Aural Pure 5ths Equal Temperament

[Richard Moody]

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
>