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

[Richard Brekne]

David.,Fred

Take the span of D3-A4 and tune them to a perfect 3:1 12th then measure 
the 3rd partial of each, and split the difference into 19 semitones with 
the 19th root of 3. Then do the same except take A3-A4 tune them to a 
perfect 6:3 octave type... then measure the 3rd partial of each (to keep 
on the same page) and split the difference into 12 semitones with the 
root of two. Compare the A3-A4 area of both. The differences ARE 
significant... tho small. They become more significant as you impose the 
P-12th priority on the rest of the piano. So much so that you just cant 
do it in the lower bass... really even on small consoles. The lower bass 
gets too narrow... high if you will, and its sounds well... to use a 
word I usually guard myself against.... bad.

In the treble the end effect is that you end up with a very moderate 
stretch and a significantly altered treble tuning curve. You can see 
this clearly on a graphic of both types of curves. The octave priorities 
most often use end up with the F5-F6 area a bit lower then the P-12th... 
yet the highest range of the P-12th is lower. In a sense you can say 
(and this is exactly what Jim Coleman commented back then...) that the 
treble stretch is quite moderate...without sounding like it is. 
(paraphrased but an accurate commentary)

If you stop to think about what partials tests we use for the treble 
this only makes sense. We use 6:3, graduate to 4:2 and then its rather 
bingo what some folks choose for that highest octave.  Especially the 
last 5-6 notes. I see folks tuning C8 regularly way above the 35 cents 
sharp limit a P-12th tuning usually imposes on it.

The bass... well in the end you use 12th types instead of octave types. 
And if you think about them... they fit "inharmonicity wise" right in 
between the various octave types. This is why (I believe) Stoppers real 
maths work on the subject support his claim and why Kent observes that 
Stopper (or rather the P-12th scheme itself) deals with inharmonicity in 
a unique way that yields a very nice tuning.

That transition area in the bass where one needs to move from 6:3's to 
something a bit wider. You can see this happening quite easy by just 
cross checking 12th types. You can do this with octave types as well.. 
but using 12th types seems to <<find>> that exact area where the stretch 
needs to begin... and how much it ends up "stretching".  Thats the 
beauty of the thing as it is a computable curve that works with the 
inharmonicity of every individual piano. You simply split an appropriate 
12th interval in the temperament area into 19 semitones using the 19th 
root of 3, Tune the next 19 notes upwards in the treble to exact perfect 
12ths of the resultant <<temperemant>> area and then extend the perfect 
12th condition all the way up using those.  (Actually you end up 
creating an exponential like curve instead of the resulting linear curve 
a strict calculation of 19 semitones with the 19th root of 3 yeilds for 
the temperament area.. but thats another matter)  In the bass.. you just 
cross check at all times the 3:1, and 6:2.. on big pianos also the 9:3 
in the lower bass.  Inharmonicity forces these to coincide at some 
point...just wide of pure.  And thats where the <<automatic stretch>> 
kicks in.

Those of use with Tunelab can do this easy. Take a blank tuning curve 
and set all tuning partials to the 3rd partial except the lowest 
octave.. which you set to the 6th but never really use and the top from 
F6 upwards to the 1st as Tunelab wont allow the 3rd in that region and 
you dont really need them anyways. Then set your tuning curve priorities 
to 3:1 in the treble and 6:3 in the bass... sample your usual notes and 
create the curve. Tune a middle area 12th range and the corresponding 
12th above to each note in this range directly to the ETD. Also tune the 
corresponding octave below to each note in this middle range. That gives 
you all but the lowest octave and the highest 12th tuned. Extend the 
perfect 12th priority to the top area using the 12th region directly 
below (their 3rds partials should be very close to right on if Tunelab 
has calculated inharmonicity correct).  For the bass you have an octave 
already tuned... as a 6:3 octave type. I retune this area and the entire 
base cross checking with 3:1, 6:2's and in the lower region with 9:3's 
on big instruments. Tunelab Pocket makes doing this cross checking 
extremely easy because of its partials switching button.

Try it... compare closely both aurally, graphically and maths wise if 
you can.  I am sure you will see a very significant difference and Dr. 
Coleman immediately noticed (and liked btw) back when he looked at 
this.  His P-5ths tuning also creates similar unique effects.  Indeed... 
all enforced tuning priorities will result in their own overall tone 
colour. This is nothing new we discuss this kind of thing all the time. 
It should surprise no one that the P-12th tuning handles inharmonicity 
of real pianos differently then other tuning priorities..

Cheers
RicB



    Richard:

    The difference between the 12th root of 2 and the 19th root of 3 is
    6.297037897993807971388553887547e-5 or
    0.00006297037897993807971388553887547.  I can't tune that precisely but
    maybe you can.  :-)

    dp


    David M. Porritt, RPT
    dporritt at smu.edu