<!doctype html public "-//w3c//dtd html 4.0 transitional//en"> <html> <body text="#000000" bgcolor="#FFFFFF" link="#0000EE" vlink="#551A8B" alink="#FF0000">   Hi list. Those of you who have followed the disscussions on para, bass tuning, longitudinal waves, and the like may find this line of interest. I have been looking at some frequency data taken from a tuned piano, graphing the behavior of partials, and of coincident partials in octaves. The piano was measured by Dr. Sanderson in 1978 and the resulting tables were included in an article published in the Piano Technicians Journal in 1978. I have to assume therefore that the tuning was pretty much up to par. The measuring included the 1st through the 8th (7th left out) partial for notes C3 to C5 and were taken in Hz. Plotting the "distance" between each pair of octave coincidents starting at C3/C4 up to C4/C5 resulted in a very interesting graph. The entire section was tuned such that the 4:2 octave type showed a 0.5 beats per second wide beat rate. The beat rate of the 6:3 type started at 0.5 plus and fell first gradually, then faster and faster off to 1.0 bps narrow. The 4:8 fell quickly off from 0.1 narrow bps to 4.7 narrow bps. None of that should suprise anyone who is familiar this kind of thing. What is a bit suprising tho is the behavior of the 2:1 in this data set. One would expect the 2:1 to show a gradual tendency towards widening, as the 4:2 is held constant. And indeed it does show this tendency. What is a bit strange tho is that it starts off at 0.3 bps wide (narrow in relationship to both the 4:2 and the 6:3 at the first octave in this sample C3/C4). It stays narrow in relationship to 4:2 until A3/A4, and then rises slowly to a faster wide beat rate then the 4:2. If this is a bit foggy, let me put this in another light. If we tune an octave using the 4:2 octave type as our anchor, we would expect that the 2:1 should always be a bit wider.(and all partial pairs with higher ratios narrower)  But here the opposite is true with regard to the 2:1 The 2:1 is less wide then the 4:2. It has a slower beat rate with both being wide of pure. By imagining a continuence of this incomplete measurement of frequencies, to include the octave down to the lowest octave on the piano, and continue to hold the 4:2 constant then we could expect this tendency to continue and increase.  We could also expect the 6:3 to go wide of the 4:2 and perhaps even the 8:4 would eventually go wide of the 4:2.  (This is of course dependent upon the scaling being consistant) Now you might be tempted to jump on this and bring out the para inharmonicity thing. But para is perceived as an anamalous phenomenon, unpredictable, following no pattern. In this case we have a very consistant development over 13 octaves. And there is the fact that the tendency towards narrowing for the 2:1 (decending) is to be expected by the fact that the 4:2 is held at a constant beat rate. I am not sure how to explain this. It prompted me to look closer at Baldersins measurements in chapter 8 of "On Pitch".  His comparisons for different octave partials hold each of the 6 octave types at beatless (as compared to the constant 0.5bps wide in the 4:2 above), comparing the remaining types beat rates in each case. He does this for each of the "C" octaves on the piano. They show the results one would expect, that each partial pair with a lower partial ration then the beatless partial is wide, and those with higher partial ratios are narrower. However looking closely at Balderssins data also reveals an oddity in relation to those partail pairs with a lower partial ratio then the one held beatless. In every case in every table the 2:1 is beating slower then the 4:2 and both are wide of pure. Further the 4:2 is narrow compared to the 6:3 when partial pairs higher then 6:3 are held beatless. In general on can observe in Balderssin table, the first Parital pair under the one being held beatless is wide, the next one down a bit wider, then they become increasingly narrow in relation to each other. They all stay wide of pure. All of this happens in a manner which follows to closely a distinct pattern to be attributed to para. This pattern suggests a factor of predictability that is not in line with previously accepted models for predicting inharmonicity. More measurements of entire pianos done in this fashion would of course be required to substantiate all this. My description of this may be a bit hard to follow, so anyone who is interested in looking at this for themselve, may mail me privately and I will send them the graph and the data which it was based on. There is of course another possibility. I might be totaly bonkers and have looked at these numbers to long.. <grin>. In that case I have no doubt that the boys on line will explain and correct my analysis. In any case, I am having a gas of a time. Richard Brekne     </body> </html>