source file: mills2.txt Date: Fri, 4 Jul 1997 22:36:44 +0200 Subject: Re: Definitions, pure fifths tuning From: "Richard Moody " Ed Remler wrote... > 3. Mean Temperament has two general meanings. First it means a system of > tuning by fifths such that four ascending fifths and two descending > octaves make up a perfect third. It also means any one of many slight > variations about this basic system. > 4. Bach's WTC refers to a mean tempered clavichord; in particular, > well-tempered means mean tempered. Since the WTC starts in the key of C, > it seems that Bach's thought of mean tempered in terms of ascending > fifths started on C. Regarding the scheme (if I understand it right) of up four fifths and down two ocataves will give a pure third. Using the notation that middle C c' and tuning from the octave below, or c Tuning this on a piano gives a c - e third that beats sharp by more than 5 beats per second, more like 7 or 8. After trying two or three times and erring on the flat side for fifth going up and sharp octaves comming down, the c - e third was still sounded faster than the rate prescribed for ET or 5.1-- bps. To check against paper, using the ratio for the 5th or 3/2, or 1.5, and using A220, (a) for conveince, four fifths gives a frequency of 220*1.5^4 (as entered on the W95 calculator, OK so ^ is the x^y key) or 1113.75. So two octaves down or /4 gives 278.4375 for c# ' . Hmm already that is sharper than 277.183 called for by ET. Where one begns the temperament is all important and I believe as Ed suggests that it should start from C, especially for the reason he gives of WTC starting at C. Continuing the temperament started above, the b fifth to e is tuned. Here is is intersting to note that in the octaves e - e', d - d', and c - c', all of the fourths and fifths sound pure, execpt f which has not been tuned, so tuning f from c' and Bb from f gives the diatonic scales in both C and F with pure fourths and fifths. If one keeps close to the circle of fifths as Ed mentions, it would be logical to tune the "flat keys" next, so from Bb tune up two octaves to bb' (I knew this notation system would be rough in ASCII), down a fifth (eb') down a fifth (ab) down a fifth (db). octave up from db (db' ) fifth down and here we are at gb and f # and the wolf is from f # to B (fifth) and f # to b (fourth) The rests of the fourths and fifths are as pure as the skill of the tuner. It is interesting to hear the thirds some pure and some well.. nasty. This also can be quickly tuned from (for the sake of our piano tuner members) C3-G3-D3-A3-E3-B3-(F#4), Then C3-F3-Bb2-Eb3-Ab3-Db4-Gb4(/F#4) and the octaves Gb4-G3, Eb4-E3, Db4-D3. and the E and B octaves. This I believe might be the temperament that the variations are based. To get rid of the wolf, it would have to be spread out, which means temper the fifths, or put in another place by tuning further into the circle of fifths, or starting the temperament from another key. Richard Moody Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 5 Jul 1997 02:47 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05237; Sat, 5 Jul 1997 02:47:40 +0200 Date: Sat, 5 Jul 1997 02:47:40 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA05210 Received: (qmail 16696 invoked from network); 5 Jul 1997 00:47:33 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 5 Jul 1997 00:47:33 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu