Hi Cy, Some comments about the graphs you sent further below, though I think Ron and others already covered a lot of it. Stéphane wrote me with some questions and observations and I've been spending about every spare moment, including on the train to work this morning, trying to comprehend the relationships I wrote about before better. Putting my vague intuitive ideas down in writing sure challenges me to improve my understanding! As far as I can tell, most of my conclusions about the relationships and the effects of the E(s) : E(t) ratio are true (and not original), but for one part I still can't understand WHY it's true. I think I just barely get the math, but what I don't get at all is the basis in physical reality behind one key relationship. I plan to post again when I've either figured something out or distilled the right question to ask on the list. About the charts you attached, it looks like these numbers include some random variations, but generally support the effect of E(s) : E(t). Here's my take, anyway. It's not surprising that the numbers are quite similar or even opposite to what you expected in the middle of the scale and above, because even a small upright has enough room to have "full length" strings in that range. For instance, A440 is often fairly close to 16" whether it's a grand or an upright. So, assuming modern piano wire is used in all, the main differences in string-based inharmonicity around and above that note would be from differing string diameters. For reasons like the ones Ron N. was talking about, there might actually be more reason to have thicker strings in the middle range of a larger piano than in a smaller one, so inharmonicity in that particular range might well be higher for the larger piano than for the smaller. Allowing for manufacturer variations in preference or tradition the spreadsheet you attached pretty much shows the expected trends from A440 on up. (I say "string-based" above, because, especially if the input impedance falls too low at some points reducing the mutual independence of the string and bridge/soundboard systems, spring/mass reactances from the bridge and soundboard could also cause some inharmonic output.) But now if you look at the range between A3 (A220) and D#3, which appears to be the lowest recorded note on the long bridge of the smallest pianos, you can see the trend reverses -- the larger pianos have lower inharmonicity. That's because the compromises in the smaller pianos start loosely about an octave below the 16-inchish strings at A440, and you can visualize this by bridge lines that stop curving and become straight from a certain point down, or in more extreme cases have a hockey-stick shape, which is just because they try to keep the strings as long as feasible until they run out of room. Someone experienced in scaling can say whether they also sometimes make these strings comparably thicker to try to compensate for the power difference. Wound strings can and need to be shorter than any neighboring plain strings. So once again, at the top of the wound strings, some small pianos have enough room for almost "full length" strings for a handful of notes. They might be actually a bit shorter, and/or a bit thinner, than the corresponding strings on the larger pianos, but in any case they aren't as grotesquely different as the strings at the low end of the long bridge, so at A2, for example, the inharmonicity numbers are pretty comparable between all the pianos (but only in the larger or better scaled ones are they close to the numbers just above the break). Finally the smaller pianos run out of room again as you get down into the lower bass until, at A0, inharmonicity figures may be twice their counterparts in the larger pianos. Like Ron said, the bad thing is not that inharmonicity goes so high on the small pianos, because indeed you can see that in the bass and midrange, overall the differences aren't that great, and it's actually some of the larger pianos that have some of the very highest inharmonicity factors in the treble, presumably for projection and good power curve reasons. (Being in the higher treble, these extreme factors are easier to deal with since the upper partials are weak and also there are few to no higher notes, or partials from lower notes, above these notes to line up with their upper partials anyway.) Rather, the problem is that the ideal is to have what some of the better scaled larger pianos have, which is inharmonicity factors which start at a moderate level at the lowest bass note and decrease smoothly and gradually up to the break; then pick up at the same level on the other side of the break and smoothly and gradually increase from there on out up through the highest treble note. So you have a gentle lopsided bowl shaped curve with only one reversal of direction, a smooth one without any jumps that centers on the break. Then it's more possible to optimize the other factors to make a scale that actually sounds even, and it's possible to find a tuning solution where everything lines up in a pleasing way. The smaller or poorly scaled pianos, and especially the small AND poorly scaled pianos, have a reversal of direction above the break, and probably a jump down across the break, and steeper climb in inharmonicity below that, so generally some very steep and hilly activity going on that makes even sounding scales and great sounding tuning solutions less likely. That's my take on that, though those experienced in scaling could no doubt speak more informatively on these points. If those Steinway Model "A" figures are correct (rising below A3 to .595 at D#3, and way down at .089 at A2) maybe somebody can explain what's going on there! (That's a 6'2" piano, right? Yet the inharmonicity at D#3 is higher than the Baldwin console, and more importantly, on the rise from the notes above? And why is it so low at A2 -- is that one of those wrapped bichords on the long bridge or something? And did they make a handful of the plain strings above super thick to try to match power or something?) But to get back to your question, the data you attached seems to me consistent and mostly predictable by the E(s) : E(t) ratio. There's a synopsis of an interesting paper by Galembo and Cuddy at http://www.geocities.com/CapeCanaveral/Lab/8779/grand_VS_upright.htmlIn It concludes, in part, "the spectral envelope had a stronger influence on perceived timbral difference than the inharmonicity . . . to improve the sound quality of small upright pianos, the spectral envelope parameters, particularly spectral width and spectral irregularity, are of great importance." That seems to generally support the idea that frequency-dependent impedance factors are more important scaling issues for overall sound than inharmonicity by itself. They describe an interesting experiment: "The second experiment, we synthesized an A0 tone, 1 second long, whose spectrum was the combination of the spectral envelope of the real tone of the small upright piano 114-cm high with the inharmonicity of a concert grand piano tone. We prepared tone pairs consisting of different combinations of three tones--the hybrid tone, an upright prototype, and a grand prototype." "Listeners were asked to scale the perceived timbral difference between the members of each tone pair. Results suggested that the hybrid tone was significantly closer in timbre to the upright prototype than to the grand prototype. This finding means that the spectral envelope had a stronger influence on perceived timbral difference than the inharmonicity." But they also say "This conclusion does not mean however, that inharmonicity is irrelevant. Support for the importance of inharmonicity is reflected in our synthesis of a hybrid tone with the spectral envelope of a grand piano tone and the inharmonicity of the upright tone. The presence of strong inharmonicity in the hybrid makes it impossible to produce an acceptable imitation of the grand piano tone." It does seem to me like you pretty much have to consider inharmonicity directly as part of figuring the wrapped string scale, though I hope somebody tells me if I'm wrong about that. I am guessing, however, that Ron and others involved in scale work might say -- for the plain strings anyway -- that if you use care and common sense to design the physical scale so the bridge has a smooth sweep and maintains % break, and your bridge/soundboard is well made so it has sufficient impedance and is free of sharp resonances, and you gauge the wires for a smooth power curve, a smooth inharmonicity curve will pretty much take of itself. Whereas, if you worked from inharmonicity and expected the rest to take of itself, you might have a big mess (for example, extreme case, you could have two neighboring notes thus: the lower-pitched one with a shorter string and a smaller gauge than the higher-pitched one. Obviously, your power curve would be all messed up, but inharmonicity wouldn't necessarily be out of line at all.) Cheers, Trent Lesher -----Original Message----- From: Cy Shuster [mailto:741662027@theshusters.org] Sent: Tuesday, May 03, 2005 9:37 AM To: Pianotech Subject: Re: Inharmonicity factors Trent, Your idea is fascinating. I think some of the best results that this list produces (when possible!) comes from standard empirical science: proposing a theory to describe Real-World observations, and testing it, including predictions of results not yet measured. I've been puzzled by a casual survey of iH readings that I've taken. I thought it would be obvious to spot the difference between a spinet and a concert grand, or to otherwise tell a "good" scale from a "bad" one, just by the numbers. (Maybe it *is* possible, and I don't know how; or I've measured wrong; help welcomed). For example, attached are some of the confusingly similar numbers I've gathered. Which of these stick out to you? Does this match your theory of ratios of different elasticity? --Cy-- ****** IMPORTANT NOTICE ****** This e-mail, and any attachments hereto, is intended only for use by the addressee(s) named herein and may contain legally privileged and/or confidential information. If you are not the intended recipient of this e-mail, you are hereby notified that any dissemination, distribution or copying of this e-mail, and any attachments hereto, is strictly prohibited. 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