> > I think inharmonicity facilitates the tuning of ET on modern > > pianos. .. I'd like to hear more on your thoughts here. Why does this > facilitate ET 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 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. The theory is that IH is related to string thickness Vs string length. That would imply that the ratio of thickness to length should correlated to IH. If the string is 2 inches at .030 inches diameter and compared to a string 40 inches at .045 inches the ratios are .030/2 = .015 compared to .045/40 = .001125. The length to thickness ratio is less in longer strings there for according to theory, IH should be less in longer strings. > 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 From: Richard Brekne <Richard.Brekne@grieg.uib.no> To: <pianotech@ptg.org> Sent: Thursday, August 22, 2002 3:13 AM Subject: Re: May the 4ths be with you > Richard Moody wrote: > > > ----- Original Message ----- > > From: Richard Brekne <Richard.Brekne@grieg.uib.no> > > > > > > > >Dr Sanderson did say right out that inharmonicity changes the > > > theoretical picture relative to 5ths and 4ths... which should > > also > > > have their contiguous relationships. > > > > I think inharmonicity facilitatates the tuning of ET on modern > > pianos. > > It seems at least in theory to give a 'natural" stretch to the > > octaves. That is if the upper partials of the lower notes are > > sharper than the lower upper partials of the upper note in all the > > intervals being tuned. > > Hmmm.. I'd like to hear more on your thoughts here. Why does this > facilitate ET ? I think (now anyways) that maybe I would agree at > least soundwise, except for the fact that tho the upper partials of > lower notes by and large stay sharper then their coincidents from > notes above, they tend to do so less and less the higher up you go. > This results in the need for to vary progressively beat rates in > different octave types. And I cant see that that particular phenomenon > is in the spirit of ET theory... or what ?. > > > > > I would say in spite of inharmonicity it can be demonstrated the > > beat rates follow closely the theoretical. In fact so closely > > that an anomaly usually indicates a mistake on the part of the > > tuner rather than the piano maker. > > Aural tuning is much like live performance, mistakes will be made, > > no two will be exactly alike and but for a rare magical few most > > can be improved on. But I bet if performers knew the checks we > > have to prevent error they would be envious. ---ricm > > How does this jive with whats been said about 4ths having rather > steady beat rates, 3rds not actually having a true contiguous > relationship, and 5ths actually moving from narrow to wide as one > moves up into the treble ? > > Certainly we are able to create an ET that by and large fits the > bill. That we eliminate HT key colour variance is evidence enough of > that. > > Maybe I misunderstand the formal definitions of ET, but it seems like > to me that inharmonicity forces us to come in conflict with some of > these rules. > > Cheers > > RicB > > -- > 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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