among physicians, the same with constitution, or a certain disposition, of the solids and fluids of the human body, by which it may be properly denominated strong, weak, lax, &c.
In every person there are appearances of a temperament peculiar to himself, though the ancients only took notice of four, and some have imagined these were deduced from the theories of the four humours or four cardinal qualities; but it is more probable that they were first founded on observation, and afterwards adapted to those theories, since we find that they have a real existence, and are capable of receiving an explanation. The two that are most distinctly marked are the sanguineous and melancholic, viz. the temperaments of youth and age.
1. Sanguineous. Here there is laxity of solids, discoverable by the softness of hair and succulence; large system of arteries, redundancy of fluids, florid complexion; sensibility of the nervous power, especially to pleasing objects; irritability from the plethora; mobility and levity from lax solids. These characters are distinctly marked, and are proved by the diseases incident to this age, as hemorrhagies, fevers, &c. but these, as they proceed from a lax system, are more easily cured.
2. Melancholic Habit. Here greater rigidity of solids occurs, discoverable by the hardness and crispature of the hair; small proportion of the fluids, hence dryness and leanness; small arteries, hence pale colour; venous plethora, hence torsequency of these, and liveliness; sensibility, frequently exquisite; moderate irritability, with remarkable tenacity of impressions; steadiness in action and slowness of motion, with great strength; for excess of this constitution in maniacs gives the most extraordinary instance of human strength we know. This temperament is most distinctly marked in old age, and in males. The sanguineous temperament of youth makes us not distinguish the melancholic till the decline of life, when it is very evident, from diseases of the veins, hemorrhoids, apoplexy, cachexy, obstructions of the viscera, particularly of the liver, dropsies, affections of the alimentary canal, chiefly from weaker influence of the nervous power. So much for the sanguineous and melancholic temperaments; the other two are not so easily explained. The choleric temperament takes place between youth and manhood.
3. Choleric, the distribution of the fluids is more exactly balanced; there is less sensibility, and less obesity, with more irritability, proceeding from greater tension, less mobility and levity, and more steadiness in the strength of the nervous power. As to the
4. Phlegmatic. This temperament cannot be distinguished by any characters of age or sex. It agrees with the sanguineous in laxity and succulence. It differs from that temperament, and the melancholic, by the more exact distribution of the fluids. Again, it differs from the sanguineous, by having less sensibility, irritability, mobility, and perhaps strength, though sometimes indeed this last is found to be great.
These are the ancient temperaments. The temperaments, indeed, are much more various; and very far from being easily marked and reduced to their genera and species, from the great variety which is observable in the constitutions of different men.
TEMPERAMENT of the Musical Scale, is that modification of the sounds of a musical instrument, by which these sounds may be made to serve for different degrees of different scales. See MUSIC, Chap. VII.
Temperament, though intimately connected with music, is not, properly speaking, a part of that science. The objects of music, as a science, are, to ascertain the laws of musical sound, as depending on the powers of the human voice. The purpose of temperament is, to regulate, in a way least adverse to these laws, a certain departure from them, rendered necessary by the imperfections of instruments.
Although the temperament of the scale of instruments be practically familiar, the true principles on which it depends have been much disputed. Various opinions have been hazarded, and systems proposed. We offer an abridged view of that which appears to us to merit a preference (A).
Before consideration of the tempered scale, a short review of the nature of the true scale is necessary.
From the conformation of the vocal organs, all nations, in singing, make use of the same inflections of voice. These inflections, called notes, are said to be grave or acute, in proportion to the degree of hoarseness or shrillness with which they are sung. The state of voice with respect to gravity or acuteness with which any one note is sung, is termed its pitch.
Two notes having the same pitch are termed unisons, or are said to be in unison to one another. The difference of pitch between any note and another is denominated an interval.
(A) Amongst the very numerous authors on the subject of temperament, we have selected, for our chief guides, the late Dr Robert Smith of Cambridge, and Professor John Robison of Edinburgh. In all attempts to sing, the ear, either unconsciously, or from the direction of recently hearing it, selects a particular note, from the previous impression of which the voice naturally forms other notes, at certain though unequal intervals. The note, thus selected, is termed the key note or fundamental. When chosen, it instantly assumes a particular and predominant character. The ear involuntarily refers to it the intonation of all other notes, readily recurrs to it during performance, and is dissatisfied unless the voice close upon it.
Where the singer has assumed a key note, and, after singing that note, sings the note nearest in acuteness to it without forcing the voice, and so on, the series of notes, thus naturally formed, constitutes what is called the natural scale. The notes of it are termed its degrees; thus the key note is the first degree of the scale; the natural note next in acuteness to it, is named the second degree, or second of the scale, and so on.
Two untaught men, attempting to sing the same scale together, always sing in unison. But a man and a woman, making the same attempt, sing naturally in such a difference of pitch, although they proceed by the same intervals, that the eighth note only of the male voice ascending, is in unison with the key note of the female voice. Were the male voice to ascend to a ninth note, it would be in unison with the second of the female voice; the tenth note of the former would be in unison with the third of the latter, and so on.
We have thus two scales in succession, perfectly similar in the relation of the degrees of each to their respective key notes; but differing in pitch by the interval between these key notes.
This interval, comprehending seven smaller intervals and eight degrees, is, from this last circumstance, called an octave; and this term is also applied, somewhat inaccurately, to the series of the eight degrees. Thus we say, that the octave formed by the female voice is an octave acuter than that which is produced by the male voice; meaning, that the eight degrees sung by the woman are acuter by the interval of an octave, than those sung by the man.
Not only are the natural octaves of the male and female voice exactly similar; but the same similarity is found in the extremes of the human voice, and, beyond them, as far as musical sounds can be produced. Many men can sing the second octave below, and most women the second octave above, a given key note common to both voices. Yet the gravest octave of such a male voice, and the acutest octave of such a female voice, are equally similar in their relations (although they differ in pitch by an interval of two octaves), as the two central octaves are.
All the different natural inflections of the human voice are thus contained in one octave, since all other octaves are only repetitions of the same inflections in a graver or acuter pitch.
The octave, then, consists of eight degrees and seven intervals. Two of these intervals, those between the third and fourth, and the seventh and eighth degrees, are sensibly less different in pitch than the others. And although we have no direct measures of the pitch of octaves and semitones, we term these smaller intervals semitones, and even the others tonic intervals, presuming the latter to be equal to each other, and a semitone interval to be equal to the half of a tonic one.
The degrees of the natural scale are, by British musicians, distinguished by the first seven letters of the alphabet. The letter C, for some reason less important than difficult to explain, has been appropriated to the note most easily assumed as a key note by both the male and female voice; the second of the scale is termed D, and the third E, and so on. As the human voice, and consequently most musical compositions, comprehend four octaves, we represent the ordinary octave of the male voice by Roman capitals, and that of the female voice by Roman minuscule letters. The gravest male octave is distinguished by Italic capitals, and the acutest female octave by minuscule Italics. The whole natural scale may therefore be exhibited thus:
| Gravest Male Octave | Ordinary Male Octave | Acutest Female Octave | |---------------------|----------------------|----------------------| | C | D | Bc | | D | EF | cd | | EF | G | ef | | G | A | g | | A | BC | a | | BC | D | bc | | D | EF | cd | | EF | G | ef | | G | A | g | | A | BC | a | | BC | D | bc |
In this exhibition, the juxtaposition of the thirds and fourths, and of the sevenths and eighths or replicates of the first degree, indicates the semitone intervals; and the asterisks represent the tonic intervals of the natural scale, or the artificial intercalary sounds, which, as we shall presently see, it becomes necessary to substitute in those intervals.
Were all voices of the same compass, and were musical feelings satisfied with the natural scale, we might rest here. Being furnished with a key note adapted to all voices, and with instruments accurately tuned to that key note, it would be unnecessary to examine whether any other note of the natural scale could be assumed as the key note of a different scale, and if it could, whether any agreeable effect resulted from the discovery.
But the use of different scales, the key notes of which are derived from the different degrees of the natural scales, has been found not only to be one of the chief sources of the pleasure imparted by musical performances, but to be indispensably necessary from the physical inequality of voices.
The central 'c' of the scale, called in music the tenor C, can be produced by every species of voice. The gravest male voices, termed bass, can form this note, but very few notes above it. The treble, or acuter female voice also produces it, but seldom descends farther. The acuter male voices, called tenor, have this 'c' scarcely above the middle of their compass, and it is not much below the middle of that of the counter-tenor or gravest female voices. Now it is obvious that an air in the natural scale, which should rise above 'c', and fall below it in the same proportions, might be sung by the tenor or counter-tenor voice, but would be too acute for the bass voice, and too grave for the treble. Either of these voices, in order to execute the same air, must assume a different key note from 'c'; and as all the degrees of the scale are regulated by the key note, the air must of course be executed in a scale different from that of 'c'.
Again, suppose a singer who can sing a given air only in the scale of B, to be accompanied by an instrument tuned in the scale of 'c'. Should the lyrist begin on his own key note, he is a semitone above the key note of the singer; and should he begin on the note which is in unison with the singer's key note, the next degree is wrong, being but a semitonic interval by the instrument, and a tonic interval by the voice. In short, all the degrees but one will be found wrong. This is an evident consequence of the inequality of the semitonic to the tonic intervals; and if the tonic intervals, which we presume to be equal, be not exactly so, the discordance will be still greater.
The remedy for this is apparently obvious. If the semitonic intervals are each equal to half of any of the tonic intervals, we need only to interpose other sounds between each two of the degrees which form the tonic intervals; and then, in place of eight degrees and seven unequal intervals, we shall have twelve degrees and twelve equal intervals, each of them equal to a semitone. An instrument thus furnished, appears to be adapted to any voice, and to resemble the modern harpsichord or organ, which have twelve seemingly equal intervals in the octave. Such were the practical resources of the Greek musicians, sanctioned by the approbation of Aristoxenus, and of all those who were satisfied with the decision of the ear alone.
But philosophers and mathematicians ascertained the existence of a certain connexion between musical intervals and mathematical proportions, and gradually opened the way to the discovery that the relations of the musical scale, as naturally formed by the human voice, depend on principles equally plain and certain with the simplest geometrical propositions.
Pythagoras is said to have discovered, that if two musical chords be in equal tension, and if one of them be half the length of the other, the short one will sound an octave above the long one; if one-third shorter, it will produce the fifth; if one-fourth shorter, it will give the fourth. Thus the relation of the key to its octave was discovered to correspond to the ratio of $2 : 1$; that of the key to its fifth to be in the ratio of $3 : 2$; and that of the key to its fourth to be in the ratio of $4 : 3$. For instance, if a chord of a given size and tension, and 12 inches long, produce 'C', another of the same size and tension, but only six inches long, will give the octave 'c'; one eight inches long will sound the fifth 'g'; and one nine inches long will produce the fourth 'f'.
Now as the string of eight inches giving the fifth, and that of six inches producing the octave, are in the ratio of $4 : 3$, which is that of the fourth; it follows, that the interval between the fifth and octave is a fourth; and as the chord of nine inches producing the fourth, and the octave of six inches, are in the ratio of $3 : 2$, the interval between the fourth and octave must be a fifth. Thus the octave 'c' c, is divided into a fifth 'c g', and a fourth 'g c', or into a fourth 'c f', and a fifth 'f c', both in succession. The two fourths 'c f', and 'g c', leave an interval 'f g', corresponding, as we have seen, to the ratio of $9 : 8$.
We have thus the ratios of the octave, of the fifth, and of the fourth; and it does not appear that the ancient theorists proceeded farther. They seem to have preferred the harmony of fourths and fifths to that of thirds nor third, and sixths, so essential in modern harmony. By pursuing the system of the mathematical ratios, we find that $5 : 4$ gives the major third 'c e'. And the fifth 'g' being already determined by the ratio $3 : 2$, we ascertain the ratio of the minor third 'e g' to be $6 : 5$, which is the difference between $3 : 2$ and $5 : 4$. In the same way, the ratio of the third 'e' being $5 : 4$, and that of the fourth 'f' being $4 : 3$, we ascertain the ratio of the semitone 'e f' to be $10 : 15$, or $4 : 3 = 5 : 4$.
A note in the ratio of $5 : 4$, or that of a major third Ratio of 'f', gives 'a', the major sixth of the neutral scale; and a note in the same ratio of $5 : 4$ to 'g' produces 'b', sixth and the major seventh of that scale. The ratio of 'g a' will thus be $10 : 9$, and that of 'a b' $9 : 8$, the same with that of 'f g'; and that of 'b c' will thus be $16 : 15$ like 'e f'.
We have in this way the mathematical ratios of all the degrees of the natural scale except that of the second 'd'. Considering, however, the second to be a perfect fourth graver than the fifth, and having ascertained the fifth 'g' to be a perfect fourth below 'c', as $2 : 1$ is to $3 : 2$; so $3 : 2$ gives $9 : 8$, which we take for the ratio of the second.
Thus have been formed two distinct systems of interpretation of the natural scale; that of mean tones and semitones, founded on the rules of Aristoxenus, and the Pythagorean practice of ancient artists, and that of the ratios deduced from the discoveries of Pythagoras, and the calculations of mathematicians.
The difference between the Aristoxenean system of circular representation of mathematical ratios, and the Pythagorean system of mathematical ratios, will best appear from the following construction. Let the circumference of a circle (fig. 1.) be divided by dotted lines (according to the principles of Aristoxenus) into five larger and equal intervals, and two smaller intervals also equal. Let it also be divided by full lines into portions determined by means of the musical ratios. Thus let the arches CD, FG, and AB be proportional to the logarithm of $9 : 8$, GA and DE to those of $10 : 9$, and EF and BC to those of $16 : 15$ (B). Let us divide another circle in the same manner; but instead of having its points of division marked C, D, &c., let them be marked 'key' 2d, 3d, 4th, 5th, 6th, 7th. This circle, which may be described on a piece of card, is to be placed on the other, and is to move round their common centre.
In whatever point of the outer circle the point 'key' of the inner one be placed, it is obvious that the other points of the outer circle will show what degrees of its scale for composition the inner circle, will serve for degrees of the scale determined by the point 'key'. By this we see clearly the insufficiency of the degrees of the natural scale, for the performance of compositions in different scales, and
(b) We may make CD=61°, 72; CE=155°, 9; CF=149°, 42; CG=210°, 38; CA=265°, 3; and CB=326°, 48. But although the errors of the Aristoxeneans were demonstrated by the certainty of the ratios, and although the dependence of musical intervals on the latter he said to have been known since the days of Pythagoras, the nature of that relation remained unknown for ages.
Galileo discovered that the ratios express the frequency of the aerial undulation, by which the several sounds are generated. He demonstrated that the vibrations of two chords, of the same matter and thickness, and of equal tension, will be in the ratio of their lengths, and that the number of oscillations made in a given time will be inversely as their lengths. The frequency of the sonorous undulations of the air is therefore inversely as the length of the string. Thus $2 : 1$ being the ratio of the octave, the undulations which produce the acuter sound are twice as frequent as those which generate the graver. The ratio of the fifth, $3 : 2$, indicates that in the same time that the ear receives three undulations from the upper sound, it receives only two from the lower. This is not peculiar to sounds produced by the vibration of strings; those produced from the vibration of bells, and from the undulation of the air in pipes, are regulated by the same law.
Thus, it is demonstrated that the pitch of musical sound is determined by the undulations of the air; and that a certain frequency of undulations produces a certain and unalterable musical note. It has been found that any noise whatever, if repeated 240 times in a second, at equal intervals, produces the tenor 'c'; if 360 times, the 'g', or fifth above. It had been imagined that musical sound was only to be produced by those regular undulations, which are occasioned by the vibrations of elastic bodies. We are assured that the same effect will be produced by any noise, if repeated not less than 30 or 40 times in a second; and that the experiment has been tried with a quill snapping against the teeth of a wheel.
By Galileo's discovery, the principles on which the just intonation of the natural scale depends, are shown to be certain and plain. To proceed in our search of an exact measure of temperament of this perfect intonation, we must consider the nature and effects of consonant and dissonant chords.
A chord is a combination of two or more simultaneous musical sounds. If the coalescence be so complete that the compound sounds cannot be distinguished, the chord is said to be consonant; if the separate sounds are distinctly heard, the chord is termed dissonant.
All consonances are pleasing, although some are more so than others. All dissonances are unsatisfactory, and some are very harsh.
In consonances, no inequality of sound is perceptible. In dissonances, the ear is sensible of an alternate increase and diminution of the strength of the sound, without variation of pitch. This is occasioned by the alternate coincidence and bisection of the vibrations of the component sounds. For example, suppose two perfect unisons produced from two pipes each 24 inches long. Each sound has 240 vibrations in a second, either exactly coincident, or exactly alternate. In either case, the vibrations are so frequent and uniform as not to be distinguishable, and the whole appears one sound. But let one of the pipes be only 23 inches and seven-tenths long, it will give 243 vibrations in a second. Therefore the 1st, the 8th, the 16th, and the 24th vibration of the longer pipe, will coincide with the 1st, the 81st, the 162d, and the 243d of the shorter. In the instant of coincidence, the aerial agitation produced by the one vibration is reinforced by that produced by the other. The deviations from coincidence gradually increase till the 49th vibration of the longer pipe, which will commence in the middle of the 41st vibration of the shorter one. The vibrations here bisecting each other, the aerial agitations of both will be weakened. The compounded sound will consequently be stronger at the coincidences and weaker at the bisections of the vibrations. The increase of strength, which is termed the beat, will recur thrice in every second. Thus the vibrations are in the ratio of $80 : 81$, or of a comma; and the compounded sound now supposed is an unison imperfect by a comma.
If a third pipe, tuned a perfect fifth to the longer of the two former, be sounded at the same time with the shorter, the dissonance will beat nine times in a second; and is thus shown to be a fifth imperfect by a comma.
The perfection or imperfection of any consonance may thus be ascertained with equal facility and precision; and by this method, any perfect consonance may be altered to any acquired state of temperament.
The theory of beats is therefore valuable, as giving both us the management of a phenomenon intimately connected with perfect harmony, as affording us precise and practicable measures of all deviations from it, and as thus forming the basis of the most accurate system of temperament.
For the preparatory process of determining the exact degrees of the scale, let us attend to the following ingenious and amusing experiment.
Let two harpsichord wires be exactly tuned in unison at the pitch of the tenor 'c', to be acted on simultaneously by a wheel rubbed with rosin, like that of a viola. Let a scale of 240 equal parts be described under one of the strings, equal in length to the sounding part of it, and numbered from the end at which the wheel is applied. Let a moveable bridge be placed under this string, but so as not to alter the tension of it in the least.
The two open strings being in perfect unison, without any beating whatever, let the moveable bridge be advanced slowly from the nut, while the wheel is applied to both strings. All kinds of chords, consonant and dissonant, will of course be successively heard. Between the consonances there will be a beating, which will increase as we approach the consonance, cease on our reaching it, appear again as we leave it, diminish as we recede from it, and again increase as we approach to the succeeding consonance.
After this general view, let us more particularly examine the several degrees of the scale.
On placing the moveable bridge at 120, we shall hear a perfect octave, without any beating. If the division be not quite exact, there will be a little beating; but by shifting the bridge very gently to either side, the increase or diminution of the beating will guide us to the true place, where it will entirely cease.
On placing the bridge at 160, the perfect concord of fifth the key and fifth will be heard. Any alteration of the bridge to either side will produce a disagreeable beating.
A rapid flutter in the vicinity of 180 will cease at that point, and give place to the consonance of the key and fourth.
On approaching 192, an angry waspish beating is succeeded at that point by the animating concord of the key and major third.
As we leave 192, the beating assumes a melancholy character, and ceases at 200, the place of the plaintive consonance of the key and minor third.
Between that point and the nut, we have only a succession of discords. As we were at a loss to ascertain the mathematical ratio of the second of the scale (art.19), so we have some difficulty in determining its just place by the theory of beats, and the experiment under consideration. We are uncertain whether we shall fix it at a minor tone, or at a major tone above the key. Both form a harsh dissonance with the key. The major tone, however, is thought less disagreeable: it admits of five more concords in the octave than the minor; and the ratio of it $9 : 8$, is that suggested by the similarity of its interval with the fifth, to the interval of the fifth and octave (art. 19). On these accounts we prefer it; and its place in the division under our precise consideration is 213.
Let the bridge now be placed near, and slowly moved to 150: the beating subsides into a consonance, slightly pleasing, that of the key and minor sixth.
At 144, we have the agreeable concord of the key and major sixth. From 144 to 120 we hear nothing but discord.
In this interval, however, we have to find the place of the sensible note or major seventh. The ear informs us, that the interval between the major seventh and the octave, must be similar to that between the major third and the fourth. Applying to the former interval the ratio of the latter, that of $16 : 15$, we place the moveable bridge at 128; for as $15$ is to $16$, so $120$ gives $128$. We also feel, that the interval between the fifth and major seventh is exactly similar to that between the key and major third, of which the ratio is $5 : 4$. Now, applying the same ratio to $160$, the place of the fifth,
we find $5 : 4 :: 160 : 128$. We thus determine 128 to be the place of the major seventh of the scale.
The interval or difference between the minor tone $10 : 9$, and the major tone $9 : 8$, is $81 : 80$, termed comma. This interval is not employed in practical music, but must be distinctly understood by theorists, and tervals, particularly in treating of temperament.
There are therefore four descriptions of simple intervals; that is, intervals which do not include more than a major tone. These are, comma, of which the ratio is $81 : 80$; hemitone, or $16 : 15$; minor tone or $10 : 9$; and major tone, or $9 : 8$ (c).
We have now to consider how far the perfect intonation of the natural scale must be departed from in keyboard instruments, such as the organ and harpsichord; so that the same sound may serve for different degrees of different scales.
These instruments have twelve sounds in every octave; that is, they have the eight natural degrees and four intercalary sounds, viz. between C and D, D and E, F and G, G and A, and A and B.
The purpose of these intercalary sounds is, that an air may be performed in any pitch; that is, that any sound may be taken for a key note, and that other sounds may be found to form the scale of that key note, at intervals corresponding to those of the natural scale.
Thus, if instead of C, the key note of the natural scale, we take B for the key note required; A, which is the seventh to B, will by no means answer for the seventh of the assumed scale; for the interval between A and B is a major tone, of which the ratio is $9 : 8$, whereas the interval between the seventh of the scale and the octave, can only be a hemitone, the ratio of which is $16 : 15$. We must therefore employ the intercalary sound between A and B, which in this employment we call A#, or A sharp. But we shall presently see that we cannot tune even this sound in the ratio of $16 : 15$ with B. For, let us take F for the key note of another scale, we find that B will not serve for the fourth of that scale, being a major tone above A the third; whereas the fourth of the scale is only a hemitone above the third. We must therefore have recourse to our intercalary sound between A and B, which
(c) The logarithmic measures of these intervals, and of the compound intervals determined in the way which we have described, are:
| Interval | Logarithmic Measure | |----------------|---------------------| | Comma | 54 | | Hemitone | 280 | | Minor tone | 458 | | Major tone | 512 | | Minor third | 792 | | Major third | 969 | | Fourth | 1249 | | Fifth | 1761 | | Minor sixth | 2041 | | Major sixth | 2219 | | Seventh | 2730 | | Octave | 3010 |
The octave being thus divided into 3010 equal parts, a circle of which the circumference is divided into 301 degrees, and a concentric moveable circle having a nonius subdividing each into ten parts, will form a convenient instrument for examining all temperaments of the scale. which we must here call Bb, or B flat, and which ought in this state to be tuned a hemitone above A, or in the ratio of \(16:15\) with that note. Now, this intercalary sound cannot be both in the ratio of \(16:15\) with A, and in the same ratio of \(16:15\) with B. This would extend the whole interval between A and B, to the ratio of about \(8:7\); whereas it should only be in that of \(9:8\). We must therefore tune the intercalary sound in such a diminished relation to A and to B, that it may serve either for A \(^\#\) or B \(b\).
But, even independent of these intercalary notes, some temperament of the natural scale is necessary.
Let the four-fifths, 'c g', 'g d', 'd a', and 'a e', be tuned all perfect. Then tune the two perfect octaves from 'c' downwards, 'e', 'e': 'e'. The major third 'c e' resulting from this process, will be too sharp by a comma, or \(81:80\), and will beat 15 times in a second. The minor third 'e g' and the major sixth 'c a' will be still more discordant.
It is therefore impossible to have perfect fifths, and at the same time perfect thirds and sixths. Now, although a perfect fifth, occasionally employed, be pleasing, yet the ear does not relish a succession of perfect fifths; such a succession not only renders the harmony languid, but creates a doubt as to the key, which is unsatisfactory. On the other hand, an alternate succession of major and minor thirds and sixths constitutes the chief and most brilliant part of our harmonies. We therefore find it necessary to sacrifice somewhat of the perfect harmony of the fifths to that of the third and sixths.
It is this accommodation which is properly called Temperament; and to this system of it, by which the fifths are diminished, and the thirds and sixths preserved perfect, we give the preference.
We have just seen that four consecutive perfect fifths compose an interval, greater, by a comma, than two octaves and a major third. But in the tuning of our instruments requiring temperament, these intervals must be rendered equal. Because, as we have seven hemitonic intervals in the fifth, twelve in the octave, and four in the major third; so the interval of four-fifths contains twenty-eight hemitonic intervals, and that of two octaves and major third contain also twenty-eight, being twenty-four for the two octaves, and four for the major third. The real difference being, however, a comma, it is plain, that if we keep the major thirds perfect, we must diminish or flatten each of the four-fifths one-fourth of a comma.
It is not easy to ascertain with perfect exactness the quarter comma by which the first fifth 'c g' is to be diminished. We shall, however, be sufficiently accurate for practical purposes if we flatten 'g' till a beating of 9 beats in four seconds is produced (D).
Having in this manner tuned 'g', we diminish the next fifth 'g d', one-fourth of a comma, by flattening d till 'g d' beat half as fast again as 'c g', or \(13\frac{1}{2}\) beats in four seconds (E).
The next fifth, 'd a', must be diminished in the same proportion by flattening a till 'd a' beat 15 times in six seconds.
Instead of tuning upward the fifth 'a e', tune downward ('e') the octave 'a', and then tune upward the fifth 'a e', and flatten it till it beat 15 times in eight seconds.
If we take 15 seconds for the common period of all these beats, we shall find
\[ \begin{align*} \text{The beats of 'c g'} &= 34 \\ G'd' &= 25 \\ 'd a' &= 37\frac{1}{2} \\ 'a e' &= 28 \end{align*} \]
On tuning downwards the octave 'e' we have the major third 'c e' perfect without any beating; and we proceed, tuning upwards a fifth flattened by one-fourth of a comma, and when the beating becomes too quick, tuning downward an octave. We may do this till we reach 'b' \(^\#\), which should be the same with c, a perfect octave above 'c'.
It will be better, however, to stop at 'g' \(^\#\), and then to tune fifths downward from 'c' and octaves upwards, when we get too low. Thus we have 'c' E, F 'f', 'f' Bb.
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(D) If any concord, whose perfect ratio is \(\frac{m}{n}\) (\(m\) being the greatest term of the smallest integers expressing that ratio), be tempered sharp by the fraction \(\frac{p}{q}\) of a comma, and if M and N be the pulses made by the acute and grave notes of the concord during any number of seconds, the number b of beats made in the same time by this concord will be \(\frac{2 q m N}{161 p + q}\) or \(\frac{2 q n M}{161 p + q}\); and if it be tempered flat, then \(b = \frac{2 q m N}{161 p + q}\) or \(\frac{2 q n M}{161 p - q}\). (Smith's Harm. 2d edit. p. 82, &c.). Now, let \(\frac{m}{n}\) be \(= \frac{3}{2}\), the ratio of the fifth; \(q = 1\), \(p = 4\); therefore, \(\frac{p}{q}\) = one-fourth of a comma, and N = 'e' or 240 pulses in a second. Therefore, \(\frac{2 q m N}{161 p + q} = \frac{2 \times 3 \times 240}{161 \times 4 + 5} = \frac{1440}{645} = 2.25\) beats in four seconds very nearly.
(E) Because fifths, being in the ratio to each other of \(3:2\), N in this fifth \(= 360\).
(F) The grave octaves of the upper terms of each of these tempered fifths may be determined with perfect accuracy, by making the grave octave beat with the lower term of the tempered fifth as often as the upper term does with it; for instance, by making G 'c' beat as often as 'c g', &c. For, it has been demonstrated by Dr. Smith, that the upper term of a minor concord beats equally with the lower term, and with the acuter octave of that term; but that the upper term of a major concord beats twice as fast with the acuter octave of the lower term, as it does with the lower term itself. Therefore, as 'g' beats twice as fast with c as with 'c', and is with its grave octave G in the ratio of \(2:1\), G 'c' beats precisely as often as 'c g'. The process of temperament thus recommended, will be greatly facilitated by employing a pendulum made of a ball of about two ounces weight, sliding on a light deal rod, having at one end a small ring. Let this pendulum be hung by the ring on a peg, and the ball adjusted so as to make 20 vibrations in 15 seconds. This done, mark the rod at the upper edge of the ball, and adjust it in the same manner for 24, 28, 32, 36, 40, 44, and 48 vibrations. Then having calculated the beats of the different fifths, set the ball at the corresponding mark, and temper the sound till the beats keep pace exactly with the pendulum.
In order to discover, should it be necessary, the number of pulses made in a second by the tuning fork, by which we tune the tenor 'c' of our instrument, let a wire be stretched by a weight till it be unison or octave below the fork; let \(\frac{1}{2}\) th then be added to the weight. Being thus tempered by a comma, the contemporaneous sounding of the fork and wire will produce a beating; and on multiplying the beats by 80, the product gives the number of pulses of the fork, and consequently of the 'c' of the instrument tuned from it. But the common 'c' tuning forks are so nearly consonant to 240 pulses, that this process is scarcely necessary.
On the system of temperament now proposed, Dr Smith makes the following useful observation and deduction. The octave consisting of five mean tones and two limmas, it is obvious that by enlarging the tones we diminish the limmas, and that the increment of the tone is two-fifths of the contemporaneous diminution of the limma. Let \(v\) represent any minute variation of this temperament: the increment of a mean tone is \(2v\), and the contemporaneous diminution of the limma \(-5v\). Again, if the tone be diminished by \(-2v\), the limma will increase by \(-5v\). Let us observe the variations of the intervals in the latter case.
The perfect fifth consisting of three tones and a limma, its variation will be \(-6v + 5v\) or \(-v\). That is, the fifth is flattened by the quantity \(v\). Consequently the fourth is sharpened by that quantity.
The second, being a tone above the key note, and being therefore flattened by \(-2v\), the minor seventh is increased by \(2v\).
The minor third consisting of a tone and a limma, its variation is \(-2v + 5v\) or \(3v\). Accordingly, that of the major sixth is \(-3v\).
The major third, or two tones, is therefore diminished by \(-4v\). Consequently the minor sixth is increased by \(4v\).
The major seventh, being the inversion of the limma, is therefore varied by \(-5v\).
The tritone being diminished \(-6v\), the false fifth is accordingly \(6v\).
On this observation, Dr Smith has founded the following geometrical construction: Divide the straight line CE (fig. 2.) into six equal parts \(Cg, gG, da, da, aE, Eb, bT\), and intersect the points of division with the six parallel lines \(gG, daD, \&c.\) representing the intervals arranged according to the system of mean tones and limmas.
Let any length \(gG\), on the first line to the right of the line \(CE\), represent a quarter of a comma, \(G\) will thus mark the place of the perfect fifth, and \(g\) that of the tempered fifth, flattened by a comma.
Take \(daD\), double of \(gG\), on the second parallel also on the right hand; \(D\) will mark the place of the perfect second, and \(d\) that of the tempered second, flattened by the half comma \(daD\).
By setting off \(aA\) on the third parallel to the left, equal to \(gG\), we have \(A'\) the perfect major sixth, and \(a\) the transferred major sixth, sharpened by the quarter comma \(Aa\).
The major third being in the system of mean tones kept perfect, the place of that degree will be \(e\).
By taking \(bB\) on the fifth line, on the right, equal to \(gG\), we find \(B\) to be the place of the perfect major seventh, and \(b\) to be that of the tempered major seventh flattened by the quarter comma \(bB\).
And by making \(dT\) on the sixth line, to the right, equal to \(daD\), we have the contemporaneous temperament of the tritone flattened by the half comma \(dT\), and of the false fifth, sharpened by that quantity.
Any other straight line \(Ct\) drawn from \(C\), across these parallels, will represent, by the intervals \(gG, daD, \&c.\) the temperaments of another system of mean tones and limmas. Since it is plain that the simultaneous variations \(gg', dd', \&c.\) from the former temperament are in the just proportions to each other. The straight line thus employed, \((Ct, or Cc')\), has therefore been termed the temperer.
As the arrangement of the sounds of keyed instruments having only twelve keys for an octave, and meant to be used in different scales, must approach nearly to a system of mean tones, or rather mean limmas, this construction of Dr Smith's is very useful. The temperer points out, not only all the temperaments of the notes with the key note, but also the temperaments of the harmonic concords. Thus it will be seen, that the temperament of the minor third forming the interval between the major third and fifth, is in all cases the same with that of the major sixth and octave, and that the temperament of the major third forming the interval between the fourth and major sixth, is equal to that of the key and major third of the scale.
It has been proposed, in order to render Dr Smith's construction still more useful, that it should be drawn of such a size as to admit of the following supplementary scales.
1. A scale of \(gG\) divided into thirteen parts and a half, expressing the logarithmic measures of the temperaments mentioned in the note (c), a comma being = 54.
2. A scale of \(gG\) divided into 36 parts, giving the beats made in 16 seconds by the notes \(e, g\), when tempered by any quantity \(Gg'\). the alterations between the fifths and major thirds, flattening the fifths and sharpening the major thirds; and making both beat equally fast along with the key; and since enlarging the fifth increases the tone, and consequently diminishes the limma, the intercalary sounds become thus better suited for their double service of the sharp of the note below, and the flat of the note above. Much, however, is lost in the brilliancy of the major thirds, which are the most effective concords. The fifths are not much improved, and the sixths are evidently hurt by this temperament (H).
These methods of tuning by beats are incomparably more exact than by the ear. We cannot mistake above one beat, that is, in the fifth \( \frac{1}{2} \) th, and in the major third \( \frac{1}{3} \) th of a comma.
We have offered a short view of what appears to us to be the preferable system of temperament. It has been deduced from the observations of the most able theorists, and will greatly assist a tuner; but to him there are farther necessary, as to a musical performer, a correct ear, patient attention, and long practice.