[CAUT] ET vs UET

[Ed Sutton]

Reading the Ellis appendix, another question occurred to me.

Jorgensen repeatedly uses his "equal beating" argument: "tuners of the past believed that equal-beating was equal- tempering, and so they mistakenly mistuned fifths that were progressively narrower rather than equal."

But Sanderson showed that due to inharmonicity and octave stretching, mid-range fifths are, in fact, equal-beating on a modern piano. At what time did piano scaling become inharmonic enough for this to happen?

Ed S.
  ----- Original Message ----- 
  From: Fred Sturm 
  To: caut at ptg.org 
  Sent: Wednesday, April 21, 2010 9:52 PM
  Subject: Re: [CAUT] ET vs UET


  On Apr 20, 2010, at 9:03 PM, Fred Sturm wrote:


    Are there any balancing pieces of data suggesting tuners "artistically altering ET" to achieve better results? If anyone knows of even the slightest hint of such data, please bring it forward to be added to the mix.


  As long as I have a head of steam up, I might as well write some more and get it over with <G>. There is, of course, one piece of evidence that the 19th century Victorian Temperament believers continually point to: the "Ellis tunings." In what I wrote above I was looking for documentary evidence (words) to show that tuners (a) had a certain intention that was not ET and (b) had a method, as the missing documentation. But I think it is time that somebody addressed those Ellis tunings head on.
  For any of you who don't know, who aren't familiar with the page (485) in the extraordinary "translator's appendix" to Helmholtz' On the Sensations of Tone, here is some background. Ellis, an amateur scientist in the 19th century tradition (somebody with money who could afford to spend his time doing such things) did a lot of original research while involved in translating Helmholtz' work. One of many projects was that of measuring pitch, using a set of 105 tuning forks (he called it a tonometer), carefully calibrated 4 Hz apart (he describes in some detail the process of tuning them and using them). He claimed to be able to calculate within one cent by counting beats. Truly due diligence would require experimenting to see what the margin of error would be for Ellis' method. One cent plus or minus what? But for now we'll assume that his measurements are exact enough. One thing he did with the forks was to measure temperaments of seven instruments: four pianos and three organs (two reed, one pipe).
  Jorgensen found WT traces in some of the recorded tunings, looking at them with his WT colored glasses. Let me play devil's advocate and look at them with ET colored glasses instead. Let's focus first on the pianos. Three of them were at Broadwood's, supposed to have been tuned by the firm's "best" tuners, the fourth being Ellis' personal piano tuned by his "ordinary" tuner "and let stand unused a fortnight." 
  [I have puzzled over that phrase, and have invented a scenario that makes sense to me: Ellis went down to Broadwood to arrange for his experiment, and while he was there, told them "While I'm thinking of it, please send the tuner to my house as well." Then, when his measuring had been done, he thought that as he had his own relatively freshly tuned piano available as well, he might as well measure it too. In any case, this has a ring of truth and probability to it for me].
  Jorgensen found traces of WT in one of the "best" tunings (called #4 because its measurements are listed in row four of the table) and in the "Ordinary" tuning. But he found that he needed to make some adjustments to each in order to make usable tunings of them. So he proceeded to "correct what had 'obviously' slipped."  In the "ordinary" tuning, he moved one note by 7.5 cents, and two others by 4 cents each. In "Best #4"  tuning, Jorgensen moved one note 2 cents, and another 3 cents.  
  I will follow the same procedure, but for Best #4 and Best #5. For B#4 I will move one note 5 cents, another 3 cents. For B#5 I will move one note four cents, one note two cents. (Note that my adjustments are more modest than Jorgensen's). The result for these two adjusted tunings: B#4 now has two 2 cent errors, two 1 cent errors. B#5 now has six 1 cent errors. Each scores 85% on the RPT exam. Not too bad, those guys could tune reasonably well.
  As for the "ordinary" tuning, I notice that most of the notes are flat, some as much as 8 to 11 cents. I suspect the owner of having neglected the instrument (he was not a musician, and it has been said he was "tone deaf"). So many notes being flat leads me to believe the piano needed a major pitch raise. We can hardly take seriously a measurement of such an instrument after two weeks. I will throw out this record as unsuitable.
  This leaves the black sheep, B#3. An amazingly bad tuning, with notes mostly sharp, by as much as 11 cents, and no apparent pattern. Someone had a bad day? I'll speculate again, and hypothesize that the shop foreman, having been told to humor this gentleman scientist, had the tuner take a new piano, with a couple chipping on it, and tune it. That would explain it being that haywire. A credible story at any rate. But, bottom line, this record also needs to be expunged (as Jorgensen also did).
  So we end up with two pianos tuned to a very reasonable ET. That leaves the organs. The pipe organ was another disaster, and I haven't come up with a story (other than the tuner being a drunkard). One of the harmoniums was pretty much spot on, with four one cent errors. It is described as having been very carefully tuned as the standard of pitch for the manufacturer (Blaikley). The other is the famous Moore & Co., which has one deviation of 4 cents, five of 2 cents, and three of 1 cent, all in the flat direction except for one of the 2 cent errors. Now, interestingly enough, if we score this for the PTG tuning test, using the pitch correction number (which is 1.1), the errors mostly fall within the 1 cent parameter (-2 +1.1= -0.9; -1 + 1.1 = 0.1: the -2 errors become -0.9 errors, within the 1 cent tolerance), and we are left with a total of 8 points in errors, for a score of 80%. Another RPT is born!
  But I will also note that tuning reed organs is not an exact science, as anyone who has done a little (I have) can attest. You listen, remove the reed, file it a bit (which heats it up), put it back, etc. Unless it is a very particular job, you get it good enough and stop.
  Bottom line, I don't see an iota of evidence of anything but ET in the Ellis data.
  This is overstating my devil's advocate case, but I don't think I am overstating anymore than Jorgensen did in favor of his pet theory. Comments?

  Regards,
  Fred Sturm
  fssturm at unm.edu
  "I am only interested in music that is better than it can be played." Schnabel

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