Was OnlyPure not P12ths Tunings

[Bernhard Stopper]

----- Original Message ----- 
From: "Stephen Birkett" <sbirkett@real.uwaterloo.ca>
To: <pianotech@ptg.org>
Sent: Saturday, April 16, 2005 4:30 AM
Subject: Re: Was OnlyPure not P12ths Tunings


> Bernhard:
>>No. this method does not ensure at all, that those four combinations sound
>>the purest possible (i.e. to get the smallest amplitude modulation).
>>You must do a nonlinear beat summation (numerically or by measurement or 
>>by ear)
>>of the whole spectra of the three notes invovlved.
>
> What is a "nonlinear beat summation"?

A aural judgment, or measured time delayed signals represented by a display,
or a numerical analysis of an amplitude sum of those three (or more) 
signals.
>
> A single slow but loud beat will be more offensive to the ear than several 
> faster almost inaudible beats, regardless of the position in

exact.

> the spectrum of coincident partials. Based on the definition of "purity"
> as:
>>the state, where the sum of the beats is minimal
> the former will be judged more "pure" than the latter, since the beat 
> amplitudes are not taken into consideration in this defintion. In

if you judge by ear or by measurement, (that is what i do in the my 
method(s)),
amplitudes are taken into account. To solve numerically over frequencies, 
one must indeed consider
amplitudes also.

> other words, "purity" in this context is a moving target. So why bother to 
> lock in on one particular artificial representation of the chimera?
>
>>And this method is new, and that is what i filed for patenting. One can 
>>not patent maths or an idea or a system, but a method you can.
>
> This is true of course. US patents are granted for claims related to 
> either methods or devices. In the old days, you couldn't patent software 
> (i.e. an algorithm) directly. You had to get around this by patenting the 
> software "installed on a computer [device]".  Of course that is now much 
> freer. However, a method is usually patentable only when its application 
> LEADS to something new. In this tuning case the patent would be for a new 
> method to arrive at a public domain tuning formula that is already 
> well-known. This isn't really any different

Before Helmholtz, tuning formulas were related to string lengths or pipe 
lengths.
Since Helmholtz, all the known tuning formulas were interpreted as frequency 
related.
Frequency is not a linear thing over time in most musical signals,
So the tuning formulas give incorrect results.

Beats are more linear over time than frequencies, since they are produced 
relative to the (variable) frequencies. Defining a tuning over time linear 
frequency formula for nonlinear signals does not lead to the same result as 
a beat oriented method.

> from me coming up with a new "method" to empty garbage cans and then 
> trying to get the city to licence my garbage slinging kinesthetics when I 
> see the guys at the kerb infringing my patent.


>>and with all that, with a usual frequency measuring ETDs is not possible 
>>to measure beats correctly.
>>Tuning the three notes for pure state is like tuning with unison tuning 
>>precision.
>>Try doing this with an ETD on one note with three strings one after the 
>>other.
>>The difference you hear, is what the ETD lies. And this lie is in every 
>>ETD temperament.
>
> Unless the ETD can directly compare frequencies without using an 
> intermediary target reference, in the same way the ear detects beats.

this is what i mean. i have developed such an ETD, working with time delayed 
signals
to visualize/analyze the beats.

>
>>Piano sounds are nonlinear in frequency AND time.
>
> What  do you mean by "nonlinear in time"?

frequency of musical signals changes over time. if measuring beats by a 
reference signal and measuring frequency,
of the signal, the results for the beats are not the same as the perceived 
one.

>
> And while I'm asking for clarifications, I hope I'm not the only one here 
> wondering about:
>>fractal symmetry of the beat-relations to the interval relations in the 
>>"stopper-tuning"

> and
>>Since due to the extension of the tuning matter from two dimensions to 
>>three (and more) dimensions the octaves/fifths problem dissappears,
> What is fractal symmetry of beats and three dimensional tuning matter?

the fractal symmetry of beat ratios to interval ratios means that the 
generating math of the 19th root of 3 ("Stopper-Tuning") produce beats 
between octaves and fifths, that have the same ratios as the intervals. 
(true for signals linear in frequency and time)
That means the beats produce Sub-Oscillations that have the same ratios of 
1/1, 3/1, ~2/1, ~3/2 etc.
These symmetric constellations occur only in the "Stopper-Tuning".
The symmetry is the base for the effect that those 3-note combinations can 
be set to a pure state.

So in the new method, he tuning formulas relates to the beats, not to the 
absolute frequencies, and can therefore be extended to nonlinear signals. 
This is the main difference to the old stuff.

>
> Stephen
>
> -- 
> Dr Stephen Birkett, Associate Professor
> Department of Systems Design Engineering
> University of Waterloo, Waterloo ON Canada N2L 3G1
> Director, Waterloo Piano Systems Group
> Associate Member, Piano Technician's Guild
>
> E3 Room 3158
> tel: 519-888-4567 Ext. 3792
> fax: 519-746-4791
> PianoTech Lab Room E3-3160 Ext. 7115
> mailto: sbirkett[at]real.uwaterloo.ca
> http://real.uwaterloo.ca/~sbirkett
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