196 HERMANN VON HELMHOLTZ 



developed, Helmholtz thought it well to undertake historical 

 studies of the development of the scale in different nations, our 

 major and minor scales having been developed very late. In 

 July, 1862, he read a paper on 'The Persian and Arabian 

 Scales' to the Nat. Hist. Med. Verein. In the system he pro- 

 poses for the construction and tuning of musical instruments, 

 in which all the keys can be played in pure consonant chords, 

 twice as many intervals were required as usual. These 

 historical studies led him to the conclusion that in Greek 

 scales, the fifth fifth from C upwards was used as the third 

 of C, from which it differs only by the minute interval 

 81/80; while if eight fifths are taken downwards from C the 

 tone F flat is reached, which only differs from the third of 

 C by something like the tenth part of the interval 81/80, so 

 that it can be substituted for it. The musicians of Persia and 

 Arabia took advantage of these substitutions to obtain pure 

 natural scales ; their system was one of seventeen fifths, from 

 which scales with Pythagorean, or natural, thirds and sixths 

 were derived. Helmholtz followed a common physical prin- 

 ciple throughout, i. e. that of the relationship of musical tones. 

 The definition chosen by him, according to which two com- 

 pound tones are related when they have some common 

 over-tones, shows that the notes most nearly related to the 

 fundamental are the octave, and then the fifth and fourth ; 

 these related musical tones occur in all ' modes '. 



After Helmholtz had ascertained from his experiments on 

 timbre that differences in quality of musical tone depend 

 principally upon the number and strength of the harmonic 

 partials that accompany the prime tone, he next had to 

 investigate the forms of the elastic vibrations executed by 

 variously sounding bodies. He published his observations on 

 the vibrations in strings produced by the bow of a violin, in 

 the Proceedings of the Glasgow Philosophical Society, on 

 December 19, 1860, with the title ' On the Motion of the Strings 

 of a Violin*. It was at once obvious that the string set in 

 motion by the bow can only vibrate in the same plane as 

 the string and the hair of the bow. He powdered the string 

 of a fine instrument with starch, illuminated it strongly, and 

 examined its motions by means of a vibration microscope 

 constructed for the purpose, the object-glass of which was 



