Decimal: Difference between revisions

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'''Decimal''' is ~~a temperament~~ in both the [[dicot]] and [[semaphore]] families of temperaments. It is also [[hemipyth]], ~~meaning that it has~~ approximations of √2, √3, ~~and~~ √(3/2)~~. It is special in that it is also an~~ [[~~exotemperament~~]].

'''Decimal''' is an [[exotemperament]] in both the [[dicot]] and [[semaphore]] families of temperaments. It is also the prototypical fully [[hemipyth]] temperament, with approximations of √2 at [[7/5]], √3 at [[7/4]], √(3/2) at [[5/4]] and √(4/3) at [[8/7]], and [[pergen]] (P8/2, P4/2), splitting all Pythagorean intervals.

More precisely, it is the 7-limit temperament that tempers out both [[25/24]], the classic chromatic semitone, and [[49/48]], the septimal diesis. These two intervals have a rather similar function separating close intervals and creating "major" and "minor" triads (either pental ones splitting the perfect fifth or septimal ones splitting the perfect fourth), and tempering them out allows [[5/4]][[~]][[6/5]] to be a neutral third approximating √(3/2) and [[7/6]][[~]][[8/7]] to be a neutral semifourth approximating √(4/3). These can be equated (far more accurately) to [[11/9]] and [[15/13]] respectively, tempering out [[243/242]] and [[676/675]] and extending this temperament to the [[13-limit]].

Revision as of 11:12, 28 October 2024 view source Jerdle (talk \| contribs) autopatrolled, emailconfirmed 212 edits m Removed a comma (as in the punctuation mark, not the musical term!) Tag: Visual edit ← Older edit		Revision as of 12:03, 28 October 2024 view source Jerdle (talk \| contribs) autopatrolled, emailconfirmed 212 edits Added more info and rewrote some text. Tag: Visual edit: Switched Newer edit →
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	'''Decimal''' is ~~a temperament~~ in both the [[dicot]] and [[semaphore]] families of temperaments. It is also [[hemipyth]], ~~meaning that it has~~ approximations of √2, √3, ~~and~~ √(3/2)~~. It is special in that it is also an~~ [[~~exotemperament~~]].		'''Decimal''' is an [[exotemperament]] in both the [[dicot]] and [[semaphore]] families of temperaments. It is also the prototypical fully [[hemipyth]] temperament, with approximations of √2 at [[7/5]], √3 at [[7/4]], √(3/2) at [[5/4]] and √(4/3) at [[8/7]], and [[pergen]] (P8/2, P4/2), splitting all Pythagorean intervals.

	More precisely, it is the 7-limit temperament that tempers out both [[25/24]], the classic chromatic semitone, and [[49/48]], the septimal diesis. These two intervals have a rather similar function separating close intervals and creating "major" and "minor" triads (either pental ones splitting the perfect fifth or septimal ones splitting the perfect fourth), and tempering them out allows [[5/4]][[~]][[6/5]] to be a neutral third approximating √(3/2) and [[7/6]][[~]][[8/7]] to be a neutral semifourth approximating √(4/3). These can be equated (far more accurately) to [[11/9]] and [[15/13]] respectively, tempering out [[243/242]] and [[676/675]] and extending this temperament to the [[13-limit]].		More precisely, it is the 7-limit temperament that tempers out both [[25/24]], the classic chromatic semitone, and [[49/48]], the septimal diesis. These two intervals have a rather similar function separating close intervals and creating "major" and "minor" triads (either pental ones splitting the perfect fifth or septimal ones splitting the perfect fourth), and tempering them out allows [[5/4]][[~]][[6/5]] to be a neutral third approximating √(3/2) and [[7/6]][[~]][[8/7]] to be a neutral semifourth approximating √(4/3). These can be equated (far more accurately) to [[11/9]] and [[15/13]] respectively, tempering out [[243/242]] and [[676/675]] and extending this temperament to the [[13-limit]].