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 PROCESS 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:
- DANTITONE: discrete or discontinuous steps.
- CANTITONE: continuous changes.
DANTITONES and CANTITONES model various processes. Here, I'll use to MODEL THE LIMIT PROCESS.
But first, I 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.