312 



MUSICAL SCALES 



The following definitions and Tables 307 and 308 are taken from the American Stand- 

 ard Acoustical Terminology Z24.1, 1949. 



Just scale. — A just scale is a musical scale such that the frequency intervals are repre- 

 sented by the ratios of small integers. 



Equally tempered scale. — An equally tempered scale is a series of notes selected from 

 a division of the octave (usually) into 12 equal intervals. 



Equally tempered semitone (half -step). —An equally tempered semitone is the inter- 

 val between two sounds whose basic frequency ratio is the twelfth root of two. 



Note. — The interval, in semitones, between any two frequencies is 12 times the logarithm 

 on the base 2 of the frequency ratio. 



Cent. — A cent is the interval between two sounds whose basic frequency ratio is the 

 twelve-hundredth root of two. 



Note. — The interval, in cents, between any two frequencies is 1200 times the logarithm 

 on the base 2 of the frequency ratio. Thus, 1200 cents = 12 semitones = 1 octave. 



TABLE 306.— FREQUENCY RATIOS AND INTERVALS FOR JUST AND 

 EQUALLY TEMPERED SCALES 



Just temperament 



\ 



Frequency Cents 



ratio from from 



starting starting 



Interval from starting point point point 



Unison 1:1 



Minor second or semitone 16:15 111.731 



Minor tone 10:9 182.404 



Major second or whole tone 9:8 203.910 



Minor third 6:5 315.641 



Major third 5 :4 386.314 



Perfect fourth 4 :3 398.045 



Augmented fourth 45 :32 590.224 



Diminished fifth 64:45 609.777 



Perfect fifth 3:2 701.955 



Minor sixth 8:5 813.687 



Major sixth 5 :3 884.359 



Harmonic minor seventh 7:4 968.826 



Grave minor seventh 16 :9 996.091 



Minor seventh 9:5 1017.597 



Major seventh 15 :8 1088.269 



Octave 2:1 1200.000 



Equal temperament 



SMITHSONIAN PHYSICAL TABLES 



