Chapter 7

Funda- | Major | Major | Perfect | Perfect | Major | Major | Oc- | mental |Second | Third | Fourth | Fifth | Sixth | Seventh | tave | | | | | | | | | 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 54--multiplying 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 (1/2), it would produce a flatter tone than the exact half.

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

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

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

Middle C (2C) 60 inches. -------------------------------------------------- O O

E (4/5 of 60) 48 inches. -------------------------------------------- O O

G♯ (A♭) (4/5 of 48) 38-2/5 inches. -------------------------------------- O O

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

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

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

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

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

Tempered thirds qualified to produce true octave: =

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

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

QUESTIONS ON LESSON XII.

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

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

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

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

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

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

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

LESSON XIII.

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

PROPOSITION II.

That the student of scientific scale building may understand fully the reasons why the tempered scale is at constant variance with exact mathematical ratios, we continue this discussion through two more propositions, No. II, following, demonstrating the result of dividing the octave into four minor thirds, and Proposition III, demonstrating the result of twelve perfect fifths. The matter in Lesson XII, if properly mastered, has given a thorough insight into the 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.

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

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

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

TABLE SHOWING RELATIVE VIBRATION OF INTERVALS BY IMPROPER FRACTIONS.

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

QUESTIONS ON LESSON XIII.

1. State what principle is demonstrated in Proposition II.

2. State what principle is demonstrated in Proposition III.

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

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

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

LESSON XIV.

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

~Beats.~--The phenomenon known as "beats" has been but briefly alluded to in previous lessons, and not 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]

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