"The normal brain is like a harp of many strings strung to perfect harmony. The transmitting conditions being perfect, are ready, at any impulse, to induce pure sympathetic assimilation. The different strings represent the different ventricles and convolutions. The differentiations of any one from its true setting is fatal, to a certain degree, to the harmony of the whole combination. If the sympathetic condition of any physical organism carries a positive flow of 80 per cent on its whole combination, and a negative one of 20 per cent, it is the medium of perfect assimilation to one of the same ratio, if it is distributed under the same conditions to the mass of the other. If two masses of metal, of any shape whatever, are brought under perfect assimilation, to one another, their unition, when brought into contact, will be instant.'' [Keely and His Discoveries, Chapter 7]

Different writers have put forth different views of what constitute a musical vibration, but their various views do not make any difference in the ratios which the notes of this sound-host bear to each other. Whether the vibrations be counted as single or double vibrations, the ratios of their relative motions are the same. Nevertheless, a musical vibration is an interesting thing in itself, and ought to be correctly defined.
A string when vibrating musically is passing and re-passing the central line of its rest or equilibrium with a certain range of excursion. Some writers have defined a vibration to be the passage of the string from one extreme of its excursion to the other, while some have preferred to define it as the passage of the string from the one extreme of its excursion to the other and back again. D. C. Ramsay has been led in his researches to define a vibration as the movement of the string from its central line of rest to the extreme of its excursion on one side, and back to the central line of rest; and from the central line of rest to the extreme of its excursion on the other side, and back again to the "right line," as he calls it, as a second vibration. His reasoning on this will be seen in what follows. (See Fig. 3, Plate IV.) [Scientific Basis and Build of Music, page 21]

Musical sounds are usually caused in the ear by certain vibrations of the surrounding air, which originate from solid bodies in a state of vibration from some force exerted upon them. Vibrations of the air require to attain a certain rate of speed before they become audible to the human ear; and they require to have certain ratios of rate of rapidity in order to constitute that beautiful host of sounds which constitutes the music of mankind. These musical vibrations may arise in the air from a vibrating organ pipe, or a vibrating tuning fork, or a bell, or a sounding glass, or a strand of wire or gut-string, or other rhythmically vibrating body; but to explain and define the nature of a musical vibration from the action upon it of an elastic string is to explain and define it for all. But before defining what a vibration of a string is, let us hear what others have said about it. Charles Child Spencer, Treatise on Music, p. 6, says- "It is customary in calculating the ratios of vibration of musical strings, and which answer to the waves of the atmosphere, to reckon by double vibrations, so that instead of saying there are 32 single vibrations in the lowest sound, C, writers on this branch of music say there are 16 double vibrations in this sound. This method of calculation, therefore, gives 256 vibrations for the fourth Octave C." Playfair, in his Outlines of Natural Philosophy, p. 282, says- "It is usual to reckon the vibrations of a string different from those of a pendulum; the passage from the highest point on one side to the highest point on the other is reckoned a vibration of a pendulum; the passage from the farthest distance on one side to the farthest distance on the other and back again to its first position, is the accounted a vibration of a musical string. It is properly a double vibration." Holden, in his Rational System of Music, says- "Mr. Emerson reckons the complete vibration the time in which a sounding string moves from one side to [Scientific Basis and Build of Music, page 22]

"The perfect character of a musical sound is the result of the harmonious workings of the vibrations of the "perfectly elastic" air with the perfectly elastic string. The forces which act on the air and string being proportional to the distances passed through, makes the times of the vibrations equal,1 and the pitch of the sound the same throughout. These varied forces and distances, with the equal times of the vibrations, and with the simultaneous compressions and expansions of the air and string, are all according to the universal laws of Continuity and Duality. [Scientific Basis and Build of Music, page 24]

"To suppose that the vibration of a string is the same as the oscillation of a pendulum is like adding equals to unequals, and supposing the wholes to be equals." [Scientific Basis and Build of Music, page 24]

"So the vibration of an elastic string and the oscillations of a pendulum have different definitions, though they do, with regard to their ratios illustrate each other." [Scientific Basis and Build of Music, page 25]

"notes which are produced by the two primes, 3 and 5. As the quadrant contains all the angles which give the different proportions in form, so does the ratio of 1:2, or the area of an octave, contain all the different notes in music. The ratio of 1:2 corresponds to unity, and, like the square and the circle in form, admits of no varieties. Half the length of a string gives an octave when the string is homogeneous and uniform; if the one half has more gravity than the other, the center of gravity of the whole string gives the octave. The ratio of 1:2 rests on the center of gravity. [Scientific Basis and Build of Music, page 27]

Well, how are we to get the true minor scale? There is a remarkable fact, and a beautiful one, which suggests the method. Such is the economy of Nature, that from one system of proportion employed in two different ways, in the one case as periods of vibrations and in the other as quantities of strings, everything in Music's foundation is produced. It is a remarkable fact that the numbers for the lengths of the strings producing the major scale are the numbers of the vibrations producing the minor scale; and the numbers for the lengths of the strings for the minor scale are the numbers of the vibrations of the notes of the major scale. Here Nature reveals to us an inverse process for the discovery of the minor scale of notes. [Scientific Basis and Build of Music, page 31]

After vibrations the next thing is musical notes, the sounds produced by the vibrations falling into the ear. Sounds arise in association. There are no bare simple sounds in music; it is a thing full of the play of sympathy. Such a thing as a simple solitary sound would be felt as a strange thing in our ears, accustomed as we are to hear affiliated sounds only. These affiliated sounds, called "harmonics," or "partials" as they have also been called, because they are the parts of which the sound is made up, are like perspective in vision. In perspective the objects lying in the line of sight, seem smaller and smaller, and more dim and indefinite as they stretch away into the distance; while nearer objects and those in the foreground are apparently larger, and are more clearly seen. This is the way of a musical sound; one of its component elements, the fundamental partial, being, as it were, in the foreground to the ear, is large and pronounced; while the other elements are less distinctly heard, and are fainter and fainter as they recede into the musical distance in the perspective of the ear. Few have any idea of the number of these weaker partials of a musical sound. Tyndal's illustrations in his very instructive work on Sound show a string spontaneously divided into twenty segments, all vibrating separately, being divided by still nodes along its length; and a vibrating string will keep thus [Scientific Basis and Build of Music, page 58]

Speaking of acute harmonics Pole says - "The first six are the only ones usually considered to be of any practical importance, and it is rarely possible to distinguish more than 10 or 12."
Mercenne (French, 1636) says - "Every string produces 5 or more sounds at the same instant, the strongest of which is called the natural sound of the string, and alone is accustomed to be taken notice of; for the others are so feeble that they are only perceptible to delicate ears . . . not only the octave and fifteenth, but also the twelfth and major seventeenth are always heard; and over and above these I have perceived the twenty-third and ninth partial tones in the dying away of the natural sound."[Scientific Basis and Build of Music, page 59]

which seems to show that not only has one part of a vibrating string sympathy with another part of it so as to go into harmonic partials, as we have just seen, but as if the very air itself had sympathy with harmoniously vibrating strings; for Tartini observed that two harmonious sounds being produced and sustained as they can be, for example, by a strong bow on the violin, a third sound will be heard. Tartini's name for it was simply "a third sound." This is not an overtone, as Helmholtz has called the harmonic partials of one sounding string, but an undertone, because it is a "grave harmonic," away below the sounds of the two strings which awaken it. The subject of these undertones has been carefully studied since Tartini's day, and more insight has been obtained since we are now able to count and register the vibration of any musical sound. Helmholtz has called these third sounds of Tartini's "difference sounds," because when awakened by two strings, for example, the vibration-number of the third tone is the difference of the vibrations-numbers of the two tones which awaken it. The note C with vibration-number 512, and another C whose vibration-number is 256, the octave, awakened no third sound, because there is no difference between the two numbers - the one is just the doubled or halved; but if we take C256 and G381, its fifth, the difference number is 128; this being a low octave of C256, it has the effect of strengthening the upper one. Helmholtz found this to be the law of the third sound as to its producing, and the effect of it when produced. This third sound, mysteriously arising in the air through the sympathy it has with all concordant things, is another among many more suggestions that the whole Creation is measured and numbered to be in sympathy one part with another. The Creation is a universe. [Scientific Basis and Build of Music, page 60]

In getting the length of a string, in inches or otherwise, to produce the scale of music, any number may be fixed on for the unit; or for the vibrations of the root note any number may be fixed on for the unit; but in the fractions which show the proportions of the notes of the scale, there is no coming and going here; this belongs to the invariables; there is just one way of it. Whatever is not sense here is nonsense. It is here we are to look for the truth. The numbers which express the quantities and the numbers which express the motions are always related as being of the same kind. The fractions bring their characters with them, and we know by this where they come from. 1/4 of a string gives a note 2 octaves above the whole string, no matter what may be its length; 2 has exactly the same character as 1; 2/4 gives the note which is 1 octave above the whole string; but in the case of 3/4 here is a new ingredient, 3; 3/4 of a string gives a note which is a fifth below the [Scientific Basis and Build of Music, page 75]

note by 2/4; and by the law of duplication, the law of the octave interval, a note which is a fifth below the note by 1/4, by 2/4, or by 1, the integer, i.e., the whole string. [Scientific Basis and Build of Music, page 76]

There is nothing extraordinary in this. It is another fact which gives this one its importance, and that is that the musical system is composed of three fifths rising one out of another; so this note by 3/4 becomes the root not only of a chord, but the root of all the three chords, of which the middle one is the tonic; the chord of the balance of the system, the chord of the key; the one out of which it grows, and the one which grows out of it, being like the scales which sway on this central balance-beam. Thus F takes its place, C in the center, and G above. These are the 3 fifths of the system on its masculine or major side. The fractions for A, E, and B, the middle notes of the three chords, are 4/5, 3/5, and 8/15; this too tells a tale; 5 is a new ingredient; and as 3 gives fifths, 5 gives thirds. From these two primes, 3 and 5, along with the integer or unit, all the notes of the system are evolved, the octaves of all being always found by 2. When the whole system has been evolved, the numbers which are the lengths of the strings in the masculine or major mode are the numbers of the vibrations of the notes of the feminine or minor mode; and the string-length-numbers of the minor or feminine are the vibration-numbers of the notes of the major or masculine mode. These two numbers, the one for lengths and one for vibrations, when multiplied into each other, make in every case 720; the octave of 360, the number of the degrees of the circle. [Scientific Basis and Build of Music, page 76]

The simplest condition of quantities and motions is in a string where half the length is double the vibrations. Next in the order of simplicity is a [Scientific Basis and Build of Music, page 79]

pendulum where fourth the length is double the oscillations. A third condition in this order is in springs or reeds where half the length is four times the vibrations. If we take a piece of straight wire and make it oscillate as a pendulum, one-fourth will give double the oscillations; if we fix it at one end, and make it vibrate as a spring, half the length will give four times the vibrations; if we fix it at both ends, and make it vibrate as a musical string, half the length will produce double the number of vibrations per second. [Scientific Basis and Build of Music, page 80]

2 - In the second sphere the tension of strings and other elastic bodies imbues them with forces operating upon the elastic air, producing vibrations quick enough to awaken sounds for the human ear. Here Nature plays on her tuneful harp the same grand fugue; from which everything in music is derived. [Scientific Basis and Build of Music, page 86]

The sympathy of one thing with another, and of one part of a thing with another part of it, arises from the principle of unity. For example, a string requires to be uniform and homogenous to have harmonics producing a fine quality of tone by the sweet blendings of sympathy; if it be not so, the tone may be miserable ... You say you wish I were in touch with Mr. Keely; so do I myself ... I look upon numbers very much as being the language which tells out the doings of Nature. Mr. Keely begins with sounds, whose vibrations can be known and registered. I presume that the laws of ratio, position, duality, and continuity, all the laws which go to mould the plastic air by elastic bodies into the sweetness of music, as we find them operative in the low silence of oscillating pendulums, will also be found ruling and determining all in the high silence of interior vibrations which hold together or shake asunder the combinations which we call atoms and ultimate elements, but which may really be buildings of wondrous complexity occupying different ranges of place and purpose between the visible cosmos and Him who built and evermore buildeth all things. The same laws, though operating in different spheres, make the likenesses of things in motion greater than the differences. [Scientific Basis and Build of Music, page 87]


At the middle of the string the stopped note and the harmonic notes are the same; but corresponding places above and below the middle give the same harmonic, although these places when stopped give different notes. [Scientific Basis and Build of Music, page 92]

Nine-tenths of a string, if stopped and acted on, gives a tone the ratio of 9:10, but if touched and acted on as a harmonic it gives a note which is three octaves and a major third above the whole string. If the remaining tenth of the string be acted on either as a stopped note or a harmonic it will give the same note which is three octaves and a major third above the whole string the ratio of 1:10, so that the stopped note of one-tenth and the harmonic of nine-tenths are the same. Indeed the bow acting on stopped note of one-tenth, on harmonic of nine-tenths, or on harmonic of one-tenth, produces the same note, as the note is the production of one-tenth in each case; for in the harmonic, whether you bow on the nine-tenths or the one-tenth, while it is true that the whole string is brought into play, yet by the law of sympathy which permeates the entire string, it vibrates in ten sections of one-tenth each, all vibrating in unison. This is what gives the harmonic note its peculiar brilliancy. [Scientific Basis and Build of Music, page 92]

"The three notes of the dominant chord resolve by each note going to the next note upward - G soars to A, B to C, D to E. The three notes of the subdominant resolve by each note going to the next note downward - C sinks to B, A to G, F to E. The two upper notes of the dominant resolve into the tonic chord according to the Laws of Proximity and Specific Levity; and the two lower notes of the subdominant resolve into the tonic chord according to the Laws of Proximity and Specific Gravity. And in this way Nature, in chord-resolution, has two strings to her bow." [Scientific Basis and Build of Music, page 96]

VIOLIN-FINGERING - Whenever the third finger is normally fourth for its own open string, then the passage from the third finger to the next higher open string is always in the ratio of 8:9; and if the key requires that such passage should be a 9:10 interval, it requires to be done by the little finger on the same string, because the next higher open string is a comma too high, as would be the case with the E string in the key of G.
In the key of C on the violin you cannot play on the open A and E strings; you must pitch all the notes in the scale higher if you want to get [Scientific Basis and Build of Music, page 99]

the use of these two open strings in the key of C, on account of the intervals from G to A and from D to E being the ratio of 9:10, the medium second in the scale. G, the third finger on the third string, to A, the open second string, and D, the third finger on the second string, to E, the open first string, being in ratio of 8:9, the large second, you must either use the fourth finger for A and E, or use all the other notes a comma higher. But if thus you use all the notes a little higher, so as to get the use of the A and E strings open, then you cannot get the use of the G and D strings open. On the other hand, in this key of C, if you use the G and D strings open, you cannot use the A and E strings open. One might think the cases parallel, but they are not; because you have a remedy for the first and second open strings, but no remedy for the other two. The remedy for the first and second open strings is to put the fourth finger on the second and third strings for the E and A; but it would be inconvenient, if not impossible, to use the other two strings, G and A, by putting the first finger a comma higher than the open string. [Scientific Basis and Build of Music, page 100]

Whenever a sharp comes in in making a new key - that is, the last sharp necessary to make the new key - the middle of the chord in major keys with sharps is raised by the sharp, and the top of the same chord by a comma. Thus when pausing from the key of C to the key of G, when F is made sharp A is raised a comma. When C is made sharp in the key of D, then E is raised a comma, and you can use the first open string. When G is made sharp for the key of A, then B is raised a comma. When D is made sharp for the key of E, then F# is raised a comma; so that in the key of G you can use all the open strings except the first - that is, E. In the key of D you can use all the open strings. In the key of A you can use the first, second, and third strings open, but not the fourth, as G is sharp. In the key of E you can use the first and second open. [Scientific Basis and Build of Music, page 100]

Whatever interval is sharpened above the tone of the open string, divide the string into the number of parts expressed by the larger number of the ratio of the interval, and operate in that part of the string expressed by the smaller number of it. For example, if we want to get the major third, which is in the ratio of 4:5, divide the string into five parts and operate on four. The lengths are inversely proportional to the vibrations. [Scientific Basis and Build of Music, page 100]


This plate sets forth the essential duality of the musical system of vibrations. It is a remarkable fact that the numbers of the vibrations of the major mode are the numbers for the string proportions of the minor mode; and vice versa, the string proportions in the major are the numbers of the vibrations in the minor. We have, however, to see that we use the proper notes and numbers; we must know the secret of Nature. This secret rests in the duality of the notes, and begins from the two D's. The center of gravity of the musical system of vibrations is found in the comma space between the |two D's as they are found in the genesis of the two modes. In these |two D's the vibration number and string proportions are nearly identical. Starting from this point as the center of gravity in the [Scientific Basis and Build of Music, page 118]

dual system, as the strings are shortened the vibrations of course are more, and as the strings are lengthened the vibrations are fewer. This is household lore now; but the new insight and the deeply interesting order of Nature is that the major and the minor contain each other and respond to each other in this striking way; and while manifesting such diversity of character are so essentially one. [Scientific Basis and Build of Music,page 119]



On harmonical parallel between tone and colour
—On the term of "rest," fifths, and the sympathy of music with life
Relativities of sounds and vibrations of strings
—The doctrines of three pairs, six tones, and the law of "two and fro"
—The germ of the system probably to be found in the adaptability of numbers
—Sudden death of Dr. Gauntlett, . . . . . 48 [Harmonies of Tones and Colours, Table of Contents4 - Harmonies]

"All things are touched with melancholy,
Born of the secret soul's mistrust
To feel her fair ethereal wings
Weighed down with vile degraded dust.
There is no music in the life
That sounds with idiot laughter solely;
There's not a string attuned to mirth,
But has its chord in melancholy."
Thomas Hood. [Harmonies of Tones and Colours, Reflections on the Scheme3, page 45]

1872.—"It gives me great pleasure to write to you on this subject. Music deals more with the imaginative faculty than any other art or science, and possessing, as it does, the power of affecting life, and making great multitudes feel as one, may have more than ordinary sympathy with the laws you work upon. You say 'from E, root of B, the fountain key-note F, root of C, rises.' There is a singular analogy here in the relativities of sounds, as traced by comparing the numbers made together by vibrations of strings with the length of strings themselves, the one is the inverse or the counterchange of the other. The length of B and E are the counterchange of F and C, hence they are twin sounds in harmony." [Harmonies of Tones and Colours, Extracts from Dr. Gauntlett's Letters1, page 48]

See Also

Ramsay - PLATE XXIV - Vibrations of the Major are the String-lengths for the Minor
String Theory
Laws of String Vibration
nine string chord
sympathetic string
harp string
Vibrating Rod Harmonics

Created by Dale Pond. Last Modification: Monday April 5, 2021 02:25:41 MDT by Dale Pond.