May the 4ths be with you

[Richard Moody]

> > 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
>
>