Bernhard Stopper wrote: >Not exactly, since you should take 440 * 3,somewhat^(1/19) (because of >inharmonicity of the 3rd harmonic what is not negligible at this point). > Yes... thats what I actually do... except that I dont use the 19th root, I use Scotts quadratic interpolation with A3 set to a just 6:3 octave in to A4. Yeilds a curved line instead a straight one. But for the purpose of example below I used 440 as a base frequency for both. Inharmonicity here should be predictable enought according to the major ETD authors that the resulting difference will be the same anyways. >But >i think we can say that the aural integration of the octave spectrum fits >much better to inharmonicity in P12 than with standard-ET octaves. (The >"acoustic octave" effect.) > >friendly wishes. > >Bernhard > > > > Cheers RicB >>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 >>> >>> >>_______________________________________________ >>pianotech list info: https://www.moypiano.com/resources/#archives >> >> > >_______________________________________________ >pianotech list info: https://www.moypiano.com/resources/#archives > > >
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