80 MUSIC is when an interval has to be inverted. The inversion of an interval less than an octave is the difference between it and an octave ; i. e., the interval which remains after the first has been subtracted from an octave. Thus, to invert the minor third we divide 2 by f ; or, in other words, we invert the vibration frac- tion of the interval and multiply by 2. This operation gives ns f ; therefore, the inversion of the minor third is the major sixth. Evi- dently there exists a mutual relation between an interval and its inversion, so that each is the inversion of the other. Thus, the inver- sion of the major sixth is the minor third. The following three pairs of consonant inter- vals, embraced within the compass of an oc- tave, have to each other the mutual relation of inversions : Minor third, . . Major sixth, . . f Major third, . . f- Minor sixth, . . Fourth, .... | Fifth, . . . . f Musical sounds of different pitch, simultane- ously emitted, form a chord. Chords formed of two notes are called binary chords ; those of three notes are called triads. A binary chord is consonant when its two notes form a consonant interval. In a triad there are three intervals: one between its lowest note and^the next higher, one between the middle and high- est note, and one between the lowest and highest. The triad is only consonant when all three of these intervals are concords. Therefore, to form consonant triads we select a note, then find the others, each of which forms with the bottom note a consonant inter- val. We then determine whether the interval between the two higher notes is a consonant one; if this be so, then the triad is conso- nant. To determine all of the consonant tri- ads contained in an octave, above any selected bottom note, we must assign to the middle and top notes every possible consonant position with respect to the fixed bottom note, and reject all such relative positions as give rise to dissonant intervals between those notes themselves. The remaining positions will constitute all the consonant triads which have for their lowest note that originally selected. The intervals at our disposal are : for the mid- dle note, from the minor third to the minor sixth ; and for the upper note, from the major third to the major sixth. In the following table the possible positions of the middle note with respect to the bottom note are shown in the left-hand vertical column, the name of each interval being accompanied by its vibra- tion fraction. The possible positions of the top note are similarly shown in the top hori- zontal line. Each space common to a hori- zontal and vertical line contains the vibration fraction of the interval formed between the simultaneous positions of the middle and upper notes named at the beginning of these lines. The intervals thus formed which are dissonant are designated by being enclosed in brackets. Whenever they are consonant the name of the interval is given. Major third. Fourth. Fifth. f Minor sixth. 1 Major sixth. Minor third. in] m M^jor third. 1 Fourth. Fourth. Major third. M] Mtaor third. UN Fourth. [] Minor third. Major third. m [tt] Fifth, t Minor sixth. 1 [If] An examinati< that the follow Middle note. Minor third. Major third. Fourth. The above co musical notati< m of the above tables shows ing are all the consonances : Upper note. Fifth, or minor sixth. Fifth, or major sixth. Minor sixth, or major sixth. usonances are thus expressed in m: We thus obtain two groups of three major and three minor triads, which may be ar- ranged thus : (a) Fifth. Major third. , xj Fifth. , . W Minor third. < Minor sixth. Minor third. sixth, third. jMaior sixth.
Fourth.
( Minor sixth. l Fourth. The above six consonant triads may be de- fined by the intervals separating the middle from the bottom note, and the top from the middle note, instead of defining these in- tervals, as we have done above, by the inter- vals formed by their middle and top notes with the bottom note. To bring about this change we perform on each one a subtraction of intervals. Thus, the difference between a fifth and a major third is f x|=, or a minor third. In this manner we find that the top and middle notes are separated by the follow- ing intervals :
