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middle

middle or central note of a three note chord

Russell
Opposite poles of a bar magnet thrust away from each other as far as they can go. That is the very purpose of the electric current which divides the universal equilibrium. If opposite poles attracted each other they would have to be together in the middle, instead of "pushing" away from each other to the very ends. [A New Concept of the Universe, page 18]


Ramsay
"The prime 5, like the prime 3, produces new notes. One of these, namely A5, is derived from unity, i.e., the note produced by the ratio of 1:2; the second note is produced from the note derived from the first power of 3, namely E15; and the third is produced from the note derived from the second power of 3, namely B45. The notes thus produced by the prime 5 are the middles, that is, the thirds of the chords. As it is the second and third powers of 3 which possess great centrifugal force, and not the first power of that number; and as it is only the first power of the number 5 which Nature employs in this business, so this makes the character of the notes produced by the prime 5 to depend on the character of the notes from which they are derived. One of the 3 notes produced by the prime 5 is derived from unity, that is the note produced by the ratio of 1:2, and like that note it is strongly acted on by the force of gravity.1 A second note produced by the prime 5 is derived from the note produced by the second power of 3, and like that note it possesses increased" [Scientific Basis and Build of Music, page 27]

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]

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]

BY THE EDITOR


The Greeks most probably constructed their musical tetrachords in a symmetrical order in analogy with their sculpture, and showed the ear identical with the eye in its love of symmetry. With them, therefore, the Dorian mode would have a certain pre-eminence. Beginning this mode on D, without knowing the musical mystery that resides in D, they had two tetrachords with the semitones symmetrically in the middle in one mode; it was next possible for them to arrange in pairs, symmetrically, the other tetrachords.

D8 E5 F9 G8 A9 B5 C9 D
E5 F9 G8 A9 B5 C9 D8 E — C9 D8 E5 F9 G8 A9 B5 C
F9 G8 A9 B5 C9 D8 F5 F — B5 C9 D8 E5 F9 G8 A9 B
C9 D8 E5 F9 G8 A9 B5 C — E5 F9 G8 A9 B5 C9 D8 E
B5 C9 D8 E5 F9 G8 A9 B — F9 G8 A9 B5 C9 D8 E5 F
A9 B5 C9 D8 E5 F9 G8 A — G8 A9 B5 C9 D8 E5 F9 G
G8 A9 B5 C9 D8 E5 F9 G — A9 B5 C8 D9 E5 F9 G8 A

[Scientific Basis and Build of Music, page 45]

The CHROMATIC SYSTEM of chords is developed from these three primitive chromatic chords, and in the course of its development one or two notes are brought in semitonic progression to the middle, one or two to the root, and one or two to the top of all major and minor tonic chords. Likewise, at one time or another in the course of the system, there is one note in common with the middle, one note in common with the root, and one note in common with the top of all the major and minor tonic chords. [Scientific Basis and Build of Music, page 57]

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]

Now we come to a remarkable arrangement of Nature. The minor does not grow in the same way out of this third chord's top. Two features come before us: first the minor chord grows out of the major, but it is taken not from the top but from the middle, from a rib out of his side. B, the middle of the major dominant chord; B, the last-born of the major genesis; B is the point of departure in the outgrowth of the minor mode. The feminine is a lateral growth from the masculine. Another feature: it grows downward, like a drooping ash or willow. Its first generated chord is its dominant, and its last is its subdominant. Its middle chord, like the middle one of the major, is its tonic. Still further, it is generated by division, not multiplication; B45 is divided by 3 and by 5 for the root and middle of this highest chord, E and G. E15 is divided by 3 and 5 for the root and middle of the tonic chord, A and C. A5 is divided by 3 and 5 for the root and middle of the lowest chord, D and F. Thus we have the whole generation of the elements of music, six generations of harmony, like the six days of creation. Up to this point the whole process and aspect is inverse; growing from a middle; growing downward; growing by division;- while the major is growing from the top; growing upward; growing by multiplication. But here the inverse aspect ends. The generating primes of the major are 3 and 5; 3 and 5 are also the generating primes of the minor. In this essential phase of their creation their comparison is direct, not inverse. [Scientific Basis and Build of Music, page 67]

The varied effect of position in chords. When a chord stands as C E G C, having its root also at the top, it has its softest, dullest, most united effect; it is undramatic, with little contrast. When it stands as E G C E, having its third at the top and bottom, it has a more ticklish, interesting, far-away effect. In reveries composers often finish thus, as if it had vanished - an unsettled effect. When it stands as G C E G, with its top at top and bottom, it has its most dominant character - loud, swelling. In the position C E G C it stands mixingly with the subdominant C E f G a C, and in this its first position its unseen filling in is chiefly from the region of gravity; hence its soft, grave, dull, heavy effect; and it passes very easily to the subdominant chord. When it stands as G C E G it stands mixingly with the dominant G b C d E G, and has its third position and most brilliant effect and uprising, for its unseen filling in is then chiefly from the region of levity; and it passes easily to the dominant chord. When in its second position, its middle position E G C E, its unseen filling in is mixingly both subdominant and dominant, E f G a b C d E; it has then its most interesting and puzzling effect; on the one hand its softest, dullest, and one-est, on the other hand its most brilliant effect, as if it would at once both sink and soar. [Scientific Basis and Build of Music, page 72]

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]

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]

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Make middles in the bass as much as possible. Roots and middles, and middles and tops, do well in arrangements; for example, F and A. [Scientific Basis and Build of Music, page 85]

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]

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]

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]

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]

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]

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]

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]

DIATONIC RESOLUTIONS, SIMPLE AND COMPOUND.


In the major system, when the tonic chord follows the subdominant one, there is one semitonic progression to the middle of the tonic, and one note in common with the root, so these two chords are linked together in different ways. When the tonic chord follows the dominant one, there is one semitonic progression to the root of the tonic, and one note in common with its top, so these two chords also are linked together in two different ways. When the tonic chord follows the compound dominant, i.e., the dominant seventh, there are two semitonic progressions, one to the middle and one to the root, and one note in common with its top, so these two are linked together in the same two ways; but the semitonic progression being double gives this resolution great urgency. And now we come to the two chords, the subdominant and dominant, which have no note in common, and must, when they succeed each other, be helped to come together. Nature teaches us how this is to be done by a process of borrowing and lending which will establish between them a similar relationship to that which keeps the continuity of the other chords in succession. We have seen that the top of the subdominant and the root of the tonic are a note in common to these chords, and so the top of the tonic and the root of the dominant also are a note possessed in common by these two chords. In like manner in this disjunct part, when the dominant follows the subdominant, the root of the subdominant is lent to the top of the dominant, and thus they come to have a note in common. The top of the [Scientific Basis and Build of Music, page 111]

PLATES XVII. & XVIII.


These two plates show the chromatic chord resolving into the twelve major and twelve minor tonic chords of the twenty-four scales. There seems to be twenty-five, but that arises from making G♭ and F# in the major two scales, whereas they are really only one; and the same in the minor series, E♭ and D# are really one scale. C in the major and A in the minor, which occur in the middle of the series, when both sharps and flats are employed in the signatures, are placed below and outside of the circular stave to give them prominence as the types of the scale; and the first chromatic chord is seen with them in its major and minor form, and its typical manner of resolving - the major form rising to the root, and falling to the top and middle; the minor form falling to the top, and rising to the root and middle. The signatures of the keys are given under the stave. [Scientific Basis and Build of Music, page 116]

See Also


center of the system
center
fulcrum
middle chord
middle in the majors
middle in the minors
middle of the dominant
middle of the major subdominant
middle of the minor dominant
middle of the subdominant minor
seesaw
tonic chord

Created by Dale Pond. Last Modification: Saturday January 2, 2021 03:07:09 MST by Dale Pond.