones corre- 

 spon<ling to the ratio of the octave. Much more elaborate tables, but without the 

 column of differences, have been published by Prony and l)y Ellis. In using the 

 table it is well to remember that the average uncertainty in pitch of public per- 

 formers in Berlin was found to be about one-tenth of a semitone. 



NAT MUS 1900 31 



