Sympathetic Vibratory Physics

Ramsay
"The third note of the octave scale, E, the center of the tonic chord in the key of C, is the center of the system. It is the note which has the least tendency either upward or downward, and it has immediately above it in the octave scale the note which has the greatest amount of specific gravity, F, the root of the major subdominant; and immediately beneath it the note which has the greatest amount of specific levity, D, the top of the major dominant. Thus the root of the subdominant chord and the top of the dominant are placed right above and below the center of the system, and the gravity of the one above, and the levity of the one below, causes each of them to move in the direction of the center. These tendencies are seen in the scale at whatever key it may be pitched, and by whatever names the notes may be called. And it is on account of this permanency of character of the notes that the third note of the scale, E, in the key of C, has a lower effect¹ than the second, D; and that the fourth note, F, has a lower effect than either the first, second, or third; the fifth note, G, has a higher effect than the fourth, F; but the sixth, A, has a.." [Scientific Basis and Build of Music, page 28]

In order to find the notes for the next major key above C, we have to multiply the vibration-number of D, which is the top of the dominant C, by 3 and 5. It is out of the key of C at this point that the new key sprouts and grows, and by the primes and method which produce the key of C itself. So if we would find the relative minor of C, let us take the note which is a minor third below D - that is, B - to produce the minor. The minor sprouts and grows from this point of the key of C; for the relative minor grows out of the major, as out of the man at first the woman is taken. Moreover, B is the last-born of the notes for the major scale; for the middles, that is, the thirds of chords, are always produced by the prime 5; and the tops, that is, the fifths of chords, are produced by the prime 3, and are born before the thirds, though placed after them in the chords. Well, because B is the last-born note of the major, as well as a minor third below the top of the highest chord of the major, it seems that the minor should have this for its point of departure. Again, we have seen that the major and the minor are found in their strings and their vibrations by an inverse process, that one going back upon the other; and, there taking Nature's clue, let us proceed by an inverse process of generating the minor. Making B45 our unit, as F1 was our unit for the major, let us divide by 3 and 5 for a root and middle to B, as we multiplied by 3 and 5 for a top and middle to F. B45 divided by 3 is 15; here then is our E, the root of the chord, just where we had found it coming upward; for, remember, we found E15 by multiplying C3 by 5. This E, then, is the same in major and minor. Now B45 divided by 5 is 9; [Scientific Basis and Build of Music, page 31]

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]

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]

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 various raisings and lowerings of notes in advancing keys, major and minor. - In each fifth of the majors ascending the top of the dominant is raised a comma. A40 in the key of C becomes A40 1/2 in the key of G; E60 in the scale of G is E60 3/4 in the scale of D; B90 in the scale of D is B91 1/8 in the scale of A. This alteration of the top of the dominant major goes on through all the twelve scales. Similarly, by the Law of Duality, each fifth in the minors descending has the root of the subdominant lowered a comma. D54 in the key of E minor is D53 1/2 in the key of A; G72 in the scale of A is G71 1/9 in the scale of D; C48 in the scale of D is C47 11/27 in the scale of G. This alteration of the root of the subdominant goes on through all the twelve minor scales. [Scientific Basis and Build of Music, page 62]

In just such a manner, only by more obvious leaps, the middle of the dominant in the advancing major scales is raised a sharp - i.e., four commas. When D27, the dominant top of the key of C, is multiplied by 5, it generates F#135; so, taking it one octave lower, F64 in C major is F#67 1/2 in the key of G. C96 in the key of G is C#101 1/4 in the key of D; G72 in the key of D is G#151 7/8 in the key of A. And this raising of the middle of the dominant goes on through all the twelve major keys.[Scientific Basis and Build of Music, page 62]

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]

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

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

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

PLATE X.

THE GROWTH-LIKE CONTINUITY OF CHORDS AND KEYS.

Under the symbol of a music plant this plate gives us to realize the growth-like continuity of chords and scales. The roots of the three chords of a key are represented in F, C, and G of the key of C. The plant might be represented as a creeping stem, like the creepers of the strawberry, with its progressive roots struck into the earth; but it is better to show an upward stem with aerial roots, for such are the roots of the musical plant. The main stem of the plant has the three chords, F a C e G b D; that is, F a c, C e g, G b d, the subdominant, tonic, and dominant. The terminal chord, D f# a, is to show that the keys as well as the chords GROW out of each other. Include the side branches which terminate with the octave notes of the chords, read thus - F a c f, G e g e, G b d g, because a chord is felt to be most complete in its unity when thus shut in by the octave note of its root. This is the reason why the great three-times-three chord does not stop at D, the top of the dominant chord, but goes on to the sixth octave of the fundamental root, shutting all in by the great peacemaker, F, in order to preserve the unity of the effect which this chord of chords produces. Before D. C. Ramsay showed that the scale of Harmonics extended to six octaves, it was by teachers of the science of music only extended to four. [Scientific Basis and Build of Music, page 110]

PLATE XXII.

THE MATHEMATICAL TABLE OF THE TWELVE MAJOR SCALES AND THEIR RELATIVE MINORS.

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]

Hughes
The modulating gamut
—One series of the twelve keys meeting by fifths through seven octaves
—Keys not mingled
—A table of the key-notes and their trinities thus meeting
—The fourths not isolated
—The table of the twelve scales meeting by fifths
—The twelve keys, trinities, scales, and chords thus meeting are written in musical clef
—The twelve meeting through seven circles, each circle representing the eighteen tones
—The keys of C and G meeting, coloured
—Retrospection of the various major developments, . . . . 29 [Harmonies of Tones and Colours, Table of Contents3 - Harmonies]

ON a keyed instrument only twelve are major key-notes, but as the double tones C#-D? and F#-G? are roots, there are fourteen different chords. The fourteen that are roots are written in musical clef. As an example of the major chords in the different keys, we may examine those in the key of C. A major fifth includes five out of the seven of its key; with the third or central note it is the threefold chord, or fourfold when the octave note is added. Including the silent key-notes, a threefold chord embraces eight, or, counting the double tones, not including E#, eleven. The first and second chords of the seven of the harmony are perfect major chords in the key of C; the central note of the third chord, being #C-?D, is a discord. The first pair of fifths in the scale, with its central note, is a chord of the key; if we include the octave, the last pair of fifths, with its central note, is the same chord an octave higher than the lowest chord of the seven. Of the chords written in musical clef of the twelve keys, the octave chord is only written to C, the seven of each having two chords and the scale one, thirty-six in all, or forty-eight if the octave chords are added. Notice how the chords of each seven and the chord of its scale are altered. [Harmonies of Tones and Colours, Diagram V - The Chords of the Twelve Major Keys, page 27a]

The keys of C and G meeting are coloured, and show the beautiful results of colours arising from gradual progression when meeting by fifths. Each key-note and its trinities have been traced as complete in itself, and all knit into each other, the seven of each rising a tone and developing seven times through seven octaves, the keys mingled. The twelve scales have been traced, developing seven times through seven octaves, all knit into each other and into the key-notes and their trinities. The chords have also been traced, each complete in itself, and all knit into each other and into the key-notes, trinities, and scales. And lastly, one series of the twelve keys, no longer mingled, but modulating into each other, have been traced, closely linked into each other by fifths through seven octaves, three keys always meeting. Mark the number of notes thus linked together, and endeavour to imagine this number of tones meeting from the various notes. [Harmonies of Tones and Colours, The Twelve Scales Meeting by Fifths, page 31a]

Ascending, begin with C in the innermost circle, F being its root. The Key-note C becomes the root of G, G becomes the root of D, and so on. In descending, begin with the octave Key-note C in the outermost circle. F, the root of C, becomes the fifth lower Key-note. F is the next Key-note, and becomes the root of B?, &c. The 12 Keys in their order are written in musical clef below. Lastly, the Keys of C and G, ascending on a keyed instrument, are written in music as descending; therefore, to shew correctly notes and colours meeting, it is necessary to reverse them, and write C below G. All are seen to be complementary pairs in tones and colours. [Harmonies of Tones and Colours, Diagram VII Continued2, page 31e]

Sympathetic Vibratory Physics

Bridging Science and Spirituality

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