Keely
"The maximum test was made to placing an iron weight of 580 lbs. on the extreme end of the long arm of the lever. To lift this weight required a pressure of 18,900 lbs. to the square inch counting the difference in the length of the two arms and the area of the piston. When Keely turned the valve-wheel leading from the receiver to the flexible tube and through it into the steel cylinder beneath the piston, simultaneously with the motion of his hand the weighted lever shot up against its stop a distance of several inches, as if the iron were cork. [Snell Manuscript - The Book, page 3]
"No conceivable power by pressure can force the atom, or the molecule, out of its spherical form. Its wonderful condition of elasticity causes it, if submitted to enormous pressure, to reduce in area, but does not change its shape. If submitted to the explosion of nitro-glycerine, which tears up the surface of the rock it is placed on, before displacing the surrounding atmosphere, it will no more effect the sphericity of the molecule than the waft of a butterfly's wing would roll a five hundred pound cannon ball up an inclined plane." [COMMENTS OF JOHN WORRELL KEELY ON DR SCHIMMELS LECTURE]
Ramsay
The simple natural scale is the fifth; the compound natural scale is the octave; the harmony scale, or chord-scale, is the three fifths; the great genetic scale is six octaves; for, like the six creation days, it takes the six octaves to give birth to the elements of which the wondrous structure of our music is built up; the birthplace of B, the seventh of the octave scale, is the sixth octave of the great genetic scale. The area of the twelve major and twelve minor scales is twelve fifths or seven octaves, the twelfth fifth being a comma and the apotome minor in advance of the seventh octave. This is a quantity so small that it can be ignored in real music; and the two notes, say E# and F, joined to close the circle of this horizon of our music world. E# is the top of the twelfth fifth, and F is the top of the seventh octave; and they are practically, though not exactly mathematically, the same note. Illustrations of this will be found among the plates of this work. [Scientific Basis and Build of Music, page 79]
This plate is a representation of the area of a scale; the major scale, when viewed with the large hemisphere, lowest; the minor when viewed the reverse way. It is here pictorially shown that major and minor does not mean larger and smaller, for both modes occupy the same area, and have in their structure the same intervals, though standing in a different order. It is this difference in structural arrangement of the intervals which characterizes the one as masculine and the other as feminine, which are much preferable to the major and minor as distinctive names for the two modes. Each scale, in both its modes, has three Fifths - subdominant, tonic, and dominant. The middle fifth is the tonic, and its lowest note the key-note of the scale, or of any composition written in this scale. The 53 commas of the Octave are variously allotted in its seven notes - 3 of them have 9 commas, 2 have 8, and 2 have 5. The area of the scale, however, has much more than the octave; it is two octaves, all save the minor third D-F, and has 93 commas. This is the area alike of masculine and feminine modes. The two modes are here shown as directly related, as we might figuratively say, in their marriage relation. The law of Duality, which always emerges when the two modes are seen in their relationship, is here illustrated, and the dual notes are indicated by oblique lines across the pairs. [Scientific Basis and Build of Music, page 106]
It will be observed that this plate represents intervals by its areas, that is, the distances between the notes; and the notes themselves appear as points. But it must be remembered that these distances or intervals represent the vibrations of these notes in the ratios they bear to each other. So it is the vibration-ratios which constitute the intervals here pictorially represented as areas. The area, as space, is nothing; the note itself is everything. [Scientific Basis and Build of Music, page 107]
The Octave being divided into 53 commas, the intervals are measured, as usual, by these, the large second having 9-commas, the medium second having 8, and the small second 5. These measures are then made each the radius by which to draw hemispheres showing the various and comparative areas of the seconds. The comparative areas of the thirds are shown by the hemispheres of the seconds which compose them facing each other in pairs. The comma-measures of the various thirds thus determined are then made the radii by which to draw the two hemispheres of the fifths. The areas of the three fifths are identical, as also the attitudes of their unequal hemispheres. The attitude of the six thirds, on the other hand, in their two kinds, being reversed in the upper and under halves of the scale, their attitude gives them the appearance of being attracted towards the center of the tonic; while the attitude of the three fifths is all upward in the major, and all downward in the minor; their attraction being towards the common center of the twelve scales which Nature has placed between the second of the major and the fourth of the minor, as seen in the two D's of the dual genetic scale, - the two modes being thus seen, as it were, revolving [Scientific Basis and Build of Music, page 113]
See Also
area of a scale
Eighteen Attributes or Dimensions
Figure 6.17 - Areas and Volumes - Relations and Proportions
Length
Propositions of Geometry
Table 12.02 - Length Area and Volume Math
Volume