Frequency temperament: Difference between revisions

Line 9:

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 generator ~~[[9/7]]~~ and period [[2/1]]. If we want to add a third interval, ~~then~~ multiply ~~9/7~~ by 2 to obtain ~~18/7~~. Since ~~18/7~~ is ~~greater than~~ an octave, subtract 1 to get ~~[[11/7]]~~. To ~~get~~ a fourth interval, multiply ~~9/7~~ by 3 ~~to get 27/7, then~~ subtract 2 to get ~~[[13/7]]~~. ~~In contrast to~~ a ~~logarithmic~~ rank-2 temperament~~, we can only produce 7 intervals this way (creating [[7ado]]) before the intervals will start repeating~~.

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.

Revision as of 01:53, 2 March 2023 view source CompactStar (talk \| contribs) 5,020 edits No edit summary ← Older edit		Revision as of 01:59, 2 March 2023 view source CompactStar (talk \| contribs) 5,020 edits No edit summary Newer edit →
Line 9:		Line 9:
	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.		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 generator ~~[[9/7]]~~ and period [[2/1]]. If we want to add a third interval, ~~then~~ multiply ~~9/7~~ by 2 to obtain ~~18/7~~. Since ~~18/7~~ is ~~greater than~~ an octave, subtract 1 to get ~~[[11/7]]~~. To ~~get~~ a fourth interval, multiply ~~9/7~~ by 3 ~~to get 27/7, then~~ subtract 2 to get ~~[[13/7]]~~. ~~In contrast to~~ a ~~logarithmic~~ rank-2 temperament~~, we can only produce 7 intervals this way (creating [[7ado]]) before the intervals will start repeating~~.		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.