THE HISTORY OF MUSIC (art and Science)

FROM THE EARLIEST RECORDS TO THE FALL OF THE ROMAN EMPIRE.

 

CHAPTER IX.

 

Basis of the science. Its fundamental laws. Earliest uses of music. Mathematical divisions of strings not alone sufficient. Minor tones introduced by Didymus, and followed by Claudius Ptolemy. Neither the Greek scale nor the modern is properly in one key. Hence the question whether Elevenths were concords. How to test intervals. The true proportions for scales. Rules for adding and deducting intervals. Scales of Didymus and of Ptolemy. Defects of the modern scale. The law of Nature the only true guide. Objections to the Fourth and minor Seventh of the present scale. Causes of Concord and Discord. Pythagorean ideas realized by modern science. Sounds too high and too low for our hearing.

 

THE discussion, of ancient and modem science must, in a measure, go hand in hand; for, as our present scale is Greek, so whatever applies to ancient times is equally applicable to the present. No science has more fixed and clearly established fundamental laws than music. The wind will teach them as it plays upon the strings of an Aeolian harp; for, although tuned to one pitch, it will cause them to emit sounds of every variety. The same law exists in the natural sounds of a trumpet, horn, or open tube of any kind, and all the notes will follow in the same succession. By blowing into the tube so slowly as just to make the sound continuous, the lowest, or fundamental note, produced by the entire length of the pipe is first heard; then, by gradually increasing the rapidity of the breath, an ascending series of notes will follow; every one of which may be predicted as they rise gradually, higher and higher, up to the extreme pitch that can be obtained from the breath of the mouth. The same rising succession of notes is heard in the harmonic sounds that follow upon one of the long strings of a pianoforte, after the fundamental note, produced by the whole length of string, has been struck, and when the string gradually subdivides itself into smaller and smaller nodes before finally coming to rest. They then follow so rapidly as to seem to run one into the other. From these laws, we may deduce both a perfect Diatonic, and a perfect Chromatic scale from any given note. The proportions of musical intervals may be measured either by the divisions of a string, or by the gradual cutting down of a pipe. Results in harmony may be foretold with certainty as either good or bad, by calculating the proportions of the intervals together with the roots of the sounds, and without any appeal to the ear. Again, the ears may be stopped, and the eye will tell, from the motions of sand scattered upon the sounding board of a pianoforte, or any other vibrating surface, whether the chord that has been struck upon the instrument has been a concord or a discord. In the former case, the movements of the sand will be symmetrical and regular; and, in the latter, they will show that discord reigns by their disturbed state, and by their seeming to battle together. The Octave is the first ascending sound, after the primary one, in the harmonic scale of nature, and all subsequent sounds are but subdivisions of it at higher pitches. The Octave system, with its included and harmonic-following Fifth and Fourth, and major and minor Thirds, is the foundation of all music. Sound, as is well known, does not exist in the atmosphere, but is an affection of the brain produced by succeeding elastic waves of air that strike upon the drum of the ear, and which, for that reason only, are called “sound-waves”.

From all this, and from much more that might be said, there can be no more evident fact than that it was the design of the Creator that music should be the companion and the solace of man; and from this we may deduce that, in the mouth of man, there can be no more fitting medium for the praise of his Maker.

The ancient heathen attributed a divine origin to their music, and, accordingly, the earliest uses to which we find it to have been applied by them are those of religious worship. At a later period, music was also cultivated for educational purposes, especially among the Greeks, and chiefly with the view of elevating the mind above its too frequently grovelling tendencies. “The first and noblest application of music”, says Plutarch, “is in offering the tribute of praise to the immortals: the next is the purifying, regulating, and harmonizing the soul.”

Speaking of times past, Plato says: “Our music was then divided according to certain species and figures. Prayers to the gods were one kind of song, to which they gave the name of hymns. Opposed to this was another species which might be called Threni (Funeral Dirges), another, Paeans (Choral Songs to Apollo or Artemis), and another, The Birth of Dionysus (the Greek Bacchus), which I hold to be the dithyrambic verse. There were also Nomes (or simple and severe chants upon a few high notes), accompanied by the Kithara, which were equally distinct. These and some others being prescribed, it was not allowable to use one kind of chant for another. But, in process of time, the poets introduced unlearned license; they, being poetic by nature but unskilled in the rules of the science, trampled down its laws. Over-attentive to please, they mixed threni with the hymns, and paeans with dithyrambs, imitated music intended for the flute upon the Kithara, and confounded each kind with every other.” (Laws, lib. 3.) Add to this Plutarch’s account. He says: “In the yet more early times, the music of the theatre was unknown to the Greeks; the whole art being then made subservient to the honour of the gods, and to purposes of education. Theatres themselves were then unknown; and their only music consisted of those sacred strains which were employed in the temples as a means of paying adoration to the Supreme Being, and of celebrating the praises of the great and good of our species. It is probable that the modem word ‘Theatre’ and the very ancient one theorein (to look at), have their derivation from Theos, the Deity. In the present day, so great is our degeneracy, that we have absolutely lost both the knowledge and the notion of that system by which youth were formerly trained up to honour and virtue. The only music now studied and listened to is that of the theatre.” (De Musica, cap. 27.)

Notwithstanding the divine origin attributed to music, it is very doubtful whether any of the civilized nations of antiquity knew the laws of Nature as to the prescribed succession of musical sounds, or, perhaps, much beyond the general observation, such as that of Aristotle, that high notes are of more rapid vibration than low ones. So far as we are acquainted with ancient systems of music, they seem to have been founded upon the divisions of a string upon some instrument of the monochord kind, with a movable bridge (hupagogeus) under it, for the purpose of measuring; or else to divide by pressing the string against a finger-board Since, then, the science of music was thus learnt from a string, it must surely offer the most simple and intelligible means of explaining it. It will give the least amount of trouble to the reader; and, although there must be figures in all cases, yet, if explained by a string, nothing more than the elementary rules of arithmetic can be required.

The Greek system is defective in one essential point, that, although the divisions of a string will show the ratios that its parts or intervals bear to the whole length, they will not point out the positions in which those intervals must be placed in a musical scale, so as to make consonances of them by keeping them within one key, or from one root. So, a scale may look well-proportioned upon paper and yet be practically bad. The same length of a string may be divided off in one part, so as to be concordant with the rest; and, in another part, to be discordant.

The defects of this origin are shown in many of the Greek scales, and, among others, in our own, it being wholly Greek.

The Octave, the Fifth, the Fourth, and the major tone, (i.e., sounding eight-ninths of a string compared to the whole length,) were included in the Pythagorean system of music; and the seemingly slight change which created true consonant major and minor Thirds, and the minor tone, (of nine-tenths of a string compared to the whole,) were improvements introduced by Didymus about the commencement of the Christian era, and followed by Claudius Ptolemy, about the year 130 or 140. Still, the Greek Diatonic scale remained a compound of sounds derived from different roots, and was, and is, therefore, strictly speaking, in different keys.

For instance, in our adopted scale of C major, one-half of the Diatonic Octave, or the notes on the long keys of the pianoforte, is in the scale of C, and the other half is derived from the scale of F. This is consequent upon its having been composed out of two Greek conjoined tetrachords, B, C, D, E, and E, F, G, A, which, when taken as parts of a major scale, and not of a minor, as of old, have their roots or key-notes the one in C and the other in F. If a minor scale were to be tested in the same way, it would show greater variety of roots, therefore greater deviation from the right path.

A comparison with the scale of Nature will presently prove this; but, in the meantime, in order that the purport of these remarks may be understood, suppose that, in the key of C major, we sound C in the base, and with it C and F in the treble, the last two being at the interval of a Fourth. The treble F makes a discord with the base C. But if we again sound the upper C with the G immediately below it, instead of the F above, and retain C as the base, it is a concord. And yet from G to C, and from C to F, are both Fourths taken from the key-note, the one above and the other below it, in our key of C. The difference is, that from C to F is an artificial interval, disavowed by Nature in her scale of C, but from G to C is Nature’s interval. The former is from the root of F, and requires F for its base. Then it will be concordant. These cases will be further exemplified in the sequel.

As my present subject is the Science of Music, I speak freely of the defects of our adopted scale. Its deficiencies may at least be made known, however improbable any change of system may be. Let us face the difficulties, and see what a dwarfed scale for melody we have to work upon, through having copied from the Greeks.

The intervals from G to C and from C to F, were two of the puzzles to writers upon Harmony, not only for several ages past, but even far into the present century. They had no rule by which they could duly account for Fourths being both discords and concords in what was considered to be one key, so they divided themselves into opposite camps; the one contending that Fourths, and what have been called ‘Elevenths’, or combined Octaves and Fourths, were concords, and the other as stoutly maintaining that they were discords. Neither of the two parties thought of appealing to the Harmonic scale for the solution of the difficulty. Harmonics were, until lately, more looked upon as a trouble to pianoforte makers that ought to be got rid of, than as containing the essence of music, and as being therefore a necessary study for a musician. There is indeed little that can be more instructive than a comparison of our scale, calculated by Greek mathematicians, with that most ancient of all scales, the scale of Nature. Every musical interval within the Octave may be so misplaced as to leave the key and become a discord, and it is from the scale of Nature only that a fitting position for each has to be determined. Mathematical scales are insufficient without it, and yet this material deficiency in them, and especially in our own scale, has been but little thought of. A choice of good intervals may suffice for varied harmony, but to be consonant in one key, they must be derived from the same root.

The Greek scale which preceded the time of Didymus, although usually coupled with the name of Pythagoras, might equally be called the scale of ancient Asia, and of ancient Egypt. It has already been shown that the Greek one-octave scale began a Fourth below the key-note, thus taking the interval of the Fourth downward in its consonant form to the key-note or Mese, and that it ended a Fifth above the key-note. Also that the Fifth above the key-note was compounded of a major tone, called diazeuctic, or disjunctive, and of another Fourth. So the skeleton of the Octave was thus complete, and there remained but to fill up the two Fourths by smaller intervals. So far the Greek scale and the scale of Nature agree, and from that point they part company. These Fourths were originally subdivided, each into two major tones and a remnant. The choice of major tones was directed by one of them being the exact interval between a Fourth and a Fifth. When two of them were included in a Fourth, the remnant became one of that kind of semitone which was in the ratio of 243 to 256, to which the name of a diesis was given by the earliest Pythagoreans, such as Philolaos, but which later Pythagoreans named limmas, meaning remnants of the interval of the Fourth, after the two tones were taken out of it. Aristoxenians and Greek practical musicians called these remnants semitones, but such semitones are different from the semitone of later Greek, and of modern music.

When the Greek scale was extended to two Octaves, by adding on a Fifth at the lower extreme of the original Octave scale, and a Fourth at its upper end, the two-octave scale began and finished at the key-note, like our own, and equally agreed with Nature’s law as to the skeleton of the Octave. Therefore, for the comparison of ancient with modern music, which is here proposed, we will take one Octave in this latter form. Suppose the key to be Hypo-Dorian, or A minor, then from A to B will be the disjunctive tone, and there will remain the two conjoined Fourths, B, C, D, E, and E, F, G, A, just as on the long keys of a pianoforte.

The way to test such intervals as the Octave, the Fifth, the Fourth, the major Third, and the minor Third, upon a string, is to stop successively the half, the third part, the fourth, fifth, and sixth parts, and to sound the remainders of the string, comparing each of these intervals with the sound of the whole length. We have no equivalent in modem music to the note produced by stopping the seventh part of a string, which is the Harmonic Seventh, but it is a natural note upon the horn. It was employed in the last century with untempered instruments, such as fiddles and basses, in small bands, as well as for ages before, when horns and trumpets had no keys or slides. It affords additional passages in melody without change of key. It is called the Harmonic Seventh in reference to its key-note, so, in C, would be called Harmonic B flat, and we might employ it where we cannot use our B flat, because the latter does effect a change of key. Swiss singers, says Spohr, in his Autobiography, employ the Harmonic Seventh in their music, as well as the Harmonic Fourth, which is the interval produced by stopping the eleventh part of a string. They are quite right to do so, because they enlarge their sphere of melody, and have Nature on their side in both cases. The Harmonic B flat divides the upper Fourth, from G to C in the key of C, into two all-but-equal parts, and these might be called Thirds, but they are of diminishing compass, and next to the minor Thirds that we employ. Natures Octave is divided into eight tones, beginning with the eighth part of a string up to the sixteenth part; but we, following the Greeks, Chaldeans, and Egyptians, with their seven planets and seven notes, have still but seven. Nature divides the interval, from G to C into the same number of parts as that from C to G.

As the seventh part of a string gives the Harmonic B flat, so the eighth part stopped gives the key-note, C, above it. I have already said that the stoppings of the ninth and of the tenth parts of a string raise its pitch by the intervals of our major and of our minor tone. From those, the moderns pass on to the sixteenth part, and by stopping it, they raise the note by what is now termed indifferently a major semitone, or a Diatonic semitone. When we pass down from C to B, or from F to E, it is by the semitone in question. Its name is from the Latin, and that of hemitone from the Greek, but they are equally improper; because, instead of being a semitone, the interval of a sixteenth part of a string is really the smallest of the eight tones of Nature. It is too wide to be the half of even our major tone. Its name should have been changed when Didymus and Ptolemy enlarged its proportions. The Pythagorean limma, or Aristoxenian semitone, was as 243 to 256, and Didymus changed it to 240 to 256, which is as 15 to 16.

A true tonal scale is from the eighth to the sixteenth part of a string, whatever the length of that string may be. Length only changes the fundamental note. The two intervals to which we give the name of tone are the largest of the eight of Nature’s. Those eight decrease progressively in the ascending Octave; and we employ but three of them, viz., the largest two, and the least. We name the first two Tones, and this least we misname a major or Diatonic semitone.

There is another, and a truer semitone, in modem music. It is produced by stopping the twenty-fifth part of a string, and therefore is much less than the Diatonic semitone. It is the true interval between G and G sharp in Nature’s scale, when C is the fundamental sound, or key-note. This semitone, like the other, has two names. It is called minor, and Chromatic, and it is employed when the name of the note remains unchanged, as from F to F sharp, or from G to G sharp.

All the before-named intervals were used by the Greeks in some one or other of their scales. Even the Harmonic Fourth and the Harmonic Seventh were thus included.

Our major and minor semitones were coupled together in the Chromatic scale of Didymus, and the two combined are equal to one minor tone. Hence, when he added the usual interval of a minor Third between the highest two strings of the Fourth or tetrachord, he made the best possible arrangement for a Greek Chromatic scale. With two such tetrachords, and the diazeuctic major tone, he completed the Octave.

His Enharmonic scale was equally good, for he divided his major semitone, 15/16, into its two best quarter-tones, 30/31, and 31/32. Then a major Third, 4/5, completed that tetrachord.

But, before referring further to figures, there are three simple rules that every incipient musician should know. It is not, however, to be assumed that all do know them; for although it must be supposed that there are books on music which contain these rules, yet it has not been my fortune to have met with any one of them. Musicians appear too generally to have thrown such information aside, and mathematicians, when writing upon music, assume that their readers know every kind of rule beforehand.

It is indispensable for a real musician, that he shall be able to tell with certainty what will be the effect of any combination of intervals, and he may often wish to ascertain it for himself when he cannot have the opportunity of testing them practically. It is well, therefore, to know how he can judge of them on paper, with even greater certainty than by ear, however good that ear may be. Indeed, it is by far the more convenient way of testing unfamiliar intervals.

The three rules are: How to add intervals; How to deduct one from another; and How to compare one with another. The answer to all may be comprehended in a line. To add, multiply; To deduct, cross-multiply; To compare, bring them to a common denominator. Still, these directions will not be, in all cases, sufficient; and, in order to be understood by all, I hope to be excused for further explaining and exemplifying them.

To add one interval to another, multiply the numerator by the numerator, and the denominator by the denominator. If we say three-eighths, three is the numerator, and we denominate eighths. Then reduce the multiplied totals to their smallest figures, by finding out what is their “Greatest Common Measure”

To do this, we must follow the ordinary rule of arithmetic, which is thus expressed: “Divide the greater by the less, and the preceding divisor by the remainder, and so on continually until there is no remainder. The last divisor will be the Greatest Common Measure.”

This will perhaps be more quickly understood by an example. The ancient Pythagorean tetrachord, or Fourth, consisted of two major tones and a limma, or remnant; in other words, of the three intervals, 8/9, 8/9, and 243/256.

Then for the two major tones and limma = 8/9 x 8/9 x 15552/20736 to be explained thus :

For the numerator, 8 times 8 are 64, and 64 times 243 are 15552. For the denominator, 9 times 9 are 81, and 81 times 256 are 20736. Divide the greater by the less, 20736 by 15552; it leaves 5184. Then 15552 by 5184, and it leaves no remainder. Therefore, 5184 is the last divisor, and the Greatest Common Measure. Divide the two original sums by 5184; it shows 15552/20736 to be equal to 3/4.

For the second rule: To subtract one interval from another, by cross-multiplication, the readiest way is to invert the figures of one of the two ratios, and to place them under the others. Then to multiply the upper by the under. This position of the figures is the more convenient for a sum. To prove the rule in the simplest way, we know that 4 to 2 is the same ratio as 2 to 1. Cross-multiply, and it will show them to be equal. Again, we know that a Fifth and a Fourth together make an Octave, as from C up to G, and from G to Octave c. Therefore, if we deduct a Fifth from an Octave, the remainder ought to be a Fourth. The ratio of the Octave is as 1 to 2 in length, or as 2 to 1 in vibrations. The interval is the same either way, so the case may be stated either way. Here, adopting the former, the Fifth is as 2 to 3, and the Fourth as 3 to 4. 

Therefore, taking the Octave as ......................1:2

Multiply by the inverted figures of the Fifth……...3 : 2

The remainder shows the Fourth, viz……............... 3:4

For the third rule: How to compare intervals. The most useful example will be to take our present scale, and to compare every interval with its keynote in C. To D is a major tone, 9/8, or as 9 vibrations to 8 of the key-note. To E, a major Third, or 5 vibrations to 4 of the key-note. To F a Fourth, in figures 4/3. To G a Fifth, 3/2. To A a Sixth, in figures 5/3. To B a major Seventh, 15/8. Lastly, the Octave is as 2 to 1 of the key-note. So the scale stands thus : 1, 9/8, 5/4, 3/2,5/3, 15/8, 16/8. As the

C, D, E, F, G, A, B, C.

Octave here includes odd numbers, as four to three, and five to three, which are two of the imperfections of our scale, we cannot have a lower common denominator than 24, where it ought to be 8. So we must multiply every ratio by such figures as will bring its under-figures to 24. For instance, 9/8 is equal to 27/24, multiplying by 3. Next, we must multiply 5/4 by 6, and so on: 24, 27/24, 30/24, 32/24, 36/24, 40/24, 45/24, 48/24.Then dropping the lower figures, we compare the proportions of our Octave scale as 24,27, 30, 32, 36, 40, 45, 48.

This rule, or multiplying the ratios by 24, is necessary for understanding Dr. Wallis’s edition of Claudius Ptolemy, and many more books.

When the principal intervals are stated in figures, according to their proportionate vibrations, the Octave is written 2/1, or 2 to 1. The Fifth as 3/2, or 3 to 2. The Fourth as 4/3. The major Third as 5/4, and the minor Third as 6/5. The major, or Diatonic semitone, as 16/15.

And now, having given the three necessary rules, I will in future state only the results, and leave them to be tested by the curious.

One of the Greek scales in which the Harmonic Seventh, or seventh part of a string, was employed, is exceedingly worthy of note, and quite an exceptional scale in Greek music. It is the Even Diatonic (Diatonon homalon) of Claudius Ptolemy, given in the 16th chapter of his first book. The remarkable part is, that he follows out the natural division of the scale in all the intervals that are included in the Fifth, from the key-note upwards. Therefore he has so far a true major scale, with its major Third, instead of the perpetually recurring minor Third that minor Third being always consequent upon the disjunctive major tone immediately above the keynote, and to the semitone of the tetrachord being next above it, as A to B, and B to C. They caused Greek scales to be always minor. Jean Jacques Rousseau’s remark, that the minor scale is not given by Nature, is a very just one. After the major Third, which is in the place of the old minor, Ptolemy employs the Harmonic Fourth, or the eleventh part of a string, being a nearly equal division between E and G. So, in the scale of C, Ptolemy has C, D, E, Harmonic F (instead of our F), and G. Next, as to the tetrachord, or Fourth, below the key-note, he first divided it into its two legitimate parts by Harmonic B flat. So far he had proceeded thoroughly according to natural laws, but as that one division of the Fourth gave him only three notes: G, Harmonic B flat, and C; and four were required for a Greek tetrachord, he altered that excellent arrangement, and repeated the intervals that he had just employed in the Fourth above the keynote, viz., for the D, E, Harmonic F, and G.

Before that change, he had adopted Nature’s scale so far as taking successively the sixth, seventh, eighth, ninth, tenth, eleventh, and twelfth parts of a string. Yet he was not led to it by any insight into Nature’s laws, but by one of the Pythagorean doctrines which neither Pythagoras nor his school had ever carried out. The doctrine was to employ only super-particular ratios, such as 10 to 9, 9 to 8, 8 to 7,7 to 6, 6 to 5, 5 to 4, 4 to 3, and 3 to 2. As Ptolemy here employed them in gradually decreasing intervals, he fell into the law of Nature.

When the Pythagoreans gave the name of limma to the proportions of 243 to 256, which are less than the half of a major tone, they called the remaining greater part an apotome, or segment. It had the awkward proportions of 2048 to 2187. The comparative sizes of the two will be made clearer if we multiply the figures of the limma by 8, thus making it 1944 to 2048. The difference between these two was called a Pythagorean comma (komma), viz., 524,288 to 531,441. Therefore, if a Pythagorean comma be added to two limmas, it makes one major tone. But there is another point to be noticed about this comma. If twelve perfect Fifths be taken from any note, say from C upwards, they will end upon B sharp, and it will be a Pythagorean comma sharper than the seven-octave C. The reason of this reversal of order is, that we make Fifths where Nature has not designed them, because the notes have to serve other purposes. Octaves are the only continuously perfect intervals.

A few other intervals with peculiar names will sometimes be met with; and, being bound to explain them, it is better at once to clear the board. A minor semitone deducted from a major semitone, leaves what is now termed an Enharmonic diesis, 125 to 128. This diesis is less than one of the Enharmonic-quarter tones of Didymus,31/32 = 124/128. The modern Enharmonic diesis is a nominal difference between C# 3 and Db. The interval between our Diatonic, or major semitone, and a major tone is 128/135.

A Greek Enharmonic diesis, or quarter-tone, is sometimes called a Tetartemorion, meaning quarter-piece of a tone, and a Chromatic diesis, or third part of a tone, is called a Tritemorion. These two intervals have not infrequently been mistaken by lexicographers for the much larger ones of a Fourth, which is two tones and a half, and of a Third, which is two tones.

A Schisma is an interval to be read of in mathematical music, but one not often brought into practice. It is the approximate half of a Pythagorean comma. A Diachisma is a similar division of the before named limma. As the interval of a Diachisma approaches to a quarter-tone, it may have been practically employed in the ancient Enharmonic scale.

Lastly, the comma of Didymus is sometimes referred to as a syntonic comma. This is an important interval in modern as well as in ancient music. It is far more so than the comma of Pythagoras. The comma of Didymus is the interval between a major and a minor tone, or between the eightieth and the eighty-first parts of a string.

So delicately organised is the human ear, that it was but this eighty-first part that worked the great revolution between the ancient scale of Pythagoras and the very present scale. First, Didymus, and, after him, Claudius Ptolemy, deducted this comma from one of the two major tones that formed the ancient Ditone, or over-sized major Third, and so changed it into our consonant major Third. Moreover, the comma thus taken away from the tone was added to the limma, and brought that interval into its present proportions as a major semitone. By these changes the Greek Diatonic scale attained its present improved proportions. So, the difference between a major and a minor tone, as well as that between a limma and a major semitone, is a syntonic comma, or comma of Didymus, or the eighty-first part of a string.

To prove the effect of this apparently small, but really very important, change, we have but to add together the two major tones of which the ancient Ditone, or Pythagorean Third, consisted, by multiplying the numbers 8/9 x 8/8 = 64/81. If it had been a true major Third, the ratio would have been 64/80, which is the same as 4/5, as will be found by dividing the two numbers by 16. Although the old Ditone did pass for a Third in melody, it would not bear the test in harmony. Every ear found it to be a harsh discord. The ear is so much more delicately organised than the eye, that even a hundred and sixtieth part of difference in vibrations, in one second of time, has a rough and unsatisfactory effect, which every ear can distinguish; whereas the quickest eye cannot distinguish, or count, more than twenty-four vibrations in the same brief period. The delicacy of the one organ is quite as eight to one of the other.

 

The improved major Third of Didymus and of Ptolemy consisted, like our own, of two tones, the one major and the other minor: 8/9 x 9/10 = 72/90 =4 /5. Then the limma being changed into a major semitone, made a true Fourth 4/5 x 15/16 = 60/80 = 3/4.

And now as to the discordance of the minor Pythagorean Third, which must also be proved; for there is nothing like proof to fix anything as a fact upon the memory. It consists of a limma and a major tone : 243/256 x 8/9 = 1944/2304 = 27/32. Twenty-seven to thirty-two are indifferent proportions that carry discord with them. They are neither multiple, as 2, 4, 8, 16, 32, nor super-particular, i.e., one number is not the unit, or one particle above the other. They want the comma to make them super-particular and consonant. The ratio is identical with our imperfect minor Third of today, as between D and F, when the scale has been tuned for the key of C; because it has then a minor, instead of a major tone in it: 15/16 x 9/10 = 135/160 = 27/32. This defect was inherited from Claudius Ptolemy’s scale. The true minor Third consists of a major tone and a major semitone : 15/16 x 8/9 = 120/144 =5/6.

One of the musical laws of Pythagoras was, that, to be concordant, all ratios must be either multiple (pollaplasioi), like the Octave, 2, 4, 8, or like the Twelfth, 3, 9, 27, 81, or else they must be superparticular (epimorioi), as 3 to 2, 4 to 3, or 5 to 4. This doctrine is referred to, among others, by Aristotle, in his 41st Problem of Section 19. We have every reason to suppose it to have been derived among other laws, from Egypt, because, although it was held as a maxim by the school of Pythagoras, it was very imperfectly acted upon either by him, or by his disciples, for a full 500 years after his death. Therefore, his followers could not have regarded it as a really essential principle in music, and as a law of Nature in the division of a string, or of a column of air enclosed in a pipe. If otherwise, they acted too inconsistently in having admitted only the Octave, the Fifth, and the Fourth, as simple consonances. They should have included intervals in the ratio of 5 to 4, and 6 to 5, which would have added the major and minor Thirds to their scales in a consonant form. When Claudius Ptolemy followed out their doctrine, and so brought true major and minor Thirds into his scales, he twitted the followers of Pythagoras with their inconsistency in that respect.  (Ptolemy, lib. 1. cap. 6.)

Neither the Octave, nor any interval within the Octave, can be divided into equal parts. The most consonant and the nearest to equal division of the Octave is into a Fifth and a Fourth, and the ratios of both are super-particular, 2/3 and 3/4 = 6/12 = 1/2. The Fifth must, in like manner, be divided into major and minor Thirds, 4/5 x 5/6 = 20/30 = 2/3. The best division of the Fourth would be by the Harmonic Seventh, making from G to C 6/7 x 7/8 = 42/56 = 3/4. The major Third would be into major and minor tones, 8/9 x 9/10 = 72/90 = 4/5.We lack the divisions of minor Third, and of major tone, in our adopted Greek scale, but we divide the minor tone into our two semitones, 15/16 x 24/25 = 360/400 = 9/10.

The first Greek who is known to have carried out the doctrine of super-particular ratios into all his scales is Didymus. He had been preceded by Archytas, and by Eratosthenes, but they did so only in part. Claudius Ptolemy followed after Didymus, but made the same one exception to this true principle as did Eratosthenes, by retaining the old Pythagorean Diatonic scale, among others, either out of respect for the name of Pythagoras, or because it was in general use. Nevertheless, each offered improvements upon it. Didymus wrote a treatise upon the differences between Aristoxenians and Pythagoreans, of which we now know only some extracts, quoted by Porphyry in his Commentary upon Claudius Ptolemy.

As a scale designed for the Diatonic system of the Greeks, that of Didymus had some advantages over Ptolemy’s arrangement, because both were intended for the minor scale. The difference between the two is but slight, the intervals being the same, and the scale of Ptolemy seemingly copied from that of Didymus, of which it is a mere transposition. In every Octave, two minor tones are necessary, one being required for each of the two Fourths, to make them consonant. Didymus placed one of his minor tones between C and D, and the other between F and G, while Ptolemy changed their places to between D and E, and between G and A, as we do now. In this last interval Ptolemy broke through the Greek law of having a full tone below Mese, or the key-note, but he could not make a novelty by any other means. Didymus obtained a perfect Fourth from A to D, a perfect minor Third from D to F, and a perfect Fifth from D to A. The imperfections of these intervals in our adopted scale have been a great perplexity to modern musicians.

But although Didymus had these advantages in a minor scale, they were outweighed by disadvantages when the key-note was changed in later ages from minor to major. To obtain due proportions for a minor scale, Didymus had made the Fifth from C upwards, and the Fourth from C downwards, both imperfect.

The advantages and the disadvantages of these two systems, which have been ranked as No. 1 and No. 2, by mathematicians for our present imperfect seven planet scale, will be best seen by placing them side by side, reminding the reader that every major Third, Fourth, and Fifth must have one minor tone, and but one, to be perfect.

In both scales, the disjunctive tone, A to B, was necessarily major, according to Greek laws, but in the major scale of C, according to Nature’s law, it ought to be a minor tone:

Scale OF Didymus.

9 to 8 - 16 to 15 - 10 to 9 - 9 to 8 - 16 to 15 - 10 to 9 - 9 to 8

A to B - B to C -- C to D - D to E - E to F - F to Gr - Gr to A

SCALE OF PTOLEMY.

9 to 8 - 16 to 15 - 9 to 8 - 10 to 9 - 16 to 15 - 9 to 8 - 10 to 9

A to B - B to C - C to D - D to E - E to F --F to Gr - G to A

The imperfections of the scale of Didymus are, that by having placed two major tones together, (G to A, and A to B,) he made a false major Third from G to B; also a false Fourth from G to C, because there was no minor tone in it; also a false Fifth from C to G, because he had two minor tones in it. Again, from B to D, and from E to G, are false minor Thirds, because they are made up with minor tones instead of major.

The imperfections of the scale of Claudius Ptolemy are, that from A to D is a false Fourth, from D to F a false minor Third, and from D to A a false Fifth. Also, that he has two different kinds of major Sixth, one from C to A, with two minor tones in it, and another from F to D, with one minor tone.

If Nature were called in to judge between the two mathematicians as to the true positions of major and minor tones, she would say that the one was right in the one place, and the other in another. Her law agrees with Ptolemy as to the intervals between C and D, and between D and E, but she wills a major tone between G and A, and a minor tone between A and B.

The above scale, by Claudius Ptolemy, to which he gave the name of “the tightly-strung Diatonic” (Diatonon syntonon), is the one adopted by the moderns. It is, perhaps, the best that has been devised for keyed instruments upon the inherently defective system of making a true Fourth from the key-note upwards. Even by Greek laws, the tetrachords began on the second note. A singer, or a fiddle player, may avoid the defects of a scale, but a pianoforte-player cannot alter the tuning' of a note for any change of key. We are so thoroughly Greek in our system of music that it seems hopeless now to get rid of the prime defect of having the half of every Diatonic scale in one key, and the other half in what is misnamed its subdominant, or just a Fourth above it. It is that Fourth which makes our scale to be in two keys instead of one. Such is, therefore, the scale in which we are immediately concerned; and, with all deference to the Greeks, we may, perhaps, venture to look into its defects, as well as its advantages. We have one infallible guide to test it by, though it has been but little subjected to that kind of analysis. A thorough knowledge of our scale is a first requisite for a composer to make good harmony.

The preceding figures will have shown that the two tetrachords, B, C, D, E, and E, F, G, A, are equal; that their proportions are identical, (16 to 15, 9 to 8, and 10 to 9,) and that the one follows immediately upon the other in fact, that they are equal conjunct tetrachords; The following scale of Nature will show that equal intervals, within two consecutive tetrachords, cannot arise from one root in a Diatonic scale, because Nature’s Octave scale diminishes proportions at each step, viz., a ninth, a tenth, an eleventh, twelfth, thirteenth, fourteenth, fifteenth, and sixteenth parts of a string.

That interval, from E to F, to which we give the name of major semitone, is the interval between a major Seventh and its Octave, and it therefore leads to its Octave, and makes F become the Octave and a new key-note. Then G becomes Second to F as its major tone, and A, which should be major, is lowered into a minor tone, to make it a Third to F. Thus the scale is changed from C to that of F.

Instead of all this, the minor Third from E to G being as 5 to 6, or 10 to 12, ought to have been divided by the true Harmonic F, an Eleventh, making the intervals 10 to 11, and 11 to 12. It is the change of the ratio of an Eleventh to a Sixteenth that brings F too near to E, and makes it touch so closely upon E sharp, that we actually omit E sharp in our scale. But E sharp is wanted in Nature’s scale to make a Fifth and a Fourth to the Harmonic Seventh. The two very wrong notes in this scale of Claudius Ptolemy’s that we have adopted, are F and B flat. The ear has always told that they are defective, as will be farther shown.

G, the Fifth, retains its place either way, but A ought to be a major tone above G. Then it would be a proper Second for the key of G, and a Fifth above D. It has been altered for the sake of making it a major Third above F, a Fourth above E, and a minor Third below C; but the alteration takes it out of the key of C. Nature does not provide a minor Third under her keynote, neither does she acknowledge such a relative minor as A. For Nature’s relative minor to C, (if any scale can be so called,) we must look a Third above it, to E. According to Nature, every minor scale has its real key-note a major Third below it, so the keynote of A minor is F. In other words, a minor scale is merely one that is made to begin on the Third of the key. This will be seen further.

The law of Nature as to sounds is well known to practical men, and very simple. When a string is moved by a gentle breeze, its whole length is sounded, and, immediately afterwards, it divides itself into its aliquot parts, with quicker and quicker vibrations. These more rapid, but comparatively feeble vibrations overtake and mix with the slowly spreading waves of sound produced by the vibrations of the whole length of the string. When the velocity of the air is greatly increased, or, as we term it, when, the wind blows hard, the string is fluttered into many sections, and these shorter lengths move with multiplied rapidity of vibration to the whole length. This sensation of fluttering in parts will be sufficiently familiar to anyone who has carried an umbrella in a high wind. The sections into which the string is then divided are caused by self-made nodes, or divisions, and these nodes are nearly quiescent points, and all equidistant. The number of sections increases as each division becomes shorter, while the pitch rises proportionably to their diminution in length. This diminution is caused by the increasing intensity of the wind. It is like the overblowing of a pipe, by which it is made to produce very high notes. As the sections become less, the united sounds become louder as well as more acute, because the higher the pitch the greater the number of sections emitting it. Supposing a string to be thus divided by nodes into sixteen parts, their pitch will be four Octaves above the fundamental note produced by its whole length. An extraordinary part of this arrangement of Nature is, that in every progression the whole of the nodes are changed. Thus, from sixteen, it divides into seventeen equal parts, from seventeen to eighteen, and so on. 

So, too, when we blow into a horn, of pipe of any kind, with gradually increasing intensity and rapidity, we subdivide the column of air within the pipe, and raise higher and higher notes, just as the wind acts upon the string. In a flute, which is blown almost at a right angle to the column of air, and so the action of the breath becomes less direct than if it were blown at the end, the player may still draw eight different sounds from one fundamental note, or generator, without removing a finger to shorten the column of air. The lower the note upon which he may commence, the larger will be the number of Harmonics he can produce before reaching the limit to possible increase of rapidity in breathing. The sounds so produced have three names. They are called “Natural Notes” upon a horn, and “Harmonics” upon a string; also, according to Helmholtz’s nomenclature, “Overtones”, because they are above the tone produced, by the whole length of a string.

These Natural Notes, Harmonics, or Overtones, rise in the same order of succession as to musical intervals, from whatever fundamental note they may be derived. They do not vary in their order because the pitch of the fundamental note has been chosen high or low; and this may be proved, even when some of the low sounds may be too low to reach the ear. The one proviso for Nature’s scale is that the string shall be uniform in size and quality, and the pipe be an open one.

For exemplification of these rising sounds the following table is subjoined. The fundamental note selected is C, two Octaves below C in the base staff, and the lowest C on a pianoforte. It is the C C C pipe of the open diapason of an organ. It is still popularly reputed to be “16 feet C”; but neither 4, 8, 16, nor 32 feet C are now so long as their names represent them to be. Owing to difference of scale and to elevation of pitch; also, perhaps, to insufficient pressure of wind for pipes of enlarged diameter, a nominal “32 feet C” is now about 28 feet 6 inches in length, with 15 inches in diameter, and “4 foot C” about 3 feet 7 inches long.

I have taken the pitch at 512 vibrations for treble clef C, as the only proper standard for musical pitch; because Octaves are the only continuously perfect intervals. Nature’s Octaves are always multiplied by 2; as 2, 4, 8, 16, 32, 64, 128, 256, 512. It is to be hoped that at some future time 512 vibrations will be made the standard pitch of Europe, by whatever name, the note may be called. If the question of pitch in. England had been left to the decision of the Royal Society, instead of the Society of Arts, 512 would undoubtedly have been the standard English pitch. In the Society of Arts, 512 was admitted to be the right pitch; but, for the accommodation of manufacturers, who feared that their stock of instruments might have been rendered unsaleable, the pitch of 528, exactly a quarter of a tone too high, was carried by a majority, and thus a temporary divorce between the science and the art of music was pronounced.

The French standard of 870 for A, and so of 522 for C, is a curious specimen of legislation. Neither of the two notes can be carried down two Octaves without fractions. Truly, we read of vibrations divided into fractions, but the art of accomplishing it has not yet been divulged. Where fractions are resorted to, the root is changed. The law excited strong remonstrance among scientific musicians against le diapason normal malheureusement fixé arbitrairement. Handel’s tuning fork gives from 499 to 500 vibrations. That of Mozart, and that of Berlin in 1772, (according to the report of the French Commission,) was 843 half-vibrations for A, instead of 853, which is the calculated pitch for A under the present system of tuning the Sixth; or 864'1/3, if the true A, (a Fifth above D,) allowing 512 for C. The later works of Haydn, and those of Beethoven, were composed for a pitch approximate to 512.

Considerations for private interests need not prevent the Society of Arts from giving notice of future change. The members know what is right, but, influenced by good nature, have not yet acted up to their knowledge. Such a reunion of art and science as might thus be made, would be of at least equal benefit to art. If pianofortes can now bear a tension of 528, (and more) they can also bear thicker strings, and so can produce a better quality of tone at 512. The same rule applies to all instruments with strings, whether of wire or catgut. The plea of extra brilliancy by high pitch is a mistake; for brilliancy is not constituted by mere acuteness, but requires the addition of richness of quality in the tone. The practical effect now is, that the instruments in an orchestra are too thinly strung, and thus richness of quality is sacrificed to acuteness. The violoncello has no longer the full tone that Lindley produced. Old violins were not made strong enough to bear the new tension, so, thinner strings must be resorted to. Thus, the works of the great masters are now inadequately represented. It is a case in which Germany and England should unite. In France, change must await the repeal of an eccentric law.

 

The scale might be carried further, into quarter-tones, but it is unnecessary to print it, because there is a simple rule by which any one may tell what the interval will be, and it applies to the division of all super-particular ratios, or such as differ only by one degree. Nature makes no fractions, but doubles the numbers, and interposes the one and only intermediate number. Thus, in the above division of the Fifth, No. 3, which is in the ratio of 3 to 2 of CC, No. 2, she doubles the ratio, viz., 6 to 4, and interposes the intermediate 5. Then, in the next Octave, this Fifth is divided into 6 to 5 and 5 to 4, minor Third and major Third All odd numbers are new sounds; all even ones have before appeared in the Octave below. The numbers of the Harmonics are of importance in many ways. First, each indicates its proportion to the whole string, so No. 5 is a fifth part of the length, and No. 27 a twenty-seventh part, vibrating twenty-seven times as fast as No. 1; then, by multiplying the 32 vibrations of No. 1 by 27, we ascertain the vibrations of the latter to be 864 per second of time, or just as they stand in the table.

Again, multiply any number by 2, and we find its Octave; multiply by 3, for its Fifths though an Octave too high; multiply by 5, for its major Third Take the ratio of one number 'to another, as 21 to 14. These are as 3 to 2 ; therefore, the notes they represent are at the interval of 3 to 2, or a true Fifth. Take 9 to 12, or 12 to 16, the ratios are as 3 to 4; therefore, either pair is at the interval of a Fourth. If 15 to 18, a true minor Third; and so on. Every number thus carries its musical ratio to all the rest.

These are mere hints of the value of the scale of Nature, all evident upon the surface. It is for the musician to point out its deeper meanings.

And now to try our adopted scale by this most ancient of all scales, and the one test of right and wrong. We find neither F, nor A as we tune it, in the Harmonic scale, when C is the root, because they belong only to a fundamental F. But we have the scale of G intimately connected with the scale of C. If our A were tuned a comma of Didymus higher than it is, viz., as a major tone instead of a minor tone above G, it would agree with Nature’s No. 27, thus proving the scale of Didymus to be correct at that point. Ptolemy has mathematically calculated a Fourth above, and a Fifth below C, where no such intervals come from the root; and he has made the imported scale of the subdominant F more perfect by one degree than that of the true key-note. For instance, F has its Sixth (D) a major tone above its Fifth, although C, the nominal key-note of the scale, has it not. Transfer the name of key-note to F, and we may derive every interval of this so-called scale of C from F, except the B natural. As to B flat that is from a third root it belongs neither to C nor to F.

Nature’s Octave scale agrees to this extent with the Greek, and therefore with our own, that each may justly be said to consist of a disjunctive or major tone immediately above the key-note, and then of two conjoined tetrachords or Fourths. From C to D is the major tone, and from D to G and from G to G are the two Fourths. The difference is in the filling up of those two Fourths. It has been said already that F and B flat are two essentially Wrong notes for the key of C. Also, that A should be a major tone above G, instead of minor, as it now is; and that E sharp has been omitted in our scale only because we have a wrong F brought too close to it. Our F is only a 64th part of a string above E sharp, and is a 33rd part below the F of Nature. Again, if we had the true instead of the artificial B flat in our scale, the semitone that we omit above E would harmonize as a true Fifth above it. Our B flat is just as much above the real note as our F is above the true semitone to E. We omit three Diatonic notes out of eight, viz., Nature’s Fourth, Sixth, and Seventh; for A is but one of the semitones between the Sixth and Seventh in Nature’s scale, and it ought to be a true Fifth above D. Our B natural would then be the eighth tone in the scale, if the key-note were still counted as No. 1, and we admitted eight, as in Nature.

The special disadvantage of our adopted F and B flat is the impossibility of having more than four consecutive notes in one key while we include them. Even to have four, we must begin with the major Seventh, as B, C, D, E. Our B flat belongs neither to the key of C nor that of F; for, just as there is no such Fourth as F from the root of C, so neither is there any such Fourth as B flat from the root of F. The Harmonic B flat that we omit has the major-toned A (No. 27) as its semitone, on the one side, and the B natural of our scale (No. 15 or 30) as a tone, on the other side, to divide it from the Octave. Its ratio of 7 to 6 of the Fifth makes it the interval next in the order of consonance to a minor Third.

And now as to the constitution of CONSONANCE and DISSONANCE, two words which, although they carry their own interpretations as “sounding with”, and “sounding apart”, have, nevertheless, been misapprehended; and one of the two causes of consonance has been but little taken into the general account.

Degrees of consonance depend upon the proportion that coincident vibrations bear to those which “sound apart”. The unison alone is perfect consonance, because therein only do all vibrations coincide. Their simultaneousness is rigidly exact, whether sounded upon the unison-strings of a pianoforte, or upon the many instruments of an orchestra, with their varied qualities of tone. Only in intervals is there any intermingling of coincident and non-coincident vibrations. The unison is not an interval.

In order to abbreviate explanations, I refer to the Harmonic scale at p. 217. Nos. 1 and 2 are an Octave apart. The first has 32 and the second has 64 vibrations per second of time. Therefore, No. 2 vibrates as 2 to 1 of No. 1, and the first of every two vibrations of No. 2 coincides with one of those of No. 1, while the remaining 32 of No. 2 “sound apart”

Again, Nos. 2 and 3, or double C and double G, are at tbe interval of a Fifth, and No. 3 vibrates 9,6 to the 64 of No. 2 or in the proportion of 3 to 2. The first of every two vibrations of No. 2 coincides, with the first of every three vibrations of No. 3. So, there Are still but 32 coincident vibrations. Divide 64 by 2, and 96 by 3 to prove it.

One more example from Nos. 3 and 4. Here the total number of vibrations is 96 to 128, but it is the first only of every three of the one that coincides with the first of every four of the other. Therefore, the number of non-coincident vibrations has progressed, while the original number of 32 coincident vibrations has remained stationary; For that reason the interval of the Fourth, or 4 to 3, is less consonant than that of the Fifth, or 3 to 2; just as the interval of the Fifth, 3 to 2, is less consonant than that of the Octave, 2 to 1.

This natural law may be carried throughout the scale, wherein dissonant vibrations increase, between consecutive numbers, at every ascending step, while the consonant remain stationary. So the lower the two numbers, the more consonant the interval. Still, it is a necessary proviso for consonance that the sounds be derived from one root, as in this scale.

To take a last example from 15 and 16. They represent the interval of a major semitone in every Harmonic scale. Here it is from b to c. The numbers of vibrations are 15 times 32 of the one, to 16 times 32 of the other. But as only the first of every 15 coincides with the first of every 16, there are still but 32 coincident vibrations to leaven the mass of dissonance. So the ear pronounces the interval from b to c, when simultaneously sounded, to be exceedingly harsh and disagreeable. Nevertheless, the two sounds are absolutely required for melody.

Hence follows a rule, that, whatever may be the aggregate number of vibrations in a second of time from the fundamental note, or entire length of a string, whether it be of such a length as to give 32, 33, 132, 133, or any other quantity, the same will be the number of consonant vibrations between every two succeeding sounds of the scale. The intervals follow invariably in the same succession, and are, therefore, represented by the same numbers in every Harmonic scale. Hence any two numbers indicate the proportions of an interval, just as every one number indicates its proportion to a whole string.

Again, a second rule. Consonant vibrations are equal to the difference in the total number of vibrations between every two succeeding sounds, for just as 32 is the number of consonant vibrations in the fundamental sound of this scale, so 32 is the difference between the vibrations of every two succeeding numbers throughout the scale. If the same interval be taken an Octave higher, the same proportion is observed, but the vibrations are completed in half the time. Thus in the Octaves, 1 and 2, with 32 and 64 vibrations, and 2 to 4, with 64 and 128; the vibrations of the later are doubled in rapidity as they are in number. So they only perform in half a second of time what the others do in a second.

Coincident vibrations are strengthened beyond others by their perfect agreement, just as in the case of two hammers striking at the same instant. The united sound is then louder than if the blow of the one were to follow immediately after that of the other.

Coincident vibrations, having thus a superior power, mark a musical rhythm combining sounds of different pitch. It is this rhythmical coincidence which constitutes the charm of harmony in its different shades, for harmony has always a certain amount of dissonance embodied in it. The unison alone is free from all dissonance. Rhythm is the first in order of the pleasures derived from music. It suffices wholly for the savage, with his monotonous tom-tom beats; and, except as to the Harmonic sounds evolved, it is the only gratification that the ear can receive from such instruments of percussion as yield but a single note, such as a drum, cymbals, or castanets. In harmony, we enjoy the effects of rhythm enhanced by a combination of various sounds that differ in pitch, and we derive further pleasure from the varied qualities of tone that are produced by the many instruments of an orchestra. The due appreciation of so many simultaneous sounds is a reward reserved for those who have cultivated their powers of hearing. A peasant will better understand the single sound of a fiddle or of a flute. Some ears remain enclosed by the perpetual sugar of successive unisons, while others have a greater appreciation for varied harmony. Of the latter, some have also a taste which indulges largely in an admixture of spice, in the form of discords.

The rhythm of coincident vibrations between two sounds is often audible in the separate form of a third sound. The conditions are, that the vibrations of the two originating sounds shall be sufficiently rapid, and they must, therefore, necessarily be high in the scale. If otherwise, they will not admit of consonant vibrations in sufficient number within a second of time to form themselves into an audible musical note. If too few, the resultant tones are indistinguishable from the general sound. Another condition is, that the two primaries shall be sufficiently loud to bring out the feeble sound of the resultant tone. A few examples of these will be cited from practical experience in the sequel.

The second source of consonance to which I have adverted is in the Harmonic sounds which follow immediately after the notes of pipes, of strings, and of voices, and which thus serve to enrich their tones. If two sounds be combined, the lower will produce greater effect, and this is particularly manifest in the case of the wider consonant intervals. Thus, between Nos. 1 and 4 of the Harmonic scale the interval is a double Octave. When No. 1 is sounded, it throws out its Harmonics, 2, 3, 4, 5, 6, and they enrich the consonance with No. 4. Upon keyed instruments, Octaves are usually the only intervals thus enriched, because, in all cases, Octaves are tuned perfectly, but, in too many cases, other intervals are tempered, i.e., put either a little, or not a little, out of tune. Unless the tuning be perfect, Harmonics militate against, instead of strengthen consonance.

I have been thus minute in detailing the causes of consonance and of dissonance, because a theory as to their partial dependence upon a fixed number of vibrations has been propounded by the learned Helmholtz, Professor of Physiology in the University of Heidelberg. His view has been widely disseminated through Lectures on Sound, delivered by Professor Tyndall at the Royal Institution of Great Britain. The lectures have been published, and having reached a second edition, in which this definition is repeated, the objections to Helmholts’s view require to be pointed out. It is the more necessary, because the lectures have been largely adopted as authoritative upon sound, just as might have been expected from the varied knowledge and the high reputation of its author.

Professor Tyndall says:  Beats, which succeed each other at the rate of 33 per second, are pronounced by the disciplined ear of Helmholtz to be in their condition of most intolerable dissonance.

In order to represent this theory, derived from Helmholtz, in the fairest way, I extract one of the paragraphs from his Tonempfindungen. The original words are at the foot of the page, and the following is a very literal translation: 

The interval, b3' c", gave us 33 fluctuations in a second of time, which make the united sound very grating to the ear. The interval of a whole tone, bb1 c2, yields nearly double the number, but these are much less grating than those of the first-named narrow interval. Finally, the interval of the minor Third, a' c", should, according to computation, yield 88 fluctuations in the second; but, in fact, the latter interval allows us to hear scarcely anything of the roughness which the fluctuations of the closer intervals produce. Now, it might be supposed that it is the increasing number of fluctuations which obliterates the impression, and makes them inaudible. For this supposition we should have the analogy of the eye, which is likewise no longer able to separate a series of quickly following impressions of light when the number is too great. Take, for example, a burning coal swung round in a circle. When it describes a circuit from 10 to 15 times in a second, the eye imagines that it sees a continuous fiery circle. So, also, with the disk of colours, the appearance of which is known to most of my readers. When such, a disk rotates more than 10 times in a second, the different colours on it are blended into one fixed impression of their mixed colour. It is only by very intense light that quicker changes of the various fields of colour must take place [to be distinguishable] 20 to 30 times in a second. Thus, in the case of the eye, a similar phenomenon takes place as with the ear. When the change between irritation and rest takes place too rapidly, the change is obliterated in the perception, and rest becomes continuous and uninterrupted.

But we may convince ourselves in the case of the ear, that the increase in the number of the fluctuations is not the only cause of their obliteration in the perception. Thus, when we passed from the interval of a semitone, b3' c", to that of a minor Third, a' c", we have not only increased the number of the fluctuations, but also the width of the interval. But we may also increase the number of the fluctuations without altering the interval, by transposing the same interval into a higher region of the scale. If, instead of b#' c", we take the same two notes an Octave higher, we obtain 66 fluctuations, and if yet an Octave higher, even 132 fluctuations, and these are actually audible in the same manner as the 33 fluctuations of b#' c", though indeed they become feebler in the very high Octaves.

I have quoted Helmholtz’s words at full length, to show how the second part of his argument militates against the first. In the second part, he gives a case in which 33, 66, and 132 fluctuations are equally dissonant; and that alone should prove that dissonance follows this interval, and does not depend upon 33, 66, or 132 fluctuations. But Helmholtz has mistaken the character of these fluctuations, and to that cause must also be attributed the indefinite name he has given them. They are nothing but the coincident and consonant vibrations. It is strange that he should have so mistaken them as attribute dissonance to consonant vibrations, instead of to the exceeding number of dissonant vibrations that are mixed in the interval from b# to c.

That I may not misrepresent Helmholtz, I again give his words. At p. 258, he says: “The number of fluctuations within a given time is equal to the difference in the total number of vibrations which the two sounds execute in the same time.” That is a precise definition of consonant vibrations, and it can be of no other. The same number runs throughout a scale in more or less rapid succession, whether the interval be Octave, Fifth, Fourth, Third, or any other.

The mistake in the character of fluctuations has led Helmholtz to propound a new doctrine as to the cause of resultant sounds, to which I shall have occasion to refer hereafter.

This eminent acoustician did not sufficiently regard the musical bearings of the Harmonic scale when he proposed to lay a basis for the theory of music. That part of the subject has been too much neglected by many writers on music. Helmholtz, through his system of numbering by overtones instead of by the lengths that produce them, has missed the advantages that the proportion-numbers of the scale would have conferred, and has himself been led into such slips as to attribute to cc and dd 18 and 20 fluctuations, instead of to dd and ee. As only the ninth and tenth vibrations coincide in the example which he has given, the numbers must be our 9 and 10, or their doubles. C cannot have 18, neither can D have 20 fluctuations, when the fundamental note throughout the book is C C C, at the German pitch of 33 vibrations.

For the reasons above given, I demur equally to the doctrine in Professor Tyndall’s Lectures, on Sound, that, while dissonance is at its maximum when the beats number 33 per second, it lessens gradually afterwards, and entirely disappears when the beats amount to 132 per second. If the full length of the string be about four feet, and give 132 vibrations, there will be 132 in every following interval, consonant or dissonant.

Again, writers upon the science of music have for a long time assumed as an admitted fact, that the numerous sounds which result from the Harmonics of a string, or pipe, are not only emitted collectively and superposed, but also simultaneously with those of the entire string. There would indeed be a jargon if it were so let any one fancy half the keys of a pianoforte down at once. Then, following out this theory, they attribute all the various qualities of tone inherent in musical instruments, whether by wind, by string, or by percussion, to differences in their Harmonics.

So very general has been the submission to these assumed laws, down to the present time, that some may be astonished that I should throw even a doubt upon them. Nevertheless, both the eye and the ear give evidence against such doctrines. The test of the ear is within everybody’s reach.

For instance, strike one of the lowest keys upon a grand pianoforte smartly, and raise the finger instantly, so that the damper may fall heavily upon the string. The harder the damper, the more patent will be the fact that the Harmonics are not simultaneous, but consecutive. Each successively rising note may be identified by a cultivated ear, upon an old grand pianoforte, and even the uncultivated can distinguish the progressively rising sounds, and that the highest note of all is the last.

This order would be reversed if the sounds were emitted simultaneously, because, the higher the note, the sooner will its rapid vibrations be completed. To prove it, touch a base and a treble string of a pianoforte at the same instant.

Again, as to the Harmonics produced by the human voice. Regnauls recent’t experiments upon propagation of sound through long water pipes may be cited to establish the same order in their succession. The results of these experiments are published in the Appendix to Professor Tyndall’s Lectures. The following is an extract:

V. Experiments made with waves produced by the human voice and by wind instruments have demonstrated these principal facts. Acute sounds propagate themselves with much less facility than grave sounds. In very long conduits, to hear well, it is necessary to employ a baritone; the fundamental sounds are heard before the Harmonics, which then succeed each other in the order of pitch. The propagation of the sound changes its timbre, which is due to the admixture of the Harmonic sounds. In very long conduits, therefore, a tune embracing a certain extent of the gamut would change its character. These long conduits are the best proof, because the sounds are concentrated by them.

So far for the ear, and next as to the eye. Not only may a quick eye see the diminishing nodes upon a pianoforte string when it changes its Harmonics, but Kundt’s experiments have proved them to demonstration. He strewed the light dust of lycopodium within a glass tube, and made the glass emit its various Harmonic notes by employing slower or quicker friction. His experiments were exemplified by Professor Tyndall in his fifth lecture, and were therefore witnessed by large audiences, composed of those who take an interest in science. With every ascending sound, the dust was seen to arrange itself into a greater number of. equal divisions. The length of every section in the tube was changed just as every sound was changed. Indeed, it might have been predicted; because Harmonics are only produced by aliquot parts of a string, or of a column of air. Every division of a string into equal parts will produce an Harmonic note, but the scale must teach where to place it.

Thus, both the ear and the eye, assisted by the pipe, the string, and the voice, bear testimony against the simultaneous projection of Harmonics.

As to the duration of sounds emitted, one important cause has not been sufficiently taken into account. It is the after-current which follows upon every displacement of air, however minute that displacement may be. The vibrations of the air thus continue, as in echoes, after the exciting cause has ceased. The longer the string, the wider is its range of vibration; and, therefore, the greater the disturbance. The effect of the displacement is felt on a grand scale in the after-current which accompanies the discharge of a cannon. Not only the concussion, but also the rush of air, are sensibly felt by all who are behind or near to it. We have again the best practical evidence of the sound-waves which pervade even the seeming stillness of the air, when we hear them concentrated and intermixed within the hard and polished windings of a shell, by raising it to the ear.

And now, as to the theory which has been supposed to account for difference of tone in numberless musical instruments.

Professor Tyndall says: It is the addition of such overtones to fundamental tones of the same pitch which enables us to distinguish the sound of the clarionet from that of a flute, and the sound of a violin from both. Could the pure fundamental tones of these instruments be detached, they would be indistinguishable from each other; but the different admixture of overtones in the different instruments renders their clang-tints diverse, and therefore distinguishable.

In the first place, a flute, a pianoforte, a violin, and a hautboy, have the same Harmonics; but very different are their tones. In the second place, pure fundamental tones are always detached in harmoniums, because they have no audible Harmonics. This is perhaps owing to their being made with tapering springs. Yet different qualities of tone are sensibly produced from the different stops of harmoniums, and every ear can distinguish between them. Again, take three wooden open pipes of an organ, of equal length, but, one of a square shape, the second with the proportions of 3 to 2 in superficies, and the third of triangular form; they have the same Harmonics, but all differ in tone. If facts of this kind cannot be gainsayed, surely the two theories must fall together.

I here touch upon acoustics only so far as they are strictly related to music, and thereby run into my path. Upon other, even allied branches, I have nothing now to say.

The practical range of the ear for adequately distinguishing musical sounds does not far extend beyond the seven Octaves of a pianoforte, or else more notes would have been commonly added by the manufacturers. An eighth Octave gives very indefinite sounds to most ears, and even the extreme notes of the seven Octaves are not easily distinguishable unless their Octaves are sounded with them, to make them definite. The advantage of an eighth Octave consists in this, that it increases the quantity of tone, and gives the richness of its Harmonics to the others.

The six-octave scale of Nature is as follows: FIRST, the note and its Octave only. SECOND OCTAVE, divided into a Fifth and a Fourth, afterwards providing an equal number of intervals for each of the two divisions. THIRD OCTAVE, divided into four Thirds, of which we employ only two, and change the character of the lesser two, by having omitted the Harmonic Seventh that divides them. FOURTH OCTAVE, eight tones of gradually diminishing interval, of which we employ only the largest two and the least, but entitle the least a Diatonic semitone. FIFTH OCTAVE, the same eight tones as before, with their eight intermediate semitones. SIXTH OCTAVE, tones, semitones, and quarter-tones.

The Harmonic scale was only developed during the last century, and was scarcely thought of in the theory of music until the present. The discovery which led to its formation was made by two graduates of Oxford, about the year 1673. It was communicated to Dr. John Wallis, the celebrated mathematician, in 1676; was first made known by him in the English edition of his Algebra, in 1685, and subsequently in the Latin edition of his Mathematical Works, in 1693.

Dr. Narcissus Marsh, founder of Marsh’s Library in Dublin, and an exemplary prelate, who was successively Archbishop of Dublin and of Armagh, was residing in Oxford in and before 1676. Dr. Marsh was a great lover of music, and especially of part-music, both vocal and instrumental. These two branches were then much cultivated by members of the University, and Marsh’s chief relaxation was in private concerts with certain of them, either at his own, or at their rooms. In 1676 he informed Dr. Wallis, the Savilian Professor of Geometry, that about three years before that date, two of his friends, William Noble of Merton College, and Thomas Pigot of Wadham College, had discovered a means of producing, at command, the Harmonics or natural notes from a vibrating string, and this to all appearance simultaneously, and without intercommunication.

Before that time, little seems to have been known beyond the facts that, if two strings are tuned in unison, and the one be struck at no great distance from the other, the second string will sound with the first; and, secondly, that the wind will produce weird sounds from the strings of a harp exposed to its effects. The same amount of information was shared by the ancient Greeks, and, among the earlier modems, by St. Dunstan.

The natural notes of a trumpet, or of a horn, could not be measured; therefore it is of some importance to have discovered that, if one of the aliquot parts of a string be touched very lightly, while the string is under the friction of a bow, it will divide itself into nodes, and give the Harmonic, instead of the fundamental, notes.

It has proved to be of more importance than Dr. Wallis seems to have anticipated; for, although he turns sensibly out of his path to record it in his Algebra lest the remembrance should perish, he states it more as a natural curiosity than as of advantage to science.

The discovery lay fallow for half a century, and was then taken up by Dr. Brook Taylor, who was the first to publish analytical researches into the vibration of strings. Thenceforward, successively, by Bernouilli, Euler, Lagrange, d’Alembert, Riccati, Dr. Matthew Young, and by the illustrious Chladni, down to the eminent mathematicians of the present century.

It will be an advantage to composers to consider the difference of the several roots in every key, when they are writing for performances in large buildings of resonant and Harmonic-giving qualities. They must often wish to avoid the conflict of discordant Harmonics, since grandeur of effect will, in a great measure, depend upon care in that respect. Every semitone, and even every quarter-tone, in the Harmonic scale, may be used in melody without preparation, and without going out of the key.

“The sense of harmony” says Sir W. Herschel, “depends upon the periodical recurrence of coincidental impulses on the ear, and affords, perhaps, the only instance of a sensation for whose pleasing impression a distinct and intelligible reason can be assigned.” This passage is quite the antithesis to the definition of Helmholtz, that coincidental impulses may be causes of dissonance.

Harmony now means, both technically and truly, a mixture erf concords with discords, both of which are included in the Greek word Harmonia. If Herschel had intended consonances only, according to the popular idea of harmony, he would have limited his definition to “coincidental impulses on the ear, derived from a common root”.

Very different are the effects of the same interval in two places. What singer has not observed how much more natural and agreeable it is to sing a Fourth either up to, or down from, the key-note, than the same interval taken from the key-note to a Fourth above it? The reason is that, in the last case, he goes from one key into another. Again, the minor Third, when in its right place, is one degree more consonant than the interval between the Fifth and the Harmonic Seventh; but, if in the key of C, we sing or play ascending C, E, G, Harmonic B flat, B natural, and C, we have an agreeable melodic passage; whereas, if we substitute for Harmonic B flat, our B flat, which is a minor Third to the G, and so play C, E, G, B flat, the ear will not allow us to ascend further we are driven back upon A by the discord of B flat.

The system of subdominants is Greek, but not Natures.’ We sacrifice too much for the sake of making one extra interval of a perfect Fourth from the key-note to its Fourth above, which Nature does not allow. Her perfect Fourths are from the Second and Fifth of the key upwards, as from D and G in the key of C. Defects of this kind were less forcibly observed by the old musicians than now, because they did not test the s6ale by that of Nature; but ears, ancient and modem, have always been protesting that these notes are wrong.

The protest against the two notes, Fourth and minor Seventh, commenced in very remote antiquity, we might say, in ancient Egypt, on the assumption that Pythagoras derived his scales from Egypt, of which there is hardly a doubt. It seems impossible to attribute the peculiarity of the Greek Chromatic scale, in its passing down from the Octave, over the Seventh; and then from the Fifth, passing over the Fourth; to any other motive than that of avoiding those intervals which their ears told them were out of the key. Again, the same two notes were picked out for omission in the Greek Enharmonic scale, which, Plutarch tells us, had its origin in the desire of Olympus to avoid the minor Seventh. It is also sure that Olympus, or whoever invented that system, equally rejected the Fourth; for no ancient Chromatic or Enharmonic scale includes either the one note or the other. Similar instances may be noticed among the moderns, as in the universal rejection of the Seventh in the ascending minor scale, and in the substitution of the major Seventh, which the ear has judged to lead so definitely to the Octave, that it received from the French the name of la note sensible or, in the words of Rousseau, parce qu'elle annonce la Tonique et fait sentir le Ton. Again, in the rejection of both Fourth and minor Seventh by the musical ears of the composers of old popular ballads, Scotch and Irish. There are many English airs of the same class, but they were not included in Popular Music of the Olden Time, because the public voice would probably have attributed them to Scotland or to Ireland; also because there was already too large a number of English airs for publication in one selection. Other countries have tunes remarkable for the same omissions. Mathematical science alone will not teach that the two intervals are wrong, but the true science of music rests upon the investigation and following out of the laws of Nature, and does not sanction any conflict with those laws.

According to one learned writer upon the mathematical branches of music, there was a true scale in use from the time of Monteverde or Caccini. It had the F lowered by a sixty-fourth part, so really changed it into E sharp, and it had the true A (27 to 16). He does not allude to the Harmonic B flat which should have followed upon this arrangement, so as to make true Fourths and Fifths with this semitone above E, because he there writes only of a Diatonic major scale. But the partial adoption of the Harmonic scale is confirmed by the ancient use of trumpets and horns without keys. They were formerly very important instruments in out-door music, and could not be played upon in any other scale than that of Nature until keys or valves were invented for them. So it appears that the moderns have really retrograded, and have gone away from Nature in the present scale. The reason for employing the semitone above E, to make an F, was evidently to keep as near as Nature would permit to the present scale. One grand objection to a tempered scale is, that it makes false Harmonics, as well as false notes. Richness of tones depends much upon Harmonics.

The mixture-stops of an organ are solely for the purpose of supplying the Harmonics which are deficient in stopped pipes, and there can be no grandeur of effect in an organ without those mixture-stops. But there are organ builders who do not seem to know that such stops are to be voiced softly, and organists who forget that they are only to be used with the full organ, so that their tones may be covered by the volume of other sounds. If made prominent, they produce a disagreeable, instead of a grand effect. 

The stopped pipe of an organ is merely a pipe with a plug at the end, or cap upon it, so that the wind has to travel to and fro to obtain an exit at the open lip, or notch. The column of air is thus doubled in length, and the note produced is therefore an Octave lower than that of an open pipe. A clarionet is of the nature of a stopped pipe, and although closed only at the end next the mouth, the effect of lowering the tone by an Octave is the same. One foot in length of the clarionet produces the same C as two feet in length on a flute. Only two Harmonics can be produced from a clarionet, viz., a Twelfth, and another Twelfth above it the latter, with difficulty, on account of its high pitch. The peculiar Harmonics of the clarionet were first brought into notice by Sir Charles Wheatstone, F.R.S. Professor Tyndall says, that the clarionet has the Harmonics 1, 3, 5, 7, by opening the holes at the sides. But to do so is to change the fundamental note.

Professor Tyndall gives a useful second rule for comparing intervals, only in terms that may not be understood by every reader without a line of explanation. He gives the notes of the scale of C thus:

Names, . . ................. c, D, E, F, G, A, B, C.

Hates of vibration, ....1, 9/8, 5/4, 3/3, 5/3, 15/8, 2.

and then says:  Multiplying these ratios by 24, to avoid fractions, we obtain the following series of whole numbers, which express the relative rates of vibration of the notes of the Diatonic scale:

24, 27, 30, 32, 36, 40, 45, 48;

To multiply the ratios, means to multiply each upper number by 24, and divide by the under, as in the case of fractions. This rule may be preferred by some to the one I have given at page 200, and, for musical purposes, the one is as efficient as the other.

But, for those who are versed in mathematics, it should be pointed out that the use of the logarithms of the intervals very much simplifies calculations, as then all the multiplication, the bringing to a common denominator, &c., is entirely dispensed with. The logarithms, in fact, exactly represent to the eye what the intervals do to the ear, and we have only to deduct or compare the logarithms on paper, just as the ear does when the corresponding intervals are heard.

For example, taking two kinds of tetrachord: their composition is at once clearly illustrated by the following simple statement, in which, it will be observed, there is nothing but addition used :

 

Pythagorean Tetrachord.	Logarithm	Ptolemy'sTetrachord.	Logarithm,
Major Tone	0.05115	Major Tone	0.05115
Major Tone	0.05115	Major Tone	0.04576
Limma	0.02264	Limma	0.02803
Fourth	0.12494	Fourth	0.12494