Tables 82-83. 

 MUSICAL SCALES. 



103 



The pitch relations between two notes may be expressed precisely (i) by the ratio of their vibration frequencies; 

 (2) by the number of equally-tempered semitones between them (E. S.); also, less conveniently, (3) by the common 

 logarithm of the ratio in (i); (4) by the lengths of the two portions of the tense string which will furnish the notes; 

 and (s) in terms of the octave as unity. The ratio in (4) is the reciprocal of that in (i); the number for (5) is 1/12 of 

 that for (2); the number for (2) is nearly 40 times that for (3). 



Table 82 gives data for the middle octave, including vibration frequencies for three standards of pitch; a = 43s double 

 vibrations per second, is the international standard and was adopted by the American Piano Manufacturers' Associa- 

 tion. The "just-diatonic scale "of C-major is usually deduced, following Chladni, from the ratios of the three perfect 

 major triads reduced to one octave, thus: 



6 

 6 4:5:6 



F A C E G B D 



i6 20 24 30 36 45 54 



24 27 30 32 36 40 4S 48 

 Other equivalent ratios and their values in E. S. are given in Table 83. By transferring D to the left and using the 

 ratio 10 : 12 : 15 the scale of A-minor is obtained, which agrees with that of C-raajor except that D= 26 2/3. Nearly the 

 same ratios are obtained from a series of harmonics beginning with the eighth; also by taking 12 successive perfect 

 or Pythagorean fifths or fourths and reducing to one octave. Such calculations are most easily made by adding and 

 suDtracting intervals expressed in E. S. The notes needed to furnish a just major scale in other keys may be found 

 by successive transpositions by fifths or fourths as shown in Table 83. Disregarding the usually negligible difference 

 of 0.03 E. S., the table gives the 24 notes to the octave required in the simplest enharmonic organ; the notes fall into 

 pairs that differ by a comma, 0.22 E. S. The line "mean tone" is based on Dom Bedos' rule for tuning the organ 

 (1746). The tables have been checked by the data in Ellis' Helmholtz's "Sensations of Tone." 



TABLE 83. 



Smithsonian Tables. 



