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 analytically 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 assist 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]

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 pass by progressive transition from one to the other, and the portion of time when exactitude 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:

Fifths too flat.

Fifths not flat enough.

Some fifths correctly tempered and others not.

Some fifths sharper instead of flatter than perfect; a condition that must be watched with vigilance.

Some or all of the strings tuned fall from the pitch at which they were left.

From a little reflection upon these causes, it is seen that the last trial may prove a correct fifth and yet the temperament be imperfect. If this is the case, it will be necessary to go all over the temperament again. Generally, however, after you have had a little experience, you will find the trouble in one of the first two causes above, unless it be a piano wherein, the strings fall as in Cause 5. This latter cause can be ascertained in cases only where you have started from a tuning pipe or fork. Sometimes you may find that the temperament may be corrected by the alteration of but two or three tones; so it is always well to stop and examine carefully before attempting the correction. A haphazard attempt might cause much extra work.

In temperament setting by our system, if the fifths are properly tempered and the octaves are left perfect, the other intervals will need no attention, and will be found beautifully correct when used in testing.

The mistuned or tempered intervals are as follows:

Intervals Flattened. Intervals Sharpened.

The Fifth, slightly. The Fourth, slightly.

The Minor Third, considerably. The Major Third, greatly.

The Minor Sixth, considerably. The Major Sixth, greatly.

Tuning the Treble. In tuning the treble, which is always tuned by exact octaves, from their corresponding tones within the temperament, the ear will often accept an octave as true before its pitch has been sufficiently raised. Especially is this true in the upper octaves. After tuning a string in the treble by its octave in the temperament, test it as a fifth. For instance, after tuning your first string beyond the temperament, 3C♯, test it as a fifth to 2F♯. If you are yet uncertain, try it as a major third in the chord of A. The beats will serve you as a guide in testing by fifths, up to about an octave and a half above the highest tone of the temperament; but beyond this point they become so rapid as to be only discernible as degrees of roughness. The beats will serve as a guide in tuning octaves higher in the treble than the point at which the beats of the fifth become unavailable; and in tuning unisons, the beats are discernible almost to the last tone.

The best method to follow in tuning the treble may be summed up as follows: Tune the first octave with the beats as guides both in the octave and in testing it by the fifth. If yet uncertain, test by chords. Above this octave, rely somewhat upon the beats in the octave, still use the fifth for testing, but listen for the pitch in the extreme upper tones and not so much for the beats except in bringing up unisons, in which the beats are more prominent.

In the extreme upper tones, the musical ear of the tuner is tried to the utmost. Here, his judgment of correct harmonic relation is the principal or only guide, while in the middle octaves the beats serve him so faithfully, his musical qualifications being brought into requisition only as a rough guide in determining pitch of the various intervals. To tune by the beats requires a sharp ear and mental discernment; to tune by pitch requires a fine musical ear and knowledge of the simpler laws of harmony.

As stated above, the tuner will fail in many cases to tune his high octaves sharp enough. Rarely, if ever, will a tuner with a good ear leave the upper tones too sharp. Now, there is one more fact which is of the utmost importance in tuning the treble: it is the fact that the extreme upper octave and a half must be tuned slightly sharper than perfect; if the octaves are tuned perfect, the upper tones of the instrument will sound flat when used in scale and arpeggio passages covering a large portion of the key-board. Begin to sharpen your octaves slightly from about the seventeenth key from the last; counting both black and white. In other words, begin to sharpen from the last A♭ but one, in the standard scale of seven and a third octaves of which the last key is C. Sharpen but slightly, and increase the degree of sharpening but little as you proceed.

Tuning the Bass. In tuning the bass, listen for the beats only, in bringing up the octaves. It is sometimes well to try the string tuned, with its fifth, but the octave in the bass should suffice, as the vibrations are so much slower here that if you listen acutely the octave beats will guide you.

It is not necessary to pull the strings higher than the pitch at which they are to stand. Learn to pull them up gradually and in a way that will "render" the string over the bridges, which is an easy thing to do, the strings being so much heavier here than elsewhere. Never leave a bass string the slightest amount too sharp. As flatness is so obnoxious in the treble, just so is sharpness in the bass, so if there must be any variation in any bass tone let it be flat; but aim at perfect octaves throughout the bass.

False Waves. We say "false waves" for want of a better name. You will find a string occasionally that will give forth waves or beats so similar to the real ones that it takes a practiced ear to distinguish the difference. Where a unison contains a string of this kind, select some other string by which to tune the interval, and leave the bad string until the last; you may then find difficulty in being able to tell when you have it in unison. The cause may be a twisted string, a fault in the string by imperfect drawing of the wire, or in the construction of the sound-board.

In the low bass tones, a kind of false waves are always present, and will annoy the tuner long after he has been in regular practice. They are, however, of a different nature from the true waves in that they are of a metallic timbre and of much greater rapidity than the latter. Close attention will generally enable the tuner to distinguish between them. They are caused by what is known as "harmonics" or "over-tones"; the string vibrating in fractional segments.

False waves will occur in an annoying degree when the tuner sets a mute on a nodal point in the string; it will cause the muted string to sound a real harmonic tone. This does not happen in the upright, as the mutes are set so near the end of the string as to preclude this possibility. In the square, however, it very frequently happens, as there are so many nodes between the dampers and the bridge, where the tuner sets his mutes. If, for instance, he is tuning an octave and has his mute set precisely in the middle of the vibrating segment, in place of muting the string it sounds its own octave, which will disturb the ear in listening for the tone from the one free string. Move the mute either way until it is found to mute the string entirely.

QUESTIONS ON LESSON XIV.

Explain the cause of the beats.

How many beats per second in a unison of two strings, one tuned to 100, the other to 101 vibrations per second?

How many beats per second in an octave, the lower tone of which is tuned to 100, the upper to 201 vibrations per second?

How many beats per second in a fifth, the fundamental of which is tuned to 100, the fifth to 151?

The fifth, 2F-3C, when properly tempered, should beat 1¾ times per second. How often should a fifth, an octave higher, beat?