Hi Bernhard Hi again Bernhard So lets put this into some more perspective... this is getting really interesting by the way... my attention is really caught ! The pythagorean comma of roughly 23.46 is usually divided into 12 parts gives 1.955 cents needed to yeild a pure 2:1 octave. If we divide the comma into 19 parts then we have as you say roughly 1.235 cents needed to end up with a pure 3:1 12th Translated into our familiar root of twelve and root of nineteen and useing 440 as an example... : 440 * 2^(1/12) twelve times will yeild 880, but 440 * 3^(1/19) twelve times will yeild 880.6278 for the 2:1 relationship. Adding abou 0.63 bps before inharmonicity is considered to the A3-A4 octave. Now.... if you tune an A3-A4 octave to a pure 6:3 octave type... then the 2:1 type gets stretched by a very comparable amount.... i.e. close to 0.63 bps for the 2:1 One of my thoughts all along about the P12th that I do is that it more or less automatically takes consideration to inharmoncity. Just a thought RicB Bernhard Stopper wrote: > some additional thoughts.... > > *Twelve fifths - pythagorean comma = seven octaves* > mathematically *(3/(2^(pc/12)))^12 = (2)^7* > is *traditional equal temperament* > > *Twelve fifths= seven octaves + pythagorean comma* > mathematically *(3/2)^12 = (2)^(pc/7)^7* > is *equal pure 5th (Cordier)* > > *Twelve 12ths = 19 Octaves + pythagorean comma* > mathematically *3^12 = 2^(pc/19)^19* > *or* > *3^12 = 3^(12/19)^19* > *is equal pure 12th *(Stopper)* * > *or "acoustic octave transformed pythagorean"* > > > the list of the "pythagorean" given below is not complete, it can be > extended to any interval combination of 3 and 2 ratio of the keyboard > > regards, > > Bernhard Stopper
This PTG archive page provided courtesy of Moy Piano Service, LLC