Piano Tuning Part 11

If we divide an octave, as from middle C to 3C, into three major thirds, each in the perfect ratio of 5 to 4, as C-E, E-G? (A?), A?-C, then the C obtained from the last third, A?-C, will be too flat to form a perfect octave by a small quant.i.ty, called in the theory of harmonics a _diesis_, which is expressed by the ratio 128 to 125.

EXPLANATION.--The length of the string sounding the tone C is represented by unity or 1. Now, as we have shown, the major third to that C, which is E, is produced by 4/5 of its length.

In like manner, G?, the major third to E, will be produced by 4/5 of that segment of the string which sounds the tone E; that is, G? will be produced by 4/5 of 4/5 (4/5 multiplied by 4/5) which equals 16/25 of the entire length of the string sounding the tone C.

We come, now, to the last third, G? (A?) to C, which completes the interval of the octave, middle C to 3C. This last C, being the major third from the A?, will be produced as before, by 4/5 of that segment of the string which sounds A?; that is, by 4/5 of 16/25, which equals 64/125 of the entire length of the string. Keep this last fraction, 64/125, in mind, and remember it as representing the segment of the entire string, which produces the upper C by the succession of three perfectly tuned major thirds.

Now, let us refer to the law which says that a perfect octave is obtained from the exact half of the length of any string. Is 64/125 an exact half? No; using the same numerator, an exact half would be 64/128.

Hence, it is clear that the octave obtained by the succession of perfect major thirds will differ from the true octave by the ratio of 128 to 125. The fraction, 64/125, representing a longer segment of the string than 64/128 (1/2), it would produce a flatter tone than the exact half.

It is evident, therefore, that _all major thirds must be tuned somewhat sharper than perfect_ in a system of equal temperament.

The ratio which expresses the value of the _diesis_ is that of 128 to 125. If, therefore, the octaves are to remain perfect, which they must do, _each major third must be tuned sharper than perfect by one-third part of the diesis_.

The foregoing demonstration may be made still clearer by the following diagram which represents the length of string necessary to produce these tones. (This diagram is exact in the various proportional lengths, being about one twenty-fifth the actual length represented.)

Middle C (2C) 60 inches.

-------------------------------------------------- O O

E (4/5 of 60) 48 inches.

-------------------------------------------- O O

G? (A?) (4/5 of 48) 38-2/5 inches.

-------------------------------------- O O

3C (4/5 of 38-2/5) 30-18/25 inches.

-------------------------------- O O

This diagram clearly demonstrates that the last C obtained by the succession of thirds covers a segment of the string which is 18/25 longer than an exact half; nearly three-fourths of an inch too long, 30 inches being the exact half.

To make this proposition still better understood, we give the comparison of the actual vibration numbers as follows:--

Perfect thirds in ratio 4/5 have these vibration numbers: =

1st third 2d third 3d third (C 256 - E 320) (E 320 - G? 400) (G? 400 - C 500) --------------- ----------------- ----------------- no beats no beats no beats

Tempered thirds qualified to produce true octave: =

(C 256 - E 322 5/10) (E 322 5/10 - G? 406 4/10) (G? 406 4/10 - C 512) -------------------- -------------------------- ---------------------- 10 beats 13-1/10 beats 16 beats

We think the foregoing elucidation of Proposition I sufficient to establish a thorough understanding of the facts set forth therein, if they are studied over carefully a few times. If everything is not clear at the first reading, go over it several times, as this matter is of value to you.

QUESTIONS ON LESSON XII.

1. Why is the pitch, C-256, adopted for scientific discussion, and what is this pitch called?

2. The tone G forms the root (1) in the key of G. What does it form in the key of C? What in F? What in D?

3. What tone is produced by a 2/3 segment of a string? What by a 1/2 segment? What by a 4/5 segment?

4. (a) What intervals must be tuned absolutely perfect?

(b) In the two intervals that must be tempered, the third and the fifth, which will bear the greater deviation?

5. What would be the result if we should tune from 2C to 3C by a succession of perfect thirds?

6. Do you understand the facts set forth in Proposition I, in this lesson?

LESSON XIII.

~RATIONALE OF THE TEMPERAMENT.~ (Concluded from Lesson XII.)

PROPOSITION II.

That the student of scientific scale building may understand fully the reasons why the tempered scale is at constant variance with exact mathematical ratios, we continue this discussion through two more propositions, No. II, following, demonstrating the result of dividing the octave into four minor thirds, and Proposition III, demonstrating the result of twelve perfect fifths. The matter in Lesson XII, if properly mastered, has given a thorough insight into the princ.i.p.al features of the subject in question; so the following demonstration will be made as brief as possible, consistent with clearness.

Let us figure the result of dividing an octave into four minor thirds.

The ratio of the length of string sounding a fundamental, to the length necessary to sound its minor third, is that of 6 to 5. In other words, 5/6 of any string sounds a tone which is an exact minor third above that of the whole string.

Now, suppose we select, as before, a string sounding middle C, as the fundamental tone. We now ascend by minor thirds until we reach the C, octave above middle C, which we call 3C, as follows:

Middle C-E?; E?-F?; F?-A; A-3C.

Demonstrate by figures as follows:--Let the whole length of string sounding middle C be represented by unity or 1.

E? will be sounded by 5/6 of the string 5/6 F?, by 5/6 of the E? segment; that is, by 5/6 of 5/6 of the entire string, which equals 25/36 A, by 5/6 of 25/36 of entire string, which equals 125/216 3C, by 5/6 of 125/216 of entire string, which equals 625/1296

Now bear in mind, this last fraction, 625/1296, represents the segment of the entire string which should sound the tone 3C, an exact octave above middle C. Remember, our law demands an exact half of a string by which to sound its octave. How much does it vary? Divide the denominator (1296) by 2 and place the result over it for a numerator, and this gives 648/1296, which is an exact half. Notice the comparison.

3C obtained from a succession of exact minor thirds, 625/1296 3C obtained from an exact half of the string 648/1296

Now, the former fraction is smaller than the latter; hence, the segment of string which it represents will be shorter than the exact half, and will consequently yield a sharper tone. The denominators being the same, we have only to find the difference between the numerators to tell how much too short the former segment is. This proves the C obtained by the succession of minor thirds to be too short by 23/1296 of the length of the whole string.

If, therefore, all octaves are to remain perfect, it is evident that _all minor thirds must be tuned flatter than perfect_ in the system of equal temperament.

The ratio, then, of 648 to 625 expresses the excess by which the true octave exceeds four exact minor thirds; consequently, each minor third must be flatter than perfect by one-fourth part of the difference between these fractions. By this means the dissonance is evenly distributed so that it is not noticeable in the various chords, in the major and minor keys, where this interval is almost invariably present. (We find no record of writers on the mathematics of sound giving a name to the above ratio expressing variance, as they have to others.)

PROPOSITION III.

Proposition III deals with the perfect fifth, showing the result from a series of twelve perfect fifths employed within the s.p.a.ce of an octave.

METHOD.--Taking 1C as the fundamental, representing it by unity or 1, the G, fifth above, is sounded by a 2/3 segment of the string sounding C. The next fifth, G-D, takes us beyond the octave, and we find that the D will be sounded by 4/9 (2/3 of 2/3 equals 4/9) of the entire string, which fraction is less than half; so to keep within the bounds of the octave, we must double this segment and make it sound the tone D an octave lower, thus: 4/9 times 2 equals 8/9, the segment sounding the D within the octave.