---------------------- multipart/alternative attachment Michael G asks: > If the second harmonic of > the lower note of the 8ve = the first harm.of the higher note then the > fundamentals are not exactly an 8ve apart??? > Hi Michael, An "octave" can be many things ! Tuners, aural or visual, don't tune a 2:1 octave anywhere in the piano. Here's why: Because piano wire does not vibrate harmonically, the partials become progressively sharper, and only one set can match at a time. This is different from an organ, the partials of which are harmonic. Take for an example an organ pipe vibrating at 100 Hz. It will be producing harmonics at 200, 300, 400, 500, and so forth. Doubling in this case = an octave, so 200, 400, 800, etc. are 8ves. Another pipe vibrating at 200 would produce harmonics at 400, 600, 800, etc., so the two tones played together would produce coincident partials at 2:1 (the 2nd partial of the lower coincident with the 1st partial of the upper), as you describe. There would also be coincidences at 4:2, 6:3, 8:4, etc. The note would be so in tune that it would actuallly sound like a brighter version of the lower note. No beats. In the piano example, I'll use made-up numbers, overstated for illustration. For the 100 Hz. fundamental, because of the inharmonicity endemic to ALL piano strings, our phony piano might have octave partials at 201, 403, and 810 (exaggerated). The upper note might be 200, 401, 806. If you raise it one Hz. to match at 2:1, or 201Hz:201Hz, you will also have 403:402 and 810:807, so the upper harmonics will beat. If you raise it more so that it matches at 4:2, it won't match at 2:1, 6:3, or 8:4. Some of these beats are of course louder than others, so the "beatless" octave that some tuners describe is a actually a "sweet spot" where a weighted average of this beating is reduced to a minimum. It is sometimes an appropriate octave choice, usually in the temperament area, is often an octave that doesn't match exactly on ANY coincidence; and is wide of 2:1, less wide of 4:2, and slightly narrow of coincidence at 6:3. Other parts of the piano require that the single octaves be stretched a little beyond "clean", so that double and triple octaves sound better. While by themselves, 4:2 or 6:3 or 8:4 octaves might all sound good, a little slow beating is intentionally introduced in the octaves, in order to make the wider intervals closer to clean. A triple octave higher in the piano can not actually be as clean as possible, without introducing objectionable beats in the singles and doubles. Octave width can also be voicing dependent. Because our ear averages the beating of the various harmonics depending upon their amplitude, tuners will hear a "beatless" octave at a different width in a brightly-voiced piano and darker one. I hope this is useful. Regards, Bob Davis ---------------------- multipart/alternative attachment An HTML attachment was scrubbed... URL: https://www.moypiano.com/ptg/pianotech.php/attachments/fa/8d/84/ac/attachment.htm ---------------------- multipart/alternative attachment--
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