(1) The art of music, which is music on its spiritual and inspirational side, has been carried to a wonderful perfection of development; while the science of music, which is music on its intellectual and logical side, has been left far behind. Works on the science of music have been a failure, not because music has not a scientific basis, but, and for the most part, because Mathematicians have dealt only with the Laws of Ratios, ignorant of other laws which play an important part in music's scientific basis and build. They have carried the Laws of Ratios beyond its legitimate sphere, and so their conclusions do not represent the method of nature truthfully.
(2) The division of the octave into twelve semi-tones:
(3) Nature's own art, not man's device, enables us to realise and represent all the harmonic progressions that belong to music. It has been nature's self to the practical and inspired musician. By the division of the octave into twelve semi-tones, he is emancipated from the restrictions of jargon which arise from misapplied mathematics. The mathematicians say that such and such things are wrong; the great masters inspirationally do these very things. 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 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 musical science. It should not be forgotten that the science of the mathematical theorists is one thing, and that the composers is another.
(4) Schubert, Beethoven, Mozart, Haydn, 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.
(5) In music, Nature begins by producing the Diatonic Octave of seven notes, derived by the mathematical ratios; and when she has gone through all her peregrinations, she is found to have produced the Chromatic scale of twelve semitones, derived from her own vital operations: so that there are no anomalies. 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 (seven) 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 mathematical basis, and when mathematicians do not recognise this they reckon without their host.
(6) The number 3 is the creative power in music, producing 5ths, but it is under the control of the Octave prime - the number 2. It is the supreme octave which forms a boundary by making twelve fifths and seven octaves unite in one note. Within this horizon lies the musical system in its threefoldness - major (sharped), minor (flatted) and chromatic (accented).
(7) Twelve divisions in the octave serve all the purposes of music. This is the master-stroke of Nature in putting so much in seemingly so little. Twelve is the most genial of all numbers; it is nature's representative of the social order in music. It is the manifold divisibility of twelve which makes the chromatic system in music possible. The equalised scale of twelve semitones in the octave and the chromatic system of music are indissolubly connected. With this the scale of mathematical intonation has neither part nor lot.
(8) The life force of the notes from the LAW OF POSITION gives them a versatility which they could never have had from fixed ratios, however numerous. If the interval of the octave be excepted, there are no two notes together in a chord, nor succeeding each other in the octave scale, having the same amount of specific levity or gravity; consequently each note has an expression and character of its own. And as nature has constituted them, these various forces all converge to the centre of the Tonic Chord, and, with exception of the interval of the octave itself, the notes of the tempered scale being a little nearer each other than the mathematically perfect notes, these converging forces and this tempering mutually assist each other, and give a greater decision to the resolution of chords.
(9) The twelve semitones being the practical fulfilment of the ratios when the life force of notes is considered, the greater masters had the ratios in their key-boards, and this along with the musical ear was the sure word of prophecy to them. In their great works they have thus been enabled to develop the science of music, and to express it in the language of art.
(10) Mathematicians have not recognised 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 semi-tones 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 may assume that the interval of the octave, like the other intervals should have a grave harmonic. But the fact that the octave interval has not a grave harmonic, while all the other intervals have, shows that the natural laws are not confined to one style of working.
(11) 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 RATIOS 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 LAW 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 notes and chords their specific levities and gravities; and these two tendencies, the one upwards 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:
"A primrose by a river's brim, A Yellow primrose was to him And it was nothing more"
(12) If the effects of notes and chords depend 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.
(12.1.1) The ratio of 2 to 1 has a pure, unmixed, invariable character, always realised in the interval of the octave. (fixed, harmonic)
(12.2) The notes produced from 1 by the first (31 = 3) , second (32 = 9), and the third powers of 3 (33 = 27) have different degrees of centrifugal force. (dispersive power) (Is Ramsay counting from 0 as first power? See next paragraph.)
(12.3) The character of the notes produced by the first power of 5 (51 = 5) depends on the character of the notes from which they are derived, namely, 1, 3, and 9. (5 derived = changeable) (See position counting from aliquot numbers of 5ths below. If a "5" is developed from a 9 then it is empowered by the 3rd power of 3. If 5 is developed from an 8 it is powered by the third power of 2. This may not be what is meant by Ramsay.)
(12.4) 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 (1) the way in which they have been derived, and from (2) their position in the system; and no matter where these notes may afterwards be placed, like chemical elements they never lose their original natures within the same key system.
(13) 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 centre 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 fifth ( 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 a fourth fifth is 8,5,9,8 = 30 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) 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 placed 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.
zeroeth power of 2 (20) = 1 first power of 2 (21) = 2 second power of 2 (22) = 4 third power of 2 (23) = 8 (cube) fixed/balanced
zeroeth power of 3 (30) = 1 first power of 3 (31) = 3 second power of 3 (32) = 9 third power of 3 (33) = 27 (cube) centrifugal
zeroeth power of 5 (50) = 1 first power of 5 (51) = 5 centripetal? second power of 5 (52) = 25 third power of 5 (53) = 125 (cube) changeable
three sequential fifths in a major scale are (each) always: 9, 8, 9, 5 = 31 commas; structure and quantity of the fourth (sequential) fifth is: 8, 5, 9, 8 = 30 commas:
F, A, C = 9, 8, 9, 5 = 31 commas; Imperfect 5th (9 circles) C, E, G = 9, 8, 5, 9 = 31 commas; Perfect 5th (9 circles) G, B, D = 9, 8, 5, 9 = 31 commas; Perfect 5th (9 circles) D, F, A = 8, 5, 9, 8 = 30 commas; Diminished 5th (8 circles)
Bb, D, G = 5, 5, 9, 8, 5, 9 = 41 commas; (225 circles)
Major Third = 9 + 8 = 17 commas; Minor Third = 9 + 5 = 14 commas; Diminished Third = 8 + 5 = 13 commas;
? to ? = 4 = , diatonic semi-tone F to G = 9 = Major Second, full tone, one step G to A = 8 = Minor Second A to B = 9 = Major Second B to C = 5 = Diminished Second, chromatic semi-tone, half step C to D = 9 = Major Second D to E = 8 = Minor Second E to F = 5 = Diminished Second, chromatic semi-tone, half step 53 commas in the Major Scale.
9 (32) x 8 (22) x 5 (51) = 360 9 (32) x 8 (22) = 72 x 5 = 360 9 (32) x 5 (51) = 45 x 8 = 360 8 (22) x 5 (51) = 40 x 9 = 360
(14) 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 openings for modulation into other keys.
(15) There are joinings, however, though at a wider limit. The system of music is not a spiral line. The minor scale is developed from the major by the LAW OF DUALITY; and when this is done, D 26.666, the root of the subdominant minor, is so near to D27, the top of the dominant major, that one note may be made to serve for both; and this joins the one extreme of the major and minor systems in this note D, which has thus duality in itself. The only other place where the dual system of major and minor stands open is at the other extreme of the two modes, between B the top of the dominant minor, and F the root of the subdominant major; these unjoined ends reach away till at three fifths below F, namely Ab, and at three fifths above B, namely G#, they come in touch of each other like the two D's. When thus three fifths below F major and three fifths above B minor have been developed, the extremes Ab and G# though standing like the two D's in duality, are so near there here again one note can be made to serve for 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.
(16) The mathematical scales, if followed out regardless of other laws which rule in music, would read like a chapter in Astronomy. They would lead us on like the cycles of the moon , for example. In 19 years we have 235 moons; but the moon by that time is an hour and a half fast. In 16 such cycles, or about 300 years, the moon is about a day fast; this, of course, is speaking roughly. This is the way seemingly through all the astronomical realm of creation.
(17) And had we only the mathematical ratios used in generating the notes of the scale as the sole law of music, we should be led off in the same way. And were we to follow up in the inaudible region of vibrations, we should possibly find ourselves where light, and heat, and chemical elective motions and electric currents are playing their unheard harmonies; or into the seemingly still region of solid substances, where an almost infinite tremor of vibrations is balancing the ultimate elements of the world.
(18) Music in this case would seem like some passing meteor coming in from among the silent oscillations of the planetary bodies of the solar system, and flashing past with its charming sound effects, and leaving us again to pass into the higher silence of those subtle vibrations to which we have referred, having no infolding upon itself, no systematic limit, no horizon.
(19) But music is not such a passing thing. Between the high silence of these intense vibrations, and low silence of oscillating pendulums and revolving planets, God has constituted an audible sphere of vibrations, in which is placed a definite limit of systematic sounds; seven octaves are carried like a measuring line round twelve fifths; and motion and rest unite in placing a horizon for the musical world, and music comes and goes within this range, with its wondrous infoldings which so charm the ear, and which symbolise so many spiritual mysteries. These twelve major keys with their twelve relative minors are the musical world, and motion in the operation of 3 is not much hampered by rest controlling it in the operation of 2; and what is lost of so called "perfect intonation" is far more than made up for in the beautiful system within system, which musical science, when fairly and fully brought into view, presents for our contemplation, and the intellect feasts along with the ear.
GRAVITY - the downward effect, to the ear, of a sound in a key LEVITY - The upward effect, to the ear, of a sound in a key PROXIMITY - The nearness of two notes to each other in small intervals RESOLUTION - The tendency and going of a note of one chord to some note of the next chord CHROMATIC- A name given to the semitonic intervals