Use of continuous mute
Use of continuous mute
The upper row of O's represents the upper row of tuning pins. To these are attached the first string of each unison. To the middle row are attached the second or middle strings, and to the lower row are attached the third strings. The diagonal lines represent the three strings of the unison (trio). The asterisk on the middle one indicates that it has been tuned.
But one mute is used in tuning these unisons. It is inserted between the trios in the order indicated by the figures 1, 2, 3, etc. When inserted in place 1, between unisons B and C, it will mute the first string of C; so the first string of the trio to tune is always the third. Then place your mute in place 2 and tune the first string of C. Then, without moving your mute, bring up third string of C♯,then third string of D and so on. By this method, you tune two strings every time you reset your mute.
When through with the temperament, the next step is usually that of tuning the bass; but while we are in the treble we will proceed to give the method of setting the mutes in the upper treble beyond the temperament. All three strings have yet to be tuned here, and we have to use two mutes. The unisons are tuned in regular succession upward the same as in the example above. The mute that is kept farthest to the left, is indicated by the letter A, and the one kept to the right, by the letter B, as in diagram below.
Use of two mutes
Use of two mutes
The mutes are first placed in the places indicated by the figures 1 and 2, thereby muting first and third strings of the first unison beyond the temperament, which is 3C♯. The middle string of this unison is now tuned by its octave below. (If you have left imperfect unisons in your temperament, rendering it difficult to tune octaves by them, it will be well toreplace your continuous mute so as to tune from a single string.) Having tuned the middle string of C♯, move mute B to place 3 and tune third string of C♯. Then, move mute A to place 2 and tune first string of C♯. Your mutes are now already set for tuning the middle string of D. After this is done, proceed to move mute B first, then mute A; tuning middle string, then third, then first, moving step by step as indicated in example above until the last unison is reached. By this system you tune three strings every time the mutes are set twice.
The over-strung bass usually has but two strings to a unison and only one mute is needed. In the extreme low or contra-bass, pianos have but one string, in tuning which the mute is discarded. Set the mute as indicated by the figures 1, 2, 3, etc., in the diagram below, always tuning the string farthest to the right by its octave above; then move the mute to its next place and tune the left string by the right. Here, again, you tune two strings every time you reset your mute. The I's represent bass strings.
Tuning the Bass
Tuning the Bass
Setting the Mutes in the Square Piano.
In setting the temperament in the square piano, simply mute the string farthest to the left and tune the one to the right until the temperament is finished, then set the mutes in the bass the same as in the upright. In tuning the treble, if the piano has three strings, the same system is used as has been described for the upright. When the piano has but two strings to a unison, as is usually the case, employ the system described for the bass of the upright, but reversed, as you are proceeding to the right instead of to the left.
Remove the shade before beginning to tune a square piano, and if necessary, lay the dampers back and trace the strings to their pins so as to mark them. Certain pins are marked to guide the tuner in placing his hammer. The way we have always marked them is as follows:
Mark both pins of each pair of C strings with white crayon. Mark only one pin of each pair of G's. Knowing the intervals of the other keys from the marked ones, you can easily calculate correctly,upon which pin to set your hammer to tune any string desired. For instance, if you are striking D♯, next above middle C, you calculate that, as D♯ is the third chromatic interval from middle C, you are to set the hammer on one or the other of the pins belonging to the third pair to the right of the pair marked as middle C. B would be first pair to the left, F♯ would be first pair to the left of the marked G, and so on. It is usually necessary to mark only those pairs near the middle of the piano, but we advise the beginner to mark throughout the scale, as by so doing he may avoid breaking a string occasionally by pulling on some other than the one he is sounding. This will occur in your early practice if you do not use caution. And for safety, some tuners always mark throughout.
QUESTIONS ON LESSON XI.
One of the first questions that arises in the mind of the thinking young tuner is: Why is it necessary to temper certain intervals in tuning? We cannot answer this question in a few words; but you have seen, if you have tried the experiments laid down in previous lessons, that such deviation is inevitable. You know that practical scale making will permit but two pure intervals (unison and octave), but you have yet to learn the scientific reasons why this is so. To do this, requires a little mathematical reasoning.
In this lesson we shall demonstrate the principles of this complex subject in a clear and comprehensive way, and if you will study it carefully you may master it thoroughly, which will place you in possession of a knowledge of the art of which few tuners of the present can boast.
In the following demonstrations of relative pitch numbers, we adopt a pitch in which middle C has 256 vibrations per second. This is not a pitchwhich is used in actual practice, as it is even below international (middle C 258.65); but is chosen on account of the fact that the various relative pitch numbers work out more favorably, and hence, it is called the "Philosophical Standard." Below are the actual vibration numbers of the two pitches in vogue; so you can see that neither of these pitches would be so favorable to deal with mathematically.
International—3C–517.3. Concert—3C–540.
(Let us state here that the difference in these pitches is less than a half-step, but is so near that it is generally spoken of as being just a half-step.)
Temperament denotes the arrangement of a system of musical sounds in whicheach onewill form a serviceable interval withany oneof the others. Any given tone must do duty as the initial or key-note of a major or of a minor scale and also as any other member; thus:
C mustserve as1,in thekey ofCmajor orCminor.""2,""B♭"B♭"""3,""A♭"A"""4,""G"G"""5,""F"F"""6,""E♭"E"""7,""D♭"C♯"
Likewise, all other tones of the instrument must be so stationed that they can serve asany memberofany scale, major or minor.
This is rendered necessary on account of the various modulations employed in modern music, in which every possible harmony in every key is used.
Rationale of the Temperament.
Writers upon the mathematics of sound tell us, experience teaches us, and in previous lessons we have demonstrated in various ways, that if we tune all fifths perfect up to the seventh step (see diagram, pages 82, 83) the last E obtained will be too sharp to form a major third to C. In fact, the third thus obtained is so sharp as to render it offensive to the ear, and therefore unfit for use in harmony, where this interval plays so conspicuous a part. To remedy this, it becomes necessary to tune each of the fifths a very small degree flatter than perfect. The E thus obtained will not be so sharp as to be offensive to the ear; yet, if the fifth be properly altered or tempered, the third will still be sharper than perfect; for if the fifths were flattened enough to render the thirds perfect, they (the fifths) would become offensive. Now, it is a fact, that the third will bear greaterdeviation from perfect consonance than the fifth; so the compromise is made somewhat in favor of the fifth. If we should continue the series of perfect fifths, we will find the same defect in all the major thirds throughout the scale.
We must, therefore, flatten each fifth of the complete circle, C-G-D-A-E-B-F♯-C♯-G♯ or A♭-E♭-B♭-F-C, successively in a very small degree; the depression, while it will not materially impair the consonant quality of the fifths, will produce a series of somewhat sharp, though still agreeable and harmonious major thirds.
We wish, now, to demonstrate the cause of the foregoing by mathematical calculation, which, while it is somewhat lengthy and tedious, is not difficult if followed progressively. First, we will consider tone relationship in connection with relative string length. Students who have small stringed instruments, guitar, violin, or mandolin, may find pleasure in demonstrating some of the following facts thereupon.
One-half of any string will produce a tone exactly an octave above that yielded by its entire length. Harmonic tones on the violin are made bytouching the string lightly with the finger at such points as will cause the string to vibrate in segments; thus if touched exactly in the middle it will produce a harmonic tone an octave above that of the whole string.
Two-thirds of the length of a string when stopped produces a tone a fifth higher than that of the entire string; one-third of the length of a string on the violin, either from the nut or from the bridge, if touched lightly with the finger at that point, produces a harmonic tone an octave higher than the fifth to the open tone of that string, because you divide the string into three vibrating segments, each of which is one-third its entire length. Reason it thus: If two-thirds of a string produce a fifth, one-third, being just half of two-thirds, will produce a tone an octave higher than two-thirds. For illustration, if the string be tuned to 1C, the harmonic tone produced as above will be 2G. We might go on for pages concerning harmonics, but for our present use it is only necessary to show the general principles. For our needs we will discuss the relative length of string necessary to produce the various tones of the diatonic scale, showing ratios of the intervals in the same.
In the following table, 1 represents the entire length of a string sounding the tone C. The other tones of the ascending major scale require strings of such fractional length as are indicated by the fractions beneath them. By taking accurate measurements you can demonstrate these figures upon any small stringed instrument.
FundamentalMajorSecondMajorThirdPerfectFourthPerfectFifthMajorSixthMajorSeventhOctaveCDEFGABC18/94/53/42/33/58/151/2
To illustrate this principle further and make it very clear, let us suppose that the entire length of the string sounding the fundamental C is 360 inches; then the segments of this string necessary to produce the other tones of the ascending major scale will be, in inches, as follows:
CDEFGABC360320288270240216192180
Comparing now one with another (by means of the ratios expressed by their corresponding numbers) the intervals formed by the tones of the above scale, it will be found that they all preserve their originalpurity except the minor third, D-F, and the fifth, D-A. The third, D-F, presents itself in the ratio of 320 to 270 instead of 324 to 270 (which latter is equivalent to the ratio of 6 to 5, the true ratio of the minor third). The third, D-F, therefore, is to the true minor third as 320 to 324 (reduced to their lowest terms by dividing both numbers by 4, gives the ratio of 80 to 81). Again, the fifth, A-F, presents itself in the ratio of 320 to 216, or (dividing each term by 4) 80 to 54; instead of 3 to 2 (=81 to 54multiplying each term by 27), which is the ratio of the true fifth. Continuing the scale an octave higher, it will be found that the sixth, F-D, and the fourth, A-D, will labor under the same imperfections.
The comparison, then, of these ratios of the minor third, D-F, and the fifth, D-A, with the perfect ratios of these intervals, shows that each is too small by the ratio expressed by the figures 80 to 81. This is called, by mathematicians, thesyntonic comma.
As experience teaches us that the ear cannot endure such deviation as a whole comma in any fifth, it is easy to see that some tempering must take place even in such a simple and limited number of sounds as the above series of eight tones.
The necessity of temperament becomes still more apparent when it is proposed to combine every sound used in music into a connected system, such that each individual sound shall not only form practical intervals with all the other sounds, but also that each sound may be employed as the root of its own major or minor key; and that all the tones necessary to form its scale shall stand in such relation to each other as to satisfy the ear.
The chief requisites of any system of musical temperament adapted to the purposes of modern music are:
1. That all octaves must remain perfect, each being divided into twelve semitones.
2. That each sound of the system may be employed as the root of a major or minor scale, without increasing the number of sounds in the system.
3. That each consonant interval, according to its degree of consonance, shall lose as little of its original purity as possible; so that the ear may still acknowledge it as a perfect or imperfect consonance.
Several ways of adjusting such a system of temperament have been proposed, all of which may be classed under either the head of equal or of unequal temperament.
The principles set forth in the following propositions clearly demonstrate the reasons for tempering, and the whole rationale of the system of equal temperament, which is that in general use, and which is invariably sought and practiced by tuners of the present.
Proposition I.
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 quantity, called in the theory of harmonics adiesis, 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 by4/5of its length.
In like manner, G♯, the major third to E, will be produced by4/5of that segment of the string which sounds the tone E; that is, G♯ will be produced by4/5of4/5(4/5multiplied by4/5) which equals16/25of 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, by4/5of that segment of the string which sounds A♭; that is, by4/5of16/25, which equals64/125of 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. Is64/125an exact half? No; using the same numerator, an exact half would be64/128.
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 than64/128(½), it would produce a flatter tone than the exact half.
It is evident, therefore, thatall major thirds must be tuned somewhat sharper than perfectin a system of equal temperament.
The ratio which expresses the value of thediesisis 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.)
This diagram clearly demonstrates that the last C obtained by the succession of thirds covers a segment of the string which is18/25longer 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:
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.
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 principal 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 otherwords,5/6of 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.
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, andthis gives648/1296, which is an exact half. Notice the comparison.
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 by23/1296of the length of the whole string.
If, therefore, all octaves are to remain perfect, it is evident thatall minor thirds must be tuned flatter than perfectin 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 andminor 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 space of an octave.
Method. Taking 1C as the fundamental, representing it by unity or 1, the G, fifth above, is sounded by a2/3segment of the string sounding C. The next fifth, G-D, takes us beyond the octave, and we find that the D will be sounded by4/9(2/3of2/3equals4/9) 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:4/9times 2 equals8/9, the segment sounding the D within the octave.
We may shorten the operation as follows: Instead of multiplying2/3by2/3, giving us4/9, and then multiplying this answer by 2, let us double the fraction,2/3, which equals4/3, 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
1.1Cto1Gsegment2/3for1G2.1G"1DMultiply2/3by4/3,givessegment8/9"1D3.1D"1A"8/9"2/3""16/27"1A4.1A"1E"16/27"4/3""64/81"1E5.1E"1B"64/81"2/3""128/243"1B6.1B"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"1F12.1F"2C"131072/177147"2/3""262144/531441"2C
Now, this last fraction should be equivalent to1/2, 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 by262144/524288, the segment producing the true octave; so the fraction262144/531441, which represents the segment for 2C, obtained by the circle of fifths, being evidently less than1/2, 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 offifths is sharper than the true octave in the ratio expressed by 531441 to 524288, which ratio is called theditonic 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.C32.32.C512.512.D36.35.92D576.574.70E40.40.32E640.645.08F42.6642.71F682.66683.44G48.47.95G768.767.13A53.3353.82A853.33861.08B60.60.41B960.966.53C64.64.C1024.1024.
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.
C128.256.512.1024.C♯135.61271.22542.441084.89D143.68287.35574.701149.40D♯152.22304.44608.871217.75E161.27322.54645.081290.16F170.86341.72683.441366.87F♯181.02362.04724.081448.15G191.78383.57767.131534.27G♯203.19406.37812.751625.50A215.27430.54861.081722.16A♯228.07456.14912.281824.56B241.63483.26966.531933.06C256.512.1024.2048.
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.
Therelationof theOctave to aFundamentalis expressed by2/1"""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.
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 correspondin 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 theratio of 2 to 1. If these two strings vary from this ratio to the amount ofonevibration, they will producetwobeats. 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 ofonevibration, there will occurthreebeats per second. In the case of the major third, there will occurfourbeats per second to a variation ofonevibration 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 theexactfifth would be, according to the instructions given beneath the table; find the difference between this and thetemperedfifth 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 mostbeautifully. 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:
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 lattercause 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: