Frequency temperament: Difference between revisions
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== Rank-1 arithmetic temperaments == | == Rank-1 arithmetic temperaments == | ||
Rank-1 arithmetic temperaments correspond to [[ADO]] systems much like how rank-1 temperaments correspond to [[EDO]] systems. | Rank-1 arithmetic temperaments correspond to [[ADO]]/ODO systems much like how rank-1 temperaments correspond to [[EDO]] systems. | ||
== Rank-2 arithmetic temperaments == | == Rank-2 arithmetic temperaments == | ||
Revision as of 01:59, 2 March 2023
WIP
Arithmetic temperaments are the arithmetic counterpart to regular temperaments.
Rank-1 arithmetic temperaments
Rank-1 arithmetic temperaments correspond to ADO/ODO systems much like how rank-1 temperaments correspond to EDO systems.
Rank-2 arithmetic temperaments
A standard rank-2 temperament has a generator interval and a period interval, and new intervals are produced by taking powers of the generator, and then reducing them logarithmically to the range from 1/1 to the period. But in arithmetic temperaments, new intervals are produced by taking multiples of the generator and reducing them arithmetically.
For example, consider an arithmetic temperament with a generator of 1.29 (440 cents) and period 2/1. If we want to add a third interval, multiply 1.29 by 2 to obtain 2.58. Since 2.58 is outside of an octave, subtract 1 to get 1.58. To add a fourth interval, multiply 1.29 by 3 and subtract 2 to get 1.87. We can continue generating new intervals this way like in a normal rank-2 temperament.