Ramsay
There are 32 notes required for each octave for the 13 major and the 13 minor mathematical scales. These 32 notes are by the law of duality arranged symmetrically from D as a center upwards to G#, and downwards to A?. D itself serves for 2 of the 32 on the piano. The first black keys on each side of D serve for nominally 3 notes each = 6. The first white key above and the first below D serve for 2 notes each = 4. The second white key above and the second below serve each for 3 notes = 6. The second black keys above and below D serve each for 3 notes = 6. The third black key above D is G#, the third below is A?; this key, for it is one, serves for 2 of the 32. There is a comma of difference between D minor and D major. Six fifths below the minor D26 2/3 is A?, the root of the subdominant of the key of E? minor; and six fifths above the major D27 is G#, the top of the dominant of F# major. The difference between this minor A? and this major G# is two commas and [Scientific Basis and Build of Music, page 85]
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]
Bruce Cathie found that by using a time based unit of 27 all of his harmonic equations 'fitted' together. Upon this discovery he was then told by someone close to NASA, that they too use a time based unit of 27 for their space program, which isn't public knowledge.
What then becomes interesting, is that the largest obelisk (twin) at Karnak has this same unit of measure built into it's length. So as the stone block tapers towards the top it's ratio is 1:27. Meaning for every 27 units it tapers by one unit based on it's height, thus giving a description in stone, of "time". [anon, unconfirmed]
See Also