Pure 12ths tuning

[Bernhard Stopper]

Comments in the text,
regards,

Bernhard Stopper

John Formsma wrote:
> These questions are for Bernhard Stopper, but I thought they might be
> interesting for the whole list.
>
> I was at your presentation Saturday at the PTG convention.  It is a
> fascinating concept, but one that I'm not quite sure about just yet.
>
> Please tell me if I understand the concept correctly. (I'm a concepts
> kind of guy, and like to first understand the logic behind a system.)
I publisehd the tuning as "Stopper temperament" in euro-piano 3/1988.
It is based on a circle of 12 duodecimos(twelfths) and 19 octaves.
The structure of this circle is inherent in the overtones spectrum, 
whereas the
usual fifhts-circle is not. The pythagorean comma is split into 19 equal 
parts and added to the
octaves.

You can express the standard fifths circle as: (p for pathagorean comma, 
some simple math, don´t worry)

(3/2)^12=2^7*p

dividing out the fifths give:

3^12/2^12=2^7*p

sorting the octaves gives:

3^12=2^7*2^12*p

or

3^12=2^19*p

which is the said 12-duodecimo-19 octaves circle that is given in the 
partial spectrum
structure. As already said, the pythagorean comma is split into 19 equal 
parts and added to the
octaves, what makes them ~+1.25 cent  (no inharmonicity taken into 
account yet).
>
> I understand your goal is pure 12ths, and you begin aurally by tuning
> D3-A4 as a pure 12th. If you have pure 12ths throughout, you will of
> necessity have wider octaves and purer 5ths  (along with wider M3s,
> P4ths, M6s, etc.). What I'm wondering is if it's possible to arrive at
> the same conclusion (Pure 12ths) by merely starting the tuning with
> wider octaves?
Possibly very close, but how would you get the needed stretch at the start?
>
> In other words, you could get the same result...just working at it
> from the other direction (wider octaves) rather than beginning with a
> P12th. Both roads would lead to the same place.
If you know exactly, how the octave must be stretched, yes. The stretch 
that is
required for getting pure duodecimos ( i prefer duodecimo, because a 
twelfth is
often misinterpreted as an octave) is ~+1.25 cent per octave without 
inharmonicity.
 But how to measure this if you start tuning aurally?

You calculate the the octave simply with 3^(12/19).
Or even with any other interval. So ~-1.25 cent for the fifth. or ~+2.5 
cent for the fourth
or double octave, etc. You can calculate any interval (n) by 3^(n/19). 
This is because any
ET is just a logarithmic scale which is defined by an interval and the 
desired number of steps.
(We do not consider inharmonicity at this moment) The desired interval 
and number of
intervals could be even be non integer numbers.
>
> It would seem that if you are comparing your Pure 12ths tuning to the
> first temperament octave of a 2:1 or 4:2 size, your method would be
> preferable in most pianos. This is because most pianos will sound
> better with a greater stretch than a 2:1 octave in the temperament
> region.
Yes, this is empirical finding and tuning practice evolution if you want,
but is this an explanation ?

An explanation for me is the mathematical interference symmetry (MIS) 
inherent in the pure 12ths tuning.
I can not describe MIS in depth here. But it has the effect, that many beats
dissappear if you play in more than 2 dimensions (voices). Such effects 
are also used intuitively
by skilled aural tuners. For me, the finding of the mathematical 
interference symmetry (found it in 2004)
is even more important than the finding of the tuning itself, since this 
explains the physical phenomene
and why the tuning sounds so nice, and is therefore a proof why to use 
exact pure twelfths and not
anything between pure fifths or pure octaves. (For example a 31th root 
of six or other averages).

>
> But many of us are already tuning with temperament octaves wider than
> 4:2, and we end up with pure 12ths as a result.
Fine if you do so, but for me not always shure and very inharmonicity 
dependent.
> We might therefore
> might have a tuning that matches yours, but have arrived there by a
> different road.
Yes of  course very close. But most tuners who tried out to start the tuning
with a pure twelfth, switch to this concept because it is more 
straightforward and easier
to use explicitely the interval, we want to get as result. I also 
described a "twelfth-gripping-device"
in my initial publication in 1988, for easier aural tuning the pure 
twelfth in he treble.
>
> Your thoughts?
Pure twelfths tuning theory and the finding of the mathematical 
interference symmetry
confirms what good aural tuners tend to do. No aural tuner should be 
offended by this concept,
but confirmed and invited to do so. As David Anderson nicely said, it´s 
time of "Sturm und Drang" for me:
I try to push my temperament to become the new tuning standard in theory 
(as is already in practice for
many tuners). So sometimes i can not be everybodys darling (excuse me 
for this).

>
> Also, could you briefly explain how your program picks out the stretch
> that a piano is supposed to have? 
I can not publish yet, it´s part of intellectual property.
> How much partial information does it
> measure before calculating the tuning? 
region dependant.
> Does it automatically adjust
> the optimum stretch, 
Yes. Just start tuning chromatically. No stretch options or 
pre-measurements required.
> or do you have stretch options like the RCT?
Maybe later, but at the moment the soft is intended to do anything 
automatically to get
a straight pure twelfts tuning and the maximum of MIS.
>
> On the Fazioli you tuned, which sounded very nice (musically), I
> noticed some 12ths that were actually wide, 
Should not be, but may occur rarely, when the inharmonicity is very 
jumpy (but this was probably not the case
on the Fazioli).
> and some double octaves
> that were also quite wide (up to around 2 bps). 
This is a part of the concept, double octaves are wider than usually 
tuned to (so indeed
there is a slight difference from tuning evolution), but this is very 
inharmonicity dependant.
Forget to get any pure octave coincident, this is not inherent in the 
pure twelfths concept and theory.
> I think the wideness
> was around G4-G6, but it might have been G3-G5. Wherever it was, there
> was also jump in the beat speed of ascending chromatic intervals
> (like the M17s). 
Rick Baldassin measured the Fazioli with the Sanderson tuner (he checked
every note for pure twelfths) immediately after i tuned it with my PDA 
software.
He said: "What you do is what you say, and this is very rare".
> Would that be attributed to the tuning program, or
> instability in the piano?
Probably instability. There was cold air condition who maybe did some 
floating to the strings
or maybe there were were also some testers before you, who pounded on 
the instrument.

The program even handles breaks on crap pianos to get smooth intervals.
>
> Thanks,
>
> John Formsma
>