[CAUT] P-12ths was: Tuning a Steinway D and aBosendorfer Imperial together

[Porritt, David]

Fred:

The 12th root of 2.0012 is the equivalent of the 19th root of 3 to 4 decimal places (1.059516053 rather than 1.059526065).

dp

David M. Porritt, RPT
dporritt at smu.edu

-----Original Message-----
From: caut-bounces at ptg.org [mailto:caut-bounces at ptg.org] On Behalf Of Fred Sturm
Sent: Tuesday, October 14, 2008 11:39 AM
To: College and University Technicians
Subject: Re: [CAUT] P-12ths was: Tuning a Steinway D and aBosendorfer Imperial together

Hi Ric,
	To some extent we have been writing at cross purposes, and some of  
that is due to unspoken assumptions. I have been writing from my own  
"practical" point of view, you from yours.
	To step back to the theoretical underpinnings, there is certainly a  
significant mathematical difference between a total system of tuning  
based on 2 divided by 12th root of 2 and 3 divided by 19th root of 3.  
The 3 system will, when expanded, be significantly wider. The equality  
of the intervals, the "half steps" in our musical world, will be "just  
as equal," but the base 3 ones will be slightly wider.
	All well and good on the theoretical front. In practice, in the real  
inharmonic world of pianos (not mentioning the psychological  
perception aspects of the human mind), we don't tune based on 2  
divided by 12th root of 2. We expand it using offsets. If we want  
"equal" half steps, we do that in mathematically smooth curves (a  
logarithmic curve on top of a logarithmic underpinning, if you like).  
In essence, this expands the 2 to something larger so that we are  
really tuning based on, say for the sake of argument, 2.1 divided by  
12th root of 2.1. So that, after expanding that system, we end up  
somewhere significantly wider than based on 2.
	A "base 3" system would come closer to what we do than a pure "base  
2" system. But it would suffer from the same type of flaw in dealing  
with the inharmonic world of pianos. 3:1 in an absolute sense (the top  
note of the 12th being 3 times the hertz of the bottom note) doesn't  
conform to the reality of piano strings any more than 2:1 does. In  
fact, being a higher partial, it is "off by more." So a system based  
mathematically on 3 would need its own set of offsets to produce what  
we refer to as 3:1 12ths (1st partial of upper note matching 3rd  
partial of lower).
	Either system would be an emulation. Is there some way in which one  
is superior to the other? I can't see that there is. Either system  
uses a logarithmic basis. At some point they will coincide so closely  
as to be indistinguishable, depending on the degree to which one is  
fudging the 2 and the 3. Perhaps the 12th root of 2.05 is the same to  
within 10 decimal points as the 19th root of 3. Perhaps it will be the  
12th root of 2.06. I don't know, and don't have the software to find  
out easily, but my point is that, when you add the necessary fudge  
factor to account for inharmonicity, there will be a point at which  
the two multiplying factors will be utterly indistinguishable from a  
practical standpoint. IOW, one can be used fairly easily to emulate  
the results of the other. They are no more different than base 10 is  
different from base 12 in arithmetic. Different ways of expressing the  
same mathematical reality, but the mathematical reality remains the  
same.
	Getting back to practical, I think that if one tunes one's initial  
temperament octave (F3/F4, A3/A4, somewhere in that area) to a pretty  
standard "somewhat wide 4:2, somewhat narrow 6:3," a compromise  
between the two, and then expands in a somewhat "standard" way, one  
comes to the 12th at pretty close to 3:1. I'm talking about tuning  
aurally, based on my own experience. If the "pure 3:1" is something  
you take as your touchstone, then you adapt to that, and go back and  
make adjustments. Maybe you start by tuning that 12th, and then fill  
in the gap. Maybe you approach it some other way. Personally, when  
tuning aurally I chose to adapt to a 6:1 19th in most cases. It's just  
another choice. (Now my most common "principle" is 8:1, fudging  
narrower in the midrange of smaller pianos).
	But I believe that if you are filling in your gaps evenly, whatever  
the outer limits of the large interval you might choose, it is quite  
easy to emulate that mathematically by measuring and calculating, and  
creating a curve of numerical offsets. And whether those are expressed  
in a base 2 or a base 3 is irrelevant. The question is simply how much  
overall stretch  you are using. Far more significant from a practical  
standpoint is "where you place the curve" (along what partial you tune).
	I guess from a practical standpoint, measuring 3rd partials and  
matching them is a more reliable way of making a tuning "pure 3:1  
throughout" in the real world. Who knows what other unintended  
consequences there will be (particularly on pianos with large  
inharmonic jumps), and whether or not those will matter to you.
	I suppose that another reason we are to some extent writing at cross  
purposes is that I have never focused on "octave types," but that's  
another issue. Lots of people do, but I don't find that to be a good  
way of "finding an overall sound" for a piano. So in that regard, we  
are more or less on the same page.
Regards,
Fred Sturm
University of New Mexico
fssturm at unm.edu



On Oct 14, 2008, at 2:28 AM, Richard Brekne wrote:

> Hi Fred... just for clarity (please excuse me copying your entire  
> post...)  I find everything you state below to be agreeable with  
> what I have been saying. No conflicts in thought per sé. I would  
> point out tho that in your first paragraph... tho it is true one can  
> <<emulate>> the results of a P 12th one in fact doesnt do this in a  
> usual temperament and octave priority tuning.  Just do your usual  
> temperament... A3-A4 is it ??.. in anycase when you have it done  
> extend the thing so that you can compare your first 12th.  Measure  
> the resultant 3:1 relationship between the two.  You'll more then  
> likely find its off a bit.. depending on your style as much as a  
> half a cent I can imagine... it also depends a bit on the piano.  
> Transfering 440 from A4-to A3 as is usually required, then tuning as  
> usual the D3:A3 5th most often puts the D3:A4 12th a bit narrow.  
> Extending this kind of priority over a whole tuning results in a  
> different tuning... naturally enough. Yes one CAN emulate a P-12th  
> tuning using octave priorities... but number one this is not in fact  
> done, and number two why would you do so when you can much more  
> easily just tune a P-12th directly ?
>
> Nice discussion !
> Cheers
> RicB