We may shorten the operation as follows: Instead of multiplying 2/3 by 2/3, giving us 4/9, and then multiplying this answer by 2, let us double the fraction, 2/3, which equals 4/3, and use it as a multiplier when it becomes necessary to double the segment to keep within the octave.

We may proceed now with the twelve steps as follows:--

Steps--

1. 1C to 1G segment 2/3 for 1G 2. 1G " 1D Multiply 2/3 by 4/3, gives segment 8/9 " 1D 3. 1D " 1A " 8/9 " 2/3 " " 16/27 " 1A 4. 1A " 1E " 16/27 " 4/3 " " 64/81 " 1E 5. 1E " 1B " 64/81 " 2/3 " " 128/243 " 1B 6. 1B " 1F? " 128/243 " 4/3 " " 512/729 " 1F?

7. 1F? " 1C? " 512/729 " 4/3 " " 2048/2187 " 1C?

8. 1C? " 1G? " 2048/2187 " 2/3 " " 4096/6561 " 1G?

9. 1G? " 1D? " 4096/6561 " 4/3 " " 16384/19683 " 1D?

10. 1D? " 1A? " 16384/19683 " 2/3 " " 32768/59049 " 1A?

11. 1A? " 1F " 32768/59049 " 4/3 " " 131072/177147 " 1F 12. 1F " 2C " 131072/177147 " 2/3 " " 262144/531441 " 2C

Now, this last fraction should be equivalent to 1/2, when reduced to its lowest terms, if it is destined to produce a true octave; but, using this numerator, 262144, a half would be expressed by 262144/524288, the segment producing the true octave; so the fraction 262144/531441, which represents the segment for 2C, obtained by the circle of fifths, being evidently less than 1/2, this segment will yield a tone somewhat sharper than the true octave. The two denominators are taken in this case to show the ratio of the variance; so the octave obtained from the circle of fifths is sharper than the true octave in the ratio expressed by 531441 to 524288, which ratio is called the _ditonic comma_. This comma is equal to one-fifth of a half-step.

We are to conclude, then, that if octaves are to remain perfect, and we desire to establish an equal temperament, the above-named difference is best disposed of by dividing it into twelve equal parts and depressing each of the fifths one-twelfth part of the ditonic comma; thereby dispersing the dissonance so that it will allow perfect octaves, and yet, but slightly impair the consonance of the fifths.

We believe the foregoing propositions will demonstrate the facts stated therein, to the student's satisfaction, and that he should now have a pretty thorough knowledge of the mathematics of the temperament. That the equal temperament is the only practical temperament, is confidently affirmed by Mr. W.S.B. Woolhouse, an eminent authority on musical mathematics, who says:--

"It is very misleading to suppose that the necessity of temperament applies only to instruments which have fixed tones. Singers and performers on perfect instruments must all temper their intervals, or they could not keep in tune with each other, or even with themselves; and on arriving at the same notes by different routes, would be continually finding a want of agreement. The scale of equal temperament obviates all such inconveniences, and continues to be universally accepted with unqualified satisfaction by the most eminent vocalists; and equally so by the most renowned and accomplished performers on stringed instruments, although these instruments are capable of an indefinite variety of intonation. The high development of modern instrumental music would not have been possible, and could not have been acquired, without the manifold advantages of the tempered intonation by equal semitones, and it has, in consequence, long become the established basis of tuning."

NUMERICAL COMPARISON OF THE DIATONIC SCALE WITH THE TEMPERED SCALE.

The following table, comparing vibration numbers of the diatonic scale with those of the tempered, shows the difference in the two scales, existing between the thirds, fifths and other intervals.

Notice that the difference is but slight in the lowest octave used which is shown on the left; but taking the scale four octaves higher, shown on the right, the difference becomes more striking.

|DIATONIC.|TEMPERED.| |DIATONIC.|TEMPERED.| C|32. |32. |C|512. |512. | D|36. |35.92 |D|576. |574.70 | E|40. |40.32 |E|640. |645.08 | F|42.66 |42.71 |F|682.66 |683.44 | G|48. |47.95 |G|768. |767.13 | A|53.33 |53.82 |A|853.33 |861.08 | B|60. |60.41 |B|960. |966.53 | C|64. |64. |C|1024. |1024. |

Following this paragraph we give a reference table in which the numbers are given for four consecutive octaves, calculated for the system of equal temperament. Each column represents an octave. The first two columns cover the tones of the two octaves used in setting the temperament by our system.

TABLE OF VIBRATIONS PER SECOND.

C |128. |256. |512. |1024. | C? |135.61 |271.22 |542.44 |1084.89 | D |143.68 |287.35 |574.70 |1149.40 | D? |152.22 |304.44 |608.87 |1217.75 | E |161.27 |322.54 |645.08 |1290.16 | F |170.86 |341.72 |683.44 |1366.87 | F? |181.02 |362.04 |724.08 |1448.15 | G |191.78 |383.57 |767.13 |1534.27 | G? |203.19 |406.37 |812.75 |1625.50 | A |215.27 |430.54 |861.08 |1722.16 | A? |228.07 |456.14 |912.28 |1824.56 | B |241.63 |483.26 |966.53 |1933.06 | C |256. |512. |1024. |2048. |

Much interesting and valuable exercise may be derived from the investigation of this table by figuring out what certain intervals would be if exact, and then comparing them with the figures shown in this tempered scale. To do this, select two notes and ascertain what interval the higher forms to the lower; then, by the fraction in the table below corresponding to that interval, multiply the vibration number of the lower note.

EXAMPLE.--Say we select the first C, 128, and the G in the same column. We know this to be an interval of a perfect fifth. Referring to the table below, we find that the vibration of the fifth is 3/2 of, or 3/2 times, that of its fundamental; so we simply multiply this fraction by the vibration number of C, which is 128, and this gives 192 as the exact fifth. Now, on referring to the above table of equal temperament, we find this G quoted a little less (flatter), viz., 191.78. To find a fourth from any note, multiply its number by 4/3, a major third, by 5/4, and so on as per table below.

TABLE SHOWING RELATIVE VIBRATION OF INTERVALS BY IMPROPER FRACTIONS.

The relation of the Octave to a Fundamental is expressed by 2/1 " " " Fifth to a " " 3/2 " " " Fourth to a " " 4/3 " " " Major Third to a " " 5/4 " " " Minor Third to a " " 6/5 " " " Major Second to a " " 9/8 " " " Major Sixth to a " " 5/3 " " " Minor Sixth to a " " 8/5 " " " Major Seventh to a " " 15/8 " " " Minor Second to a " " 16/15

QUESTIONS ON LESSON XIII.

1. State what principle is demonstrated in Proposition II.

2. State what principle is demonstrated in Proposition III.

3. What would be the vibration per second of an exact (not tempered) fifth, from C-512?

4. Give the figures and the process used in finding the vibration number of the _exact_ major third to C-256.

5. If we should tune the whole circle of twelve fifths exactly as detailed in Proposition III, how much too sharp would the last C be to the first C tuned?

LESSON XIV.

~MISCELLANEOUS TOPICS PERTAINING TO THE PRACTICAL WORK OF TUNING.~

~Beats.~--The phenomenon known as "beats" has been but briefly alluded to in previous lessons, and not a.n.a.lytically discussed as it should be, being so important a feature as it is, in the practical operations of tuning. The average tuner hears and considers the beats with a vague and indefinite comprehension, guessing at causes and effects, and arriving at uncertain results. Having now become familiar with vibration numbers and ratios, the student may, at this juncture, more readily understand the phenomenon, the more scientific discussion of which it has been thought prudent to withhold until now.

In speaking of the unison in Lesson VIII, we stated that "the cause of the waves in a defective unison is the alternate recurring of the periods when the condensations and the rarefactions correspond in the two strings, and then antagonize." This concise definition is complete; but it may not as yet have been fully apprehended. The unison being the simplest interval, we shall use it for consideration before taking the more complex intervals into account.

Let us consider the nature of a single musical tone: that it consists of a chain of sound-waves; that each sound-wave consists of a condensation and a rarefaction, which are directly opposed to each other; and that sound-waves travel through air at a specific rate per second. Let us also remark, here, that in the foregoing lessons, where reference is made to vibrations, the term signifies sound-waves. In other words, the terms, "vibration" and "sound-wave," are synonymous.

If two strings, tuned to give forth the same number of vibrations per second, are struck at the same time, the tone produced will appear to come from a single source; one sweet, continuous, smooth, musical tone. The reason is this: The condensations sent forth from each of the two strings occur exactly together; the rarefactions, which, of course, alternate with the condensations, are also simultaneous. It necessarily follows, therefore, that the condensations from each of the two strings travel with the same velocity. Now, while this condition prevails, it is evident that the two strings a.s.sist each other, making the condensations more condensed, and, consequently, the rarefactions more rarefied, the result of which is, the two allied forces combine to strengthen the tone.

In opposition to the above, if two strings, tuned to produce the same tone, could be so struck that the condensation of one would occur at the same instant with the rarefaction of the other, it is readily seen that the two forces would oppose, or counteract each other, which, if equal, would result in absolute silence.[G]

[G] When the bushing of the center-pin of the hammer b.u.t.t becomes badly worn or the hammer-f.l.a.n.g.e becomes loose, or the condition of the hammer or f.l.a.n.g.e becomes so impaired that the hammer has too much play, it may so strike the strings as to tend to produce the phenomenon described in the above paragraph. When in such a condition, one side of the hammer may strike in advance of the other just enough to throw the vibrations in opposition. Once you may get a strong tone, and again you strike with the same force and hear but a faint, almost inaudible sound. For this reason, as well as that of preventing excessive wear, the hammer joint should be kept firm and rigid.

If one of the strings vibrates 100 times in a second, and the other 101, there will be a portion of time during each second when the vibrations will coincide, and likewise a portion of time when they will antagonize each other. The periods of coincidence and of antagonism pa.s.s by progressive transition from one to the other, and the portion of time when exact.i.tude is attained is infinitesimal; so there will be two opposite effects noticed in every second of time: the one, a progressive augmentation of strength and volume, the other, a gradual diminution of the same; the former occurring when the vibrations are coming into coincidence, the latter, when they are approaching the point of antagonism. Therefore, when we speak of one beat per second, we mean that there will be one period of augmentation and one period of diminution in one second. Young tuners sometimes get confused and accept one beat as being two, taking the period of augmentation for one beat and likewise the period of diminution. This is most likely to occur in the lower fifths of the temperament where the beats are very slow.

Two strings struck at the same time, one tuned an octave higher than the other, will vibrate in the ratio of 2 to 1. If these two strings vary from this ratio to the amount of _one_ vibration, they will produce _two_ beats. Two strings sounding an interval of the fifth vibrate in the ratio of 3 to 2. If they vary from this ratio to the amount of _one_ vibration, there will occur _three_ beats per second.

In the case of the major third, there will occur _four_ beats per second to a variation of _one_ vibration from the true ratio of 5 to 4. You should bear this in mind in considering the proper number of beats for an interval, the vibration number being known.

It will be seen, from the above facts in connection with the study of the table of vibration numbers in Lesson XIII, that all fifths do not beat alike. The lower the vibration number, the slower the beats. If, at a certain point, a fifth beats once per second, the fifth taken an octave higher will beat twice; and the intervening fifths will beat from a little more than once, up to nearly twice per second, as they approach the higher fifth. Vibrations per second double with each octave, and so do beats.

By referring to the table in Lesson XIII, above referred to, the exact beating of any fifth may be ascertained as follows:--

Ascertain what the vibration number of the _exact_ fifth would be, according to the instructions given beneath the table; find the difference between this and the _tempered_ fifth given in the table.

Multiply this difference by 3, and the result will be the number of beats or fraction thereof, of the tempered fifth. The reason we multiply by 3 is because, as above stated, a variation of one vibration per second in the fifth causes three beats per second.

_Example._--Take the first fifth in the table, C-128 to G-191.78, and by the proper calculation (see example, page 147, Lesson XIII) we find the exact fifth to this C would be 192. The difference, then, found by subtracting the smaller from the greater, is .22 (22/100). Multiply .22 by 3 and the result is .66, or about two-thirds of a beat per second.

By these calculations we learn that the fifth, C-256 to G-383.57, should have 1.29 beats: nearly one and a third per second, and that the highest fifth of the temperament, F-341.72 to C-512, should be 1.74, or nearly one and three-quarters. By remembering these figures, and endeavoring to temper as nearly according to them as possible, the tuner will find that his temperament will come up most beautifully.

This is one of the features that is overlooked or entirely unknown to many fairly good tuners; their aim being to get all fifths the same.

~Finishing up the Temperament.~--If your last trial, F-C, does not prove a correct fifth, you must consider how best to rectify. The following are the causes which result in improper temperament:

1. Fifths too flat.

2. Fifths not flat enough.

3. Some fifths correctly tempered and others not.