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Note

A single-frequency sound (pitch) usually within context of a musical scale. A music note is usually designated by the letters A through G with appropriate accents, sharps or flats.

Keely
"The experiment illustrating "chord of mass" sympathy was repeated, using a glass chamber, 40 inches in height, filled with water, standing on a slab of glass. Three metal spheres, weighing about 6 ounces each, rested on the glass floor. The chord of mass of these spheres was B flat first octave, E flat second octave and B flat third octave. Upon sounding the note B flat on the sympathetic transmitter, the sphere having that chord of mass rose slowly to the top of the chamber, the positive end of the wire having been attached, which connected the covered jar with the transmitter. The same result followed the sound of the other spheres, all of which descended as gently as they rose, upon changing the positive to the negative. J.M. Wilcox, who was present remarked: "This experiment proves the truth of a fundamental law in scholastic philosophy, that when one body attracts or seeks another body, it is not that the effect is the sum of the effects produced by parts of one body upon parts of another, one aggregate of effects, but the result of the operation of one whole upon another whole." [Snell Manuscript - The Book, page 3]

In all molecular dissociation or disintegration of both simple and compound elements, whether gaseous or solid, a stream of vibratory antagonistic thirds, sixths, or ninths, on their chord mass will compel progressive subdivisions. In the disintegration of water the instrument is set on thirds, sixths, and ninths, to get the best effects. These triple conditions are focalized on the neutral center of said instrument so as to induce perfect harmony or concordance to the chord note of the mass chord of the instruments full combination, after which the diatonic and the enharmonic scale located at the top of the instrument, or ring, is thoroughly harmonized with the scale of ninths which is placed at the base of the vibratory transmitter with the telephone head. The next step is to disturb the harmony on the concentrative thirds, between the transmitter and the disintegrator. This is done by rotating the siren so as to induce a sympathetic communication along the nodal transmitter, or wire, that associates the two instruments. When the note of the siren becomes concordant to the neutral center of the disintegrator, the highest order of sympathetic communication is established. It is now necessary to operate the transferable vibratory negatizer or negative accelerator, which is seated in the center of the diatonic and enharmonic ring, at the top of the disintegrator, and complete disintegration will follow (from the antagonisms induced on the concordants by said adjunct) in triple progression, thus: First thirds: Molecular dissociation resolving the water into a gaseous compound of hydrogen and oxygen. Second: sixths, resolving the hydrogen and oxygen into a new element by second order of dissociation, producing what I call low atomic ether. Third: ninths, the low atomic ether resolved into a new element, which I denominate high or second atomic harmonic. All these transmissions being simultaneous on the disturbance of sympathetic equilibrium by said negative accelerator. [Snell Manuscript - The Book, page 4]

"The human ear cannot detect the triple chord of any vibration, or sounding note but every sound that is induced of any range, high or low, is governed by the same laws, as regards triple action of such that govern every sympathetic flow in Nature. Were it not for these triple vibratory conditions, change of polarity could never be effected, and consequently there could be no rotation. Thus the compounding of the triple triple, to produce the effect would give a vibration in multiplication reaching the ninth, in order to induce subservience, the enumeration which it would be folly to undertake, as the result would be a string of figures a mile in length to denote it. [Snell Manuscript - The Book, DISTURBANCE OF MAGNETIC NEEDLE, page 8]


Russell
"The reason that each succeeding tonal note, which constitutes the elements of matter, are one octave higher is one entirely of increased pressure. You can better understand that by compressing some air in an enclosed box or tube in which you have inserted a whistle. The more air you pump in, the higher the whistle will sound when you open its valve. If you tighten a wire the same thing will happen. The same harp string can give you many tones if you turn the tuning pins higher, or lower. By the time the first octave string has multiplied in cube ratio nine times, the speed of vibration frequencies and intensity of pressure have reached the incredible proportions of 1,073,741,824 times greater in the 9th string than the first string." [Atomic Suicide, page 30-31]

Ramsay
"Of these three chords, which constitute a scale or key, Nature next proceeds to generate, in a similar way, a family of scales or keys, and these in two lines, the Major and the Minor. The twice twelve-fold family of keys is brought forth in much the same way as were the chords which constitute them, and as were the notes which constitute the chords. There is a beautiful growth-like continuity in the production of all." [Scientific Basis and Build of Music, page 20]

"centrifugal force. A third note produced by the prime 5 is derived from the note produced by the first power of 3, and this note by the first power of 5 having being slightly acted on by the force of gravity, and the first power of 5 having only a little centrifugal force, the result is that this note E in the scale of C, derived from the first power of 3 by the prime 5, is balanced between the two forces. It is the only note in the system which in the octave scale has not a large interval on the one side of it nor on the other, and consequently it is the only note which attracts and is attracted by two notes from proximity. Thus it is that the musical system is composed of three notes having specific gravity and three having specific levity or bouyancy, and one note, E, the center of the tonic chord, balanced between these two forces. As the attractions of notes from proximity take place when the notes with downward tendency meet the note with upward tendency, had the notes been animated by only one of these forces there could have been no system of resolutions of the notes either in melody or harmony; they would all have been by gravity weighing it downwards, or by levity soaring upwards." [Scientific Basis and Build of Music, page 28]

this is the middle of our chord, E, G, B; and remember that this also is G as we found it coming upward, C3 multiplied by 3 being G9. This is another note of the minor, the same in its quantity as that of the major. Now for another chord downward we must divide the root of the one we have found, namely E15, by 3, which will give us A5, the root of a center chord for the minor, and the very key-note of the relative minor to C. And remember that this A5 is just as we found it in coming upward, for F multiplied by 5 gave us A5. Now divide E15 by 5 and we have C3, the middle to our minor chord, A, C, E. Still we must remember that this C3 is just as we found it coming upward, for F multiplied by 3 is C3. Behold how thus far major and minor, though inversely developed, are identically the same in their notes, though not in the order in which they stand in the fifths thus generated. [Scientific Basis and Build of Music, page 32]

Either the one or the other must be at fault. Had the dictates of the mathematicians and the scale of mathematical intonation wholly ruled, the advent of the great masters would have been impossible. It was well said by one writing in The Choir - "Theory should be made from music, and not music from theory . . . the final judge of music is the Ear." The Great Masters are the exponent artists of what is true in the Science of Music, though it may differ from what has been taught by the merely mathematical-intonation advocates of music science. It should not be forgotten that the science of the mathematical theorists is one thing, and that of the composers is another. Schubert, Beethoven, Mozart, Haydin, Mendelssohn, and such inspired musicians, who walked in the liberty wherewith Nature made them free, are sufficient authority against the bondage of the one-law theorists who would tie us down to the mathematical command which comes from without, but who know nothing of the life within music which is the law unto itself.1
With twelve divisions in the Octave, each note is adapted to serve in any capacity, and does serve in every capacity by turns. It is quite clear that this cannot be said of the mathematically perfect notes. And this is where it is seen that what is perfect in mathematical ratios becomes imperfect in the Musical System. Indeed, the mathematical intonation does not give a boundary within which to constitute a System at all, but goes off into never-ending cycles.
In music, Nature begins by producing the Diatonic Octave of seven notes, derived by the mathematical ratios2; [Scientific Basis and Build of Music, page 34]

It is a degradation of the mathematical primes to apply them to the getting of the semitones of the chromatic scale, as even Euler himself mistakenly does. The mathematical ratios lead the way in getting the notes of the diatonic scale, and that is all that is required of them. The true praise to the ratios is that they have constituted an organic structure with form and life-powers adapted for self-development. It would be little credit to the mother if the child required to be all its life-long pinned to her apron-strings. As the bird when developed so far leaves the shell, and is afterwards fully developed in new conditions; so the System of music when developed so far leaves the law of ratios, its mathematical shell, and is afterwards fully developed by other laws. Music has an inspirational as well as a mathematical basis, and when mathematicians do not recognize this they reckon without their host. [Scientific Basis and Build of Music, page 35]

Mathematicians have not recognized the life-power of the notes, and so they have misapplied their calculations, though these were perfect in themselves. Assuming the place of dictators, they say with an air of authority that, "strictly speaking, nothing could be more scientifically and musically untrue than the chromatic scale of twelve equal semitones as played on a tempered instrument; for in it, as in the diatonic scale, the same natural law prevails that no two tones of equal mathematical relations can melodically succeed each other." Saying that the same natural law prevails implies that they are reasoning from analogy; but in this assumption they are dictating to Nature. In a similar way they might assume that the interval of the octave, like the other intervals, should have a grave harmonic.2 But the fact that the octave interval has not a [Scientific Basis and Build of Music, page 36]

When the major scale has been generated, with its three chords, the subdominant, tonic, and dominant, by the primary mathematical ratios, it consists of forms and orders which in themselves are adapted to give outgrowth to other forms and orders by the law of duality and other laws. All the elements, orders, combinations, and progressions in music are the products of natural laws. The law of Ratio gives quantities, form, and organic structure. The law of Duality gives symmetry, producing the minor mode in response to the major in all that belongs to it. The laws of Permutations and Combinations give orders and rhythms to the elements. The law of Affinity gives continuity; continuity gives unity; and unity gives the sweetness of harmony. The law of Position gives the notes and chords their specific levities and gravities; and these two tendencies, the one upward and the other downward, constitute the vital principle of music. This is the spiritual constitution of music which the Peter Bell mathematicians have failed to discern: [Scientific Basis and Build of Music, page 37]

If the effects of notes and chords had depended entirely on their mathematical ratios, then the effect of the subdominant, tonic, and dominant would have been alike; for these three chords have exactly the same ratios. It is the law of position which gives the tonic chord its importance, and not any special ratios embodied in its structure. The ratio of 2 to 1 has a pure, unmixed, invariable character, always realized in the interval of the octave. The notes produced from 1 by the first, second, and third powers of 3 have different degrees of centrifugal force. The character of the notes produced by the first power of 5 depends on the character of the notes from which they are derived, namely, 1, 3, and 9. The final character of the notes and chords derived by the same ratios is determined by the amount of force which they have acquired from the way in which they have been derived, and from their position in the system; and no matter where these notes may afterwards be [Scientific Basis and Build of Music, page 37]

The extremes of the levities and gravities of a key-system are always at the extent of three fifths; and whatever notes are adopted for these three fifths, the center fifth is the tonic. As there never can be more than three fifths above each other on the same terms, so there can never be more than one such scale at the same time. A fourth fifth is a comma less than the harmonic fifth1; and this is Nature's danger-signal, to show that it is not admissible here. Nature does not sew with a knotless thread in music. The elements are so place that nothing can be added nor anything taken away without producing confusion or defect. What has been created is thus at the same time protected by Nature. [Scientific Basis and Build of Music, page 38]

B, namely G#, they come in touch of each other like the two D's. When this three fifths below F major and three fifths above B minor have been developed, the extremes A♭ and G#, though standing like the two D's in duality, are so near that here again one note can be made to serve both. The major series of scales and the minor series at these limits are thus by two notes which have duality in themselves hermetically sealed; but not till Nature has measured off for any one of these scales a sphere of twelve keys in which to move in perfect freedom of kinship by softly going modulations. [Scientific Basis and Build of Music, page 39]

RESOLUTION - The tendency and going of a note of one chord to some note of the next chord. [Scientific Basis and Build of Music, page 40]

It is according to the Law of Duality that the keys on the piano have the same order above and below D, and above and below G# and A♭, which is one note. In these two places the dual notes are given by the same key; but in every other case in which the notes are dual, the order above the one and below the other is the same. The black keys conform to the scale, and the fingering conforms to the black keys. On that account in the major scale with flats, for the right hand the thumb is always on F and C; and as the duals of F and C are B and E in the minor scale with sharps, for the left hand thumb is always on B and E. [Scientific Basis and Build of Music, page 44]

It is in their inverse relations that the major and the minor are equal. Every note, chord, and progression in the one has its reciprocal or corresponding note, chord, and progression in the other. This is the Law of Duality. And this general law of Nature is so deeply rooted in music, that is the numbers which represent the vibrations in the major system be made to represent quantities of string, these quantities will produce the minor system (beginning, of course, with the proper notes and numbers); so that when the quantities are minor the tones are major, and when the quantities are major the tones are minor.1[Scientific Basis and Build of Music, page 44]

There was, then, something of truth and beauty in the Greek modes as seen in the light now thrown upon them by the Law of Duality, at last discerned, and as now set forth in the genesis and wedlock of the major and minor scales. The probably symmetrical arrangement of the modes, all unwitting to them, is an interesting exhibition of the true duality of the notes, which may be thus set in view by duality lines of indication. We now know that B is the dual of F, G the dual of A, C the dual of E, and D minor the dual of D major. Now look at the Greek modes symmetrically arranged:

D EF G A BC D
C D EF G A BC EF G A BC D E
A BC D EF G A G A BC D EF G
F G A BC D EF BC D EF G A B


Thus seen they are perfectly illustrative of the duality of music as it springs up in the genetic scales. The lines reach from note to note of the duals. [Scientific Basis and Build of Music, page 46]

It should not be supposed that this division of the notes into semitones, as we call them, is something invented by man; it is only something observed by him. The cutting of the notes into twelve semitones is Nature's own doing. She guides us to it in passing from one scale to another as she builds them up. When we pass, for example, from the key of C to the key of G, Nature divides one of the intervals into two nearly equal parts. This operation we mark by putting a # to F. We do not put the # to F to make it sharp, but to show [Scientific Basis and Build of Music, page 47]

that Nature has done so.1 And in every new key into which we modulate Nature performs the same operation, till in the course of the twelve scales she has cut every greater note into two, and made the notes of the scale into twelve instead of seven. These we, as a matter of convenience, call semitones; though they are really as much tones as are the small intervals which Nature gave us in the genesis of the first scale between B-C and E-F. She only repeats the operation for every new key which she had performed at the very first. It is a new key, indeed, but exactly like the first. The 5 and 9 commas interval between E and G becomes a 9 and 5 comma interval; and this Nature does by the rule which rests in the ear, and is uttered in the obedient voice, and not by any mathematical authority from without. She cuts the 9-comma step F to G into two, and leaving 5 commas as the last interval of the new key of G, precisely as she had made 5 commas between B and C as the last interval of the key of C, she adds the other 4 commas to the 5-comma step E to F, which makes this second-last step a 9-comma step, precisely as she had made it in the key of C.2 [Scientific Basis and Build of Music, page 48]

This subtracting and adding process of Nature by which she so freely handles the notes is the way she gives us the materials of the Chromatic Scale, in which an entirely new series of chords with strikingly different effects, and with exceedingly interesting, subtle, and at the same time easy progressions, is put in possession of the practical musician. This new series of chords forms, in fact, materials for the Chromatic System, which D. C. Ramsay has discovered, and which he has elaborated, as his custom was, exhaustively - his last labor in the interests of music science and art. [Scientific Basis and Build of Music, page 48]

The common chord is a group of notes which come together by generic affinity, much like the chemical combinations of our system of atoms. The common chord is a triplet, and in the progression from one chord to another these triplets have always something in common; by the law of continuity one of the notes of the chord first is also found in chord second; and chord second also finds one of its notes in chord third. This is the way Nature gives them to us [Scientific Basis and Build of Music, page 48]

in their birthplace - F A C, C E G, G B D. Indeed in their birth not only is it so, but still further, the top note of the first chord is the root and generator of the third. They are linked in generative continuity. [Scientific Basis and Build of Music, page 49]

In the progression - that is, the going on from one to another - of these triplets in harmonizing the octave scale ascending, Nature goes on normally till we come to the passage from the sixth to the seventh note of the scale, whose two chords have no note in common, and a new step has to be taken to link them together. And here the true way is to follow the method of Nature in the birthplace of chords.1 The root of the subdominant chord, to which the sixth of the octave scale belongs, which then becomes a 4-note chord, and is called the dominant seventh; F, the root of the subdominant F, A, C, is added to G, B, D, the notes of the dominant, which then becomes G, B, D, F; the two chords have now a note in common, and can pass on to the end of the octave scale normally. In going down the octave scale with harmony, the passage from the seventh to the sixth, where this break exists, meets us at the very second step; but following Nature's method again, the top of the dominant goes over to the root of the subdominant, and F, A, C, which has no note in common with G, B, D, becomes D, F, A, C, and is called the subdominant sixth; and continuity being thus established, the harmony then passes on normally to the bottom of the scale, every successive chord being linked to the preceding note by a note in common. [Scientific Basis and Build of Music, page 49]

There are two Diatonic systems in Music - the major and the minor. With the exception of one note, all the notes of the one system are identical with those of the other. The major key C has all the notes of the minor key A excepting D, the root of the minor subdominant; and the minor has all the notes of the major exception D, the top of the major dominant. These twain are one music, the masculine and feminine of a twofold unity; one system in duality rather than two systems. [Scientific Basis and Build of Music, page 50]

There are two semitones in each system, B-C and E-F. But when the notes of the two systems are being generated simultaneously, the two semitonic intervals originate separately. While the major is generating the semitone E-F, the third and fourth of the major scale, the minor is generating the semitone B-C, the second and third of the minor scale. So E-F is the semitone which belongs genetically to the major, and B-C to the minor.1 These two semitones are the two roots of
THE CHROMATIC SYSTEM,
and they are found together in what has been called the "Minor Triad," and by other names, namely, B-D-F. [Scientific Basis and Build of Music, page 50]

Some of the elements of the Chromatic System were known 200 years ago. The Diatonic scale, being called the "Natural scale," implied that the chromatic chords were consider to be artificial; but the notes of the chromatic chords, from their PROXIMITY to the notes of the tonic chord, fit to them like hand and glove. Nothing in music is more sweetly natural and pleasingly effective than such resolutions; and hence their extensive use in the hands of the Masters. The chromatic chords have close relations to the whole system of music, making the progressions of its harmonies easy and delectable, and producing effects often enchanting and elevating, as well as often subtle and profound; and while they are ever at hand at the call of the Composer, they are ever in loyal obedience to the laws of their own structure and system. When a diatonic chord precedes another diatonic chord belonging to the same scale, it has one note moving in semitonic progression;1 but when a chromatic chord precedes a diatonic chord, it may have three semitonic progressions.2 The primary chromatic chord resolves into 8 of the 24 diatonic tonic chords, with 3 semitonic progressions. These identical notes of the chromatic chord, with only some changes of names, resolve into another 8 of the 24 tonic chords, with 2 semitonic progressions and one note in common; and when they resolve into the third and last 8 of the 24 tonic chords, they move with one semitonic progression and 2 notes in common. So to the chromatic chord there are no foreign keys.3 And as it is with the first chromatic chord, so with the other two. [Scientific Basis and Build of Music, page 51]

The triplet B, D, F, has been called the imperfect triad, because in it the two diatonic semitones, B-C and E-F, and the two minor thirds which they constitute, come together in this so-called imperfect fifth. But instead of deserving any name indicating imperfection, this most interesting triad is the Diatonic germ of the chromatic chord, and of the chromatic system of chords. Place this triad to precede the tonic chord of the key of C major, and there are two semitonic progressions. Place it to precede the tonic chord of the key of F# major, and there are three semitonic progressions. Again, if we place it to precede the tonic chord of the key of A minor, there are two semitonic progressions; but make it precede the tonic chord of E♭ minor, and there are three semitonic progressions. This shows that the chromatic chord has its germ in, and its outgrowth from the so-called "natural notes," that is notes without flats or sharps, notes with white keys; and that these natural notes furnish, with only the addition of either A♭ from the major scale or G# from the minor, a full chromatic chord for one major and one minor chord, and a secondary chromatic chord for one more in each mode. [Scientific Basis and Build of Music, page 52]

the major, the root of the major chord is the middle of the relative minor, and the middle of the major chord is the top of the relative minor; and as the note which has a semitonic progression downward to the top of the major has a semitonic progression upward to the root of the relative minor, so the same three notes which move in semitonic progression to the top, root, and middle of the major chord, likewise move by the semitonic progression to the root, top, and middle of the relative minor. In both cases the progressions are upward to the roots and downward to the tops; but in the major the movement is downward to the middle, while in the minor it is upward. So each one of these three of the four notes of the chromatic chord has two various movements.1 [Scientific Basis and Build of Music, page 53]

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In the above line it will be readily observed that these three chromatic chords, having each of their intervals a minor third, involve, as necessary elements of their build, every one of the twelve semitones at one point or another of the line. This is a second witness to the legitimacy of the chromatic scale of twelve semitones. We have our first witness to the same in the evolution of the semitones progressively in the course of the modulations by which, in growth-like continuity, Nature links the successive keys; whether developed upward, as is the natural way of the majors; or downward, as in the natural way of the minors; or half upward and half downward, which is an expedient in order to simplify the signatures. In whichever direction the modulation is effected, one note is always divided, and must, true to Nature, be signified by placing a sharp or a , as the case may be, to the 4-comma altered interval, and this always leaves a 5-comma interval to occupy the place to which Nature has assigned such interval in the original scale. When this operation has been twelve times performed, we have the chromatic scale of twelve semitones. Thus by two witnesses the thing is established. By further examination of these chromatic chords we find other interesting features beside their witness to the twelve semitones of the octave. [Scientific Basis and Build of Music, page 56]

By taking four minor thirds upward from G# or downward from A♭, we have the first chromatic chord in its twofold form. Its central note is D, the top of the dominant major, and the root of the subdominant minor, being its own dual, that is to say, its minor birth being dual to its major birth.1 On the keyboard it has the same order of keys above it and below it; this dual D [Scientific Basis and Build of Music, page 56]

being also the center of the diatonic triplet, B, D, F, which is the diatonic germ of the chromatic system. Four minor thirds upward or downward from C# we have a second chromatic chord, its central note being G. The dual1 of C# is E♭; and there is the same order of keys on the keyboard2 below C# as there is above E♭. Four minor thirds from E♭ upward or downward forms a third chromatic chord, the central note of which is A. The dual A, the center of the third chromatic, is G, the center of the second; and these two notes, by their duality, and by the duality of the two chords throughout, balance each other exactly on the keyboard on either side of the first chromatic chord, which contains all its own duals, and by this self-duality sits in the center, like the tonic chord among the diatonic three. [Scientific Basis and Build of Music, page 57]

The notes of each primitive chromatic chord are all different from the notes of the other two, so that there is no mistaking of the one for the other; any one note decides the chromatic chord to which it belongs. [Scientific Basis and Build of Music, page 57]

It is the function of the chromatic chord, in the course of its systematic development, to bring two notes, the one above and the other below, in semitonic progression to each of the three notes of the tonic chord; and likewise, without interfering with the imperial character of the tonic chord, it brings two notes in semitonic progression to each note of the other two chords; so that, although at first the subdominant and dominant chords are only like satellites to the tonic chord, they, with their chromatic chords, are now raised to the dignity of planets as it were, having the chromatic chords as their satellites. [Scientific Basis and Build of Music, page 57]

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]

Guiseppe Tartini, 200 years ago, while practicing on his violin, observed a very interesting phenomenon in music in the matter of notes or sounds, [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]

The individual character of any note, and the comparative degree of contrast between any two notes in the system, depends on at least three different causes. The first is the genetic relation of the two notes. If the one note has 2 vibrations and the other 3, or the one 4 and the other 5, or the one 5 and the other 8, because of this, and because of the excess of the vibration of the one over the other, "a third sound" or "grave harmonic" being awakened between them, the different ratios have different degrees of complexity, and, in a general way, the greater the complexity the greater the [Scientific Basis and Build of Music, page 60]

A second cause of difference in degree of contrast between two notes and other two notes in which the ratios are the same lies in this - whether the two notes belong to one chord or to different chords. Two notes in the subdominant chord have a different contrast from two in the dominant chord which have the same ratio. [Scientific Basis and Build of Music, page 61]

F is soft, grand, and solemn;
C is melodious and soft;
A is interesting and soft.

C is melodious and soft;
G is melodious and vigorous;
E is interesting and melodious.

G is melodious and vigorous;
D is interesting and vigorous;
B is light, airy, and vigorous.


Although the system is composed of only three ratios, which in themselves moreover, are of a very fixed character, yet mobility and variety are chief features among the notes of the system. Great changes are effected by small means. By lowering the second of the major D one comma, the ratio of 80:81, [Scientific Basis and Build of Music, page 61]

lastly it is altered again and becomes, by the power of 3 once more, F#,#, and serves in four keys. But this carries us beyond the horizon of our musical world of twelve keys; for in B#, the top of the tonic E, we have reached our twelfth fifth, and it here coalesces with C of the seventh octave, and closes the circle. This is the way that all notes become alternately altered, either by commas and sharps in the upward genesis of scales, or by commas and flats in the downward genesis, by the alternate powers of 3 and 5. In the upward genesis in this illustration, notes by the power of 5 serve in three keys, and those by the power of 3 serve in four keys. In the minors it is just the inverse on this by the Law of Duality. But no note serves for more than either three or four keys, as the case may be. [Scientific Basis and Build of Music, page 63]

Affinity - The genetic relation of notes. [Scientific Basis and Build of Music, page 63]

We have gone from vibrations to musical notes; from notes to chords; and now we proceed to scales - that is, groups of notes or chords in succession, which are bound and unified in some clear and definite way. [Scientific Basis and Build of Music, page 64]

embedded; and in the minor tonic chord A C E in the same way is embedded its own semitone; and in these chords they appear in their proper places as third and fourth, C, d, E f, G; and second and third, A, b C, d, E. It is these first five notes of the octave scale which in a very distinctive way constitute the natural scale, which can be harmonized consecutively in one manner. The octave is seen in this view to be a compound scale, inasmuch as a compounding method of harmonizing has to be resorted to in passing consecutively from the sixth to the seventh. Similar compounding has to be done in the minor as well.1 [Scientific Basis and Build of Music, page 65]

where there stands an open door between the sixth and the seventh, these two having no note in common, it is easy and natural to slip out of the key into another, either in ascending the major or descending the minor octave; and in order to keep in the key, the two chords of these notes have to reach out to each other a helping hand, and compound in order to affiliate. This, however, by the law of sympathy and assimilation, which reigns in this happy, realm, they are always ready to do.1 [Scientific Basis and Build of Music, page 66]

This great genetic scale, the all-producer, the all-container, extends over six octaves on each side; for it is not till high in the sixth octave we get B in the major, and it is not till low in the sixth octave that we get F in the minor. It is in the fifth octave, however, that the note which is the distinctive mark of the masculine and feminine modes is generated. D27 in the major, and D26 2/3 in the minor, distinguishes the sex of the modes, and shows which is the head and which the helpmeet in this happy family.2 On the major side F, the root of the subdominant chord, that is the chord which is a fifth below the key-note C, is the root of all. This is the beginning of this creation. If we call the vibration-number of F one, for simplicity's sake, then F1 is multiplied by 3 and by 5, which natural process begets its fifth, C, and its third, A; this is the root, top, and middle of the first chord. From this top, C3, grows the next chord by the same natural process, multiplying by 3 and by 5; thus are produced the fifth and third of the second chord, G and E. From the top of this second chord grows the third and last chord, by the repetition of the same natural process; multiplying G9 by 3 and by 5 we [Scientific Basis and Build of Music, page 66]

generate D and B; D is the highest note in the chord-scale, but B is the last-born note, and has a higher vibration-number in the genesis. [Scientific Basis and Build of Music, page 67]

Chords in a harmony are not at liberty to succeed each other in the way that single notes in a melody may. The notes in a melody may succeed in seconds, or they may succeed in larger intervals, any interval in the octave, even sometimes very effectively there may be a leap or fall of a whole octave. Chords cannot follow each other in this free way; they are under law, and must succeed accordingly. Their law is that they must be linked together either by having something in common in their elements, or have small intervals, semitonic progressions, between them. The former way, by notes in common, is the most usual way in diatonic succession of chords, the latter way, by semitonic progression, is a chief feature and charm in chromatic succession; but both in diatonic and chromatic progression of chords in harmony, notes in common and semitonic progression are usually found together. [Scientific Basis and Build of Music, page 68]

All compound and double compound chords are made up of notes already developed for the simple chords; there is no genetic developing of compound chords. Simple chords are all begotten in the genesis; they are true species; compound chords are only varieties of them. [Scientific Basis and Build of Music, page 70]

common, to mingle with more chord-society. So those added thirds which constitute compound chords are like accomplishments acquired for this end, and they make such chords exceedingly interesting. The dominant assumes the root of the subdominant, and so becomes the dominant seventh that it may be affiliated with the subdominant chords. Inversely, the subdominant assumes the top of the dominant chord that it may be affiliated with the dominant. The major tonic may exceptionally be compounded with the top of the minor subdominant when it comes between that chord and its own dominant; and the minor tonic may in the same way assume the root of the major dominant when it comes between that chord and its subdominant. The minor subdominant D F A, and the major dominant G B D, are too great strangers to affiliate without some chord to introduce them; they seem to have one note in common, indeed, but we know that even these two D's are a comma apart, although one piano-key plays them both, and the F G and the A B are as foreign to each other as two seconds can be, each pair being 9 commas apart, and G A are 8 commas apart. In this case, as a matter of musical courtesy, the tonic chord comes in between; and when it is the minor subdominant that is to be introduced, the major tonic assumes the top of that chord, and then turns to its own major dominant and suavely gives the two to enter into fellowship; for the tonic received the minor subdominant through its semitonic E F, and carries it to the major dominant through its semitonic B C, along with C in common on the one side and G in common on the other. When it is the major dominant that is to be introduced to the minor subdominant the minor tonic fulfills the function, only the details are all reversed; it assumes the root of dominant, and by this note in common, and its A in common with its own subdominant, along with the semitonic second B C on the one hand and the semitonic E F on the other, all is made smooth and continuous. The whole of this mediatorial intervention on the part of the tonic is under the wondrous law of assimilation, which is the law of laws all through creation; but when the tonic chord has fulfilled this graceful action, it immediately drops the assumed note, and closes the cadence in its own simple form.1 [Scientific Basis and Build of Music, page 71]

How far does this compounding process go? The dominant seventh has the first note of the subdominant; the dominant ninth has the second; if we should add a third note, where are we? G B D F A C; here would be the dominant with the whole of the subdominant welded to it; it would have to be called the dominant eleventh, and it has brought us right through to the root of the tonic C. What would be the use of such a chord? We might, in a similar way, add the dominant to the subdominant till we should be through to the tonic on the other side; it would be G B D F A C, and so we should have reached the top of the tonic G. This process shows us, however, that there is just a certain length that we can go, and there is satisfaction in seeing exhaustively that so it is. When the beautiful becomes the useless, it ceases to be the beautiful. [Scientific Basis and Build of Music, page 72]

less variety of effect than we find in the diatonic chords; for although these chords may appear with their notes diversely named, there are still only the three. On account of their cosmopolitan character they need, and they have, no compounding with anything else. They are themselves at home everywhere; like a universal joint, they can turn any way, and affiliate in all directions. Being 4-note chords, and all of minor thirds, their effect is always minor, and they fall with loving softness to the diatonic chords to which they resolve. How this chord in its germ is found in the diatonic chord-scale; how it becomes a 4-note chord of minor thirds; how it variously resolves, each one of the three, in three manners with 24 tonic chords - all this is so fully set forth in the pre-note and treatise on the chromatic chord that it need not be more discussed in this place. See also Plates XVI., XVII., XVIII., XIX., and XX. [Scientific Basis and Build of Music, page 73]

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 root of the subdominant is F, in the key of C major; and the top of the dominant is D. The difference between these two notes at the top and bottom of the chord-scale, is the quantity which two octaves is more than three fifths; it is the ratio of 27 to 30, a comma less than the minor third whose ratio is 5 to 6. [Scientific Basis and Build of Music, page 76]

Taking the seven notes of the octave, we have among them 42 intervals in all, without including the octave. When two notes are sounded together, [Scientific Basis and Build of Music, page 76]

the excess of the vibrations of the one note over the other makes one or more sounds which are called "grave harmonics;" e.g., in the interval of the fifth, in the ratio of 2:3, the excess of 3 over 2 is 1, so the grave harmonic is an octave below the lowest of the two notes, that is, the ratio of 1:2. This reinforces the lowest note, 2, and gives it a solid effect. In this way the octave is incorporated into the fifth, and unity with variety is combined with the law of continuity at the very threshold of harmony. In 32 of the 42 intervals the grave harmonics are notes which belong to the natural scale. In the 10 remaining intervals which have not the exact number of vibrations found anywhere in the natural scale, 6 of them are from the number 7, thus - 7, 7, 7, 21, 21, 35; the remaining 4 are from 11, 13, 13, and 19. [Scientific Basis and Build of Music, page 77]

At the first, in the laws of quantities and motions adjusting musical vibrations, there is one chord of the three notes, F, A, C, the root, middle, and top of the five notes which compose the true natural scale; this one chord can be reproduced a fifth higher, C, E, G, in the same mathematical form, taking the top of the first for the root of the second chord. In like manner this second can be reproduced another fifth higher, G, B, D, still in the same mathematical form, and so fit to be a member of the chord-scale of a key. But the law does not admit of another reproduction without interfering with the first chord, so that a fourth fifth produces no new effect; but the whole key is simply a fifth higher, i.e., if the fourth fifth has been properly produced by multiplying the top of the third fifth by 3 and by 5, the generating primes in music. That this carries us into a new scale is seen in that the F is no longer the F♮ but F#, and the A is no longer A♮ but A,. But if we suppose the fourth fifth to be simply the old notes with their own vibration numbers, then D, F, A would not be a fifth belonging either to the major or the minor mode, but a fifth a comma less. The letters of it would read like the minor subdominant, D, F, A; but the intervals, as found in the upward development of the major genesis, instead of being, when expressed in commas, 9, 5, 8, 9, which is the minor subdominant, would be 8, 5, 9, 8, which is not a fifth of the musical system; these having always, whether major or minor, two 9's, one [Scientific Basis and Build of Music, page 77]

In compound chords there are no new notes created; they are found by combining the notes of the simple chords. The dominant sevenths, major and minor, are compounded by adding one from the subdominant. [Scientific Basis and Build of Music, page 79]

All the inverse and reciprocal orders in music, whether of notes or chords, are produced by the law of Duality, and this gives them inevitably their symmetrical form. [Scientific Basis and Build of Music, page 79]

The simple natural scale is the fifth; the compound natural scale is the octave; the harmony scale, or chord-scale, is the three fifths; the great genetic scale is six octaves; for, like the six creation days, it takes the six octaves to give birth to the elements of which the wondrous structure of our music is built up; the birthplace of B, the seventh of the octave scale, is the sixth octave of the great genetic scale. The area of the twelve major and twelve minor scales is twelve fifths or seven octaves, the twelfth fifth being a comma and the apotome minor in advance of the seventh octave. This is a quantity so small that it can be ignored in real music; and the two notes, say E# and F, joined to close the circle of this horizon of our music world. E# is the top of the twelfth fifth, and F is the top of the seventh octave; and they are practically, though not exactly mathematically, the same note. Illustrations of this will be found among the plates of this work. [Scientific Basis and Build of Music, page 79]

G# as it occurs in the scales of A, E, and B major, and A♭ as it occurs in the scales of F and B♭ minor, are only distant the apotome minor, and are well represented by one key of the piano. It is only G# as it occurs in the scale of F six sharps major, and A♭ as it occurs in the scale of E six flats minor, that is not represented on the piano. These two extreme notes F# and E♭ minor are at the distance of fifteenth fifths and a minor third from each other. This supplies notes for 13 major and 13 minor mathematical scales; but as this is not required for our musical world of twelve scales, so these far-distant G# and A♭ are not required. The piano is only responsible for the amount of tempering which twelve fifths require, and that is never more than one comma and the apotome minor. [Scientific Basis and Build of Music, page 80]

If we may add a note to a chord, we may also vice versa add that chord to that note. If we may add F to the dominant chord G B D f, we may add the dominant chord to F, thus, g b d F, in some other [Scientific Basis and Build of Music, page 80]

Having found the framework of the major scale by multiplying F1 three times by 3, find the framework of the minor by dividing three times by 3. But what shall we divide? Well, F1 is the unbegotten of the 25 notes of the great genetic scale; B45 is the last-born of the same scale. We multiply upward from F1 for the major; divide downward from B45 for the minor. Again, B45 is the middle of the top chord of the major system, a minor third below D, the top of that chord, and the top of the whole major chord-scale, so B is the relative minor to it. Now since the minor is to be seen as the INVERSE of the major, the whole process must be inverse. Divide instead of multiply! Divide from the top chord instead of multiply from the bottom chord. Divide from the top of the minor dominant instead of multiply from the root of the major subdominant. This will give the framework of the minor system, B45/3 = E15/3 = A5/3 = D1 2/3. But as 1 2/3 is not easily compared with D27 of the major, take a higher octave of B and divide from it. Two times B45 is B90, and two times B90 is B180, and two times B180 is B360, the number of the degrees of a circle, and two times B360 is B720; all these are simply octaves of B, and do not in the least alter the character of that note; now B720/3 is = E240/3 = A80/3 = D26 2/3. And now comparing D27 found from F1, and D26 2/3 found from B720, we see that while E240 is the same both ways, and also A80, yet D26 2/3 is a comma lower than D27. This is the note which is the center of the dual system, and it is itself a dual note befittingly. [Scientific Basis and Build of Music, page 81]

So the dual system of Music revolves round a mathematical point which is in none of its notes, but in the empty space between the two D's. Like the earth, it is hung upon nothing. This is an exceedingly interesting musical phenomenon. In that comma of vacant space is music's center of gravity.

1 3 9 27
F C G D
D A E B
26 2/3 80 240 720 [Scientific Basis and Build of Music, page 82]

There are two octaves in the key of C, as it is called. Now for the scale of a fifth higher than C, that is G, multiply the top of the dominant, that is the highest note of the chord-scale, by 3 and by 5, and the two new notes for the scale of G will be found; the rest of the notes are the same mathematically as those of C. [Scientific Basis and Build of Music, page 82]

third; and in order to do so, A has been mathematically raised a comma, which makes G A B now a 9-8 comma third. E F G, which in the scale of C was a 5-9 comma third, must now take the place of A B C in the scale of C, which was a 9-5 comma third; and in order to do so, F is mathematically raised 4 commas, and must be marked F#; and now the interval is right for the scale of G. E F# G is a 9-5 comma interval. This mathematical process, in the majors, puts every scale in its original form as to vibration-numbers; but since the same letters are kept for the naming of the notes, they must be marked with commas or sharps, as the case may be. In the Sol Fa notation such marks would not be necessary, as Do is always the key-note, Ray always the second, and Te always the seventh. [Scientific Basis and Build of Music, page 83]

When higher or lower octaves of any note or scale are wanted for convenience of comparison, multiply or divide by two, the octave-producer. [Scientific Basis and Build of Music, page 83]

The effects of different positions of chords. The first position is clear and solid; the second is light and pretty. The notes which have the most varied effect in a stream of harmony are the upper and under notes, the edges of the stream; the two outlines determine the effect:-

Image-page-84

[Scientific Basis and Build of Music, page 84]

There are 32 notes required for each octave for the 13 major and the 13 minor mathematical scales. These 32 notes are by the law of duality arranged symmetrically from D as a center upwards to G#, and downwards to A♭. D itself serves for 2 of the 32 on the piano. The first black keys on each side of D serve for nominally 3 notes each = 6. The first white key above and the first below D serve for 2 notes each = 4. The second white key above and the second below serve each for 3 notes = 6. The second black keys above and below D serve each for 3 notes = 6. The third black key above D is G#, the third below is A♭; this key, for it is one, serves for 2 of the 32. There is a comma of difference between D minor and D major. Six fifths below the minor D26 2/3 is A♭, the root of the subdominant of the key of E♭ minor; and six fifths above the major D27 is G#, the top of the dominant of F# major. The difference between this minor A♭ and this major G# is two commas and [Scientific Basis and Build of Music, page 85]

In respect of harmony, the natural scale of five notes is like the scale of man's five senses; as the other notes can be compounded so as to form the octave of harmony, so sensation is joined by reflection, and new elements of knowledge come into existence in the process of reasoning. But the knowledge we have in our logical deductions is knowledge on different terms from sensation, which is intuitive; though if the logical process be rightly done, it is knowledge as certainly as the compound chords of the octave scale are harmony, quite as much, and a little more, perhaps, though on more complex terms, as that of the five notes of the natural scale. [Scientific Basis and Build of Music, page 86]

It is a remarkable fact that the numbers for the lengths of strings producing the major scale are the number of the vibration of the notes of the minor scale; for example, string-length as 26 2/3 will give the vibrations for [Scientific Basis and Build of Music, page 87]

When the chords follow each other placed in such a way as to bring the upper note of the one as near the upper note of the next as it can be, the upper surface of this stream of harmony is the melodic scale and the harmony is the bottom and the body of the stream.

Ramsay image page 88

[Scientific Basis and Build of Music, page 88]

Seven notes in the Octave are required for the major scale, e.g., the scale of C. All the notes of the relative minor A are the same as those of the scale of C major, with exception of D, its fourth in its Octave scale, and the root of its subdominant in its chord-scale; thus, one note, a comma lower for the D, gives the scale of A minor. [Scientific Basis and Build of Music, page 88]

The six successive major scales with sharps require two new notes each; and so with the six successive scales with flats, they also require two new [Scientific Basis and Build of Music, page 88]

notes each; that is, 24 new notes, which, with the seven original major notes and the one different minor note, make 32 in all; and 32 notes in the Octave are all that belong to 13 major and 13 minor mathematical scales.1 [Scientific Basis and Build of Music, page 89]

The six successive major scales with sharps require 2 new notes each, and the six successive minor scales with sharps require also 2 new notes each; but one of these new notes for each minor scale is supplied from the scale of the relative major, and the other from the sub-relative major, i.e., the scale one-fifth lower than the relative. So when the major scales with sharps have been developed they furnish all the new notes needed for the minors. The six successive minor scales with flats require 2 new notes each, and the six successive major scales with flats require each 2 new notes; but one of these is supplied from the scale of the relative minor, and the other from the scale of the super-relative, i.e., the scale one fifth higher than the relative. So when the minor scales with flats are developed they furnish all the new notes require by these majors.[Scientific Basis and Build of Music, page 89]

The reason why there are 13 mathematical scales is that G♭ and F# are written separately as two scales, although the one is only a comma and the apotome minor higher than the other, while in the regular succession of scales the one is always 5 notes higher than the other; so this G♭ is an anomaly among scales, unless viewed as the first of a second cycle of keys, which it really is; and all the notes of all the scales of this second cycle are equally a comma and the apotome higher than the notes of the first cycle; and when followed out we find that a third cycle is raised just as much higher than the second as the second is higher than the first; and what is true of these majors may be simply repeated as to the D# and E♭ of the minors, and the new cycle so begun, and all successive minor cycles. Twelve and not thirteen is the natural number for the mathematical scales, which go on in a spiral line, as truly as for the tempered scales, which close as a circle at this point. [Scientific Basis and Build of Music, page 89]

The two notes required for the scale of
E minor are the F# of G, and the D of C major;
for B minor, the C# of D, and the A of G major;
for F# minor, the G# of A, and the E of D major;
for C# minor, the D# of E, and the B of A major;
for G# minor, the A# of B, and the F# of E major;
for D# minor, the E# of F#, and the C# of B major. [Scientific Basis and Build of Music, page 90]

In a similar and responsive way Duality provides for the six major scales with flats.
The two new notes required for the scale of
F major are the B♭ of D, and the D of A minor;
for B♭ major, the E♭ of G, and the G of D minor;
for E♭ major, the A♭ of C, and the C of G minor;
for A♭ major, the D♭ of F, and the F of C minor;
for D♭ major, the G♭ of B♭, and the B♭ of F minor;
for G♭ major, the C♭ of E♭, and the E♭ of B♭ minor.1 [Scientific Basis and Build of Music, page 90]

It must be noted that the two new notes needed for each new scale are quite a different affair from the sexual note, that is the note which is different in major and minor, e.g., the two D's in major C and minor A. The [Scientific Basis and Build of Music, page 90]

In a musical air or harmony, i.e., when once a key has been instituted in the ear, all the various notes and chords seem animated and imbued with tendency and motion; and the center of attraction and repose is the tonic, i.e., the key-note or key-chord. The moving notes have certain leanings or attractions to other notes. These leanings are from two causes, local proximity and native affinity. The attraction of native affinity arises from the birth and kindred of the notes as seen in the six-octave genesis, and pertains to their harmonic combinations. The attraction of local proximity arises from the way the notes are marshalled compactly in the octave scale which appears at the head of the genesis, and pertains to their melodic succession. In this last scale the proximites are diverse; the 53 commas of the octave being so divided as to give larger and lesser distances between the notes; and of course the attraction of proximity is strongest between the nearest; a note will prefer to move 5 commas rather than 8 or 9 commas to find rest. Thus far PROXIMITY. [Scientific Basis and Build of Music, page 91]

By affinity the notes group in chords. The tonic is the center chord, the key of the harmony; the dominant is the fifth above it and the subdominant the fifth below it, and these two are balanced on the center chord as the scales on a balance beam. The dominant chord is vigorous and active, tending to soar; the subdominant is solemn, soft, and grave, tending to sink; the tonic is melodious and restful, and in it the harmony finds equilibrium. This far AFFINITY. [Scientific Basis and Build of Music, page 91]

Thus chord tends toward chord, and note leans toward note, and it has to be considered whether the attraction of affinity or proximity be strongest while working out the charming effects of a composition. [Scientific Basis and Build of Music, page 92]

HARMONICS ON THE VIOLIN.

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]

In the opening of the third measure the tune returns to its own key by striking the tonic. This case is a very simple illustration of how a composition will move with perfect naturalness in more keys than one, the keys so grow out of each other, and may either merely snatch a passing chord from a new key, or pass quite into it for a phrase or two, or for a whole measure, then return as naturally, either by a smooth and quiet or by a strongly contrasted turn, according to the chords between which the turn takes place. In such modulation there may or there may not be marked a #, , or , in the air itself; the note which Nature raises in the new key may occur in one of the other parts of the harmony. In Watchman it is A, the fourth, which is altered; from being it is made . The change which takes place in the sixth of the scale, which is C in Watchman, is only one comma, the ratio of 80 to 81, and it slips into the new key as if nothing had happened. No mark is placed to it, as the comma difference is never taken notice of, although it is really and regularly taking place, with all the precision of Nature, in every new key. It is, however, only the note which is altered four commas, which is marked by a #, , or , as the case may be. [Scientific Basis and Build of Music, page 94]

"There are three chromatic chords, and each of these three is related to eight particular tonic chords. When one the these chromatic chords goes to any one of its eight tonic chords, three of its notes move in semitonic progression, and the other note moves by the small tone, the ratio of 9:10. There is exception to this rule, whether the key be major or minor. But when the chromatic chord which should resolve to the tonic of C is followed by the subdominant, or the tonic of F (the example in Mr. Green's book), only two of its notes move in semitonic progress. Your friend describes the chord as if it had gone to the tonic of B; and what he said about it, and about D going to C, is what is supposed to be [Scientific Basis and Build of Music, page 94]

"The organic structure of music is formed by the three ratios of 1:2, 1:3, and 1:5, from the laws of quantities and motions; but as it is only the ratio of 1:2 that has a pure, unmixed, invariable character, and as the notes produced by the first, second, and third powers of THREE have different degrees of centrifugal force, and the character of the notes produced by the first power of FIVE depends on the character of the notes from which they are derived, so the final character of the notes and chords is determined by the amount of force which they have acquired from the way in which they have been derived, and from their position in the system; and no matter how these notes may be afterwards placed, like chemical elements, they never lose their original force. [Scientific Basis and Build of Music, page 95]

"The notes as they naturally arise from unity have different degrees of development, and according to the degree of development of each note is its specific levity or gravity. The three notes which form the subdominant chord have different degrees of gravity; the three which form the dominant chord have different degrees of levity. The remaining note is the center of the tonic chord -

Subdominant - F, A, C E G, B, D - dominant

[Scientific Basis and Build of Music, page 95]

Subdominant - F, A, C E G, B, D - dominant


- and it is balanced between the two forces. If the effects of notes or chords depended solely on their ratios, then the effect of the subdominant, tonic, and dominant would have been alike, for these chords have exactly the same ratios. The centrifugal force of the notes of the dominant chord would take if away from the tonic chord; but Nature, in her skill to build and mix, has in the octave scale placed the middle of the dominant B under the root of the tonic C, and the top of the dominant D under the middle of the tonic E; so that these two rising notes are inevitably resolved into the tonic chord. The gravitating tendencies of the notes of the subdominant would take it also away from the tonic; but in the octave scale Nature has placed the middle of the subdominant A above the top of the tonic G, and the root of the subdominant F above the middle of the tonic E; so that these two falling notes also are inevitably resolved into the tonic chord. In this way two notes resolve to the center of the tonic, D upwards and F downwards; one to the top, A to G, and one to the root, B to C. Nature has thus placed the notes which have upward tendencies under the notes having downward tendencies; she has also related them by proximity, the distance from the one to the other being always either a semitone or the small tone of the ratio 9:10. [Scientific Basis and Build of Music, page 95]

"There are two distinct laws which rule in astronomy - viz., masses and distances; and there are two distinct laws which rule in music - affinities and proximities. The notes produced by simple ratios as 1:2, 2:3, 3:4, etc., are attracted to each other by the law of affinity; notes which are beside each other in the octave scale and have moderately complex ratios as 9:10 and 15:16, are attracted to each other by their proximities. F and C, and C and G, and G and D are related to each other by affinity. C is related to the fifth below and the fifth above; G is related to the fifth above and the fifth below. F and C, C and G, and G and D are never nearer to each other than a fifth or a fourth, and in either case they [Scientific Basis and Build of Music, page 95]

are attracted to each other by affinity. But the case is quite different with F and G and C and D. The second fifth above F is G (F a c, C e g), and G becomes the interval above F in the octave scale; and these two notes are neither attracted by affinity nor proximity nor gravitating tendency. F sinks away from G, being heavier, and under it; and G soars away from F, being above it, and lighter. In a similar way the second fifth above C is D (C e g, G b d), and D in the octave scale becomes the interval of the second above C, and C and D, like F and G, are not attracted by either affinity or proximity. C is heavier than D, and being under it would sink away from it; D is lighter, and being above it would soar away from it, and so neither are they attracted by gravitating tendency. [Scientific Basis and Build of Music, page 96]

"All the bodies in the Solar System, in a general way, are attracted to the sun according to the Law of Masses; but all the satellites are attracted to their planets according to the Law of Distance. The subdominant and dominant chords in the Musical System, in a general way, are attracted to the tonic center; but each note in the octave scale is attracted to its nearest note by the Law of Proximity. [Scientific Basis and Build of Music, page 96]

"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]

Three times three are nine this would give nine notes; but as the top of the first chord serves for the root of the second one, and the top of the second for the root of the third, in this way these three chords of three notes each are formed from seven different notes. [Scientific Basis and Build of Music, page 96]

The middle one of these three chords is called the tonic; the chord above is called the dominant; and the chord below is called the subdominant. The order in which these three chords contribute to form the octave scale is as follows:- The first note of the scale is the root of the tonic; the second is the [Scientific Basis and Build of Music, page 96]

In the first six chords of the scale the tonic is the first of each two. The tonic chord alternating with the other two produces an order of twos, as - tonic dominant, tonic subdominant, tonic subdominant. The first three notes of the octave scale are derived from the root, the top, and the middle of the tonic dominant and tonic; the second three are derived from the root, top, and middle of the subdominant, tonic, and subdominant. The roots, tops, and middles of the chords occurring as they do produce an order of threes, as - root, top, middle; root, top, middle. The first, third, fifth, and eighth of the scale are from the tonic chord; the second and seventh from the dominant; and the fourth and sixth from the subdominant. In the first two chords of the scale the tonic precedes the dominant; in the second two, the subdominant; and in the third two the tonic again precedes the subdominant; and as the top of the subdominant chord is the root of the tonic, and the top of the tonic the root of the dominant, this links these chords together by their roots and tops. The second chord has the top of the first, the third has the root of the second, the fourth has the root of the third, the fifth has the top of the fourth, and the sixth has the root of the fifth; and in this way these successive chords are woven together. The only place of the octave scale where there are two middles of chords beside each other is at the sixth and seventh. The seventh note of the octave scale is the middle of the dominant, and the sixth is the middle of the subdominant. These two chords, though both united to the tonic, which stands between them, are not united to each other by having a note in common, inasmuch as they stand at the extremities of the system; and since they must be enabled to succeed each other in musical progression, Nature has a beautiful way of giving them a note in common by which to do so - adding the root of the subdominant to the top of the dominant, or the top of the dominant to the root of the subdominant, and this gives natural origin to compound chords. The tonic chord, being the center one of the three chords, is connected with the other two, and may follow the dominant and sub- [Scientific Basis and Build of Music, page 97]

dominant; and either of these chords may also follow the tonic; but when the dominant follows the subdominant, as they have no note in common, the root of the subdominant is added to the dominant chord, and this forms the dominant seventh; and when the subdominant follows the dominant, the top of the dominant is added to the subdominant, and this forms the subdominant sixth. The sixth and seventh of the octave scale is the only place these two compound chords are positively required; but from their modifying and resolvable character they are very generally used. When the dominant is compounded by having the root of the subdominant, its specific effect is considerably lower; and when the subdominant is compounded by having the top of the dominant, its specific effect is considerably higher. In the octave scale the notes of the subdominant and dominant chords are placed round the notes of the tonic chord in such a way was to give the greatest amount of contrast between their notes and the tonic notes. In the tonic chord the note which has the greatest amount of specific gravity is its root; and in the octave scale it has below it the middle and above it the top of the dominant, the two notes which have the greatest amount of specific levity; and in the octave scale it has above it the middle and below it the root of the subdominant - the two notes which the greatest amount of specific gravity. The third note of the scale, the middle of the tonic chord, is the center of the system, and is the note which has the least tendency either upwards or downwards, and it has above it the root of the subdominant, the note which has the greatest amount of specific gravity, and it has below it the top of the dominant, the note which has the greatest amount of specific levity. Thus the root of the subdominant is placed above, and the top of the dominant below, the center of the system; the specific gravity of the one above and the specific levity of the one below cause them to move in the direction of the center. [Scientific Basis and Build of Music, page 98]

"Each note in the scale is attracted to the note above it or the one below it. B is attracted to C. If F and G were attracted to each other by proximity, then A would be left alone without a note to attract or by which to be attracted by proximity. All the [Scientific Basis and Build of Music, page 98]

notes attracted by proximity are attracted in the direction of the center of the tonic chord, major or minor. But if D in the major is attracted by C, the root of the tonic, then it would be moving away from the center. Two notes which have the ratio of 8:9, as C and D, or two notes which are produced by the same ratio as C and D, or two notes where each of them is either a root or a top, as C and D, never resolve to each other by proximity. It is an invariable order that one of the notes should be the middle of a chord. [Scientific Basis and Build of Music, page 99]

"If it had been the case that D resolved to the root of the major tonic, the resolving notes to the tonic would then have been one upwards and three downwards, instead of two upwards and two downwards, according to the Law of Duality. [Scientific Basis and Build of Music, page 99]

"What we have thus said about the resolving notes to the major tonic has been allowed in the case of the minor. No one ever said that the second of the minor scale resolved to the root of the tonic. Notwithstanding the importance of the tonic notes, the semitonic interval above the second of the scale decided the matter for the Law of Proximity; and no one ever said that D, the root of the subdominant minor, did not resolve to C, the center of the tonic minor, on the same terms that two notes are brought to the center of the tonic major; with this difference, that the semitonic interval is above the center in the major and below it in the minor. The other two notes which resolve into the tonic minor are on the same terms as the major; with this difference, that the semitonic interval is below the root of the tonic major and above the top of the tonic minor. And the small tone ratio 9:10 is above the top of the tonic major and below the root of the tonic minor. If it has been the case that D resolved to the root of the tonic major, then, according to the Law of Duality, there would have been another place where everything would have been the same, only in the inverse order; but, fortunately for itself, the error has no other error to keep it in countenance. This error has not been fallen into by reasoning from analogy. [Scientific Basis and Build of Music, page 99]

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]

The intervening chord between the Diatonic and Chromatic systems, B, D, F. - This chord, which has suffered expatriation from the society of perfect chords, is nevertheless as perfect in its own place and way as any. From its peculiar relation to both major and minor, and to both diatonic and chromatic things, it is a specially interesting triad. F, which is the genetic root of all, and distinctively the root of major subdominant, has here come to the top by the prime 2. D, here in the middle, is diatonically the top of the major dominant, and the root of the minor subdominant; and on account of its self-duality, the most interesting note of all; begotten in the great genesis by the prime 3. B, the last-begotten in the diatonic genesis, top of the diatonic minor, middle of the dominant major, and begotten by the prime 5, is here the quasi root of this triad, which in view of all this is a remarkable summation of things. This B, D, F is the mors janua vitae in music, for it is in a manner the death of diatonic chords, being neither a perfect major nor a perfect minor chord; yet it is the birth and life of the chromatic phase of music. In attracting and assimilating to itself the elements by which it becomes a full chromatic chord, it gives the minor dominant the G# which we so often see in use, and never see explained; and it gives the major subdominant a corresponding A♭, less frequently used. It is quite clear that this chromatic chord in either its major phase as B, D, F, A♭, or its minor phase as G#, B, D, F, is as natural and legitimate in music as anything else; and like the diatonic chords, major and minor, it is one of three, exactly like itself, into which the octave of semitones is perfectly divided. [Scientific Basis and Build of Music, page 101]

Six Octaves required for the Birth of the Scale

EXPLANATION OF PLATES.
[BY THE EDITOR.]


THIS plate is a Pendulum illustration of the System of musical vibrations. The circular lines represent Octaves in music. The thick are the octave lines of the fundamental note; and the thin lines between them are lines of the other six notes of the octave. The notes are all on lines only, not lines and spaces. The black dots arranged in these lines are not notes, but pendulum oscillations, which have the same ratios in their slow way as the vibrations of sounding instruments in the much quicker region where they exist. The center circle is the Root of the System; it represents F1, the root of the subdominant chord; the second thick line is F2, its octave; and all the thick lines are the rising octaves of F, namely 4, 8, 16, 32, and 64. In the second octave on the fifth line are dots for the three oscillations which represent the note C3, the Fifth to F2, standing in the ratio of 3 to 2; and the corresponding lines in the four succeeding Octaves are the Octaves of C3, namely 6, 12, 24, and 48. On the third line in the third Octave are 5 dots, which are the 5 oscillations of a pendulum tuned to swing 5 to 4 of the F close below; and it represents A5, which is the Third of F4 among musical vibrations. On the first line in the fourth Octave are 9 dots. These again represent G9, which stands related to C3 as C3 stands to F1. On the seventh line of the same octave are 15 dots; these represent the vibrations of E15, which stands related to C3 as A5 stands to F1. On the sixth line of the fifth Octave are 27 dots, representing D27, which stands related to G9 as G9 stands to C3, and C3 also to F1; it is the Fifth to G. And last of all, on the fourth line of the sixth Octave are 45 dots, representing B45, which, lastly, stands related to G9 as E15 stands to C3, and A5 to F1; it is the Third to this third chord - G, B, D. The notes which arise in each octave coming outward from the center are repeated in a double number of dots in the following Octaves; A5 appears as 10, 20, and 40; G9 appears as 18 and 36; E15 appears as 30 and 60; D27 appears as 54; and last of all B45 only appears this once. This we have represented by pendulum oscillations, which we can follow with the eye, the three chords of the musical system, F, A, C; C, E, G; and G, B, D. C3 is from F1 multiplied by 3; G9 is from C3 multiplied by 3; these are the three Roots of the three Chords. Their Middles, that is their Thirds, are similarly developed; A is from F1 multiplied by 5; E15 is from C3 multiplied by 5; B45 is from G9 multiplied by 5. The primes 3 and 5 beget all the new notes, the Fifths and the Thirds; and the prime 2 repeats them all in Octaves to any extent. [Scientific Basis and Build of Music, page 102]

When these representative dots are arranged on these six Octaves of lines, at regular distances marked out by the proportionate degrees of the circle, they present to the eye this beautiful symmetrical picture of the Diatonic System of Musical Vibrations. They represent all that mathematically belongs to Music. When the notes are strung [Scientific Basis and Build of Music, page 102]

together on radial lines from the center they appear grouped in various chords and combinations, dropping out and coming in in such succession as to constitute what Ramsay, whose genius was given to set this thus before us, calls "Nature's Grand Fugue." Beginning at F in the center at the top, and moving either to the right or to the left, after a run of 7 notes we have 4 consecutive Octaves, and then comes the Minor fifth, A-E, followed by the Major fifth, G-D; and this by another Major fifth, F-C; the combinations keep changing till at the quarter of the circle we come to F, A, C, E, G, a combination of the subdominant and tonic Major; and after another varied series of combinations we have at the half of the circle the elements of 2 minor chords, D, F, A and A, C, E, and one Major chord, C, E, G; at the third quarter we have a repetition of the first quarter group; and the various chords and combinations dropping out and coming in, fugue-like; finally we return to where we began, and end with the three-times-three chord, in which the whole 25 notes are struck together, and make that wondrous and restful close of this strange Fugue. No one can hear the thrice-threefold chord of this close and ever forget it; it is "the lost chord" found; and leads the saintly heart away to the Three in One who is the Lord of Hosts; Maker of Heaven and Earth, and all the host of them. [Scientific Basis and Build of Music, page 103]

THE GENESIS.


In Fig. 1, the mathematical framework of the scales major and minor, is shown the genesis of the scale. F1, in the top figure, is multiplied by 3, and that by 3, and that by 3, which brings us to D27, top of the major dominant. F1 is the root of the whole system. C3 is the top of the first chord, and from that grows the next, and from that the next; and so we have F, C, G, and D, the tops and roots of the major system of chords. When these 3 roots are each multiplied once by 5, the middles of the chords are found, as shown - A, E, and B; so B is the last-born of the major family. When B is taken 4 octaves higher at the number 720 and divided by 3, and that by 3, and that by 3, we get the notes E, A, and D, which are the roots and tops of the minor system of chords. Dividing B, E, and A each by 5 once, we get the middles of the 3 minor chords, as shown. [Scientific Basis and Build of Music, page 103]

In the lowest Fig. this framework is filled up with all the octaves of the notes found as in the top Fig., the major being evolved upward and the minor downward, and the two systems centered at the D's, which are shown a comma distant from each other. [Scientific Basis and Build of Music, page 103]

The middle Fig. is the same, the central D's being made a pivot point on which to turn the minor down against the major in an inverse relation, setting the dual notes in file. The multiplying primes are set over the roots. The arrows point out the notes so found. The two D's are here seen right opposite of each other, because the intervals between C D in the major and E D in the minor are equal. [Scientific Basis and Build of Music, page 103]

save the octave, and made them into one, so that in its proximate meetings during its period it seems composed of the ratio 2:3 twelve times, and 3:4 seven times; twelve times 2 and seven times 3 are 45; twelve times 3 and seven times 4 are 64. This long period of 45 to 64 by its proximate meetings divided itself into 19 short periods, and oscillates between the ratios of 2:3 and 3:4 without ever being exactly the one or the other; the difference being always a very small ratio, and the excess of the one being always the deficiency of the other. This fifth, B to F, has been misnamed an "imperfect fifth." When these two notes in the ratio of 45:64 are heard together, the oscillating proximately within it of the two simple ratios gives this fifth a trembling mysterious sound. [Scientific Basis and Build of Music,page 106]


This plate is a representation of the area of a scale; the major scale, when viewed with the large hemisphere, lowest; the minor when viewed the reverse way. It is here pictorially shown that major and minor does not mean larger and smaller, for both modes occupy the same area, and have in their structure the same intervals, though standing in a different order. It is this difference in structural arrangement of the intervals which characterizes the one as masculine and the other as feminine, which are much preferable to the major and minor as distinctive names for the two modes. Each scale, in both its modes, has three Fifths - subdominant, tonic, and dominant. The middle fifth is the tonic, and its lowest note the key-note of the scale, or of any composition written in this scale. The 53 commas of the Octave are variously allotted in its seven notes - 3 of them have 9 commas, 2 have 8, and 2 have 5. The area of the scale, however, has much more than the octave; it is two octaves, all save the minor third D-F, and has 93 commas. This is the area alike of masculine and feminine modes. The two modes are here shown as directly related, as we might figuratively say, in their marriage relation. The law of Duality, which always emerges when the two modes are seen in their relationship, is here illustrated, and the dual notes are indicated by oblique lines across the pairs. [Scientific Basis and Build of Music, page 106]

It will be observed that this plate represents intervals by its areas, that is, the distances between the notes; and the notes themselves appear as points. But it must be remembered that these distances or intervals represent the vibrations of these notes in the ratios they bear to each other. So it is the vibration-ratios which constitute the intervals here pictorially represented as areas. The area, as space, is nothing; the note itself is everything. [Scientific Basis and Build of Music, page 107]

When Leonhard Euler, the distinguished mathematician of the eighteenth century, wrote his essay on a New Theory of Music, Fuss remarks - "It has no great success, as it contained too much geometry for musicians, and too much music for geometers." There was a reason which Fuss was not seemingly able to observe, namely, that while it had hold of some very precious musical truth it also put forth some error, and error is always a hindrance to true progress. Euler did good service, however. In his letters to a German Princess on his theory of music he showed the true use of the mathematical primes 2, 3, and 5, but debarred the use of 7, saying, "Were we to introduce the number 7, the tones of an octave would be increased." It was wise in the great mathematician to hold his hand from adding other notes. It is always dangerous to offer strange fire on the altar. He very clearly set forth that while 2 has an unlimited use in producing Octaves, 3 must be limited to its use 3 times in producing Fifths. This was right, for in producing a fourth Fifth it is not a Fifth for the scale. But Euler erred in attempting to generate the semitonic scale of 12 notes by the use of the power of 5 a second time on the original materials. It produces F# right enough; for D27 by 5 gives 135, which is the number for F#. D27 is the note by which F# is produced, because D is right for this process in its unaltered condition. But when Euler proceeds further to use the prime 5 on the middles, A, E, and B, and F#, in their original and unaltered state, he quite errs, and produces all the sharpened notes too low. C# for the key of D is not got by applying 5 to A40, as it is in its birthplace; A40 has already been altered for the key of G by a comma, and is A40 1/2 before it is used for producing its third; it is A40 1/2 that, multiplied by 5, gives C#202 1/2, not C200, as Euler makes C#. Things are in the same condition with E before G# is wanted for the key of A. G# is found by 5 applied to E; not E in its original and unaltered state, E30; but as already raised a comma for the key of D, E30 3/8; so G# is not 300, as Euler has it, but 303 3/4. Euler next, by the same erroneous methods, proceeds to generate D# from B45, its birthplace number; but before D# is wanted for the key of E, B has been raised a comma, and is no longer B45, but B45 9/16, and this multiplied by 5 gives D#227 13/16, not D225, as Euler gives it. The last semitone which he generates to complete his 12 semitones is B♭; that is A#, properly speaking, for this series, and he generates it from F#135; but this already altered note, before A# is wanted for the key of B, has been again raised a comma [Scientific Basis and Build of Music, page 107]

WITH THEIR RATIO NUMBERS.


In the center column are the notes, named; with the lesser and larger steps of their mathematical evolution marked with commas, sharps, and flats; the comma and flat of the descending evolution placed to the left; the comma and sharp of the ascending evolution to the right; and in both cases as they arise. If a note is first altered by a comma, this mark is placed next to the letter; if first altered by a sharp or flat, these marks are placed next the letter. It will be observed that the sharpened note is always higher a little than the note above it when flattened; A# is higher than ♭B; and B is higher than ♭C, etc.; thus it is all through the scales; and probably it is also so with a fine voice guided by a true ear; for the natural tendency of sharpened notes is upward, and that of flattened notes downward; the degree of such difference is so small, however, that there has been difference of opinion as to whether the sharp and have a space between them, or whether they overlap, as we have shown they do. In tempered instruments with fixed keys the small disparity is ignored, and one key serves for both. In the double columns right and left of the notes are their mathematical numbers as they arise in the Genesis of the scales. In the seven columns right of the one number-column, and in the six on the left of the other, are the 12 major and their 12 relative minor scales, so arranged that the mathematical number of their notes is always standing in file with their notes. D in A minor is seen as 53 1/3, while the D of C major is 54; this is the comma of difference in the primitive Genesis, and establishes the sexual distinction of major and minor all through. The fourth of the minor is always a comma lower than the second of the major, though having the same name; this note in the development of the scales by flats drops in the minor a comma below the major, and in the development of the scales by sharps ascends in the major a comma above the minor. In the head of the plate the key-notes of the 12 majors, and under them those of their relative minors, are placed over the respective scales extended below. This plate will afford a good deal of teaching to a careful student; and none will readily fail to see beautiful indications of the deep-seated Duality of Major and Minor. [Scientific Basis and Build of Music, page 109]


There are 42 intervals exclusive of the octave interval with ratio 1:2. There are seven seconds of three magnitudes, so determined in the genesis of notes - two in the ratio of 15:16; two, 9:10; and three, 8:9. There are seven corresponding sevenths - two in the ratio of 8:15; two, 5:9; and three. 9:16. There are seven thirds - one in the ratio of 27:32; three, 5:6; and three, 4:5; and there are seven corresponding sixths - three in the [Scientific Basis and Build of Music, page 109]

ratio of 5:8; three, 3:5; and one, 16:27. There are seven fifths - one in the ratio of 45:64; one, 27:40; and five, 2:3; and seven corresponding fourths - five in the ratio 3:4; one, 40:54; and one 32:45. These are the ratios of the intervals in their simplest expressions as given in the second outer space above the staff in the plate. In the outer space the intervals are given less exactly, but more appreciable, in commas. The ratios of the vibration-numbers of each interval in particular, counting from C24, are given in the inner space above the staff. These vibration-numbers, however, are not given in concert pitch of the notes, but as they arise in the low audible region into which we first come in the genesis from F1, in the usual way of this work. The ratios would be the same at concert pitch; Nature gives the numbers true at whatever pitch in the audible range, or in the low and high silences which lies out of earshot in our present mortal condition. [Scientific Basis and Build of Music, page 110]

When Ramsay gave a course of lectures in Glasgow, setting forth "What constitutes the Science of Music," his lecture-room was hung round with great diagrams illustrating in various ways his findings; an ocular demonstration was also given of the system of musical vibrations by his favorite illustration, the oscillations of the Silent Harp of Pendulums. A celebrated teacher of music in the city came to Mr. Ramsay's opening lecture, and at the close remained to examine the diagrams, and question the lecturer, especially on his extension of the harmonics to six octaves. Having seen and heard, this teacher went and shortly after published it without any acknowledgment of the true authorship; and it was afterwards republished in some of the Sol-Fa publications, the true source unconfessed; but our plagarist stopped short at C, the top of the tonic, instead of going on to F, the sixth octave of the root of all; the effect of this was to destroy the unity of the great chord. The 22 notes instead of 25, at which this teacher stopped, allowed him, indeed, to show the natural birthplace of B, which Ramsay had pointed, but it beheaded the great complex chord and destroyed its unity. If C, the root of the tonic, be made the highest note, having quite a different character from F, it pronounces its character, and mars the unity of the great chord. Similar diversity of effect is produced by cutting off only two notes of the 25 and stopping short at D, the top of the dominant; and also, though in a weaker degree, by cutting off only one note of the 25 and stopping at E, the middle of the tonic; this, too, disturbs the unity of the fundamental sound. [Scientific Basis and Build of Music, page 111]

THE 24 MAJOR AND MINOR SCALES IN #s AND ♭s AND THEIR MUTUAL PROVIDINGS.


When the major and minor scales are generated to be shown the one half in #s and the other half in ♭s, it is not necessary to carry the mathematical process through the whole 24, as when the majors are all in #s and the minors all in ♭s; because when six majors have been generated in #s, they furnish the new notes needed by the six relative minors; and when six minors have been generated in ♭s, they furnish the new notes for the six relative majors. This plate begins with the major in C and the minor in A. The notes of these two are all identical except the D, which is the sexual note, in which each is not the other, the D of the minor being a comma lower than the D of the major. Going round by the keys in #s, we come first to E minor and G major. G major has been mathematically generated, and the relative minor E gets its F# from it; but the D of C major must also be [Scientific Basis and Build of Music, page 112]

given to this scale, as the D of A minor would be a comma too low; it would make a 9-comma interval between D and E, the seventh and eighth, where the minor mode has an 8-comma one. So its two new notes are thus found in the relative and sub-relative majors. This is the way of their mutual providing in the region of the #s; the # seventh of the major is given to be the # second of the minor, and the comma-higher second of the sub-relative becomes the seventh of the minor; and then we have a true written representation of what Nature has done. [Scientific Basis and Build of Music, page 113]

Starting again at C major and A minor and going round by the keys in ♭s, we come first to D minor and F major. The major gets its fourth from the ♭ sixth of the relative minor; and as the interval between D-E, the major sixth and seventh, must be a 9-comma interval, and its own D-E is only an 8-comma one, it must take the D of A minor, which is a comma lower, and this will correctly show the 9-comma interval between D and E. This is the way of their mutual providing in the region of ♭s; the ♭ sixth of the minor is given to be the ♭ fourth of the relative major; and the comma-lower fourth of the sub-relative minor becomes the correct sixth of the major. The arrows indicate the source from which, and the place to which; the new notes come and go. [Scientific Basis and Build of Music, page 113]


When Plate XIII. is divided up the middle of the column, as in Plate XIV., so as that one side may be slipped up a fifth, representing a new key one-fifth higher, its subdominant made to face the old tonic, the two new notes are then pictorially shown, the second being altered one comma and the seventh four commas. The key at this new and higher pitch is by Nature's unfailing care kept precisely in the same form as the first; and wherever the major scale is pitched, higher or lower, the form remains unaltered, all the intervals arranging themselves in the same order. The ear, and the voice obedient to it, carry Nature's measuring-rule in them, and the writing must use such marks as may truly represent this; hence the use of sharps, flats, and naturals; these, however, be it observed, are only marks in the writing; all is natural at any pitch in the scale itself. All this is equally true of the minor mode at various pitches. These two plates are only another and more pictorial way of showing what the stave and the signature are usually made to express. [Scientific Basis and Build of Music, page 114]


One purpose of this plate is to show that twelve times the interval of a fifth divides the octave into twelve semitones; and each of these twelve notes is the first note of a major and a minor scale. When the same note has two names, the one has sharps and the other has flats. The number of sharps and flats taken together is always twelve. In this plate will also be observed an exhibition of the omnipresence of the chromatic chords among the twice twelve scales. The staff in the center of the plate is also used as to show the whole 24 scales. Going from the major end, the winding line, advancing by fifths, goes through all the twelve keys notes; but in order to keep all within the staff, a double expedient is resorted to. Instead of starting from C0, the line starts from the subdominant F0, that is, one key lower, and then following the line we have C1, G2, etc., B6 proceeds to G♭ instead of F#, but the signature-number continues still to indicate as if the keys went on in sharps up to F12, where the winding line ends. Going from the minor end, the line starts from E0 instead of A0 - that is, it starts from the dominant of A0, or one key in advance. Then following the line we have B1, F#2, etc. When we come to D#5, we proceed to B♭ instead of A#6, but the signature-number continues as if still in sharps up [Scientific Basis and Build of Music, page 114]

In the festoons of ellipses the signatures are given in the usual conventional way, the major F having one flat and minor E having one sharp. The major and minor keys start from these respective points, and each successive semitone is made a new keynote of a major and a minor respectively; and each ellipse in the festoons having the key shown in its two forms; for example, in the major F, one flat, or E#, eleven sharps; in the minor E, one sharp, or F♭, eleven flats. Thus is seen all the various ways that notes may be named. The four minor thirds which divide the octave may be followed from an ellipse by the curved lines on which the ellipses are hung; and these four always constitute a chromatic chord. [Scientific Basis and Build of Music, page 115]

SYSTEM OF THE THREE PRIMITIVE CHROMATIC CHORDS.


The middle portion with the zigzag and perpendicular lines are the chromatic chords, as it were arpeggio'd. They are shown 5-fold, and have their major form from the right side, and their minor form from the left. In the column on the right they are seen in resolution, in their primary and fullest manner, with the 12 minors. The reason why there are 13 scales, though called the 12, is that F# is one scale and G♭ another on the major side; and D# and E♭ separated the same way on the minor side. Twelve, however, is the natural number for the mathematical scales as well as the tempered ones. But as the mathematical scales roll on in cycles, F# is mathematically the first of a new cycle, and all the notes of the scale of F# are a comma and the apotome minor higher than G♭. And so also it is on the minor side, D# is a comma and the apotome higher than E♭. These two thirteenth keys are therefore simply a repetition of the two first; a fourteenth would be a repetition of the second; and so on all through till a second cycle of twelve would be completed; and the thirteenth to it would be just the first of a third cycle a comma and the apotome minor higher than the second, and so on ad infinitum. In the tempered scales F# and G♭ on the major side are made one; and D# and E♭ on the minor side the same; and the circle of the twelve is closed. This is the explanation of the thirteen in any of the plates being called twelve. The perpendicular lines join identical notes with diverse names. The zigzag lines thread the rising Fifths which constitute the chromatic chords under diverse names, and these chords are then seen in stave-notation, or the major and minor sides opposites. The system of the Secondary and Tertiary manner of resolution might be shown in the same way, thus exhibiting 72 resolutions into Tonic chords. But the Chromatic chord can also be used to resolve to the Subdominant and Dominant chords of each of these 24 keys, which will exhibit 48 more chromatic resolutions; and resolving into the 48 chords in the primary, secondary, and tertiary manners, will make 144 resolutions, which with 72 above make 216 resolutions. These have been worked out by our author in the Common Notation, in a variety of positions and inversions, and may be published, perhaps, in a second edition of this work, or in a practical work by themselves. [Scientific Basis and Build of Music, page 115]

advance by semitones, the keys with ♭s and #s alternate in both modes. The open between G# and A♭ in the major, and between D# and E♭ in the minor, is closed in each mode, and the scale made one. The dotted lines across the plate lead from major to relative minor; and the solid spiral line starting from C, and winding left and right, touches the consecutive keys as they advance normally, because genetically, by fifths. The relative major and minor are in one ellipse at C and A; and in the ellipse right opposite this the relative to F# is D#, and that of G♭ and E♭, all in the same ellipse, and by one set of notes, but read, of course, both ways. [Scientific Basis and Build of Music, page 117]


This is a twofold mathematical table of the masculine and feminine modes of the twelve scales, the so-called major and relative minor. The minor is set a minor third below the major in every pair, so that the figures in which they are the same may be beside each other; and in this arrangement, in the fourth column in which the figures of the major second stand over the minor fourth, is shown in each pair the sexual note, the minor being always a comma lower than the major. An index finger points to this distinctive note. The note, however, which is here seen as the distinction of the feminine mode, is found in the sixth of the preceding masculine scale in every case, except in the first, where the note is D26 2/3. D is the Fourth of the octave scale of A minor, and the Second of the octave scale of C major. It is only on this note that the two modes differ; the major Second and the minor Fourth are the sexual notes in which each is itself, and not the other. Down this column of seconds and fourths will be seen this sexual distinction through all the twelve scales, they being in this table wholly developed upward by sharps. The minor is always left this comma behind by the comma-advance of the major. The major A in the key of C is 40, but in the key of G it has been advanced to 40 1/2; while in the key of E, this relative minor to G, the A is still 40, a comma lower, and thus it is all the way through the relative scales. This note is found by her own downward genesis from B, the top of the feminine dominant. But it will be remembered that this same B is the middle of the dominant of the masculine, and so the whole feminine mode is seen to be not a terminal, but a lateral outgrowth from the masculine. Compare Plate II., where the whole twofold yet continuous genesis is seen. The mathematical numbers in which the vibration-ratios are expressed are not those of concert pitch, but those in which they appear in the genesis of the scale which begins from F1, for the sake of having the simplest expression of numbers; and it is this series of numbers which is used, for the most part, in this work. It must not be supposed, however, by the young student that there is any necessity for this arrangement. The unit from which to begin may be any number; it may, if he chooses, be the concert-pitch-number of F. But let him take good heed that when he has decided what his unit will be there is no more coming and going, no more choosing by him; Nature comes in [Scientific Basis and Build of Music, page 117]

with her irrevocable proportions to measure his scales for him. The stars at the C of the first scale and at the B# of the last show the coincidence of 12 fifths and 7 octaves. The number of B# is 3113 467/512; C24 multiplied 7 times by 2 brings us to the number 3072; these two notes in the tempered system are made one, and the unbroken horizon of the musical world of twelve twofold keys is created. The very small difference between these two pitches is so distributed in the 12 tempered scales that no single key of the 12 has much to bear in the loss of perfect intonation. [Scientific Basis and Build of Music, page 118]


This diagram shows pictorially the open in the spiral of the mathematical scales, in which, if written in sharps only, B# is seen a little, that is, a comma and the apotome minor, in advance of C, and as the first scale of the new cycle; for it is a violation of Nature's beautiful steps to call it a thirteenth scale of this order, since every scale in the order is 31 commas in advance of the preceding, whereas B# is only one comma and a small fraction in advance of C. If the scales be written in ♭s and #s for convenience of signature, then G# is seen a comma and apotome in advance of A♭; while the whole circle of keys advancing by fifths are each 31 commas in advance of the preceding. We may therefore cast utterly from us the idea of there being more than twelve mathematical scales, and view the so-called thirteenth as simply the first of a new round of the endless spiral of scales. There is, however, in this note a banner with the strange device, "Excelsior," for it leads us onward into ever-advancing regions of vibrations, and would at last bring us to the ultimate and invisible dynamic structure of the visible world. The tempered system of 12 keys, as in Fig. 1, is by causing the G# and A♭ to coalesce and be one, as the two D's are already literally one by Nature's own doing. [Scientific Basis and Build of Music, page 118]


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]

Another remarkable thing is that these dual numbers, when multiplied into each other, always come to 720. Now this number, as we see in the great genesis, corresponds to 1 in the major, being the point of departure for the development of the feminine mode, as 1 is the point of departure in the masculine mode. This 720 is the octave of 360, which is the number of the degrees of the circle, so divided in the hidden depths of human antiquity; and when F1 becomes F2, then B360 is the answering note and number in the dual system. All the notes in the masculine development are above F2; and all the notes in the feminine development are below B360. The unoccupied octave between F1 and F2 and that between B720 and B360 may be counted as the octave heads or roots of the two modes, and then F2 and B360 as the points from which the development of music's diversity begins; and it is noteworthy that the number of the degrees of the circle should be found in this connection. When was the circle so divided? Who divided it so? And why did he, the unknown, so divide it? Was Music's mystery known in that far-off day before the confusion of man's sinking history had blotted out so much of the pure knowledge of pristine days? [Scientific Basis and Build of Music, page 119]


This plate, in the outer stave, has the 32 notes which arise with mathematical development of twelve scales in advancing fifths. The notes are marked with sharps, flats, and commas. The flats and commas of lowering are placed on the left of the notes, in the order in which they arise, reading them from the note downward; the sharps and commas of rising on the right, also reading from the note upward. The whole of these 32 notes are brought within the compass of an octave. [Scientific Basis and Build of Music, page 119]

The inner stave contains the chromatic scale of twelve notes as played on keyed instruments. The flat and sharp phase of the intermediate notes are both given to indicate their relation to each other; the sharpened note being always the higher one, although seemingly on the stave the lower one. The two notes are the apotome minor apart overlapping each other by so much; ♭D is the apotome lower than C#; ♭E the apotome lower than D#; F# the apotome higher than ♭G; G# the apotome higher than ♭A; and A# the apotome higher than ♭B. The figures for the chromatic scale are only given for the notes and their sharps; but in the mathematical series of notes the numbers are all given. [Scientific Basis and Build of Music, page 120]

Fig. 2. - In this figure the two modes are placed in their inverse relation, in order to show the notes standing opposite each other in their duality. Here the two D's come also opposite each other, inasmuch as in the two modes the C-D-E interval is inverted, becoming C D E in the one and E D C in the other. And so the 9-comma second between C and D in the major comes opposite the 9-comma second between D and E in the minor, and the two 8-comma seconds, of course, come opposite each other also. [Scientific Basis and Build of Music, page 120]


The curved lines enclose the three chords of the major mode of the scale, with the ratio-numbers for the vibration in their simplest expression, counted, in the usual way in this work, from F1, the root of the major subdominant. The chords stand in their genetic position of F F C A, that is F1 by 2, 3, and 5; and so with the other two. The proportions for a set of ten pendulums are then placed in file with the ten notes from 1 to 1/2025 part of 1. Of course the one may be any length to begin with, but the proportions rule the scale after that. [Scientific Basis and Build of Music, page 121]

See Also


Frequency
Interval
Key
note in common
Pitch
Scale
Tone
triad
Triplet

Created by admin. Last Modification: Monday January 18, 2021 04:15:36 MST by Dale Pond.