Hi Stephane, Ron, Cy, and everybody, I've always thought of it like this: a. Elastic forces restore the string back towards a straight line after any deflection (with overshoot from momentum, causing repeated cycles). b. Within these restoring forces, there is a ratio I'll call E(s) : E(t) between the transverse elastic behavior from the bending stiffness of the string material itself [E(s)], and the transverse elastic behavior due to longitudinal tension [E(t)]. c. E(t) increases proportionally to the mode number (partial number). Conversely and simultaneously, mass per mode decreases proportionally with mode number. Since pitch is proportional to any simultaneous and equal reduction in mass and increase in stiffness (or to the square or inverse to the square of any one factor alone), the relative pitch is proportional to the mode number so the partials would be harmonic were they determined only by mass and E(t). The frequency (pitch) of the partials would follow a harmonic series (1x,2x,3x,4x,5x...). d. However, E(s) varies more than proportionally with mode shape due to beam mode stiffness behavior. I think maybe it varies as the 3rd power instead of directly, and I think it has to do with bending radius or leverage effects (anybody help me here?). In any case, if partials were determined by E(s) alone, they would be more like the modes of a beam, or like Cy mentioned, a rigid rod. If you look at beam modes, the frequency of the 2nd mode is often around 4 times the first mode and the third mode is often around 9 times the first mode. Regardless of the exact math or whether the example beam is cantilevered or whatever, the point holds that modes (partials), above the first one, if determined only by mass per mode and material stiffness per mode can be way sharp of a 1x,2x,3x,4x harmonic series. In short, E(s) causes partials to be sharper and sharper compared to a standard harmonic series the higher they go, while E(t) doesn't cause any effect like that. So inharmonicity goes up when the ratio E(s) : E(t) goes up. e. The overall spring constant for bending stiffness is set by the material and diameter. When string diameter is increased, wire stiffness increases more than cross-sectional area because of beam geometry, so when string diameter is increased, the spring constant goes up more than the mass, which is proportional to cross-sectional area, goes up. To keep the same pitch for a given length, tension must be increased proportionally to mass, so for the same pitch and length, E(s) goes up more than E(t) when the diameter is increased. Conversely, decreasing diameter while maintaining pitch and length reduces the E(s): E(t) ratio. f. If you increase the pitch without changing the string diameter or material or length, you also reduce the E(s): E(t) ratio. (Although the bending stiffness of the material is not affected, E(t) goes up.) g. If you keep the diameter and the mass and the pitch and thus also tension about the same, but use a less stiff or 'softer' wire, like a vintage type wire or something, then E(t) is proportionally higher relative to E(s). For a more extreme case, to maintain the same pitch with bronze wire, which is denser, tension would actually have to increase, or diameter be reduced, so again the E(s) : E(t) ratio is reduced. h. In the last case, however, I can see how sustain might be reduced as softer materials are used, if softer materials increase the ratio of internal friction to total E and mass. Just guessing. I'm leaving out all kinds of things like the effect of a non-rigid termination (bridge motion) on mode harmonicity, but I'm also wondering if maybe the E(t) : E(s) ratio is a really big part of the inharmonicity equation. I think these factors explain part of the apparent paradox between things like, on the one hand, Del's interest in lower tension scale pianos for the home, where I think he mentioned (among other things) a sweeter tone with less inharmonicity, and, on the other hand, Stephane's remarks about increasing the tension on an old Pleyel leading to lower inharmonicity. In one case, lower inharmonicity is associated with lower tension, in the other case it's associated with higher tension. But in the first case I believe that Del mostly had smaller wire gauges in mind while maintaining comparable scale lengths and of course pitch; in the second case Stephane had the same wire gauge and length in mind while increasing the pitch. Note in both cases, the ratio of pitch to diameter is increased while keeping length and material constant. The two approaches aren't equivalent in all ways because loudness and other aspects of tone are affected differently, and sustain is affected by the ratio of energy stored in mass and stiffness vs. energy dissipated by internal string friction and in the bridge and soundboard, and I think inharmonicity is also affected by the ratio of these energies vs. the mass/spring reactions of the moving bridge. (In Stephane's case of tuning up in pitch, if you extend it to a whole semi-tone, you can also see that it is equivalent to keeping pitch and material constant while generally reducing diameter and increasing scale length.) I started thinking about this some time back when I was trying to understand why the % of breaking strain should be an important tonal factor. I couldn't understand unless there was some transformation the material went through as it got stressed to a certain point. Maybe there is such a transformation zone, but I eventually concluded maybe it also had something to with achieving a certain ratio of E(t) to E(s), and also maybe, as Stephane mentioned, of E(t) to internal friction. Maybe for many string materials -- whether like steel, nylon, and gut, whose strength : density ratios are roughly equal, or steel and bronze where they are not -- strength is roughly proportional to stiffness and so % breaking strain is a handy guesstimate, within reasonable gauges, for the E(t) : E(s) ratio. Anybody have any ideas which parts of this might make sense, or where my understanding might have gone off on a detour? Cheers, Trent Lesher -----Original Message----- From: Stéphane Collin [mailto:collin.s@skynet.be] Sent: Sun 5/1/2005 6:00 PM To: Pianotech Cc: Subject: Re: Pitch in Paris ca. 1860 Apparently not : If we agree that flexibility is the ability of some stuff to change its shape easily, and elasticity is the ability of some stuff to recover it's original shape after having been altered, then for a string, more tension increases elasticity and decreases flexibility, and lowers the inharmonicity. I believe that more tension in the string reduces somewhat the internal frictions, which I believe are responsible for the inharmonicity. Not sure of that, though. But I believe that in a soundboard, it is the same, but with more pressure instead of tension. More compression in a board, less internal frictions, and better tone. Best regards. Stéphane Collin. ----- Original Message ----- From: "David Vanderhoofven" <david@vanderpiano.com> To: "Pianotech" <pianotech@ptg.org> Sent: Sunday, May 01, 2005 11:37 PM Subject: Re: Pitch in Paris ca. 1860 > But doesn't less flexibility lead to increased inharmonicity? > > David Vanderhoofven > > At 04:17 PM 5/1/2005, you wrote: >>more tension, less flexibility, methink. >> >>anon. >>----- Original Message ----- >>From: <mailto:imatunr@srvinet.com>Joe And Penny Goss >>To: <mailto:pianotech@ptg.org>Pianotech >>Sent: Sunday, May 01, 2005 10:57 PM >>Subject: Re: Pitch in Paris ca. 1860 >> >>More tension less elasticity >>Joe Goss RPT >>Mother Goose Tools >><mailto:imatunr@srvinet.com>imatunr@srvinet.com >>www.mothergoosetools.com >>----- Original Message ----- >>From: <mailto:collin.s@skynet.be>Stéphane Collin >>To: <mailto:pianotech@ptg.org>Pianotech >>Sent: Sunday, May 01, 2005 2:32 PM >>Subject: Re: Pitch in Paris ca. 1860 >> >>Hello Frank. >> >>You said : >> >>"Raising the pitch however will increase load and inharmonicity." >>I thought opposite (raising pitch will decrease inharmonicity, because of >>increase in elasticity of the string). Am I wrong ? >> >>Best regards. >> >>Stéphane Collin. > > > > _______________________________________________ > pianotech list info: https://www.moypiano.com/resources/#archives > > ****** 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. If you have received this e-mail in error, please immediately notify me at (312) 207-1000 and permanently delete the original and any copy of any e-mail and any printout thereof.
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