As all this continues, I have some time to contemplate the design. Playing around with the two-string "monochord" has been extremely informative. By placing the monochord on the soundboard of the real instrument, I've been able to probe and analyze the resonance of the different regions of soundboard. But perhaps even more importantly, I've been able to experiment with the tuning and tone quality of this iron string. This has caused me to reconsider my scaling factor on the instrument: my original plan was to go for close to the maximum possible scaling length, but now I am not so sure. At least with this one particular length of note, I believe I like the sound better when it is tuned significantly lower than the highest "safe" pitch.
The monochord speaking length is about 54 cm, which is the max scale length for the note E4, with this type of iron string. This iron is very fragile, and I got some good lessons in this as I attempted to string up the monochord. I broke the first string I tried, probably due to abrasion near the tuning pegs. I very carefully brought a second string up to pitch (E4), and kept it there for a few days as I played around with it, monitoring among other things how quickly it went flat and how often I had to re-tune it until it seemed to stabilize: in this regard, performance was satisfactory. However, on one of the re-tunings, I broke the string again. Did I go too sharp over the note? Did I change the pitch too abruptly? Had the string been damaged somehow, from all of the handling and banging on it with different implements? Hmmmm. I don't know, and I don't like it. I want these strings to hold up to rather inhospitable conditions, and after all, I am planning a percussive action. The strings need to not break at the drop of a hat.
So I replaced the string again (not too happy to use up my supply of this specialty instrument wire, not even on the real instrument, but clearly more testing was needed). This time, I decided to leave it below pitch for a while, just so I could bang on it with less fear of the string breaking. I started out with the pitch at B3: way below the intended pitch. I was surprised at how good this tone sounded: nice harmonics, loud volume, and none of the pitch instability or other symptoms of way-too-low string tension. Well, that's nice, but I had to know if the string could really take its intended design tension, or not: I might have a systemic problem in my calculations. So I brought it up to E4, carefully. The string didn't break, it held the pitch fine, everything seemed perfect -- except, something was lacking in the tone quality. I had hoped to hear the quality get better, and certainly the volume to increase, as I brought the pitch way up from B3 to E4. But actually, the increase in volume was not dramatic, and frankly the tone quality seemed to markedly decrease, at least in my subjective opinion. Hmmmmmmmmmmmmmmmm. So where's the sweet spot, then? B3 is perhaps a little too low, though it does sound better than E4. I tried C4, and D4. Both sound better than E4. I think that overall, I like C4 more than D4, but it's a tough call: too much of "whatever it is" that tuning flatter brings in, starts to sound bad, a muddy overload of harmonics. But not enough of it sounds bad in the other way, too thin and sparse, not enough of that delicate spidery tone which I greatly prize. And there is no particular reason to assume that the same degree of flat-tuning which sounds good here, in the middle of the range, will have the same positive effect in the far bass, or the far treble... Darn, the world got more complicated again, I hate when that happens! Actually I love it, this is the real design process in action.
Basically, my string lengths are pretty much irreversibly defined at this point, but not the pitches. Instead of running from E2..E6, I could run from C2..C6, or D2..D6. I don't really have to decide absolutely, until quite late in the construction of the keyboard and action. Indeed, I could probably design the keyboard to come apart and be rearranged, so that (with significant effort) the pitches could be changed after the fact. But really, I just want to get it right in the first place. I'm hoping I can be happy with C2..C6, because that makes for a nice keyboard layout. Starting and ending on D, I don't like. I don't like arrangements that "break up" the groups of two and three sharps in the layout. So, starting and ending on B, C, E, or F are the best choices. (Yet another thing that I don't like about pianos.) One will note that my choices of pitch-compass are deriving from considerations like how the pattern looks on the keyboard. Others would be concerned with the notes required to play specific pieces from The Literature, but I am mostly interested in improvisation and new compositions: whatever pitches and tone quality this experimental instrument turns out to have, I will adjust my playing to fit, and I will find ways to use whatever sounds the instrument can make.
My realization that a shorter scale length may sound "better", by my definition, is made purely from subjective observations. My simplistic understanding of the math and physics, had me convinced that "longer is better" when it comes to string-scaling. Trying to reconcile the math with the reality that my ears are telling me, I have re-visited the "Inharmonicity Equation"; and I think I can see some explanation for what I'm hearing. Basically, if the aim is to reduce the inharmonicity in the string vibrations (which is not a "given", this in itself is one of my working assumptions which should be carefully examined), then there are three parameters over which the instrument designer has control, assuming the string material is already decided: string length, string diameter, and string tension. Increasing the tension does reduce the inharmonicity, but it does so by the least dramatic amount of the three parameters. Increasing the length helps more, and most dramatically helpful is decreasing the string diameter. I have been aware of all these factors, but I have probably tended to treat them as all about equal in importance. In fact, the string diameter seems to be the real key.
The Inharmonicity Equation defines a value, B, which is the coefficient of inharmonicity. The higher B is, the more inharmonic are the partials (they always pull sharp from where they should be). Distilling out the constants into a value "K", the equation for B reduces to:
B = K * d^4 / (L^2 * T)So B depends on the 4th power of the diameter, a dramatic sensitivity. Increasing T divides down B, but only in a directly inverse relationship. Increasing L obtains results much more effectively, by a power of 2.
This does not explain why strings seem to sound better at less than maximum tension; however, it shows that, if one subjectively feels that a lower tension leads to a better tone, one doesn't need to feel "guilty": the inharmonicity is probably still quite low, and the change in tone quality is probably coming from other factors, affecting the balance of amplitudes of the partials, not necessarily their frequencies.
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