How accurate are inharmonicity measurements?

[Robert Scott]

To those with visual tuning devices:

  I would like to confirm some research I have been doing on 
measurement of inharmonicity for the TuneLab97 program.  A few 
months ago Jim Coleman, Sr. and I had some exchanges on this list 
regarding the issue of how voicing might affect inharmonicity.  
While that particular question may have been left undecided, the 
more general question that interested me was "How accurately can you 
measure inharmonicity anyhow?"  A related question is "How 
accurately do you need to measure inharmonicity in order to use 
those measurements to construct good tunings?" 

  Although I have not had personal experience with either the SAT or 
the RCT, I gather that the operation of these two devices is 
different regarding the measurement of inharmonicity.  It seems that 
the SAT relies on the tuner to push buttons to stop the display.  
The RCT, on the other hand, "listens" for a period of time and then 
automatically makes calculations of all significant partials.  It 
seems that each of these methods has its own advantages.  The 
automatic method is obviously more convenient for the tuner, since
only one sound sample needs to be taken and the computer calculates 
all partials from that sample.  But false beats can confuse such 
algorithms, making the measurement unreliable.  (I have heard that 
the RCT detects this condition and warns the tuner to take a better 
sound sample.)  With the more manual method used in the SAT, the 
tuner has the opportunity to see the irregular movement of the 
lights caused by false beats and either pick another string or use 
his judgement about when the lights are the closest to being 
"stopped".  My question is, which of the two methods is most 
reliable on the average. 

I have attempted to implement the automatic method in the next 
version of the TuneLab97 program and have gotten mixed results.  It 
seems that when I pick very good and stable strings (no false 
beats), I get excellent repeatability in the inharmonicity 
measurements.  My measurements are based on a 3 second sound sample.  
But when I try to measure strings with a little wobble in them, the 
inharmonicity measurements are more variable.  The following 
repeated inharmonicity measurements were made.  The notes and the 
partials that I selected just happen to correspond to the FAC 
measurements in order to make it easy to compare my measurements to
SAT measurements.  The measurements were made on a Kawai 650.


Note   2nd   3rd   4th   5th   6th   7th   8th  8th-4th
---   ----  ----  ----  ----  ----  ----  ----  -------
F3   -0.42  1.25  2.20  3.20  4.45  5.89  8.01    5.81
F3   -0.23  1.30  2.06  3.22  4.46  5.86  7.95    5.89
F3   -0.45  1.29  1.98  3.23  4.44  5.82  7.92    5.94
F3   -0.38  1.33  2.04  3.32  4.46  5.88  7.94    5.90
F3   -0.41  1.25  1.99  3.32  4.40  5.86  7.91    5.92
F3   -0.19  1.33  2.03  3.33  4.44  5.89  7.93    5.90
F3   -0.19  1.34  2.01  3.33  4.42  5.89  7.95    5.94
F3   -0.28  1.29  1.89  3.22  4.34  5.79  7.93    6.04
    ...a different unison:....
F3    0.64  1.84  2.38  3.72  5.11  6.78  7.97    5.59
F3    0.63  1.80  2.37  3.70  5.07  6.74  8.65    6.28
F3    0.63  1.79  2.37  3.69  5.08  6.73  8.02    5.65
F3    0.64  1.73  2.38  3.67  5.08  6.67  8.61    6.23
F3    0.65  1.84  2.39  3.73  5.11  6.78  8.02    5.63
F3    0.56  1.67  2.30  3.61  4.96  6.61  8.55    6.25

Note   2nd   3rd   4th   4th-2nd
---   ----  ----  ----   -------
A4    1.20  3.27  6.72   5.52
A4    1.12  3.27  6.74   5.62
A4    1.11  3.24  6.72   5.61
A4    1.22  3.36  6.88   5.66
A4    1.00  3.19  6.73   5.73
A4    1.34  3.58  7.00   5.66
A4    1.31  3.69  6.99   5.68

Note   2nd
---   ----
C6    3.34
C6    3.23
C6    2.86
C6    2.55
C6    2.26
C6    2.80
  ..a different unison:...
C6    5.95
C6    6.00
C6    6.55
C6    6.15
C6    5.81

I would appreciate any confirming or contradicting information on 
this subject.  In particular, I would like to know how much 
variation there would be if two or three different tuners measured 
the FAC numbers on the same piano.  I want to know how much a factor 
individual judgement plays when strings are less than perfect.

Bob Scott
Ann Arbor, Michigan
(Detroit-Windsor PTG)