MATH the part of mathematics that deals with the relationships between lines, angles, and surfaces (also relative or proportional volumes)

SINGULAR the way in which different things are arranged in relation to each other

**Ramsay**

The relations which music has to mechanics, the sphere of centers; to **geometry**, the sphere of measures; and to arithmetic, the sphere of numbers, show how deeply seated music is in the nature of things, and how independent it is of the will or choice of the musician. His composition may take any form his inspiration may suggest; they are subject to him; but as to the nature of music and its laws, he must keep himself subject and obedient to them. Music is of the aesthetic; but the aesthetic is of the *nature of things*. [Scientific Basis and Build of Music, page 92]

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]

**R. A. Schwaller de Lubicz**

"Proportion belongs to **geometry** and harmony, measurement to the object and to arithmetic; and one necessitates the other. Proportion is the comparison of sizes; harmony is the relationship to measures; **geometry** is the function of numbers." [R. A. Schwaller de Lubicz, The Temple in Man, page 61]

See Also

**Algebraic Values of Trigonometric Functions**
**Area**
**Circle**
**Constructive Cubes**
**Corner Cube Prisms**
**Corner Cube Retro-Reflectors**
**Corner cube retroreflectors**
**Cube**
**Cube Sphere**
**cube-sphere**
**Chord**
**Figure 3.16 - Idea Preceeds Manifestation in Material Form using Cubes and Cones**
**Figure 3.26 - Formation of Spheres along Six Vectors of Cubes**
**Figure 3.4 - Focalizing Lenses at nested Cube faces**
**Figure 6.0.5 - Cube with Vortices showing Structural Relations**
**Figure 6.1 - Orthogonal Vortex Motion as Structural base of Cubes**
**Figure 6.3 - Cube with Orthogonal Vectors**
**Figure 6.5 - Triple Planes - May Underlay some Sacred Geometry or Religious Concepts**
**Figure 6.8 - Resulting in a Cube mutually assimilating to a Common Center**
**Figure 6.10 - Wave Dynamics between Cube Corners**
**Figure 6.11 - Cube Corner Reflectors Dissipating and Concentrating**
**Figure 6.12 - Spheres and Cubes are Gods Only Tools**
**Figure 6.14 - Triple Three Cubes**
**Figure 6.15 - The Neutral Cube**
**Figure 6.16 - Juxtaposed Corner Cubes**
**Figure 6.18 - Sphere Circumscribed by Cube**
**Figure 6.19 - Sphere to Cube - Relations and Proportions**
**Figure 10.06 - Vortices in Cube extending in to and out from Center**
**Interval**
**One More Step Toward Building The Cube-Sphere Wave-Field**
**Part 06 - Formation of Cubes**
**Proportion**
**Propositions of Geometry**
**Rainbow**
**Ratio**
**This Three Dimensional Cube Universe of Nine**
**We Now Build the Nine Equators of Cube-Sphere Wave-Fields**
**6.0.5 - Space seen as Constructive Cubes**
**6.10 - Nineness of Cubes**
**6.11 - Neutral Cubes**
**6.12 - Corner and Face Cubes**
**6.14 - Sphere and Cube**
**6.14.1 - Mirror Cube**
**6.2 - Development of Cubes**
**6.4 - Sacred Geometry**
**6.5 - Cubes divide into six tetrahedrons**
**6.6 - Cube Corner Retroreflectors**
**6.7 - Corner receivers from corners of cubes**
**6.7.5 - Compound Cubes**