Making P12ths Acoustically

[Richard Brekne]

Bill Ballard wrote:

> The acoustic version of P12ths is a little simpler than the
> electronic. To tune, say, A4 a P121 from D3, start off by playing
> that P12th using the L string of A4. Listen to the beat rate on that
> P12th, occurring as RicB points out, between the 3d partial of D3 and
> the 1st partial of A4. Reproduce that beat rate by mis-tuning the
> center string of A4 (up or down, as needed) against the L string.
> Now bring that L string into unison with the center string. Check
> again for drift, and correct again by shimming the unison.
>

I know two really good aural tests for the 12th.  One uses the major
sixth below the lower note of the 12th. The beat rate of that against
both notes in the 12th should be the same. As you get higher you more
"sense" the beat rate then you do try and count it,,, but they should
"sound" the same.

Also you can ghost the coincident by sounding the fourth below the lower
note of the 12th.  Hold the 12th and hit the fourth below and listen for
beating.  Both of these are for the 3:1 12th type.

>
> Want to preview how the amount by which A4 as determined by this
> P12th, may be stretched sharp of the octaves it is normally
> referenced to?  (That is the 2:1 single octave with A3, and the
> double, 4:1 octave with A2.) Play the chord formed by A2/D3/A3. When
> A4 is a P12th from D3, the double octave beat rate between A4 and A2
> will be equal to the 4ths beat rate between A2 and D3. The single
> (2:1) octave will likewise have a beat rate equivalent to  the 3:2
> 5ths beat rate between D3 and A3. Also if the 6:4 beat rate in that
> same 5th is something other than a reasonable approximation of double
> the 3:2's beat rate, this is a clear indication of poor scaling.
> When this P12ths pattern is carried up an octave, say in the example,
> A5, there will be disagreements between A5's triple octave and it's
> double octave. Be ready with the fudge brush.

Course all this depends a bit on what you do inbetween D3, and A4. If you
start out with a P12th here, and have evenly split the difference for all
the notes intbetween, and then continue with P12ths for every note then
the whole stretch of the tuning changes. Sticking to 3:1 12th types the
bass quickly becomes a bit sharper then most ears would like, so you have
to move on to the 6:2.  A 9:3 in the lowest notes could be for large
grands, but would be far to low for most instruments.  I like to listen
for the 9:3 beat rate in aural tuning tho, and keep it as slow as I can
get away with.

The midrange of the tuning gets stretched more then a usual octave
priority approach. Typically the distance between D3 and A4 increases by
about 0.3 to 0.5 bps over the range of the 12th, and this tendency goes
on upwards for a while and then evens out with octave tunings around A6.
After that most octave tunings will stretch way beyond the P12ths. The
net effect of the 12ths tuning in the tenor/treble is that it feels a bit
more tense then its total stretch numbers would normally hint at.

> Yep, I like P12ths. I've been tuning the top half of the piano that
> way for years. Or at least as high up as I can hear the 3rd partial
> in the lower, reference note, which is usually most of the way
> through the 5th octave. After that, the tuned note is at the bottom
> of the 7th and from there on up it's 2:1s.

Actually, as it turns out, From the midrange up to about A6, most tunings
end up with very close to P 12ths. Its above that area and below roughy
G2 that things change significantly. Variations in stretch for these
ranges could just as easily be accomplished by moving to a different 12th
ratio type as they are moving to different octave types.

One of the nice things about using P12ths as a priority is that it
requires almost no calculations or assumptions about the inharmonicity of
piano strings.  The octaves fall into place as a result of holding the
12ths pure, and examining them reveals a very smooth curve, albiet not
quite the same kind of curve we usually get.  In essence this means we
have a procedure for truly combining ear tuning with the ETD's cabability
for listening to frequencies. Since nearly all  12ths are to be set to a
3:1 pure relationship, the phase display needs to stop for both the 3rd
partial of the one note, and the 1st of the other.


> Bill Ballard RPT
> NH Chapter, P.T.G.
>
>

Thanks for piping in Bill

--
Richard Brekne
RPT, N.P.T.F.
UiB, Bergen, Norway
mailto:rbrekne@broadpark.no
http://home.broadpark.no/~rbrekne/ricmain.html