in their birthplace - F A C, C E G, G B D. Indeed in their birth not only is it so, but still further, the top note of the first chord is the root and generator of the third. They are linked in generative continuity. [Scientific Basis and Build of Music, page 49]

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]

A third cause of difference of contrast in notes is the individual character which belongs to them according to their place in the genetic scale - that is, their birthplace character - the amount, namely, of centrifugal force which they have inherited. [Scientific Basis and Build of Music, page 61]

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]

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]

When Ramsay gave a course of lectures in Glasgow, setting forth "What constitutes the Science of Music," his lecture-room was hung round with great diagrams illustrating in various ways his findings; an ocular demonstration was also given of the system of musical vibrations by his favorite illustration, the oscillations of the Silent Harp of Pendulums. A celebrated teacher of music in the city came to Mr. Ramsay's opening lecture, and at the close remained to examine the diagrams, and question the lecturer, especially on his extension of the harmonics to six octaves. Having seen and heard, this teacher went and shortly after published it without any acknowledgment of the true authorship; and it was afterwards republished in some of the Sol-Fa publications, the true source unconfessed; but our plagarist stopped short at C, the top of the tonic, instead of going on to F, the sixth octave of the root of all; the effect of this was to destroy the unity of the great chord. The 22 notes instead of 25, at which this teacher stopped, allowed him, indeed, to show the natural birthplace of B, which Ramsay had pointed, but it beheaded the great complex chord and destroyed its unity. If C, the root of the tonic, be made the highest note, having quite a different character from F, it pronounces its character, and mars the unity of the great chord. Similar diversity of effect is produced by cutting off only two notes of the 25 and stopping short at D, the top of the dominant; and also, though in a weaker degree, by cutting off only one note of the 25 and stopping at E, the middle of the tonic; this, too, disturbs the unity of the fundamental sound. [Scientific Basis and Build of Music, page 111]

See Also


Created by Dale Pond. Last Modification: Monday December 28, 2020 04:19:17 MST by Dale Pond.