Our word "tone", as in a "step" of a musical scale, is from the Greek word "tonus" meaning "order" or "ordering".

The term "isotone" means TWO ORDERINGS IN CADENCE AND DIRECTION -- BOTH INCREASING OR BOTH DECREASING, and PERHAPS BOUNDED, so do not "go on forever". The term "antitone" means TWO ORDERINGS IN CADENCE in OPPOSITION -- ONE INCREASING, THE OTHER DECREASING, such that ONE ORDERING IS BOUNDED.

I label as "maxtone" THE INCREASING ORDERING since each TONE AFTER THE FIRST MAXIMIZES PRECEDING TONE OR TONES. I label as "mintone" the DECREASING ORDERING, since each TONE AFTER THE FIRST MINIMIZES THE PRECEDING TONE OR TONES.

Obviously, BOUNDING ONE ORDERING INDUCES A BOUND ON THE OTHER, SINCE THEY MUST REMAIN IN CADENCE.

(I've articulated the ANTITONE or ANTITONIC PROCESSING as a SIMPLE ORDERING of THE GALOIS CONNECTION or CORRESPONDENCE which can be between PARTIAL ORDERINGS or LATTICES as in Galois' Proof of Failure to Solve all 5th Degree Algebraic Equations by Radicals. There the MAXTONE was DEGREE of ROOT FIELD; MINTONE, DEGREE OF UNSOLVED EQUATION.)

I distinguish two types of ANTITONES:

1. DANTITONE: discrete or discontinuous steps.
2. CANTITONE: continuous changes.

DANTITONES and CANTITONES model various processings. Here, I'll use to MODEL THE LIMIT PROCESSING.

But first, I note how it can be simply explained to children or teen-agers in terms of daily processings.

From something I read in Cybernetics (1948) by Norbert Wiener, I realized that CLIMBING UP OR DOWN STAIRS is ANTITONIC. In climbing, THE MAXTONE IS THE SET OF RISERS UPSTAIRS; THE MINTONE IS THE DISTANCE FROM THE TOP. (In descending, the roles are reversed.) Each RISER CORRESPONDS TO A UNIT DISTANCE FROM THE TOP. The NUMBER OF RISERS IS BOUNDED, so THIS INDUCES THE NUMBER OF DECREASES.

Finding a sock in a drawer, or a folder in a filing cabinet, similarly is DANTITONIC.

In a (CANTITONIC) LIMIT PROCESSING:

• the ORDERING is, to repeat, CONTINUOUS or ANALOGIC, not DISCRETE or DIGITAL, as in the cases above;
• the MAXTONE consists of the INCREASING SEQUENCE OF TERMS;
• the MINTONE consists of "distance from the limit";
• by CHOOSING an EPSILON-DISTANCE from LIMIT L, the MAXTONE IS TENTATIVELY BOUNDED;
• this INDUCES a DELTA-BOUND on THE MINTONE of SEQUENTS.
• the SPECIFICATION ALLOWS REPEATED CHOICES ON THE MAXTONE, NECESSARILY INDUCING BOUND ON THE MINTONE.
• This "realizes" THE LIMIT of THE SEQUENCE.
• This MODEL can be adapted for all LIMIT PROCESSINGS IN ANALYSIS.

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THE ANTITONIC PROCESSING PATTERN TABLE (DEFAULT PATTERN: AB = CONSTANT)

TYPE	MAXTONE	MINTONE	BOUND	I-BOUND	CONSTANT
Galois connection	Extension fields	Galois subgroups	Identity subgroup	Irreducible root field	?
PYTHAGOREAN MONOCHORD	VIBRATING STRING LENGTH	TONE PITCH (FREQUENCY)	STRING LENGTH	"HIGHEST" PITCH	?
Lever	Load	Length of lever arm	Maxtone	Mintone	?
Stairway	Risers	Distance to top	Mintone	Maxtone	?
Finding sock in drawer	Number retrievals	Number remaining socks	Mintone	Maxtone	?
Finding file in cabinet	Number retrievals	Number files remaining	Mintone	Maxtone	?
De Broglie equation	Wavelength	Momentum	Mintone	Maxtone	Planck's h-bar constant
Planck- Einstein equation	Wavelength	Energy	Mintone	Maxtone	c x h-bar
Division algorithm	Partial quotients	Partial remainders	0 <= R < d	Total quotient	?
Wheel-&- axle	wheel radius	axle radius	either radius	other radius	?
Boyle's Law	Volume	Pressure	either	other	gas constant
Charles' Law	1/temperature	Volume	Mintone	Maxtone	gas constant
Pulley	Rope strands going-in	Strands going-out	Mintone	Maxtone	?
Hydraulic press	Area going-in	Area going-out	Mintone	Maxtone	constant
Limit L	Operation	L - [result]	0 <= epsilon	Operation	?
Hooke's Law	Stress	1/Strain	Mintone	Maxtone	?
Bernoulli's Principle	Fluid speed	Internal pressure	Maxtone	Mintone	?
Doppler Effect	Distance/Fre- quency	F/D	Distance	Frequency	?
Spectrum	Wavelength	Power	either	other	?
Bee-dance	Nectar- distance	Dancing speed	Maxtone	Mintone	?
Arith.-Geo. Mean Processing	Geometric mean	Arithmetic mean	either	other	?
Archimedean circle-exhaus- tion	Inscribed polygon	Circumscribed polygon	either	other	p
Information content	Entropy	Information measure	Mintone	Maxtone	Maximum entropy
Mathematical Expectation	Event value	Event probability	Mintone	Maxtone	?
Arches	Arch height	Horizontal force	either	other	?
Mathematical modeling of reality	Tractability of models available	Reality	Maxtone	Mintone
Extension and intension of words	Intension	extension	Mintone	Maxtone	?

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PHYSICS: ANTITONIC GROUP (MAXTONE*MINTONE=1) TABLE

PHYSICS FIELD	NAME	ANTITONE	GROUP	COMMENT
OPTICS	LIGHT PROPOGATION	u l = c	(u/c)( l/c) = 1	u = frequency l = wavelemgth c = vacuum light speed
QUANTUM THEORY	PLANCK-EINSTEIN FORMULA	E l = hc	(E/hc)( l)=1)	E = energy
QUANTUM THEORY	DE BROGLIE FORMULA	pl = h	(p/h)(l/h) =1	p = momentum vector
QUANTUM THEORY	SCHRÖDINGER'S EQUATION (see literature)	par. diff. eqn. satisfying Planck-Einstein & De Broglie antitonic eqns	xxxxxx	Quantum Physics, Eisberg&Resnick, pp. 144-5
MECHANICS (Work = Force x Distance)	MACHINE's activation amplifies antitonically input force or torque.	MIN: input Force MAX: transmitter distance	(WorkInput/WorkOutput = 1)	3 basic antitonics: lever, incl. plane, hydraulic press
MECHANICS	BASIC MACHINE: THE LEVER	MAX: moment arm Lin; MIN: moment arm Lout	Theor. Mech. Advantage= MAX/MIN; Actual Mech. Adv. = Fin/Fout; Efficiency = AMA/TMA	xxxxx
MECHANICS	BASIC MACHINE: INCLINED PLANE	MAX: xxx MIN: yyyy	zzzzzzzzzz	xxxxx
MECHANICS	BASIC MACHINE: HYDRAULIC PRESS	MAX: xxxxx MIN: yyyy	zzzzzzzzzz	xxxxxxx
DYNAMICS	(Newton's) SECOND LAW OF MOTION	xxxxx	xxxxxx	The Variational Principles of Mechanics, Lanczos.
DYNAMICS	(Newton's) THIRD LAW OF MOTION	xxxxx	(ACTION)(INVERSE REACTION) = 1	xxxxx