Mans ability to invent and develop language and number assured a high degree of sophistication for communication and clarifying phenomena. What is the basis for the impulse that drives an individual’s action? One most appropriate answer would be choice. Choice suggests a use of language, an action and in the broadest sense, a setting of standards. Each choice becomes part of a series of events comprising periods of time that collectively describe experience.

In any situation you might be bid. Give me one! It is presumed that you understand a ‘unit’ as the circumstance demands. Each discipline has its own unique set of units or standards that we may use for communication. It is much easier to use available standards than create new ones. Still, gaps exist between disciplines that limit communication. It could prove useful to have a standard that provides a unit for exchange. Such a system would need to possess universal standards. It is quite easy to add apples and oranges, as long as apple remains apple per unit orange or otherwise. The same principle holds graphically, when changing for example certain black pixels red in a color map. Here reciprocity provides a means for non-conventional comparison. This thought is the basis for interdisciplinary communication and the standard making all this possible lies in the application of Harmonics.

The fabric of Harmonics is ratio and proportion. In elementary mathematics we are introduced to fractions and some of the following different ways of saying them and their reciprocals: one half, one to two, one divided by two, one part and two parts, two divided by one, two to one and further to the singular two or one. Harmonic ratios are referenced best as expressions of unity such as two to three (2/3 or 3/2). This reference aims at a comparison of unity or amounts of one element to another. The selection of elements can be quite tricky because we do not normally compare apples and oranges. The use of a computer allows these comparisons in the DATA world with both graphic and sound outputs. Probing with instruments is dependent upon the factor of resolution. We’ll continue to separate theory and standard. Now would be a good time to show the particulars of Harmonics in an outline form based upon Hans Kayser’s Handbook of Harmonics.

I. Introduction – History of Harmonics.

II. Theorems of Tone Numbers.

A. Use of the monochord.

B. Fractions, decimals, base 2 logarithms, cents, frequency, polar coordinates, tone angles, string lengths and tone values.

C. Reciprocity of frequency and string length, the geometric arrangement of time and space.

D. Equality within musical proportions – principles of equal distance.

E. Acoustics – rhythm, periodicity, resonance and interference.

F. The state of simultaneous conflicting attitudes; continua: right/left; material/psychic; major/minor; static/dynamic and so forth.

G. Examination of the senses and an analogy of scientists probing blindly with instruments – reliability and resolution.

III. Theorems of Tone Rows.

A. The overtone row, undertone row, Plato’s tetraktis.

B. Whole numbers.

C. The intervals, octave multiplication and reduction.

D. Calculating base 2 logarithms, the number ‘e’, the I Ching, Weber-Fechner law.

E. Convergence, divergence, perspective, symmetry.

1. The rows of Archimedes and Leibnitz.

2. The straight and the curved.

3. Reciprocal relationship of the eye and ear,

(equal distance & perspective).

IV. Theorems of Tone Groups.

A. The lambdoma of Albert von Thimus. The partial tone diagram – interpolation of overtone and undertone rows (“T” diagrams & the rationing of coordinates).

B. Major & minor tone rows, Thimus rows, the binomial theorem.

C. Polarity – further discussion of reciprocity. A three dimensional model of the partial tone diagram (lambdoma).

D. Equal tone lines, harmonic rays – tones with the same character but different degrees (3/1, 3/2, 3/4…) or identical tones (2/1, 4/2, (6/3…)

E. Index and generators of the partial tone diagram.

F. Parabolas, hyperbolas and ellipsis (conic sections) as derived from the partial tone diagram.

G. Arithmetic, harmonic and geometric proportions (partial tone diagram). Irrational numbers comprehended geometrically. Harmonic number as a mediator between concept, idea and sensuality (Haptik).

H. Harmonic proportions in architecture.

I. Trinities: 1/n – 1/1 – n/1; morning – noon – evening; right – middle – left, (world religions).

V. Structure of Tone Groups.

A. Variations of the partial tone diagram (triangular, quadratic and circular).

B. Combinations of three variation forms (triangular, quadratic and circular).

C. Polar diagrams – the circle as a monochord sl:ring divided by whole numbers.

D Tone spirals and tone curves, logarithmic and decimal – the tone cycloid, planets and orbits, a proof for irregularities in the harmonic system and a further discussion of perspective and reciprocity.

E.- The completed partial tone diagram – coordinates in four directions (not considered as a combination type). Examination of the free fall law of Galileo and Newton’s gravity law as compared with the consistency of time and space in perspective and equal distant proportion. A discussion concerning the concept of gestalt.

F. Logarithmic structure – a logarithmic graph of the completed partial tone diagram.

G. Tonal space – a cubic model made from partial tone diagrams, the three dimensional spiral (screw), tone sphere, platonic solids, other examples from crystallography, discussion of the multifunctionality of the ear.

H. Sound Pictures.

1. A discussion of seeing and hearing – geometric form and bridging gaps between disciplines comparing the Wolfram model.

2. Harmonic proportions of the human gestalt – changing proportions for the human form in fine art.

3. Combination types to index 12 – examination of melodic moments within each index (counterpoint).

VI. Selections.

B. Chords –

1. Polar geometry.

2. Major and Minor.

3. Genesis of chords within the partial tone diagram – interval and chord construction.

C. Melodic and chordal examples.

D. Cadence – step-dialectic – a parallel to traditional harmony.

E. Counterpoint – reciprocals and a comparison to drama. Traditional counterpoint and the partial tone system.

F. Direction – linear, polar vectors.

1. Fourier theorem.

2. A discussion of causality.

G. Interval exponents and constants.

1. Successions of same interval or same tone.

2. Color analysis.

H. Symmetry

1. Harmonic.

2. Organic.

3. Artistic – Beethoven’s 5th.; examples from other disciplines.

I. Time and Space.

1. Harmonic view – string length/frequency, perspective/equal distance, major/minor.

2. Lorenz contraction, Einstein, and philosophic viewpoints.

J. Enharmonic – study of tone proportions projected into the space of an octave.

1. Interval exponential rows.

2. Syntonic Comma and Pythagorean Apotome.

K. Temperament – concept of compromise extended to octaves with less or greater than 12 tones.

L. Number Symbology – methods of harmonic number analysis.

M. Harmonics examples.

1. Norms.

2. Hierarchy.

3. Tolerance.

4. Harmonic cosmology.

*sic*], music.’ That is the point where tone takes on the attribute of data and communication. Note the use of irrational numbers on the top and left side of the diagram. The Greeks labeled the 0/0 as Eidos, that translates as God or Out of Nothing. This point is known as the origin and is the point from which the Harmonic Cosmology emanates in multidimensions. All values originate in this point. Without this origin the monochord on the right could not be cut perfectly by the ratios in the table, when using only those ratios to the right of the daigonal to cut the string length. From left to right the tonal progression is the overtone row and from top to bottom the undertone row. This cross-hatching of overtones and undertones forms the ratios that comprise the Lambdoma. Both major and minor (musical) universes are in evidence here. Below is an Index 8 Lambdoma table that also shows the 3 place logarithm associated with each harmonic.