192 



Tables 144 and 145 

 MUSICAL SCALES 



The pitch relations between two notes may be expressed precisely (i) by the ratio of their vibra- 

 tion 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 (1); (4) by the lengths of the two portions 

 of the tense string which will furnish the notes; and (5) in terms of the octave as unity. The ratio 

 in (4) is the reciprocal of that in (1); the number for (5) is 1/12 of that for (2); the number for 

 (2) is nearly 40 times that for (3). 



Table 144 gives data for the middle octave, including vibration frequencies for three standards of 

 pitch; A 3 = 435 double vibrations per second, is the international standard and was adopted by the 

 American Piano Manufacturers' Association. The " just-diatonic scale " of C-major is usually 

 from the ratios of the 



deduced, following Chladni, 

 octave, thus: 



40 



three perfect major triads reduced to one 



4:5:6 



6 4 



F A C EG 



16 20 24 30 36 



24 27 30 32 36 



Other equivalent ratios and their values in E. S. are given in Table 145- 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-major 

 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 subtracting 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 145. Disregarding the usually negligible differ- 

 ence of 0.02 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 144. — Data for Middle Octave 



Smithsonian Tables. 



