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

[Richard Brekne]

Hi Fred

    Fred writes:
    ......
    I don't think of a pure 12th as particularly wide as a basis for 
    tuning. Take a 3:2 5th above a 2:1 octave. Tuning conservatively,
    the  3:2 5th will be about 2 cents narrow and the 2:1 octave will be
    about  2 cents wide. Result: pure 3:1 12th. Someone looking to
    increase  stretch might shade the 3:2 5th a bit wider, say 1 cent
    narrow, and  shade the 2:1 octave wider as well, say 3 cents wide
    (meaning the 10th/ 17th test would have the 17th beating just less
    than 1 bps in mid  range). Not at all unusual, and the 12th will now
    be wide by 2 cents.  With "pure 5ths" as Jim Coleman wrote about a
    few years back, the 12th  will be wider than that. I would say that
    a tuning based on pure 3:1  12ths in the mid range would be fairly
    standard sounding, close to the  original FAC or an RCT 5 or 6.

I'd say you are absolutely correct. I think you will find that octave 
types and 12th types have an un-noticed relationship relative to their 
<<wideness>>. We all know that in general the higher order of octave 
type you use, the wider the interval... but 12th types fit right 
inbetween as you go.  2:1, 3:2, 4:2, 6:3, 6:2, 8:4, 10:5, 9:3. Seems to 
follow that the larger the spread between the to numbers in the ratio of 
the interval type... the greater the spread. I haven't got this 100% 
confirmed... but its something I've been noticing more and more. Pocket 
Tune Labs partial switching button along with my P-12ths template curve 
makes it real handy to cross check all interval these types on the fly. 
Its particularly useful in getting a beautiful big base sound to cross 
check interval types... particularly 4:2 6:3, 6:2 and 9:3.  No matter 
what instrument you have (para-inharmonicity aside) the lower order 
intervals will tend to converge at a point slightly wide (between a half 
and one bps) where some higher order will come nearly pure.  That tends 
to be the cleanest bass spread for any piano.  On some pianos it will be 
the 6:3 that is clean... others the 6:2... on big grands down in the 
lowest bass the 9:3 is my choice quite often.


    If the 3:1 12ths are used consistently throughout the piano, top to 
    bottom, I would expect the result to be rather more narrow than
    most  of us tune, particularly in the bass. Your samples didn't
    sound that  way in the bass, though I thought the high treble
    sounded a bit  flatter than what I would want personally. I suspect
    the VTD does a  6:1 19th for the bottom couple octaves or so, based
    on what I heard.

    .........

In the US... yeah.. most folks end up stretching the high area way more 
then this.  F6 has its 3rd partial typically around 35 cents offset... 
which puts C8's fundamental there. For the bass.. Jim Coleman and I 
exchanged several emails about the narrowness of the 12th in the base 
when I first put my Tune Lab P 12ths tuning public.  Eventually thats 
what got me onto 12th types instead of octave types.  Now I cross check 
to see that several key interval types are all acceptable. It makes Tune 
Lab a functional multi-partial ETD really... tho it is set up as a 
single partial apparatus. 

A word on our German friend Stopper.  He has done some deeper maths to 
justify the use of the 12th interval quite a ways back... along the 
kinds of lines of older maths done for different temperaments through 
time. That said... the basic idea of tuning P-12ths clearly out dates 
his work. The aural test is the major 6th and the double octave major 
3rd. Been around for a long time.  I generally run a 5:4,5:3,5:2,5:1 
test aurally. With the 5 being the reference note. All 4 should have a 
similar beat rate with the 5:4 and 5:1 being just a tad faster then the 
other two, and the 5:2 being even less so faster then the 5:4.  This 
yeilds a very slightly wide 4:2 for the lower pair of the double octave 
and a consistantly wide 4:1 2:1 for the double octave itself and the 
higher octave. You can keep an eye on the 4:3 and 3:2 inbetween the 
lower pair as well as you go if you like. All both easy and quick to do 
both aurally and using TL's partial switching button.

Cheers
RicB