A good description and a useful approximation of the Stopper temperament on instruments with lower inharmonicity in the temperament and lower treble region. As you figured out correctly in step 7, the 4/1 double octave A3-A5 has about the same beat rate as the fourth A3-D4 (in abscence of inharmonicity, it is exactly the same), it is important to know that one is using a different world view in this temperament, we are definitely leaving the pure octave concept. We should not have aurally or naturally (i.e. inharmonicity compensated) pure octaves or double octaves. I still want to supplement, that inharmonicity requires a duodecime (twelfth) which is slightly stretched beyound the 3/1 coincident, to target the singular symmetry (discovered 2004) of this temperament at a maximum, which turns the temperament compromise into a pure tuning on the chords level, by canceling beats through symmetric harmonic interference. In fact this temperament solves the pythagorean problem finally, as the "problem" (the pythagorean comma) fits exactly into a symmetric lattice where it simply "dissappears". Also of importance to notice that this has nothing to do with solving the pythagorean comma by inharmonicity. Bernhard Stopper Am 26.03.2009 um 03:03 schrieb Ed Sutton: > Here's a set of relationships that makes treble octave tuning simple. > > 1) Assume a pure twelfth as the standard "width control" interval. > > 2) Consider the following notes as an example: A3, D4, A4, A5. > > 3) All have coincident partials at A5. > > 4) All can be tested at A5 using test note F3. > > 5) All "beat rates" will be occuring at pitch of A5. > > 6) To describe the relationships, assume F3-A3 is 7 bps and A3-D4 is > 1bps. Then F3-D4 is 8 bps. > > 7) Since D4-A5 is pure (0 bps), F3-A5 will be 8 bps, the same as F3- > D4. (Easy to hear) A3-A5 will be 1 bps at 4/1. > > 8) Since D4-A4 is a fifth, about a half bps slower than A3-D4, then > F3-D4 is 8 bps and F3-A4 is 7.5 bps. This also means A3-A4 is about > one half bps at 4/2. > > 9) Thus we can say that the tempering of the fourth A3-D4 determines > the pitch of A5, and since the 3rd/10th/17th test requires that F3- > A4 be intermediate of F3-A3 and F3-A5, the fourth also determines > the fifth and single octave. > > 10) Using the one test note, and listening to all 4 notes at A5 is > very easy to hear and do. This relationship is generally good from > at least the lowest plainwire notes and at least as high as A6. > > IMHO. > > Ed Sutton -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://ptg.org/pipermail/pianotech_ptg.org/attachments/20090326/ccf99449/attachment.html>
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