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Major thirds beat rates
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Sound samples
These are demonstration sound samples to accompany the book Theory and practice of piano tuning (Brian Capleton PhD).
These samples are made available independently by the author as a free educational resource, and do not form part of any contract or sale.
Licensing information
These samples are licensed for single user non-commercial educational use only and may not be redistributed.
Major thirds beat rates
The following samples demonstrate "traditional" theory beat rates for major thirds between G35 (G3) and F-sharp46 (F4).
1. G to B
2. A-flat to C
3. A to C-sharp
4. B-flat to D
5. B to D-sharp
6. C to E
7. D-flat to F
8. D to F-sharp
The isolated beating partials for major thirds between F33 (F3) and G47 (G4) are given in the following samples.
1. F3 to A
3. G to B
4. A-flat to C
5. A to C-sharp
6. B-flat to D
7. B to D-sharp
8. C to E
9. D-flat to F
10. D to F-sharp
11. E-flat to G4
Examination of beat rates
The following is the major third C40 - E44 on a Steinway model M (1930). The beat rate averaged over 2 seconds is 10 Hz.
Next is the isolated beating partial (the lowest adjustable partial) from the major third above.
Isolated partial at 10 Hz (average)
Note that the beat rate is not constant. Also, the beat amplitude is not constant, but varies irregularly. Beat rates in the major thirds can typically vary over the decay time by at least 0.5 Hz, which is the same order of magnitude as the change from the "traditional" specified beat rate from one major third to the next.
"Traditional" tuning theory gives a beat rate of 10.38 Hz for this partial. Compare the above real partial with a mathematically generated 10.38 Hz beat. Which one seems fastest?
Does the beat rate above seem 0.38 Hz faster than the rate in the Steinway sample? Compare that difference to the difference of just 0.12 Hz (less than 1/3 of the first difference) between the above beat rate and the one below.
General note on beat rates
The beat rates given by the "traditional" theory are calculated from a theoretical model that is rudimentary and idealised. As can be seen from the example above, actual beating is typically unstable in both beat rate and beat amplitude. The "beat rate map" of "traditional" theory is in practice therefore only a starting guide, not a definitive prescription. The major features that the "traditional" model does not take into account are (1) inharmonicity, (2) false beating, and (3)
bridge and soundboard effects.
The usual practical objective for beat rates in equal temperament tuning by master tuners is progression of beat rates. Contrary to urban myth, this is not done for the sake of the theory, but for the sake of progression itself, and its effect on the overall pitch intonation and tone throughout the instrument. It is never acoustically possible to tune just any chosen beat rate for a major third, for each and every major third. The beat rates are inter-dependent, so progression cannot be achieved for just any, arbitrary set of rates. If there were no inharmonicity or false beating, then progression would indeed occur when the stated "traditional" rates were tuned. Deviation from any of these rates would then destroy progression, i.e. these would be the only rates at which progression would occur.
In practice, the presence of inharmonicity, false beating, and bridge and soundboard effects, may mean that the appearance of the complete set of beat rates stated by "traditional" theory is acoustically impossible. Progression may only be achievable with different rates.
It is not inharmonicity per se that presents the greatest difficulties in achieving progression. It is the rate of change of inharmonicity over the part of the compass in which the scale is tuned, that causes the greatest problem. The rate of change may result in a situation in which continuous progression is physically impossible. There will nevertheless be a small range of "optimum progression" scenarios that are possible.
Note on beat rate progression and electronic tuning
A correctly applied electronic tuning will not produce best progression if the actual inharmonicity values of the individual notes, used in creating the target frequencies, are not accurate.
