> JIM writes: > << at least be an equal opportunity 'temperament basher' and > continue this thought by saying that "unequal" temperaments are also "out of tune" but unequally so! :-) >> > > Greetings, > This seems to be a slight oversight that I would like to respond to. > It is difficult to say that a temperament is "out of tune", without first > defining what "in tune" actually means. .......Or does "in tune" refer to realized >super particular ratios and Just intonation, such as one might find in a >meantone? Ed Foote I think we can agree that "In tune" (as opposted to "tempered") means no beats in the interval being tuned. That is usually called a Just interval and whose coincident partials are at the same frequency, ie no beats. (I think this term should be in caps so it won't be confused with 'just' as in "just another interval".) Such an interval is also called pure, meaning 'absence of beats'. A superparticular ratio is one whose numerator exceeds the demoninator by one. Thus 3/2, 4/3 and 5/4 are superparticulars. (the 5th, the 4th, the 3rd) There is a belief that the simplest superpaticular ratio of an interval is what defines 'Just'. Others include the simplest ratio as in the 6th, (5/3). The act or art of tempering is tuning intervals slightly away from Just. These tempered intervals of course not "in tune" as the orchestra would play them but are considered in tune if they meet the criteria of the temperament. Thus a pure 5th would be considered a mistake in ET, however in Pythagorean or many of the Wells it is acceptible if in the right place. In the scheme of temperaments some intervals "come out" Just. For example in Pythagorean, the diminished 4ths sound as a pure 3rd. The augmented 2nds sound as a pure minor 3rd. But this is not by design. Since Pythagorean is only a series of pure fifths, it might be debated that it is a temperament at all. In Meantone theory, the objective is for 3rds to come out pure, however in practice it is practical to tune them pure, rathar than tuning a series of 4 flat 5ths hoping for a Just 3rd. ---ric
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