Hi Jeff, may I call you Jeff ? Everyone here calls me Ric, grin at least
when they are not calling me something less then flattering. Comments below
Mr. Brekne:
I have researched the many threads in the archives regarding P12th
tuning and also Mr. Stoppers tuning. There is a feature of P12th
tuning that seems contrary to “Modern Tuning Theory”. Supposedly,
fifths become wide of just intonation in the higher treble. But it
doesn’t seem possible to have wide fifths and also have pure 3:1
twelfths. Can you comment on this?
I've never looked closely at the behavior of 5ths in the treble myself,
but I have heard this mentioned before. I started an article back in
1999 I never finished that went along these lines as it related to
octave types. That was stimulated by an article I'd read where I believe
it was Dr. Sanderson, who'd published a set of tables of frequencies for
different partials for the area of C3 to C5. Fundemental through 8th
partial (7th left out) I graphed these chromatically two ways. In
parallel fashion so as two see each individual curve and comparatively
so as to see the spread between different octave types as it developed.
In this tuning the 4:2 octave type was held to a constant 0.5 bps wide
length. At C3-C4 the 6:3 was also at 0.5 bps wide, the 2:1 was at 0.25
bps wide, and the 8:4 was just shy of pure on the narrow side. At C4-C5
the 2:1 had gone to about 0.7 bps /wide, /passing the 4:2 where as the
6:3 had developed towards the narrow and was at 1bps narrow. The 8:4
went south so to speak and was just shy of 5 bps narrow. There was a
smooth development in how each of the octave types moved in their
respective directions.
I inferred from this at the time that if you held a given octave type at
a constant beat rate over any given range then higher order octave types
would become more narrow as you moved treblewise and that lower order
types would become wider. If the 6:3 had been held at a constant 0.5 bps
wide in that same two octave range then both the 4:2 and the 2:1 would
have become wider then they started out, and both would have ended up
wider then the 6:3. The 8:4 would still have gotten more narrow... but
not as much.
Soooo.. in answer. It depends on your stretch. If you use a moderate
enough stretch then the 5ths would (I imagine for I have not actually
run these particular numbers) not go wide. If you hold a 3:1 12th to a
constant width throughout the entire then octave types become bound to
that and you find close order octave types to the 3:1 12th type (i.e the
6:3 and 4:2) change directions (with respect to their respective
widths). The 4:2 actually changes directions twice in the one sample I
did myself. With the 3:1 held pure, the 4:2 octave started at about 0.5
bps increased to about 1.5 bps half way up and then decreased again to a
minimum of 0.3 bps before changing directions again and staying in an
increasingly wide development. The 6:3 started about 0.5 bps narrow and
developed increasingly narrow until the exact same spot where the 4:2
first changed directions. At this point the 6:3 started developing wider
kept developing in that direction. The 8:4 had an abrupt point at the
same place where the 6:3 and 4:2 changed directions. But instead of
changing directions it changed how quickly it got increasingly narrow.
The 2:1 flattened out at that same point for a while, before
re-establishing its tendancy towards getting wider.
One nearly always ends up with a C8 fundamental pitch of around 35
cents... which equates to a quite moderate stretch. Yet the area around
f5-f6 usually is offset just a tad higher then I usually see in Octave
priority stretches. In other words, the shape of the curve is different
when using a P-12ths. It gets steeper sooner and doesn't get quite as
steep towards the high treble as an Octave priority stretch does. I cant
say I ever found the time to look so much closer into exactly why this
all happens as it does... that is to say how different interval types
effect others when you hold one type constant. So I cant fully answer
your question. But in 12ths the 5th does not ever get really wide in the
treble as you observe. It stays at a fairly constant just narrow of pure
all the way up.
If anyone is interested, I can post what I managed to get done from that
article. I'd just gotten done with Baldersins On Pitch, had gotten well
into using CyberTuner and looking at some of the graphs (which I can
recommend for those interested in all this) for how different intervals
widths developed for whatever stretch scheme had been selected... and
was just getting into the idea of P-12ths, which was because of a few
articles and columns I'd read from early ptg journals and because of an
increasing reliance on a few aural tests that went in the direction of
P-12ths.
Hope this didnt get too long... tho I suspect it did. I apologize ahead
of time if I got something or another turned backwards in the above...
its easy to do when describing these things. Much easier to just look at
graphs and data.
Cheers
RicB
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