The investigations undertaken by this eminent maker convinced him that the ideal striking point lay between one-seventh and one-ninth of the speaking length of the string. Now, our investigations have shown us that the most harmonious and agreeable compound tone is that which is formed by the combination of the first eight partials. It would seem, therefore, that one-eighth of the speaking length would be more correct than the approximation that was arrived at by Broadwood. Theoretically, indeed, the latter is nearer to the ideal point; is, in fact, the ideal point. For obvious mechanical reasons, however, it is usually impossible to hit this point with exactitude, and the approximation suggested and used by Broadwood has been proved, by the practice of the best makers, to offer the nearest practical solution.
We may, then, lay it down as a rule to be followed that a point as nearly as possible midway between one-seventh and one-ninth of the speaking length of the string should be chosen and adhered to as the proper place where the blow of the hammer should be struck. If this rule be faithfully followed the greatest obstacle to purity of tone is removed and the most harmonious and agreeable combination of partials is in a fair way to be secured. Nevertheless, it is necessary to make an exception for the highest notes on the piano. Practical experience has shown that one-tenth is a better striking point for the very highest and shortest strings.
Thus we have been able to enunciate and discuss the principal laws that govern the activities of sounding strings, particularly those of the pianoforte. As the argument is developed, it will often appear that the theoretical exactitude of the rules here laid down must be modified in practice. Such a condition is always inevitable as between a body of laws and the application thereof. It will be found, however, that the variations to be recorded are not generally very important, and the reader will be well advised to make the rules enunciated in this chapter his continual leaning post and guide.
The most conspicuous difference is, perhaps, that which exists between the theoretical and practical results of halving string lengths to obtain octaves. In practice it is found that pianoforte strings generally sound a little flat of the octave when divided at exactly the middle point. But the variation is the fault of the steel wire and not of the rule.
CHAPTER V.
THE MUSICAL SCALE AND MUSICAL INTONATION.
We have now considered as much of the phenomena of musical sounds as may be considered to have a bearing upon the purpose of our investigations. We may then devote some space to the matter of the expression of musical ideas, and the intonation which has been devised in order to reduce the mental products of composers to the limitations of musical instruments. Music is expressed through the medium of a scale of tones, all of which bear definite relations to each other as to pitch. The “diatonic scale,” which is the foundation of musical intonation, is composed of a series of eight tones which are named after letters of the alphabet, the last tone having the same name as, and being the octave to, the first. The frequencies of these tones always bear the same ratios, one to another, whatever may be their positions within the compass of any instrument. Now, considering the frequency of the first tone to be unity, the frequencies of the others are in the following proportions:
| C | D | E | F | G | A | B | C |
| 1 | 9⁄8 | 5⁄4 | 4⁄3 | 3⁄2 | 5⁄3 | 15⁄8 | 2 |
If we now divide these proportionate numbers each by the other we have the proportionate intervals that separate them. Doing this, we have the following result: