It seems that if an interval were pure all of the "eligible" partials would also be beatless. But you raise an interesting point by noting that there are more than one series of partials effecting the interval being tuned, on paper anyhow. The fifth for instance, (how did I think of that?) the third partial of the lower note beats with the second partial of the upper note. Take fifth F to middle C. The theoretical rate is .590 bps. That is the 3:2 series. Now look at the 6:4, (double) and the rate is 1.180 .... exactly double. These rates can be seen in Reblitz's tables 4-12 ,, 4-14 (first edition). However he doesn't comment on these "extra rates" and no one else has mentioned them tmk. Surely these rates must be audible because we can hear the beats of minor thirds which are caused by the 6:5 series. However no one must hear them separately, else they would be dwelt with in tuning instructions. It would seem that they would reinforce the .590 beats. There is much to be done with theory on paper. Of course in practice things are not always as predicted. Some pianos are easy to get triple octaves, while others are impossible. Also even with meter type electronic tuners one can observe the A octave stretched automaticaly when set by ear. (1/2 cps +/- ) (again this varies from piano to piano). Since all of this happens because of inharmoanicity (sic) ( :) the stretching is accomplished automatically when tuning aurally. And since it is better to err on the sharp side and since the strings tend to pull flat and we have to be sharp anyway to set the pins, octaves get even more stretching just from considerations of accurate and stable tuning. Another reality of tuning esp in temperament, if you can hear fifths beating they are too narrow. On the other hand they can be too pure. The sound they make is not like a beating, rather a tonality. That is to say fifths sound purer than they look on paper. Maybe because the double beats from the upper partials smoothes them out? Richard Moody > From: Robert Scott <rscott@wwnet.com> > To: pianotech@byu.edu > Subject: Re: Aural Pure 5ths Equal Temperament > Date: Wednesday, May 21, 1997 6:47 AM > > Doesn't the same consideration hold true for pure 5ths? There > can be a 3:2 pure fifth (where the 3rd partial of the lower note > zero beats with the 2nd partial of the higher note) or a 6:4 pure > fifth (where the 6th partial of the lower note zero beats with > the 4th partial of the higher note). I suppose you could even > talk about 9:6 and 12:8 pure fifths too, but by the time the > partials get that high they are probably too weak to need > special attention. How does this impact the discussion of > tuning by pure 5ths? > > Bob Scott > Ann Arbor, Michigan >
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