Ramsay
"lower effect than the fifth; the seventh, B, has a higher effect than the sixth; but the eighth, C, has a lower effect than the seventh. If the effects of notes or chords depended wholly on the mathematical primes by which they are measured and located, or the ratios inherent in them, then the effects of the tonic, subdominant, and dominant chords would have been alike, for these chords are measured by exactly the same primes, and have exactly the same ratios. It is the position of the tonic chord which gives it its importance and not any special primes by which it is produced, nor any special ratios inherent in it. Notes by the power of 2 have a pure unmixed and invariable character. Notes by the first, second, and third powers of 3 have different degrees of centrifugal force; and the character of the notes produced by the first power of 5 depends on the character of the notes from which they are derived. The final character of 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 where these notes may be afterwards placed, like chemical elements, they never lose their original forces and tendencies. What Tyndal says of the inorganic chemical elements of the brain is true of the inorganic notes of music, "They are all dead as grains of shot." It is the organic state which gives the notes and chords their gravities and (levity|levities, and these two tendencies, the one upward and the other downward, constitute the vital principle of music. It is true that the mathematical operation is required to give birth and life to music, and that the mathematical system gives the knowledge of causes down to the law of gravitation, yet the artistic effects are fully realised from the tempered system deriving its organic harmony from this vital principle of music. The centrifugal tendencies of the notes of the subdominant, are too strong to be at all disturbed by the system being tempered. The enormous power of these chords corrects the effect which might otherwise arise from tempering, as the enormous power of the sun corrects the perturbations of the planets." [Scientific Basis and Build of Music, page 29]
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]
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]
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 sharp, ?, 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 sharp, ?, or ?, as the case may be. [Scientific Basis and Build of Music, page 94]
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]
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]
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
We here trace the twelve harmonies developing in succession. Notice how exactly they all agree in their mode of development; also the use of the chasms between E and F, B and C. Remark also the beautiful results from the working of the double tones, especially C#-D?, and E#-F?, causing the seven tones of each harmony, when ascending, to rise one tone, and, descending, to reverse this movement. F#-G? is the only double tone which acts as F# when a key-tone, and G? when the root of D?. The root of each harmony is the sixth and highest tone in each succeeding harmony, rising one octave; when it is a double tone, it sounds according to the necessity of the harmony. The intermediate tones are here coloured, showing gradual modulation. The isolated fourths (sounding sevenths) were the previously developed key-tones; these also alter when they are double tones, according to the necessity of the harmony. Beginning with B, the isolated fourth in the harmony of C, the tones sound the twelve notes of a keyed instrument, E# being F?, and the double tones, some flats, some sharps. [Harmonies of Tones and Colours, Combinations of dissonance, rests, page 24]
In the musical clef, the sixth and seventh notes from the fundamental key-note C (F and #F) are repeated, so that the use of the two poles (#F and ?G) may be clearly seen, and that the notes and colours precisely agree. [Harmonies of Tones and Colours, Diagram III - The Major Keynotes Developing by Sevens, page 25a]
Below, the 6th and 7th Key-notes are repeated, to shew the use of the poles F#, G?. [Harmonies of Tones and Colours, The First Circle are 7 Keynotes, page 25c]
AS an example of the twenty-four, compare A major, developing, in Diagram II., with A minor, Diagram IX., taking the notes in the order which they sound in trinities. The three notes of the primaries sounded by A minor are, first, the same root as the major; the two next are the fourth and seventh higher notes (in the major, the fifth and sixth); the secondaries only vary by the sixth and seventh notes being a tone lower than in their relative major. Observe the order in which the pairs unite; the fourth in depth, sounded seventh, isolated. A and its root do not rise from the chasms. The fundamental key-note C was seen not to be interfered with, neither is the fundamental minor key-note A; G# on the one side, and B? on the other, being the key-notes. The seven of each minor harmony embrace only seventeen tones. C major and A minor are the only two keys which sound the seven white notes of keyed instruments. The minor scale and chords of A are not included in this remark. [Harmonies of Tones and Colours, Diagram IX - The Minor Keynote A and Its Six Notes, page 34a]
In the musical clef the sixth and seventh notes from A, the fundamental minor key-note, are repeated, in order to show the use of the poles D#-C?, and that the colours agree. The use of the two poles, both in the major and minor series, is strikingly evident. [Harmonies of Tones and Colours, Diagram X - Minor Keynotes Developing by Sevens, page 35a]
In the Minor Scale, the Trinities and Scale develope five pairs; the last pair become the fifth higher key-note and its root, consequently the sixth pair would develope the higher key.[Harmonies of Tones and Colours, The Seven of each Harmony with its Scale, page 59]
See Also