Designs on a grand scale

[Overs Pianos]

>At 11:17 pm +1000 5/10/06, Overs Pianos wrote:
>
>>Substituting the figures for the above speaking lengths into the 
>>above multiplier equation we get;
>>
>>=(135/94)^(1/(26-1))
>>= 1.014584831
>>
>>You can now take this multiplier and create your log-style speaking 
>>lengths from Bfl26 to A1.
>>
>>Bfl26 = 94 cm
>>A25 = 94 * 1.014584831  =  95.4  cm
>>G#24 = 95.4 * 1.014584831 = 96.8 cm
>>. . . etc. all  the way to A1 at 135 cm
>

JD wrote;

>So to complete the series we would have:
>
>94.0, 95.4, 96.8, 98.2, 99.6, 101.1, 102.5, 104.0, 105.5, 107.1, 
>108.6, 110.2, 111.8, 113.5, 115.1, 116.8, 118.5, 120.2, 122.0, 
>123.8, 125.6, 127.4, 129.3, 131.1, 133.1, 135.0

That's correct.

>Three questions:
>
>1. I suppose that by "log-style" you mean you are aiming to produce 
>an exponential curve.  If so, how can you do that by simply 
>multiplying the numbers by x rather than raising them to a certain 
>power?

I'll show you by including a sample spreadsheet as an attachment, 
with the same figures as used in my previous post. The term for 
deriving the speaking lengths in this way is 'geometric progression'.

>2. Your example produces a line that bulges in the middle _towards_ 
>the strike line as shown in the chart below, where the red line is 
>straight and the blue is the graph of your lengths.  I hardly think 
>this your intention.

Yes John, it bulges if you view it from the back of the piano, and is 
hollowed if you view it from the keyboard. And that is exactly my 
intention. If you go through the excercise of choosing core and wrap 
diameters for such a scale, you'll find that its a whole lot easier 
to arrive at a satisfactory set of numbers, when compared to 
traditional bass scales, in which the bridge curve runs the other 
way. I know it might look unusual, but it works. Have a look at my 
280 concert grand line drawing on the 'for sale' section of my 
website. The 280 has a bass bridge designed with exactly this 
procedure. It really does make a lot of scaling sense once you get 
used to it. Del Fandrich was, to my knowledge, the first designer to 
scale basses in this manner.

>3. What is the purpose of a curved bass bridge, no matter what the 
>equation for the curve, other than to give extra length to the 
>second octave and enhance the tone quality where it matters more?

I'm not trying to enhance one section over another at all. I want to 
produce something which has uniform characteristics for the entire 
instrument. The best results, on the computer, seem to result when 
the bass bridge is scaled in a similar fashion to the treble bridge. 
The traditional bass bridge would seem to be a kind of hockey stick 
shape, similar to the hockey stick treble.

>   To my mind there is no mathematical objection to a straight bass bridge.

I have no doubt that a straight bridge can be made to work well. But 
I think there are advantages to the inwardly curved bass bridge.

Firstly, I have absolutely no time for using suspended bass bridges 
in any piano, regardless of how short the piano under consideration 
might be. Suspended bridges expose the soundboard panel to 
undesirable torsional forces which cannot be doing anything to help 
the sound board operate as a uniform panel. Furthermore, just from a 
listening standpoint, suspended bridges seem to sound inferior to 
straight connected bridges.

Secondly, as mentioned earlier, if you scale the bass log-style, 
you'll end up with a scale which makes it somewhat easier to select 
wrap and core diameter combinations which work. Furthermore, the lay 
of the bridge at the low end will allow you to undercut the last few 
notes if necessary to achieve a similar effect to a suspended bass 
bridge, without introducing undesirable torsional forces to the 
panel. So it seems to be a way of designing a bass bridge which has 
the benefits of a suspended bridge without the undesirable side 
effects.

Lets now look at the attached sample spreadsheet. Have look around 
the sheet and you'll notice the following.

I've entered the actual chosen speaking lengths, for this sample 
scale, into cells C2 and C27. The multiplier equation is filled into 
cell D2. It reads =(C2/C27)^(1/25). In cell D3 you enter the equation 
=D2. You then use the Fill Down function for cells D3 through to cell 
D26. This sets up all necessary cells in column D so that they have 
the multiplier number ready for use. All multiplier cells in column D 
will update if either of the speaking lengths for A1 or Bfl26 are 
altered. Now, in cell C3 we place the formula =C2/D2. This calculates 
the speaking length for cell C3 by dividing the speaking length of 
note A1 (cell C2) by the multiplier in D2 (because I have written the 
fomula to calculate the speaking lengths from A1, reducing in 
speaking length down to Bfl26, I have used divide, since the speaking 
lengths are getting smaller from the first original cell used for the 
calculations. I could have run the calculation from Bfl26, by 
increasing the speaking lengths to achieve the same results using 
multiply). The cells C3 through to C26 are filled using the Fill Down 
function so that each new speaking length is calculated from the 
results of the speaking length immediately above. Don't fill down to 
cell C27 or you will create a circular reference.

The beauty of setting up a scaling sheet in this way is that you can 
change the speaking length values for either A1 or Bfl 26 and the 
sheet will update all the intermediate speaking lengths to form a 
perfect geometric progression.

Try changing the values for the speaking lengths for A1 and Bfl26 
(just type in a new speaking length and hit the 'enter' or 'return' 
key), and watch the sheet update both the multiplier and the 
intermediate speaking lengths automatically. When I'm designing the 
treble plain wire section/s, I use the same scaling techniques to 
derive the intermediate speaking lengths between those notes which 
I've selected for the plain wire gauge changes. The chosen notes for 
the wire gauge changes are determined by best matching the 
inharmonicity and impedance to ideal curves which I generate for 
these variables.

The mathematics required is quite simple, but the results seem to work nicely.

Best regards,
Ron O.



>
>
>


-- 
OVERS PIANOS - SYDNEY
    Grand Piano Manufacturers
_______________________

Web http://overspianos.com.au
mailto:ron at overspianos.com.au
_______________________
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