historical calculator (was temperamentals)

[Richard Moody]

Stephen Birkett writes
	> No algebraic calculation is necessary to work with these numbers. 
> Constructive geometry is all you need.

	For some reason from Pythagorus to Aristoxenes, to Ptolmy, to Mersenne, they all
chose to solve the proportional spacing problem with math. Mersenne does not mention
this method of "constructive geometry" for fret spacing, or organ pipe dimensions
strng lengths etc. But that doesn't mean it wasn't used......
	This is very Interesting.   references? Do you have any more on this? I got lost
when you said make a logarithmic
curve using 13 equally spaced lines.  Does the bent wood determine the curve?   Why
wood, would not a steel band work.  Does the species of wood matter? Green or dry?
How thin? Is this before the "mesoblaum"  I can't remember the spelling. What era
was this "solution" proposed?   

you wrote...
	> In practice I believe piano builders actually splined their curves onto 
> the soundboard of a piano under construction directly, based on a chosen 
> reference value for one string. Subtle scaling alterations could easily 
> be incorporated in this technique.

 	Is this how they got the curve of the bridge?  Harps had a curve.  Hmm in Mersenne
the bridges of the spinet are straight, those of the harpsichord are curved.......
Also on the upper bearing line of the spinet is V shaped.  

While we are on the subject of spinets, do you make any or service any?  I never
liked square grands, but the rectangular shape of the spinet seems 'natural" way to
make a plucked strung keyboard instrument of four or five octaves...... I have never
seen one.  ric     


----------
> From: Stephen Birkett <birketts@wright.aps.uoguelph.ca>
> To: pianotech@ptg.org
> Subject: historical calculator (was temperamentals)
> Date: Wednesday, August 11, 1999 12:02 AM
> 
> Richard wrote:
> > BTW for those who are curious as to how the "ancients" figured out
> > something as complicated and abstract as the 12th root of 2, before log
> > tables, slide rules or calculators, it might have been done thus......The
> > mean or the square root of the ...
> >
> No algebraic calculation is necessary to work with these numbers. 
> Constructive geometry is all you need. First make a logarithmic curve of 
> arbitrary size using 13 equally spaced parallel lines (12 spaces). Mark 
> off a length on the first and double on the second, the meam on the 6th. 
> Spline a curve with a thin piece of wood to connect these three points. 
> Transfer the 13 lengths to the same line. This can be used to mark off 
> ticks on yoru logarithmic ruler. This ruler and your proportional 
> dividers is all you need to transfer to make a logarithmic curve of any 
> base size. Not only does this avoid calculation, it is actually much 
> easier to apply in practice than transferring a bunch of funny numbers. 
> In practice I believe piano builders actually splined their curves onto 
> the soundboard of a piano under construction directly, based on a chosen 
> reference value for one string. Subtle scaling alterations could easily 
> be incorporated in this technique.
>