MATHEMATICS OF THE TEMPERED SCALE.
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 pitch which 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 which each one will form a serviceable interval with any one of 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 must | serve as | 1, | in the | key of | C | major or | C | minor. |
| " | " | 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 as any member of any 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 greater deviation 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 by touching 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.
| Fundamental | Major Second | Major Third | Perfect Fourth | Perfect Fifth | Major Sixth | Major Seventh | Octave |
| C | D | E | F | G | A | B | C |
| 1 | 8/9 | 4/5 | 3/4 | 2/3 | 3/5 | 8/15 | 1/2 |
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:
| C | D | E | F | G | A | B | C |
| 360 | 320 | 288 | 270 | 240 | 216 | 192 | 180 |
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 original purity 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, the syntonic 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 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 (½), 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.)
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:
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.
- Why is the pitch, C-256, adopted for scientific discussion, and what is this pitch called?
- 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?
- What tone is produced by a 2/3 segment of a string? What by a 1/2 segment? What by a 4/5 segment?
- (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? - What would be the result if we should tune from 2C to 3C by a succession of perfect thirds?
- 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 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 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 space 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.
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.
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.
- State what principle is demonstrated in Proposition II.
- State what principle is demonstrated in Proposition III.
- What would be the vibration per second of an exact (not tempered) fifth, from C-512?
- Give the figures and the process used in finding the vibration number of the exact major third to C-256.
- 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 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?
LESSON XV.
MISCELLANEOUS TOPICS PERTAINING TO THE PRACTICAL WORK OF TUNING, REGULATING, AND REPAIRING.
Comparison of the Different Systems. Up to this time, we have given no account of any system of tuning except the one recommended. For the purpose of making the student more thoroughly informed we detail here several different systems which have been devised and practiced by other tuners. It is a matter of history that artisans in this profession and leaders in musical science have endeavored to devise a system of temperament having all the desirable qualifications.
The aims of many have been to invent a system which uses the fewest number of tones; working under the impression that the fewer the tones used in the temperament, the easier the tuner's work. These have reduced the compass of the temperament to the twelve semi-tones from middle C to B above; or from F below, to E above middle C. This system requires the tuner to make use of both fourths and fifths. Not only does he have to use these two kinds of intervals in tuning, but he has to tune by fourths up and fourths down, and, likewise, by fifths up and fifths down. When tuning a fifth upward, he flattens it; and when tuning a fifth downward he sharpens the lower tone; when tuning a fourth upward, he sharpens it; when tuning a fourth downward, he flattens the lower tone.
It is readily seen that by a system of this kind the tuner's mind is constantly on a strain to know how to temper the interval he is tuning, and how much to temper it, as fourths require a different degree of tempering from the fifths; and he is constantly changing from an interval upward to one downward; so, this system must be stamped as tedious and complicated, to say the least. Yet this system is much followed in factories for rough tuning, and also by many old professional tuners.
The table on the following page gives the succession of intervals generally taken by tuners employing this system using the tones within the F octave mentioned above. Middle C is obtained in the usual way, from the tuning fork.
SYSTEM A.
| By middle | C | tune | F | fifth below. | Temper | sharp. |
| By | F | " | B♭ (A♯) | fourth above. | " | " |
| " | C | " | G | fourth below. | " | flat |
| " | G | " | D | fifth above. | " | " |
| " | D | " | A | fourth below. | " | " |
| " | A | " | E | fifth above. | " | " |
| " | E | " | B | fourth below. | " | " |
| " | B | " | F♯ | fourth below. | " | " |
| " | F♯ | " | C♯ | fifth above. | " | " |
| " | C♯ | " | G♯ | fourth below. | " | " |
| " | G♯ | " | D♯ | fifth above. | " | " |
| Then try | D♯ | with | A♯ | previously tuned for "wolves." | ||
We think a little study and trial of this system will produce the conviction that it is a very difficult and precarious one, and that it has every disadvantage but one, namely, that it uses the smallest possible number of tones, which is really of little value, and does not compensate for the difficulty encountered and the uncertainty of the results.
Another system which has many advantages over the above, is one which employs fifths only and covers a compass of an octave and a half. This system is similar to ours in that it employs fifths in the same succession as far as G♯, the most of them, however, being an octave higher. From this G♯ there is a break in the succession, and the tuner goes back to middle C from which he started and tunes by fifths downward until he reaches the G♯ at which he left off. This system employs the tones from F below middle C to C, octave above. Below is the succession, starting upon 3C, whose pitch is determined as usual.
SYSTEM B.
| By | 3C | tune | 2C | octave below. |
| " | 2C | " | 2G | fifth above. |
| " | 2G | " | 1G | octave below. |
| " | 1G | " | 2D | fifth above. |
| " | 2D | " | 2A | fifth above. |
| " | 2A | " | 1A | octave below. |
| " | 1A | " | 2E | fifth above. |
| " | 2E | " | 2B | fifth above. |
| " | 2B | " | 1B | octave below. |
| " | 1B | " | 2F♯ | fifth above. |
| " | 2F♯ | " | 1F♯ | octave below. |
| " | 1F♯ | " | 2C♯ | fifth above. |
| " | 2C♯ | " | 2G♯ | fifth above. |
| " | 2G♯ | " | 1G♯ | octave below. |
| By | 2C | tune | 1F | fifth below. Temper sharp. |
| " | 1F | " | 2F | octave above. |
| " | 2F | " | 1B♭ | fifth below. Temper sharp. |
| " | 1B♭ | " | 2B♭ | octave above. |
| " | 2B♭ | " | 2E♭ | fifth below. Temper sharp. |
| Now by | 2E♭ | try | 1A♭ (G♯) | fifth below for the "wolf." |
| Note that this last trial brings you back to the last tone tuned before the break. | ||||
This system is used by a great number of very successful tuners, and it has but one appreciable disadvantage, which is that involved in changing from fifths upward to fifths downward. This difficulty is easily overcome, if it were all there is to encounter; but in practice, we find that after tuning the intervals in the above succession down to the last step in the first series, middle C will often have changed pitch somewhat, and the last five tones with their octaves tuned from it will not be in true harmony with the intervals tuned in the first series. For this reason it is better to go on through, as in our system, tuning by fifths upward, and if there is any change of pitch in the first tones tuned, they may be more easily corrected by going over them in the same way as at the start; also, the amount of difficulty in locating discrepancies is greatly lessened.
SYSTEM C.
The following system is one that is followed by many good tuners of the present day and has many advantages. To use this system successfully, however, one must be familiar with the number of beats necessary in each interval used.
Take 1F as a standard.
By 1F, tune 2C, fifth above.
By 1F, tune 1Bm, fourth above.
By 1F, tune 1A, major third above.
By 1F, tune 2D, sixth above.
By 1F, tune 1Am, minor third above.
By 1F, tune 2F, octave above.
By 2C or 2D, tune 1G, fourth or fifth below.
By 1G, 1A or 2C, tune 2E, sixth, fifth or third above.
By 1G or 2E, tune 1B, third above or fourth below.
By 1A or A♯, tune 2C♯, major or minor third above.
By 1A♭, 1B♭ or 1B, tune 2E♭, fifth, fourth or major third above.
By 1B♭, 1B, 2C♯ or 2E♭, tune 1F♯, major third, fourth, fifth or sixth below.
As each step is taken in this system, the tone tuned is tested with any or all of the tones previously tuned.
You will notice that six tones are tuned by the first standard, F. Therefore, if any error is left in any one of the intervals it exists in this only and is not transmitted to other tones, if corrected before such other tones are used to tune by.
The numerous tests possible, early in the system, and the small compass used, one octave, may be said to be the chief advantages of the system.
The intervals used are the minor and major third, perfect fourth and fifth, and major sixth. The thirds and sixths beat from about 7 to nearly 12 per second. The exact number of beats for each step in the system may be calculated from the "Table of Vibration Numbers" in Lesson XIII. For instance, take middle C (2C) at 256, and its major third, 2E. The exact third, determined by multiplying 256 by 5/4, is found to be 320. By reference to the table, we find the tempered third vibrates 322.54. The difference then is 2.54 vibrations per second, and, knowing that a difference of one vibration from the exact major third produces 4 beats, we simply multiply 2.54 by 4 and we have 10.16, the number of beats we should hear per second when this third is tempered correctly. Other intervals may be figured out in like manner by reference to the various tables given.
It is very doubtful if a beginner could succeed with this system. He should tune by an easier system until he can hear the beats very distinctly and judge quite accurately the rapidity of them. Having acquired this ability, he may try this system and follow it in preference to others.
In any system used it is well to test your work in the following manner:
Begin with your lowest major third and strike each third in succession, ascending chromatically. Of course, each third should beat slightly faster than the one below it. For instance, in our system of two octaves, take 1C-E; this third should beat about 5 per second. Next, take 1C♯-F, which should beat about 5-1/2 per second. The beats should increase each test nearly a half beat, or the amount of 5 beats in this octave; hence, 2C-E will beat about 10 per second; or, using the exact figures, 10.16. After arriving at the last-named test, 2C-E, you may test the remainder of the two octaves by tenths, beginning with 1C-2E. The tenth is similar to the third mathematically, and its beats are even more distinct.
We may remark here that our system may be reduced to the compass of an octave and a half by simply not tuning the octaves upward which reach beyond 2F♯; and if anything were to be gained and nothing lost by shortening the compass of the temperament, we would advise using only the octave and a half. But in many years of experience in tuning all imaginable types, styles and kinds of pianos, and by all systems, we have found good reasons for adopting the two-octave temperament as laid down in Lesson VIII, for universal application. Its advantages may be summed up as follows:
Simplicity. But two kinds of intervals are employed: the fifth and the octave. The fifth is always tuned to a fundamental below and hence always flattened, which relieves the tuner of any mental operation to determine which way he is to temper. Being a regular succession of fifths and octaves, without a break, the system is easily learned, and can be followed with little mental strain.
Uniformity. After the tuner has become well trained in tempering his fifths, there is little danger of an uneven temperament, as the various intervals used in trials will prove a false member in some chord in time to correct it before he has gotten so far from it as to make the correction difficult. When a correction is necessary, the offending point is most easily found.
Precision. In our experience, we have never known another system by which we could attain the absolute precision gained by this.
Stability. Stability is the feature wherein rests the paramount reason for employing two octaves. From what has been said in previous lessons concerning the liability of some strings to flatten or sharpen by reason of altering the tension of other strings, the student will readily see that the temperament should cover a sufficient portion of the instrument, if possible, to insure that it will stand while the remaining portion is being tuned. Our two octaves cover nearly all the strings between the over-strung bass and the brace in the metal plate. This being the case, any reasonable alteration of the strings beyond, or outside, the braces from the temperament, will rarely, if ever, affect it noticeably.
Final Inspection. Always test every key on the piano, or especially those of the middle five octaves, for bad unisons. Upon finding one, search for the string that has stood in tune, by testing each string of the unison with its octave. This being done, simply bring the other to it. Go over the whole key-board, striking octaves, and correct any that might offend. One extremely bad tone or octave may disparage your reputation, when in reality your work merits commendation.
Loose Pins. You will occasionally find pianos in which the tuning pins have become so loose that they will not resist the pull of the strings. If many of them are in this condition it is better, before you begin to tune, to take a hammer of considerable weight and drive them a little. Commence at one end of the row of pins and aim to strike all the pins with the same force. Those which are tight enough will not yield to the blow, while those which are loose may require two or three blows to tighten them sufficiently. This defect is generally found in very old squares or cheap uprights wherein the pin-block is of poor material or defective in manufacture or in pianos which have been abused.
Split Bridges. Even in pianos of the highest grade, we sometimes find a string sounding as if there was a pin or some metallic substance bearing against it. In such cases, find the string and examine the place where it crosses the bridge. You will often find the bridge split at that point or the bridge-pin, having yielded to the pressure of the string, vibrates against the next pin, giving rise to the singing effect. You can do little if anything toward repairing a split bridge. You may, however, stop the singing by inserting the point of your screw-driver between the close pins and pressing them apart. This will generally stop the difficulty for the time being at least.
Strings crossing the bridge near a split will not stand in tune well, and will, perhaps, have to be gone over two or three times. The same may be said of a broken metal plate. Many old squares have broken plates; generally found near the overstrung bass, or within the first octave of the treble. All the tuner can do is to apprise the owner of the defect and inform her that it will not stand well at this point, as the intense strain is thrown largely upon the wooden frame, which will have a tendency to yield gradually to it.
Stringing. Strings break while the tuner is drawing them up, sometimes because he does not pull them gradually, gives them an abrupt turn or draws them too far above the pitch at which they are intended to stand. More often, however, they break from being rusty at the point where they pass over the bridge or around the tuning pin. The best instruction concerning putting on new strings is, follow appearances. Make the string you put on look just like those on the instrument. In most modern pianos the string is wound with three coils around the pin.
You will, of course, have to take out the action; not the key-board, however, unless it be one of those rare cases where the key-board and upper action are built to come out together. In the square it is only necessary to remove the shade over the dampers, and the dampers, which are all removed easily by taking out the screw at the left. This allows the whole set of dampers with their support to come out together.
Treble strings are nearly always passed around the hitch-pin, one wire thus forming two strings. Take out the old string, noticing how it passes over and under the felt at the dead end. After removing the string always give the pin about three turns backward to draw it out sufficiently so that when a new string is put on, the pin will turn into the block as far as it did originally. Run one end of the string barely through the hole in the tuning pin and turn it about twice around, taking pains that the coils lie closely; then unwind enough wire (of the same size of course) from your supply to reach down to the hitch pin and back. Place the string on the bridge pins properly, draw it as tight as you can by hand and cut it off about three fingers' width beyond the pin upon which it is to be wound. This will make about three coils around the pin. Place the end in the hole and turn up gradually, watching that the string is clear down on hitch pin and properly laid on the bridge. New strings will require drawing up two or three times before they will stand in tune, and even then they will run down in a short time. It is well on this account to leave them slightly sharp, calling the owner's attention to the fact.
Knot for Splicing Wire.
When a bass string breaks at the point where it starts around the tuning pin, it can nearly always be spliced and the trouble of sending it away to have a new one made be avoided. Take a piece of new wire as large or larger than the old string and splice it to the broken end by a good secure knot. A knot called the square or ruft knot is the best for this purpose. When a bass string breaks too far from the pin to permit of a splice, the only resort will be to send the broken string to some factory and have a new one made from it.
QUESTIONS ON LESSON XV.
- Name the advantages and disadvantages of system A.
- Name the advantages and disadvantages of system B.
- What are the important points to be desired in any system of setting temperament?
- State three or four items of importance in the operation of putting on a new string.
- Why do pianos get out of tune?