<!doctype html public "-//W3C//DTD W3 HTML//EN"> <html><head><style type="text/css"></style><title>Re: Designs on a grand scale</title></head><body> <blockquote type="cite" cite>At 11:17 pm +1000 5/10/06, Overs Pianos wrote:</blockquote> <blockquote type="cite" cite><br> <blockquote type="cite" cite>Substituting the figures for the above speaking lengths into the above multiplier equation we get;</blockquote> <blockquote type="cite" cite><br> <blockquote>=(135/94)^(1/(26-1))</blockquote> <blockquote>= 1.014584831</blockquote> <blockquote><br></blockquote> </blockquote> <blockquote type="cite" cite>You can now take this multiplier and create your log-style speaking lengths from Bfl26 to A1.</blockquote> <blockquote type="cite" cite><br> <blockquote>Bfl26 = 94 cm</blockquote> <blockquote>A25 = 94 * 1.014584831  =  95.4  cm</blockquote> <blockquote>G#24 = 95.4 * 1.014584831 = 96.8 cm</blockquote> <blockquote>. . . etc. all  the way to A1 at 135 cm</blockquote> </blockquote> </blockquote> <blockquote type="cite" cite><br></blockquote> <div><br></div> <div>JD wrote;</div> <div><br></div> <blockquote type="cite" cite>So to complete the series we would have:</blockquote> <blockquote type="cite" cite><br></blockquote> <blockquote type="cite" cite>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</blockquote> <div><br></div> <div>That's correct.</div> <div><br></div> <blockquote type="cite" cite>Three questions:</blockquote> <blockquote type="cite" cite><br></blockquote> <blockquote type="cite" cite>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?</blockquote> <div><br></div> <div>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'.</div> <div><br></div> <blockquote type="cite" cite>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.</blockquote> <div><br></div> <div>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.</div> <div><br></div> <blockquote type="cite" cite>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?</blockquote> <div><br></div> <div>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.</div> <div><br></div> <blockquote type="cite" cite>  To my mind there is no mathematical objection to a straight bass bridge.</blockquote> <div><br></div> <div>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.</div> <div><br></div> <div>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.</div> <div><br></div> <div>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.</div> <div><br></div> <div>Lets now look at the attached sample spreadsheet. Have look around the sheet and you'll notice the following.</div> <div><br></div> <div>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<font color="#0000FF"> (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)</font>. 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.</div> <div><br></div> <div>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.</div> <div><br></div> <div>Try changing the values for the speaking lengths for A1 and Bfl26<font color="#0000FF"> (just type in a new speaking length and hit the 'enter' or 'return' key)</font>, 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.</div> <div><br></div> <div>The mathematics required is quite simple, but the results seem to work nicely.</div> <div><br></div> <div>Best regards,</div> <div>Ron O.</div> <div><br></div> <div><br></div> <div><br></div> <blockquote type="cite" cite><br></blockquote> <blockquote type="cite" cite><br></blockquote> <blockquote type="cite" cite><img src="cid:a06110411c14c9d131eb6@[192.168.0.4].1.0"></blockquote> <div><br></div> <div><br></div> <x-sigsep><pre>-- </pre></x-sigsep> <div><font face="Times New Roman" size="+1" color="#0000FF"><b>OVERS PIANOS - SYDNEY</b></font></div> <div><font color="#000000"><i>   </i></font><font color="#0000FF"><i>Grand Piano Manufacturers</i></font><font color="#000000"><br> _______________________</font></div> <div><font size="-2" color="#000000"><br> </font><font color="#000000">Web http://overspianos.com.au</font></div> <div><font color="#000000">mailto:ron@overspianos.com.au<br> _______________________</font></div> </body> </html>