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

[Fred Sturm]

On Oct 14, 2008, at 5:06 PM, Porritt, David wrote:

> I was pointing out the small difference in the 12th root of 2 and  
> the 19th root of 3.  Differences which are mathematically  
> significant (if you’re looking at decimals to the 5th digit) but  
> insignificant if you’re tuning real pianos with real inharmonicity  
> with contemporary tools.  The pitch change from start to 4 seconds  
> later is much wider than that.  When in the decay cycle are you  
> measuring this to 5 digit accuracy?

	The difference is very small when applied to individual half steps.  
It becomes somewhat significant when applied across the entire range  
of a piano. How significant? Well, if you start both "base 2" and  
"base 3" at the same note, the divergence is approximately 2 cents  
when you reach a 12th. Expand that over the entire range of the piano,  
and it becomes about 9 cents. "Somewhat significant." But you only  
have to change base 2 to base 2.0125 or thereabouts for the  
convergence to be absolute.
	In any case, this is nowhere near the "stretch reality" of the real  
inharmonic piano. I think that we can agree that the spread from top  
to bottom of a fairly conservative tuning would be in the neighborhood  
of 50 cents. (It is difficult to measure the first partial of notes as  
low as A0, so I am estimating and extrapolating a bit, but I think we  
can agree that this is a reasonable estimate). For those of us who  
stretch more, that divergence can easily be 100 cents.
	So to my way of thinking, the 19th root of 3 is irrelevant. In real  
tuning, one might take an expanded 3 as the basis, a 3 that  
corresponded to the measured distance between the 1st and 3rd partial  
of a sample note. This is simply, to my mind, nothing more or less  
than selecting some degree of stretch to use.  If your choice for  
stretch is to accept a 3:1 partial match, that's a perfectly  
legitimate choice. So is a 4:1. Or a 4:1 expanded by 0.5 cents. Or 3:1  
expanded by 0.5 cents. All of these will "account naturally for the  
inharmonicity of the piano." All "fall along a continuum" of  
increasing or decreasing stretch.
	In an inharmonic real piano, base 2, base 3, base 4, or any other  
whole number doesn't work except as a starting point. In any of these  
cases, it will be necessary (for purposes of calculation) to establish  
an expanded whole number: 2.15, 3.1, 4.06, something along these lines  
(these are imaginary numbers, not based on experience or calculation,  
chosen for example purposes only), on the basis of which to make the  
division of the octave. The same principle is involved: a single  
proportional multiplier to produce proportionally equal semitones.  
(And if you want to do unequal, this provides a very useful basis for  
WT offsets). There are other mathematical paths to the same result,  
but the result is identical.
	I am not for a minute arguing for or against 3:1 as a "stretch  
basis." I think, personally, that it is quite acceptable for the mid  
range. I prefer a stretch based on wider intervals, either 6:1 or 8:1  
for the outer ranges. That is simply my own personal preference based  
on my own taste and experience. I don't argue against the tastes of  
others. (I do think that most of us would agree that a bass tuning  
based on 3:1 would be too narrow).
	What I am trying to do is to point out that, IMO, "there is nothing  
magic" about the 19th root of 3 as a basis for tuning. It is simply  
indistinguishable from other mathematical ways of establishing equal  
half step relationships in the real, inharmonic world of piano tuning.
	Stopper argues otherwise (see his article, referenced in a post I  
sent previously). I don't find his arguments at all compelling. Others  
may. He makes the 19th root of 3 division the basis for the "Stopper  
comma," which he makes great claims for. He does say that to the  
"additional stretch" produced by beginning with a pure 12th must be  
added the inharmonicity of the piano, though his explanation of how  
this is done is VERY vague, and doesn't demonstrate a very good grasp  
of the complexities involved. An example of his explanation of  
inharmonicity and tuning:
  "The inharmonicity itself pushes the whole scale away from the  
theoretical frequencies derived by the scale functional formula. The  
inharmonicity is already considered when tuning aurally, since the ear  
makes an integration of the harmonics to a “virtual pitch.”  If an  
aural tuner tunes a slight beat-rate-narrow fifth, that fifth remain  
about the same amount beat-rate-narrow in instruments with different  
inharmonicity, wheras the absolute frequency deviation is up to some  
cents on stiff strings in the treble."
	He claims "the recent discovery of the Supersymmetry between the  
beats and the frequencies" based on his tuning. Perhaps if it is  
demonstrated to me, I will be blown away. I am skeptical. Actually, he  
seems more focused on electronic and other "essentially harmonic"  
instruments than on acoustic pianos.
Regards,
Fred Sturm
University of New Mexico
fssturm at unm.edu


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