See this accessible talk by Bradley Lehman for informal details, or visit his site for articles and details.
The key idea is that the squiggle at the top of the Well-tempered Clavier notates how Bach tuned his Clavier "well". Bach didn't use equal temperament; rather he tuned his keyboards in such a way that every one of the 24 keys had its own character, and all sounded OK. It looks very convincing!
My graph below shows the resulting pitches, relative to equal temperament. The two 'Bach' graphs show the two results obtained for the two possible definitions of a 'comma' (PC and SC). The red graph shows 'mean tone' tuning, one of many tunings that were standard before equal-temperament came along and made all keys sound the same.
Thanks to Patrick Welch for pointing me to this theory.
Perform a listener preference experiment: create performances of all 24 preludes from Book 1 of W.T.C., in all 12 tunings obtained by using the Bach tuning shifted by x semitones (x=0..11); also include a few other temperaments such as equal, and mean tone (perhaps using computer+midi to capture the original performance of an expert, then re-generate the performance). For each of these 336 (=24 times 14) performances, and for multiple listeners (all randomized and blinded (i.e. listener is unaware what tuning is being used)), ask each listener
|1. 'How did you like that performance overall?' (Answer: (dull) 1/2/3/4/5 (fantastic))|
|2. 'If you've heard this prelude before, please compare this performance of it to others you've heard before.' (Answer: (worse) 1/2/3/4/5 (better))|
|3. 'Did the performance sound in tune?' (Answers: no, it was awful most of the time / no, it quite often sounded bad / no, sounded out of tune on a few occasions (give the number) / yes, sounded fine / yes, tuning sounded exquisite.)|
|[The answer to this question could ideally also be recorded much more precisely during the performance, by giving each listener a button to press whenever they think a particular bar sounds out-of-tune. This would pinpoint exactly which combinations of notes produced complaints.]|
|4. 'Please record any thoughts you have about this performance'. (Free response.)|
|A. rank the two performances overall. (Answer: First was much better / a little better / Second was a little better / much better.)|
|B. rank the two performances for tuning. (Answer: First was much better / a little better / Second was a little better / much better.)|
As a control of sorts, we could include a couple of pieces not composed by Bach -- for example, pieces known to have been composed for equal-tempered performance.
The prediction of an 'extreme Lehman theory' would be that every Bach prelude sounds great in its designed tuning, and that it does not sound significantly better in any other tuning. The extreme Lehman might predict a slight contamination of these predictions by novelty effects: some listeners might find 'bad' tunings entertaining, fun, or refreshing. Or they might believe that a baroque performance sounds more authentic if it sounds aged and decrepit. We should ensure the listener surveys give them the chance, if possible, to indicate what sort of 'liking' was going on. One listener incentive might be this: we could promise to give them a CD recording of their favourite version of each prelude, so their responses would indicate what they would like to listen to again and again.
If the above 336-performance experiment (lasting roughly 14 hours per subject) is deemed too long, it can easily be cut down to a smaller size by reducing the number of preludes studied or reducing the number of tunings. Reducing the number of preludes would be my preference. If the experiment studied just 4 preludes, for example, it would last about 2 or 3 hours per subject; listening trials could easily be split up into 10-minute sessions to avoid fatigue. Experiments could be conducted over the internet, with rewards for any volunteer who completes a specified number of comparisons (eg a free CD, or your name in lights on the Bach experiment fan-site).
Lehman's existing comparisons cover the special case of comparing 2 tunings for a few pieces.
[There is a Wiki for discussion of this experimental proposal. The password for editors is 'bayes'.]
Thanks to Patrick Welch, Ian Cross, Iain Murray, Robert MacKay, Dan Tidhar, Margaret MacKay, Olivia Morris, and Graeme Mitchison for helpful discussions.
Here are the numbers in terms of absolute frequencies. Mean tone pitches were obtained directly from a web-page by Bradley Lehman. The Bach pitches were worked out by following his recipe.
|Pitches relative to C|
The precise gnuplot command I used to make the Bach graph was
semi = 2.0**(1.0/12.0) plot 'tuning.dat' u 0:($4/semi**$0) t 'Bach(P)' w linesp 3 4where the data file is tuning.dat, which in turn was made by using the tuning recipe as follows:
# pythagorean comma comma = 531441.0/524288.0 F = 4.0/3 * comma**(1.0/6.0) C = 1/1 G = 2.0/ ( 4.0/3 * comma**(1.0/6.0) ) D = 2.0/ ( 4.0/3 * comma**(1.0/6.0) )**2 A = 4.0/ ( 4.0/3 * comma**(1.0/6.0) )**3 E = 4.0/ ( 4.0/3 * comma**(1.0/6.0) )**4 B = E * 1.0 / (2.0/3.0) Fs = B / (4.0/3.0) Cs = Fs / (4.0/3.0) Gs = Cs / (2.0/3.0 * comma**(1.0/12.0) ) Ds = Gs / (4.0/3.0 * comma**(1.0/12.0) ) As = Ds / (2.0/3.0 * comma**(1.0/12.0) )
Here are graphs of the numbers from the Lehman FAQ ('deviations from equal temperament' in Lehman's Mathematical details page).