<head><style>body{font-family: Geneva,Arial,Helvetica,sans-serif;font-size:9pt;background-color: #ffffff;color: black;}</style><style> st1\:*{behavior:url(#default#ieooui) } </style><style> <ZZZ!-- /* Font Definitions */ @font-face {font-family:Georgia; panose-1:2 4 5 2 5 4 5 2 3 3;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {margin:0in; margin-bottom:.0001pt; font-size:12.0pt; font-family:"Times New Roman";} a:ZZZlink, span.MsoHyperlink {color:blue; text-decoration:underline;} a:visited, span.MsoHyperlinkFollowed {color:purple; text-decoration:underline;} span.EmailStyle17 {mso-style-type:personal-compose; font-family:Arial; color:windowtext;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in;} div.Section1 {page:Section1;} --> </style></head><body id="compText">I'm looking at White's explanation in his book Piano Tuning and Allied Arts; the paragraph that follows is part of his explanation , whose context  hypothetically supposes that we can cause an impulse on one end of a string at the same instant that the previous wave reflects off the other end: "When impulses traveling in opposite directions and in opposite phases of motion (i.e., in positions on the cord in opposite phases of each other) meet at precisely the center of the string, neither can pass each other.  At their meeting place, the exact middle of the string, the forces will be equal and opposite, so that a node or point of greatly diminished amplitude of motion, will be formed. The two pulses therefore, will have no option but to continue vibrating independently, thus dividing the string into two independently vibrating halves, each moving of course at double the original speed. Meanwhile... new impulses will be successively traveling along the cord.  As these meet the already segmented halves, there is further reflection and subdivision, at each successive impulse, so that finally, if the impulses can be kept up long enough, the result will be the division of the cord into three, four, five, six, up to perhaps ten or even more of there "ventral segments", separated by nodes, or points of minimum motion." It seems to me that there is an interaction, or a correlation, between the string's ability to vibrate along equally divided segments (as can be conveniently demonstrated by plucking a guitar string while simultaneously touching the string over, but not fretting it at, the twelfth, seventh, and fifth frets, which represent the 1/2, 1/3, and 1/4 marks), the disparity of the volume (amplitude) the harmonics produced by each of the various nodal divisions, and the tendency of the string to form nodes at points of collision of opposite-moving, out-of-phase impulses, that causes this. Theoretcally, nodes representing some division could be found forming at almost any point on the piano string -- for example, a division of the string into 562, or ten million, segments -- and I am sure that in some way the string does "divide" into these kinds of nodal divisions as competing waves meet along the string, but their effect on the overall harmonic series would be so little against the whole mass of vibration that is taking place, that they would not be noticed compared to the louder harmonics and the fundamental. The more equal segments the string divides into, the less audible the harmonic produced from this division tends to be, as can be heard in the guitar pinch-harmonic demonstration. To attempt to answer to the student's specific question, I'm not sure that it is the case that the string actually "divides up into different segments" per se; if it actually did that, I think we would see the string almost stop vibrating at the 1/2- and 1/3- points, etc., and would audibly hear the loss of the fundamental tone, as can be heard when we pinch off a harmonic on the guitar, where we can actually see the non-vibration at the nodal point and can detect the absence of the fundamental. It seems more likely that it is the case that each of those fractional segments is represented in the overall vibrational pattern of the string because of this tendency to form nodes at "wave-collision points" all along the string, whose harmonic effects can be most readily detected in each of the strongest-amplitude string divisions -- the 1/2, 1/3, 1/4, 1/5, 1/6, etc. So if we were to freeze the string in mid-motion, we might see the overall curve from end-to-end that causes the fundamental tone, and also something along the lines of  slight indentations in the string curve at the 1/2, 1/3, 1/4, etc., nodal points, each perhaps becoming less pronounced than the other, ad-almost-infinitum. Perhaps the use of the word "divides" causes some confusion because it carries an unintended implication that the string actually stops (or almost stops) moving at each of the major nodal points on its own accord, rather than continuing to vibrate at the fundamental, full-string-length frequency throughout the course of the tone. ...could be wrong, just my .02 Pfennig... Matt --------------------------------------------------- Allan Gilreath writes: I had a question from my apprentice that someone on the list may be able to help me with. We all know that vibrating strings divide up into segments with lengths approximately equal to fractional portions, i.e. ½, 1/3, ¼, 1/5, 1/6, etc. (we’re not even taking inharmonicity into account at this level.) His question is, “Why does the string divide into all of the different available fractional segments and not just even multiples of two?” I was hoping for a much better answer than just, “Because it does” but Benade, Helmholtz and Rayleigh, the best I can tell, all assume this to be a fact and I don’t really find the “why.”   Any thoughts?</body>