---------------------- multipart/alternative attachment Ed, Its from Bradley Lehman. http://www.larips.com/ http://www-personal.umich.edu/~bpl/freview.html Part one is published in "Early Music"...but you get it ALL here online = free... You can download and look at all the stuff FREE from his website, rather = than get a Early Music.=20 Richard See below: A (0), Bb (+3.9), B (0), C (+5.9), C# (+3.9), D (+2), Eb (+3.9), E (-2), F = (+7.8), F# (+2), G (+3.9), G# (+3.9).=20 =20 LaripS.com, =A9 Bradley Lehman, 2005=20 Practical instructions to set Bach's temperament by ear To set up an entire harpsichord in 15 minutes (for an experienced = harpsichord tuner!), using Bach's tuning method:=20 Recipe Get A from the tuning fork or other appropriate source.=20 Tenor F-A (below middle C) beating wide 3 beats per second. It has 3 beats = per second exactly (or double the metronome speed of 89) if we are = starting from A=3D440; or 2.8 beats (double the metronome 84) if starting = from A=3D415.=20 Tenor F fourth down to C temporarily pure.=20 Tenor A fifth down to D temporarily pure.=20 Tenor G from that C and D so the fourth D-G beats 1.5 times as fast as the = fifth C-G. (Triplets against duplets.)=20 Middle C from the tenor F and that G, likewise: the fourth G-C beating as = triplets against the duplets of the fifth F-C.=20 Middle D from tenor G and A likewise.=20 Correct the lower D and C to match these two.=20 Middle E from tenor A slightly flat, same quality as these other fifths. = It should beat as a major third from middle C at 4.5 times per second. = Also check it as a 10th from tenor C, the same 4.5 times per second. An = excellent test for this particular F-A and C-E is found at bar 21 of the = WTC book 1 C major prelude. These two major thirds should have exactly the = same character as one another, although the beat rate is different (since = the pitch is different). Coincidentally (?) this is at the golden mean of = this piece....=20 Copy middle E down to tenor E.=20 The above procedure has put all the naturals (except B) into regular 1/6 = comma temperament, the same as the first half of "Vallotti". Bach's = temperament finishes differently, as follows:=20 Pure fifths E-B-F#-C#.=20 Pure fifths F-Bb-Eb. This is a step to set the Eb exactly; we will move = this temporary Bb soon.=20 Tune G# pure to C# and then slightly lower it so the C#-G# fifth and the = Ab-Eb fifth have the same quality: very slow beat from each.=20 Lower Bb slightly so it has a slow beat from Eb like that of C#-G# and = Ab-Eb.=20 Make all octaves and unisons pure to finish the instrument: all the bass, = then all the treble.=20 Summary 1/6 comma fifths F-C-G-D-A-E=20 pure fifths E-B-F#-C#=20 1/12 comma fifths C#-G#-D#-A#=20 1/12 comma wide fifth resulting at A#-F=20 This is coincidentally (?) an ideal layout for all tonal music! It sounds = like "equal" temperament in effect of its smoothness, but it has vivid = color everywhere and is perceptibly different in every key.=20 See a mathematical analysis of the results.=20 The complete explanation is in my Early Music article, February and May = 2005.=20 Similarly, the tempering can begin from C or any other convenient note, = instead of A. The important thing is to ensure that the geometric = relationships of the results turn out correctly. It is just as easy to = start the setup of Bach's temperament from C as it is from A; the required = result is simply that the naturals F-C-G-D-A-E all end up in regular 1/6 = comma positions, relative to one another. ---------------------- multipart/alternative attachment --____WHPEPQYSAQXEHDGESJXG____ An HTML attachment was scrubbed... URL: https://www.moypiano.com/ptg/caut.php/attachments/79/9a/9b/fb/attachment.htm --____WHPEPQYSAQXEHDGESJXG____ A non-text attachment was scrubbed... Name: not available Type: image/gif Size: 10785 bytes Desc: not available Url : https://www.moypiano.com/ptg/caut.php/attachments/b6/88/28/88/attachment.gif --____WHPEPQYSAQXEHDGESJXG____-- ---------------------- multipart/alternative attachment--
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