. Science . tone to that of the lower. It followsthat on the tempered scale this ratio is thesame for any two adjacent tones. The numer-ical value of this interval is , since thesum of twelve such intervals is 2, the numer-ical value of the octave interval. These considerations coupled with the fun-damental law of string vibrations, to the effect which 0c/00=0C/0d=0d/0D = etc., thevalue of this ratio being by construc-tion. If this diagram is drawn on the top of asonometer, or a table-top across which a stringis stretched, and bridges are placed under thestring opposite 0 and c,
. Science . tone to that of the lower. It followsthat on the tempered scale this ratio is thesame for any two adjacent tones. The numer-ical value of this interval is , since thesum of twelve such intervals is 2, the numer-ical value of the octave interval. These considerations coupled with the fun-damental law of string vibrations, to the effect which 0c/00=0C/0d=0d/0D = etc., thevalue of this ratio being by construc-tion. If this diagram is drawn on the top of asonometer, or a table-top across which a stringis stretched, and bridges are placed under thestring opposite 0 and c, it forms a completefinger board for running the major, minor andchromatic scales. The device lends itself to the demonstrationof the following relations: (1) Comparison of the major and minorscales. (2) Comparison of the major andminor chords. (3) To show that on the tem-pered scale any note may be taken as key note,and all scales are equally good. For this pur-pose choose any point as starting point, call-. that, for a string of given weight and tension,the frequency of a vibrating segment is in-versely proportional to its length, suggest asimple method of finding those string lengthswhich will give the successive tones of thetempered scale. Draw two intersecting straight lines includ-ing any convenient angle (see accompanyingdiagram). From the point of intersection layofi on one line any convenient length Oc = L,on the other a length 00 = L^ Jointhe points Cc by a straight line. Locate the corresponding points B and dand join by a dotted straight line. Now drawthe series Cd, dD, Be, etc., and the dottedseries, parallel to Bet and cG. By this meansthe points c#, d, d$, e, etc., are determined atwhich a string of length L (==0c) must bestopped to give the successive tones of thetempered (chromatic) scale. This will be evi-dent from the construction of the figure in ing it point 1. Number the points from point1 upward. Sound in succession the tonesgiven by the string w
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