[CAUT] P-12ths Fred

[Richard Brekne]

Hi Fred. 

There is an awful lot worth tossing back and forth from your post.  But 
of course thats going to take us off on so many tangents we'll not have 
time for Halloween... or something like that.  I think I see the overall 
direction you are going and there are two points I'd like to take up... 
as perhaps we view things a bit differently....yet in another way the 
same after all.  I'll take a couple quotes I found interesting and toss 
them back at you with my own take.

    "...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."

I think, that on the one hand we can say that any stretch system suffers 
from the lack of ability to <<conform>> to the reality of piano strings 
just a about as much as any other stretch system does. At least from the 
subjective standpoint of what any given tuner decides to select as 
his/her preference.  Yet if you get a bit more specific from the get go 
concerning your definition of what this reality is then perhaps one 
finds an arena of a more objective nature.  If one says that the best 
stretch system is that in which the greatest number of the most audible 
coincidents in all intervals are most evenly spread, matched, and 
sonorous (a term which also can take on an objective meaning if so 
defined, and needs to be in this case) then one can indeed find both 
mathematically and aurally a system which best fits that definition from 
the set of available systems at any given time.  I dont really think 
that definition is per sé allllll that off from what our primary 
objectives are in tuning.... at least in a far more general sense

A second quote:

    "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. "

Actually... that was more or less where I was pointing when I posted my 
reply to Davids root of 2 post.  I'm a long ways from sure things work 
out quite like you are getting at. Essentially, if I read you right you 
are saying that when folks more or less split the difference of a 6:3 
and a 4:2 octave one ends up with whats inbetween... and I'll agree that 
dead center of these two is the 3:1... even tho thats not directly 
relevant in terms of an octave.... the resultant semitone spread is... 
yes ?  But what is it that aural tuners actually do once this octave is 
set ? They proceed to set the temperament inbetween and there are few 
that dont end up fudging one of the A's in the end to get the circle 
right... and they do so with out paying attention tooo awfully much to 
their original octave spread... at least not in the sense needed for 
your reasoning to work out as much as I think you want it to. Then 
too... you take your finished say A3/A4 tempered octave and proceed to 
tune downwards to D3...which most folks make as a compromise fitting the 
D3/A3 5th to an acceptable D3/D4 octave.  I'd bet 10 dollars to one if 
you measured the resultant 12th D3 / A4 that the greatest majority of 
resultant tunings would be well a half a bps off a pure 3:1... and I'll 
go out on a limb here a bit and predict you'd find them narrow of a pure 
3:1 at that 12th.  Remember...the whole way in octave priority schemes 
one is psychologically glued to the Octave as of being primary 
importance... with 5ths 4ths secondary and things like evenly decreasing 
beat rates for 3rds and 6ths... doubling of beat rates for 3rds that 
share the same <middle>> note (I forget that term at the moment) and the 
like.  One is not considering the 12th at all. 

If however one DOES simply glue oneself to the P-12th... and fudge 
everything inbetween to fit then it is exactly these other factors that 
have to give a bit. And given our propensity for guarding acceptable 
5ths and  4ths...  evenly increasing beat rates for the 3rds... then it 
is the Octave itself that becomes a bit ... random if you will. As it 
turns out its hardly random... they are just aligned a bit differently 
because they are being steered indirectly by the 12ths being held pure.  
This is at the heart of Bernhards math if I am not mistaken btw.  A 
graphic repesentation of the spread of octave types of both priorites 
shows clearly the difference.... and it shows up with the f5-f6 area 
being a bit more stretched in the 12ths then in an octaves priorites and 
less so in the highest area (for most octave priority stretches in 
popular use.

Indeed, you can see this in the inharmonicity values ETD's yield for 
their curves if you compare Reyburns octave type stretches to what 
results if you construct manually a strict P-12th tuning in Tunelab.

One final note about a strict P-12th tuning.  Seems like I am hearing 
that this is not something that would work.  I do it all the time and 
since I started I have both myself enjoyed the resultant tunings more 
and scored quite a bit of points among critical customers as well. I do 
agree with Jim Coleman about the bass... past D2 at the most even in 
small pianos.... a 3:1 is <<too>> narrow. <<Too>> being admittedly 
subjective but statistically overwhelmingly agreed upon. However.... 
looking at that area one notices that there is a point where 3:1 types 
and 6:2 types converge... just wide of pure.  And in a big enough piano 
where 9:3's start to converge as well.  Finding and tuning to that 
convergence yields a the nicest bass I have ever achieved. As deep a 
flavour as you can achieve on any piano... because it uses the pianos 
own inharmonicity in the first place, while as smooth a transition from 
tight to stretched as you can get.  In my opinion going from 6:3, to 8:4 
to 10:5's and then to 12:6's in some cases just muddies the pool with 
too much to think about even if one is approaching these octave types 
from the angle of convergence rather then trying to transition gradually 
in some other fashion..  Doing the 12th type convergence act is 
simpler... and works every time.

Ok... more then long enough as it is.... And perhaps much of this is 
simply different ways of expressing similar thoughts... who knows.  In 
anycase....

Cheers
RicB