Recently I received a photocopy of Fred Sturm's article "The Bach Temperament", from _Piano Technicians Journal_ November 2006, pp22-24. (Thanks to the PTG member who sent me this!) I appreciate the coverage in this article, and the enthusiastic response it is to my research! In Sturm's section "The Tuning Sequence", where he explains the bearing sequence as a series of beat rates, there is an important and necessary correction. He consistently asserts that the major 3rd F3-A3 should beat at "about 3.5 bps". I don't know where he got that, because the correct value there should be "3.0 bps" for that, when we start from A4=440 Hz. The rate of "about 4.5 bps" for major 3rd C4-E4 (middle C and the neighboring E) is correct. Since all the 5ths F-C-G-D-A-E are *regular* in their tempering, the beat rate for F3-A3 needs to be 2/3 as fast as whatever the C4-E4 is...since the frequencies here are a 5th lower. Namely, F3-A3 is 3.0, and not 3.5. The "about 3.0" value is shown at my web page http://www-personal.umich.edu/~bpl/larips/math.html about halfway down, in the beat-rate chart of everything from A=440. 2.98 for F3-A3, and 4.46 for C4-E4. It's also described as follows in my original article: - "Establish the naturals F-C-G-D-A-E in regular 1/6 PC; that is, set up the first half of 'Vallotti', which is already familiar to harpsichordists. The most practical single thing to know is that the major 3rd of F-A (both in the tenor octave below middle C) has about 3 beats per second. A second checkpoint is middle C up to E, with about 4.5 beats per second." - Endnote #29 clarifies that further: "For a'=440 Hz: c'-e' has 4.5 and f-a has 3.0 beats (double the metronome speeds of 134 and 89). For a'=415 Hz: c'-e' has 4.2 and f-a has 2.8 beats (double the metronome speeds of 126 and 84)." [_Early Music_ February 2005, pages 18 and 20] With that correction of 3.0 to Sturm's tuning instructions, the rest of them work fine. ===== Still, I feel that all of this numerical beat-counting (both my own and Sturm's!) is wholly overkill in practice. None of it is necessary, in setting up this temperament for practical use! The numbers are simply a measurement of results. I've set this up well over 200 times since spring 2004, on harpsichords at A=440 and always working entirely by ear, including 15 times in the past two weeks (for local Christmastime gigs including a "Messiah" yesterday....). The whole temperament doesn't even take me ten minutes for a whole 8-foot stop (approximately 50-60 notes) on a harpsichord, anymore. The bearings take less than four minutes. My usual bearing sequence goes as follows, not counting ANY numerical beat rates as I do it! I just listen for "single" or "double" quality among the tempered 5ths, as opposed to the pure 5ths/4ths: - A4 from 440 tuning fork. - Copy pure octave down to A3. - Set F3 as pure major 3rd below A3, and then lower F3 slightly until this interval develops the same speed as my memorized tempo for the F major prelude of WTC book 1. The left hand's eighth notes in that piece (which happens to be 3.0, nearly enough!). - Set pure F4 octave from F3, and quickly check that its beat rate F4-A4 is the *right* hand's sixteenth-note figuration from that same piece, i.e. about 6.0 per second. Now we have our boundaries of both these F's and both these A's. - Set F3-C4, C4-G3, G3-D4, and test A3-D4: all having the same 1/6 comma *quality* to them. (I also have a trick to get the G3 *exactly* where it belongs as midpoint between F3 and A3 -- see reference below --, but half the time anymore I don't bother using it; I just do all three of the notes C4, G3, and D4 quickly from experience and check that all of this F-C-G-D-A cycle sounds consistent in quality.) - G3 up to G4, pure octave; and test that the C4-G4 5th has this same quality, along with the D4-G4 4th. By "this same quality" I mean that the 4th played at this position has beats sounding like triplets, against the duplets of the 5th played here. That's always true in any *regular* set of naturals, and playing the notes that fall under fingers 5-1 vs 4-1 in the left hand, testing the 5th vs the 4th: the 4th makes triplets (at whatever speed) against the duplets (at whatever speed). - Play the G3-D4 5th and get its quality in mind. Now tune the A3-E4 5th with that same quality. (Slightly faster beat rate, but listen for consistent *quality*!) Test that our E4 makes the right quality with our middle C. The beat rate happens to be 4.5 there, but I just go from experience hearing its character (which admittedly is much easier on harpsichords than on pianos). *If* I'm in the mood to be stricter about this particular beat rate, I get it from another memorized tempo: this time, the sixteenth notes in the C major fugue, WTC book 1. - We now have all of F-C-G-D-A-E as regular 5ths, 1/6 comma each. - Copy octaves from C4, D4, and E4 down to C3, D3, E3. - Test the C3-E4 major 10th to be that same speed of 4.5 (sixteenths of that fugue), where it's easier to hear than the major 3rd was; proving that the octave C4 to C3 was done correctly. If in the mood, test the C3 to E3 major 3rd there to be about 2.25, i.e. the eighth notes of that same fugue subject! - Test the D3 up to A3 5th for this same 1/6 comma quality. If in the mood, compare it with the C3-G3 5th next to it; and to the F3-C4 and G3-D4. - E3 up to B3, pure 5th: and test it also as a pure 4th coming down from E4. - B3 up to F#4, and B3 down to F#3, each pure. - F#3 to C#4, and testing also F#4 down to C#4, pure. - C#4 down to C#3, pure octave. Play the C#3 up to F4 major 10th, just to make sure we haven't made anything too raunchy. It will of course be noticeably wider than it is in equal temperament. Another test to do here, if in the mood, is: C#4 is *exactly* at the midpoint between A3 and F4. They should be of the same quality as one another. - Working from both F4 and F3, make a temporarily pure Bb3. We are using this only as a stepping-stone to get the Eb correctly on both sides of it! - Bb3 up to Eb4, and down to Eb3, both pure as 5th/4th. - C#3 up to G#3, and testing also G#3 up to Eb4, both as 5ths: we want to put this note exactly as midpoint so these two 5ths have the same quality as one another (which happens to be 1/12 comma); very slight tempering from both of them. Ensure that neither of these 5ths is pure, but that we're going down the middle of them. (And if there's too much room leftover in here, or not enough, to install two reasonably 1/12 comma 5ths, we know we've messed up somewhere earlier; go back and fix the problem, and come back to this point!) - G#3 up to G#4, pure octave; and do a test play of Ab major and E major triads. The Ab major triad should sound slightly calmer in character than E major, but both decently smooth. - Go back to Bb3 and knock it down ever so slightly, so it begins to develop a slight beat from Eb3 (below it); the same character as the C#3 to G#3 5th that we just finished installing/testing; 1/12 comma here. The test above this is Bb3-F4, that our 5th here is now slightly *wide* but not enough to be bothersome. The Bb is "really" an A#, and the Eb is "really" a D#, according to the way I read Bach's diagram. - Test the F# major triad there: bright shimmering quality, more so than in equal temperament, but never uncomfortably so. Cadence it up into the B major triad...also bright but not nasty. - That's the whole thing. The results are: F-C-G-D-A-E 1/6 comma narrow each; E-B-F#-C# pure; C#-G#-D#-A# 1/12 comma narrow each; A#-F residual 1/12 comma *wide*. - Do all the octaves down through the bass, and then up through the treble, testing with 5ths/4ths on the above-noted qualities to ensure that the octaves are being done accurately. - Again, with practice (at least on harpsichords!) the bearing takes less than four minutes, and the whole register less than ten. The harpsichord is going to start drifting before half a day anyway.... And, octaves can always be touched up anytime, accurately, as we know the proper quality of the 5th and 4th played as tests within them. I usually spend two or three minutes before each playing session checking/fixing anything that has drifted. ===== All of these instructions probably make it look more difficult in print than it actually is, sitting there. My bearing sequence is so ingrained in memory, by now, that my hands do it automatically while my mind tends to drift elsewhere.... I don't glance at any beat-rate charts. That's all to emphasize again: I believe the key to understanding this temperament (and Bach's drawing) has nothing to do with counting *any* beats; but simply listening for comparative quality. We inject single, double, or no bits of beating twang into the 5ths and 4ths as we go along, and Bach's diagram (I believe) indicates the proper spots where those little bits of impure twang belong. Any counting of "x-per-second" beating might be useful to us moderns, from habit or training or experience, but it's not *necessary* to this method. Get the whole harpsichord done in 15 minutes or less, and then spend all the rest of the session playing music. ===== My older sets of instructions are here, including a handy trick for anyone interested in getting the G *exactly* centered between F and A: http://www-personal.umich.edu/~bpl/larips/practical.html http://www-personal.umich.edu/~bpl/larips/twang.html http://www-personal.umich.edu/~bpl/larips/basic.html And, my current conceptual model for the thing is in this article: http://www-personal.umich.edu/~bpl/larips/art.html Cheers, and enjoy.... Bradley Lehman http://www.larips.com
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