source file: m1410.txt Date: Sat, 9 May 1998 14:56 +0100 (BST) Subject: More on instrument aperiodicity From: gbreed@cix.compulink.co.uk (Graham Breed) Paul Erlich wrote: > I'm not getting equivocal at all. The "simplified models", as Gary > observed, seem to handle the vast majority of cases. Dave Hill did find > specific values for the inharmonicities, which I would ascribe to > methodological errors, etc. But even if you don't believe me, and take > Dave's values for the inharmonicities, they are clearly of a different > order of magnitude (typically < 1 cent) than the departures of the > series of resonant frequencies from a harmonic series (typically dozens > of cents). That's what I've been trying to say. Hang on. In TD 1392, Gary Morrison started the discussion with: > Another consideration that deserves some thought along these lines is > that very few acoustic instruments have their overtones within 1 cent of > exact harmonics. Then you said > Last time we had this discussion, we agreed (I thought) that bowed > strings, winds, and brass instruments had exact integer harmonics for as > long as they sustain a consistent tone. That is a physical fact, true of > any system with a 1-component driver, which oscillates periodically and Firstly, I took "exact" to mean an integer to an infinite number of decimal places. "How can anyone assert such accuracy?" I thought. Also, the implication is obviously of far _higher_ accuracy than the <1 cent previously suggested. I also queried how you could be so sure that a reed or whatever is exactly a 1-component driver, but Gary made the point for me. You replied with a model for the operation of reeds/lips that assumed laminar airflow, and concluded with: > Hence the amplitude of > the partials can vary, sometimes wildly, rendering the waveform > aperiodic. However, for the time domains relevant for the ear's > analysis, this does not lead to any relevant alterations in the > frequencies of the partials. So, you're presumably still with exact integer ratios. Maybe, though, you mean that pitch fluctuations place a lower limit on the pitch precision that is meaningful, and that inharmonicity is below this threshold. It isn't clear. I presumed, with all the uncertainties in the construction of the reed, that resonance with the air column was the main factor governing inharmonicity, and was corrected on that point. If you're now saying that inharmonicities of a fraction of a cent are not physically impossible, I'm happy to concur with that. My guess would be less than 0.1 cents, although I have no proof of that, and would like some. Hopefully, we can stop theorising and wait for some more data. This is interesting, though: >>Any noise in the input will cause peaks at the resonant modes. > They are not really "peaks" so much as "hills" since there are no > constructively interfering standing waves to sharpen them. While the > noise energy will be spread throughout the spectrum, the energy from the > driver (reed, lips, bow) will manifest in Dirac-delta-function-like > peaks at harmonic overtones in the spectrum. Why couldn't they be reinforced by standing waves? If the noise provides a constant impulse at all frequencies, that should lead to peaks at the resonances like with a flute, maybe still corrected by an order of magnitude towards harmonicity. Whether this means two peaks occur close together, or the original peak is shifted slightly, I don't know. Definitely smaller than the 0.1 cents for most instruments, though, because of the low noise level. Maybe not even the only source of inharmonicity, but the easiest one to quantify. I'm not deliberately causing trouble here, only I usually agree with you and it distresses me when I don't. Hopefully, this was just a problem of expression, and we can put it all behind us, marching defiantly towards a better future. > Listen, Gary and I have been defining systematic inharmonicity as cases > where, after the noise is removed, the partials deviate from a harmonic > series. Since noise doesn't exhibit interference effects, the only > inharmonicity relevant to JI is systematic inharmonicity. I can't find this definition anywhere. I was assuming "systematic" meant the deviation from integers had to be constant from one cycle to the next. >>The resonances of flutes were also given relative to 12-equal. They were >>up to 20 cents out! It's only the flautist's skill that adjusts them to >>the desired scale. >The standard fingerings take this into account, in case you didn't know >that already. So, if knowing the standard fingerings is considered part >of "skill", then you are right. I can't go back and check now, but I think both cases were covered. Harmonics played by the performer were out by less than however else they were measured. There is still a deviation with the usual fingerings, though, of quite a few cents. Enough to scupper the difference between 12-equal and meantone. Instrument manufacture may have improved since then, of course. The inharmonicity in the resonances is caused by the holes being there, and there's some complex relationship between this and the holes producing slightly the wrong pitches. I didn't read it for long enough to be sure. I usually ignore acoustic instruments, but thought I'd better clue up for this discussion.