Sympathetic Vibratory Physics

Joseph Brye
A scale in which the half steps occur between the third and fourth and the seventh and eighth tones.

The major scale consists of consecutive tones (half and whole steps) from one letter name to its repetition above or below, such as C D E F G A B C, in which all are whole steps except those from E to F (3 to 4) and B to C (7 to 8) in succession, which are half steps. This is called a diatonic scale, because the scale steps follow the letter names in succession without alteration and include five whole steps and two half steps in a definite pattern. [Brye, Joseph; Basic Principles of Music]

Ramsay
When we have got F1, and from it C3 and A5 by the primes of 3 and 5 multiplying 1, then all the octaves of the these three notes will be found by the prime 2, multiplying by it for the higher, and dividing by it for the lower octaves. When from C3 we have got G9 and E15, multiplying by the primes 3 and 5, then the octaves of these are also found by the prime 2, used again in the same way. And when from G9 we have got D27 and B45 by the primes 3 and 5, the octaves of these are also found by the prime 2. The prime 2 has an unlimited use; the prime 3 is used in the first power, the square, and the cube; the prime 5 in the first power only. Thus is evolved the true major scale, and no need for a B? or any other tinkering. [Scientific Basis and Build of Music, page 31]

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

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]

The major scale is composed of three fifths with their middle notes, that is to say, their thirds. And as three such fifths are two octaves, less the small minor third D to F, taking the scale of C for example, so these three fifths are not joined in a circle, but the top of the dominant and the root of the subdominant are standing apart this much, that is, this minor third, D, e, F. Had they been joined, the key would have been a motionless system, with no compound chords, and no opening for modulation into other keys. [Scientific Basis and Build of Music, page 38]

The structure and quantity of the three fifths in a major scale are always 9, 8, 9, 5 = 31 commas; but the structure and quantity of the fourth fifth is 8, 5, 9, 8 =3 0 commas; F, A, C, = 31 commas; C, E, G, = 31 commas; G, B, D, = 31 commas; d, f, a, 8, 5, 9, 8=30 commas. - Editor. [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]

Through the whole system, in the progression of major scales with sharps and minor scales with flats, the new sharp is applied to the middle of the major dominants, and the new flat to the middle of the minor subdominants. In the progression of major scales with flats and minor scales with sharps, the new flat is applied to the root of the major subdominant, and the new sharp to the top of the minor dominant.² [Scientific Basis and Build of Music, page 43]

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]

There is nothing too small or remote to get beyond the reach of the Law of Duality; it follows the major and minor scales through all their inverse and reciprocal progressions, and by-and-by it appears at the extremity of complex music in the shape of inverse fugue. [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]

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]

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]

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]

Moreover, it is only from one to five, that is from C to G in ascending, which is its proper direction in the genesis, that the major in being harmonized does not admit of minor chords, but if we descend this same natural major scale of the fifth from five to one, that is from G to C, the first chord is C E G; the next chord is F A C; if this is succeeded by the minor chord A C E, there are two notes in common and one semitonic progression, as very facile step in harmony; and the following two notes are most naturally harmonized as minor chords. So modulation into the minor, even in this major scale, is very easy in descending, which is the proper direction of the minor genesis.² In a similar way, it is only from five to one, that is from E to A in descending, which is its proper genetic direction, that the minor in being harmonized does not admit of major chords; but if we ascend this same minor scale of the fifth from one to five, the first chord is A C E, the next is E G B, and if this chord be followed by the major C E G, there are here again two notes in common and one semitonic progression; and the two notes following are then most naturally harmonized as major chords. So modulation into the major, even in this minor scale, is very natural and easy in ascending, which is the proper direction of the major genesis.³ The dominant minor and the tonic major are, like the subdominant major and the tonic minor, very intimately related in having two notes in common and one semitonic progression. [Scientific Basis and Build of Music, page 65]

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]

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]

D27 of the major scale; and the number 27 as string-length will give the vibrations of D26 2/3 in the minor scale, and so all through; they stand thus:-

Lengths 30 26 2/3 24 22 1/2 20 18 16 15 Vibrations
Vibrations 24 27 30 32 36 40 45 48 Lengths
[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]

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]

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]

The scales march on following each other methodically, whether they be written with sharps or flats, and

"Not a step is out of tune, as the tides obey the moon."

The most natural, because the genetic, way to write the scales is to make the major scales all in sharps, after C, because the major genesis is upward in ratios ascending; and to make the minor scales all in flats, after A, because the minor genesis is downward in ratios descending. Let the young student, however, always keep in mind that the sharps and flats are simply marks to show how Nature, at whatever pitch we are taking the scales, is securely keeping them in the same form as when they are first generated; and in their birthplace no sharps or flats are needed. [Scientific Basis and Build of Music, page 90]

PLATE VI.

THE AREA OF A SCALE.

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]

The Plate shows the Twelve Major and Minor Scales, with the three chords of their harmony - subdominant, tonic, and dominant; the tonic chord being always the center one. The straight lines of the three squares inside the stave embrace the chords of the major scales, which are read toward the right; e.g., F, C, G - these are the roots of the three chords F A C, C E G, G B D. The tonic chord of the scale of C becomes the subdominant chord of the scale of G, etc., all round. The curved lines of the ellipse embrace the three chords of the successive scales; e.g., D, A, E - these are the roots of the three chords D F A, A C E, E G B. The tonic chord of the scale of A becomes the subdominant of the scale of E, etc., all round. The sixth scale of the Majors may be written B with 5 sharps, and then is followed by F with 6 sharps, and this by C with 7 sharps, and so on all in sharps; and in this case the twelfth key would be E with 11 sharps; but, to simplify the signature, at B we can change the writing into C, this would be followed by G with 6 flats, and then the signature dropping one flat at every new key becomes a simpler expression; and at the twelfth key, instead of E with 11 sharps we have F with only one flat. Similarly, the Minors make a change from sharps to flats; and at the twelfth key, instead of C with 11 sharps we have D with one flat. The young student, for whose help these pictorial illustrations are chiefly prepared, must observe, however, that this is only a matter of musical orthography, and does not practically affect the music itself. When he comes to the study of the mathematical scales, he will be brought in sight of the exact very small difference between this B and C?, or this F# and G?; but meanwhile there is no difference for him. [Scientific Basis and Build of Music, page 108]

PLATE XII.

THE 24 MAJOR AND MINOR SCALES IN sharp AND ?s AND THEIR MUTUAL PROVIDINGS.

When the major and minor scales are generated to be shown the one half in sharp 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 sharp and the minors all in ?s; because when six majors have been generated in sharp, 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 sharp, 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]

PLATE XIV.

THE VERTEBRAL COLUMN IN MODULATION.

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]

PLATE XV.

THE CHROMATIC SCALE AND THE CHROMATIC CHORDS.

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]

Hughes
THE same laws are followed here as in the development of the major scales. In that of A, F, the sixth note, has risen to F#, in order to meet B, which has previously sounded. In descending, the seventh note, B, falls to B?, in order to meet F, which has also previously sounded. The notes, ascending or descending, always follow the harmony of their key-note, except when rising higher or falling lower to meet in fifths. We may here trace the twelve, the ascending scale sounding the fifth harmony higher than its key-note, and, in descending, sounding the fifth lower harmony. The four pairs of each scale are written at the end of the lines. If we strike the twelve scales as they follow in succession, the thirteenth note being the octave of the first, and leader of a higher twelve; having gained them six times, at the seventh they gradually rise (though beyond the power of a keyed instrument) into the higher series of seven octaves, and again, in descending, they fall lower, and are linked into the lower series of seven octaves. Nine notes of any ascending minor scale may be struck without the necessity of modulating beyond the fifth harmony. For example, in the scale of A, its tenth note, C#, rises to meet the sixth note, which has previously sounded. In descending, E?, the eleventh note, meets B?, the seventh note, which has previously sounded. The scale of A may be traced veering round by reference to Diagram IX., beginning with A, and carrying the four lowest notes an octave higher, F rising to F# in ascending, B falling to B? in descending. [Harmonies of Tones and Colours, Diagram XI - The Twelve Minor Keynotes with the Six Note of Each, page 36a]

The Major Scales are the type of Creation perfected—man being created, and the Almighty resting—every Major Scale developing the sixth and seventh notes, and the eighth the octave of the first. Therefore, every Major Scale includes the Sabbath, or Rest. [Harmonies of Tones and Colours, Supplementary Remarks, page 54]

The same laws are followed here as in the development of the Major Scales, except that the Minor Scales only develope five notes. [Harmonies of Tones and Colours, The Seven of each Harmony with its Scale, page 59]

Sympathetic Vibratory Physics

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