Hi Ric I have a few comments below and a couple links for you to look at if you would Richard Moody wrote: > The line of thinking I have followed goes like this. Tune A440 > (A4) from the fork. Tune A220 |(A3) to A440. In doing so we are > tuning the second partial of A3 to the first partial of A4. If > the 2nd partial of A3 is sharp (say it is 441) , then you have to > lower the string to get to 440 for the 2nd partial. this means the > fundamental will be more like 199.5 instead of 220 thus making a > wider octave with 440. If you tune up from A440 or tune A5 to A4 > then the second partial of A4 is being tuned to the first partial > of A5. Now if the second partial of A4 is say 881 then you must > raise the fundamental of A5 to 881. So the octave of A440 to A881 > is obviously a little wide from the theoretical 2:1. Now of course > every thing doubles in the next octave. A5 (now 881 whose second > partial is perhaps 1764), A6 will be tuned to this. The > fundamental of A6 will have to be 1764 to be beatless. But this > fundamental (2 octaves above A440) is 4 cycles per second SHARP > from the theoretical A440 x 2 x 2 which is 1760. The link below shows a set of calculated frequencies for some coincidental partials based on measured inharmonicity constants. http://home.broadpark.no/~rbrekne/images/Inharmonicities.htm This would be a pretty tight tuning in my book. A3(2) is nearly beatless to A4(1). Looking further tho the 4:2 for these two beats at about 0.6bps narrow. Add A5(1) in and its a whole beat per second narrow of A3(4). Looking at A3 and A4 again, the 6:3 relationship is already at over 2bps narrow if we tune this way. And the 8:4:2 for this double octave in the middle of the piano is already pretty wild. But thats what happens if you tune a 2:1 at A3 / A4. Most tuners seem to go for something between a pure 6:3 and 4:2 for A3/A4, and doing this will force both the 2:1 and 4:2 wide. I suppose we are in agreement so far ? > RB: > >the upper partials of > > lower notes by and large stay sharper then their coincidence > >from notes above, they tend to do so less and less the higher up > you > >go. > > Actually I thought the higher you go the higher the inharmonicity > gets. It does, but because of the way we tune what I said is more or less true... tho I admitedly worded myself abit clumsy. Take a look at the following two links. http://home.broadpark.no/~rbrekne/images/actualharmonicseries.gif http://home.broadpark.no/~rbrekne/images/octpartsdgm1.gif This is data from a piano Dr Sanderson measured in 1978 for an article in the journal. The first graph shows that general inharmonicity increases as we get higher up the scale, and it also shows clearly what this means, namely that the growth rate of the individual partials curves are different, resulting in an increasing distance between partials for any note the higher up we go. The second graph shows the consquences of adjusting for this problem, and is reflective of what I said above. In this same tuning (the 1978 one) the 4:2 octave type is held at a constant 0.5 bps over these 25 notes. This results in the 2:1 actually getting wider as we progress upwards. And it would continue to get wider higher up until the tuner decided to start placing priority on slowing them down. The 6:3 and 8:4 move towards the narrow as the lower notes' coincident out-paces the higher notes in both these octave types. If you tune this same range such that the 6:3 are held beatless, or at 0.5bps as they start out in the example above, then you'd hold the 8:4 up a while longer... this would allow for better matching of those triple octaves and would represent quite a stretch. Though the 4:2's and 2:1's would probably beat obnoxiously. Now as far as I know, there is no way to really resolve any of this such that my understanding of ET rules and regs are all perfectly kept in tact. Hence we have to compromise things. 5ths for example can and will go from narrow to wide up the scale. And the width of some octave types are going to reverse direction as you change from one type to the other. Thats what I meant by the disparagy between theory and practice. Still, as I said in an earlier post. We do manage to create a tuning that is ET in more ways then not, and the fact that we kill all key color is good evidence of that. I think this echos what was stated in the SAT manuals appendixs that theory and practice are not quite the same when he mentioned that the contiguous 4ths and 5ths dont really exist in real life tuning. > > > This results in the need for to vary progressively beat rates in > > different octave types. And I cant see that particular > phenomenon > > is in the spirit of ET theory... or what ?. > > I don't know what you mean beat rates vary by different octave > types. If you tune C5 to C4 you are in a simple 2:1 ratio. You > can't tune it to anything else unless you have extra large hands > or trust only the 5th below. You can TEST the C5--C4 with all > kinds of tests, but and this is the reason inharmonicity > facilitates ET all of those tests will show progression of beats > up the scale. The 10ths get faster, the octave and 10ths get > faster, the octave and 5ths while on paper should get faster will > never yowel, the same for the double and triple octaves, they only > beat but very little (|if at all) and that is > rogressive. ---ricm > > ---ric Grin... back at ya ! -- Richard Brekne RPT, N.P.T.F. UiB, Bergen, Norway mailto:rbrekne@broadpark.no http://home.broadpark.no/~rbrekne/ricmain.html
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