Octaves in the Bass

[Richard Brekne]

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Hi list.

Those of you who have followed the disscussions on para, bass tuning,
longitudinal waves, and the like may find this line of interest.

I have been looking at some frequency data taken from a tuned piano, graphing
the behavior of partials, and of coincident partials in octaves. The piano was
measured by Dr. Sanderson in 1978 and the resulting tables were included in an
article published in the Piano Technicians Journal in 1978. I have to assume
therefore that the tuning was pretty much up to par.

The measuring included the 1st through the 8th (7th left out) partial for notes
C3 to C5 and were taken in Hz. Plotting the "distance" between each pair of
octave coincidents starting at C3/C4 up to C4/C5 resulted in a very interesting
graph.

The entire section was tuned such that the 4:2 octave type showed a 0.5 beats
per second wide beat rate. The beat rate of the 6:3 type started at 0.5 plus and
fell first gradually, then faster and faster off to 1.0 bps narrow. The 4:8 fell
quickly off from 0.1 narrow bps to 4.7 narrow bps. None of that should suprise
anyone who is familiar this kind of thing. What is a bit suprising tho is the
behavior of the 2:1 in this data set.

One would expect the 2:1 to show a gradual tendency towards widening, as the 4:2
is held constant. And indeed it does show this tendency. What is a bit strange
tho is that it starts off at 0.3 bps wide (narrow in relationship to both the
4:2 and the 6:3 at the first octave in this sample C3/C4). It stays narrow in
relationship to 4:2 until A3/A4, and then rises slowly to a faster wide beat
rate then the 4:2.

If this is a bit foggy, let me put this in another light. If we tune an octave
using the 4:2 octave type as our anchor, we would expect that the 2:1 should
always be a bit wider.(and all partial pairs with higher ratios narrower)  But
here the opposite is true with regard to the 2:1 The 2:1 is less wide then the
4:2. It has a slower beat rate with both being wide of pure.

By imagining a continuence of this incomplete measurement of frequencies, to
include the octave down to the lowest octave on the piano, and continue to hold
the 4:2 constant then we could expect this tendency to continue and increase.
We could also expect the 6:3 to go wide of the 4:2 and perhaps even the 8:4
would eventually go wide of the 4:2.  (This is of course dependent upon the
scaling being consistant)

Now you might be tempted to jump on this and bring out the para inharmonicity
thing. But para is perceived as an anamalous phenomenon, unpredictable,
following no pattern. In this case we have a very consistant development over 13
octaves. And there is the fact that the tendency towards narrowing for the 2:1
(decending) is to be expected by the fact that the 4:2 is held at a constant
beat rate.

I am not sure how to explain this. It prompted me to look closer at Baldersins
measurements in chapter 8 of "On Pitch".  His comparisons for different octave
partials hold each of the 6 octave types at beatless (as compared to the
constant 0.5bps wide in the 4:2 above), comparing the remaining types beat rates
in each case. He does this for each of the "C" octaves on the piano. They show
the results one would expect, that
each partial pair with a lower partial ration then the beatless partial is wide,
and those with higher partial ratios are narrower. However looking closely at
Balderssins data also reveals an oddity in relation to those partail pairs with
a lower partial ratio then the one held beatless. In every case in every table
the 2:1 is beating slower then the 4:2 and both are wide of pure. Further the
4:2 is narrow compared to the 6:3 when partial pairs higher then 6:3 are held
beatless.

In general on can observe in Balderssin table, the first Parital pair under the
one being held beatless is wide, the next one down a bit wider, then they become
increasingly narrow in relation to each other. They all stay wide of pure. All
of this happens in a manner which follows to closely a distinct pattern to be
attributed to para. This pattern suggests a factor of predictability that is not
in line with previously accepted models for predicting inharmonicity. More
measurements of entire pianos done in this fashion would of course be required
to substantiate all this.

My description of this may be a bit hard to follow, so anyone who is interested
in looking at this for themselve, may mail me privately and I will send them the
graph and the data which it was based on.

There is of course another possibility. I might be totaly bonkers and have
looked at these numbers to long.. <grin>. In that case I have no doubt that the
boys on line will explain and correct my analysis. In any case, I am having a
gas of a time.

Richard Brekne



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