Ric: Of course you can call me Jeff. Thank you for your reply and for posting the graph (long ago) that you mentioned. I remember looking at it a number of times before the concept sunk in. It was very enlightening! No, your post was not too long. I was able to follow what you said, and it confirmed what I suspected. Fifths may or may not get wide in the treble. It depends on the stretch. And it also answered an unspoken question. “Modern Tuning Theory” does not produce the same results as P12 tuning. On Fri, Mar 6, 2009 at 1:58 PM, Richard Brekne <ricb at pianostemmer.no> wrote: > 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 > > > > > > > -- Regards, Jeff Deutschle Please address replies to the List. Do not E-mail me privately. Thank You.
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