May the 4ths be with you

[Robin Hufford]

Ric,
     You are right on, something which I have for quite some time been
myself aware of,  as to the actual ratio of  beating of equally tempered
major thirds necessarily  being the same as the ratio of the tempered
frequencies of the major thirds, that is 2^1/3 as you point out.  For
that matter, this applies,  similarly,  to all other  tempered
intervals.   I think it does behoove us professional "Piano Tuners" to,
at least, understand this correctly, something which, at the moment
appears to be, in general, somewhat misconstrued  particularly as
represented by postings to this list.
     While your lucid, articulate, post on this same subject eariler in
the year was lamentably  ignored by the general readership of this list,
it was very much appreciated by myself as it made a point which was very
much in need of being made, as does the one below, and was right on the
money, even though it evoked no response.  I hope such will not be the
case this time and this would, indeed, demonstrate the great utility of
this list if such a matter could be clarified and the general perception
and reference to this phenomenon be made more accurate.
     I think the 5/4 beat rate comparison has, indeed,  been a persisent
and erroneous illusion, even if very useful, which it is,  and given the
limits of human perception, it may not be possible to accurately
determine a ratio that is only 8/10s of a percent faster, if my
calculation is correct,  than the 5/4 ratio commonly referred to.  To be
correct we all should use the phrase "very slightly faster than a 5/4
ratio" when comparing contiguous major thirds.
Thanks and regards, Robin Hufford

Richard Moody wrote:

> The beat rates of contiguous 3rds if tuned in ET beat at the ET
> ratio, which is NOT 5/4.  Take the beat rate of C--E  and then
> E--G#   and you will see the ratio is not the EXACT ratio of 5/4.
> Actually the ratio of beat rates of  two contiguous 5/4 3rds are
> zero because those two thirds 3rds have no beat.   Dr Sanderson
> must have been giving a generalized explanation rather than a
> mathematically correct one.   The proof of the ratios of 3rds in
> ET is that ratio cubed, (or ^3)  equals 2.     1.25992105  cubed
> equals 2
>
> The ratio of the frequencies of ET 3rds is 1.25992105  ,  the
> ratio of their beat frequencies is 1.25992105  .    How can it be
> otherwise? -----ric
>
> ----- Original Message -----
> From: Richard Brekne <Richard.Brekne@grieg.uib.no>
> To: PTG <pianotech@ptg.org>
> Sent: Saturday, August 17, 2002 10:47 AM
> Subject: May the 4ths be with you
>
> > List.
> >
> > The following is an except from appendix F of the SAT
> > manual. It gives an explanation by Dr Sanderson > "Two
> contiguous musical intervals are intervals that touch
> > each other, in other words, share the note in the middle.
> > Tests that use contiguous intervals are easy to learn and
> > use, and tell the tuner explicitly which notes are at fault
> > and what to do to correct them.
> > Contiguous major thirds will beat in the ratio of four to
> > five because the major third itself consists of two notes
> > whose frequencies are in the ratio of four to five.
> > Displacing any interval up the keyboard will speed it up
> > theoretically in the ratio of the frequencies of the two
> > root notes involved.  Therefore two contiguous major thirds
> > should beat in the ratio of four to five, two contiguous
> > minor thirds in the ratio of five to six.Similarly, two
> > contiguous fourths should beat in the ratio of three to four
> > and two contiguous fifths in the
> > ratio of two to three.  However, on the piano this
> > theoretical relationship holds well only for the major and
> > minor  thirds.    The  fourths  and  fifths  are  so
> > strongly  affected  by  inharmonicity  that  these
> > contiguous intervals beat at almost the same speeds"
> >
> >
> > Cheers !
> >
> > Ricb