> >Personally, I don't find it at all handy to use decimal for intervals. Yes I had that problem, but that is what spread sheets and hand held calculators spit back out. I wonder if the Greeks who originally established the ratios of intervals would have used decimals if they were in the base ten system. >The following table is of Werkmeister chain as you describe above, with octave reduction. >C 1/1 >G 643/430 (=[(3/2)^(81/80)]*1/1), quarter comma fifth I would be interested to see the arithemetic that gets 643/430. My spread sheet gives 1.5076218 from what you give. So decimals sometimes are useful to prove equations and compare results. 643/430 gives 1.495348837209 which agrees to 8 places when the quarter comma fifth is computed with a calculator. To get a meantone (quarter comma) Fifth from ratios, the Fifth 3/2 must be reduced by one quarter of the syntonic comma (81/80). Because ratios in music are actually exponential rather than additive, it is not as simple as dividing the syn comma by four and using that to reduce the fifth. It is the 4th root of that comma. This is revealed because fourth fifths equal a Third (two octaves up). represented in math by (3/2)^4 To take it down two octaves divide it by 4 or (2/1)^2. With ratios you end up with 81/64. Now the pure Third (5/4) is the same as 80/64. The "difference" (proportional difference) between these two are 81/80. or that is the number needed to multiply 80/64 by to get 81/64. So the ratio of the syntonic comma is 81/80. Now the quarter comma is 81/80^1/4 or 1.0031105. This is rather difficult to do with fractions? Now if you divide (3/2) by this number you get the equiv of 643/430. >D 341/305 (=643/430*643/430*1/2) >A 1075/643 (=341/305*643/430) 341/305 = 1.1180328 643/430*643/430*1/2 = 1.1180341 OK close enough sorry all my spread sheet gives me are decimal equivs. For me it is easier to go with (3/2)/(81/80)^(1/4) x times. Besides it pastes directly into my spread sheet. >In fact the latter is the _raison d'etre_ for 1/4 comma syntonic meantone. The reason for quarter comma meantone is that they want pure thirds from four "evenly" tempered fifths rather than a third "beating rather high" from four pure fifths. >Try a this table for 12tET fifths and then calculate beat ratios for A and E. All I need is the beat rate. For that you need the ratio of the pure interval. The ratio of an ET fifth is 3/2/((3/2)^12/2^7)^(1/12) or simply 2^7^(1/12) Hmm seems like there should be one over that. or yet 2^(7/12) =1.49830...... What fractional number this would be I don't know how to compute other than look up in a table of fractional equilivants. If it is the fourth it should be 2^(5/12) ---ric
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