<HTML><HEAD> <META http-equiv=Content-Type content="text/html; charset=iso-8859-1"= > <META content="MSHTML 6.00.2800.1458" name=GENERATOR></HEAD> <BODY style="MARGIN: 4px 4px 1px; FONT: 10pt Tahoma"> <DIV>Ed,</DIV> <DIV> Its from Bradley Lehman.</DIV> <DIV> <DIV><A href="http://www.larips.com/">http://www.larips.com/</A></DIV> <DIV> </DIV> <DIV><A href="http://www-personal.umich.edu/~bpl/freview.html">http://www= -personal.umich.edu/~bpl/freview.html</A></DIV> <DIV> </DIV> <DIV>Part one is published in "Early Music"...but you get it ALL here = online free...</DIV></DIV> <DIV> </DIV> <DIV>You can download and look at all the stuff FREE from his website, = rather than get a</DIV> <DIV>Early Music. </DIV> <DIV> </DIV> <DIV> </DIV> <DIV>Richard</DIV> <DIV> </DIV> <DIV>See below:</DIV> <DIV> </DIV> <DIV><STRONG>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)</STRONG>. <DIV> </DIV> <DIV> <A href="http://www-personal.umich.edu/%7= Ebpl/larips/index.html"><IMG alt="Bach's schematic, as it appears on the = page" src="cid:LPHMXLZMXOMR.spiral-640.gif" width=640 border=0></A><B= R> <DIV align=center><A href="http://www-personal.umich.edu/%7Ebpl/larips/= index.html">LaripS.com</A>, © Bradley Lehman, 2005 </DIV> <P></P> <H2>Practical instructions to set Bach's temperament by ear</H2><SPAN = style="FONT-SIZE: 12px">To set up an entire harpsichord in 15 minutes = (for an experienced harpsichord tuner!), using Bach's tuning method: <H3>Recipe</H3> <UL> <LI>Get A from the tuning fork or other appropriate source. <LI>Tenor F-A (below middle C) beating wide 3 beats per second. <SPAN = style="COLOR: rgb(0,0,255)">It has 3 beats per second exactly (or double = the metronome speed of 89) if we are starting from A=440; or 2.8 beats = (double the metronome 84) if starting from A=415.</SPAN> <LI>Tenor F fourth down to C temporarily pure. <LI>Tenor A fifth down to D temporarily pure. <LI>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.) <LI>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. <LI>Middle D from tenor G and A likewise. <LI>Correct the lower D and C to match these two. <LI>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. <SPAN style="COLOR: rgb(0,0,255)">An excellent test for this = particular F-A and C-E is found at bar 21 of the <I>WTC</I> 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). <I>Coincidentally (?) this is at the golden mean of this = piece....</I></SPAN> <LI>Copy middle E down to tenor E. </LI></UL> <P></P> <UL> <LI><B>The above procedure has put all the naturals (except B) into = regular 1/6 comma temperament, the same as the first half of "Vallotti".</B= > Bach's temperament finishes differently, as follows: <LI>Pure fifths E-B-F#-C#. <LI>Pure fifths F-Bb-Eb. This is a step to set the Eb exactly; we will = move this temporary Bb soon. <LI>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. <LI>Lower Bb slightly so it has a slow beat from Eb like that of C#-G# and = Ab-Eb. <LI>Make all octaves and unisons pure to finish the instrument: all the = bass, then all the treble. </LI></UL><SPAN style="COLOR: rgb(255,0,0)"> <H3>Summary</H3> <UL> <LI>1/6 comma fifths F-C-G-D-A-E <LI>pure fifths E-B-F#-C# <LI>1/12 comma fifths C#-G#-D#-A# <LI>1/12 comma <I>wide</I> fifth resulting at A#-F </LI></UL></SPAN> <P><B style="COLOR: rgb(0,0,255)">This is coincidentally (?) an ideal = layout for all tonal music! </B>It sounds like "equal" temperament in = effect of its smoothness, but it has vivid color everywhere and is = perceptibly different in every key. </P> <P>See a <A href="http://www-personal.umich.edu/%7Ebpl/larips/math.html">= mathematical analysis of the results</A>. </P> <P>The complete explanation is in my <I><A href="http://em.oupjournals.or= g/" target=_blank>Early Music</A></I> article, February and May 2005. = </P> <P>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 <I>results</I> 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 <I>result</I> is simply that the naturals F-C-G-D-A-E all end up = in regular 1/6 comma positions, relative to one another. </P></SPAN><BR></D= IV></DIV></BODY></HTML>