Ed writes >you tune the G, then D, then A and E. These > four fifths must be tempered by 1/4 comma each if the C-E third is to remain > pure. > just make all those four fifths equally out of tune, have one no better > than another while keeping the C-E pure. Today I took the "next step" in the meantone odysey and tried flattening the fifths to get to (arrive at) a pure C--E third, as you and Jim posted. It seems I was just about to think of that, it appears so intuitive. And it also re-enforces my theory that tuners were able to come close to ET by doing that in the circle of fifths, or "closing the circle" as I have seen it mentioned. They would have to tune six slightly flat fifths one way and five the other. So far that does not seem to me to be harder than tuning the four fifths "very" flat to get the C--E pure third. When trying your suggestion of the four flat fifths to a pure third I was struck by the distinct beating of the fifths, (and fourths of course), and yes, these fifths should beat the same. Of course eventually they beat faster as they go up, but they do sound the same when they are the first four. When tuning the fifths to "close the circle" you don't get these distinct beats. However you get very fast thirds which to the ear if used to pure thirds must have sounded "strange" But the distinct beating of the fifths in Meantone sound strange to me. After tuning these four fifths, pure thirds can then be tuned off of them. For instance F--A the F could be tuned pure. It should beat against the C. They must have figured out that the fourth on the bottom beats the same as the fifth on top. And it should have been noticed that fourth on top beats faster than the fifth on the bottom. Twice as fast actually. At any rate all of the white keys are tuned. Since every black key is a third away from a white key the it seems now you want to choose what key signatures you want. For the key of D you need F# that leaves Bb for the key of Bb. Now you have to choose between G# or Ab. If you want the key of A you will choose C# instead of Db. That leaves a choice of the key of B or the key of Eb. The wolf would lie at Eb--G# (with E--G# pure)or Ab--C# with Ab--C pure. > It is one thing to move the music from pianoforte to the modern grand, but it > is entirely more distructive to destroy the intonation in the process! > I suggest that use of its original style temperament for a piece of music > restores information which the composer installed in the piece. The > serenity of pure intervals and the abruptness of contrasting modulations are > aspects that ET doesn't have. The unequal tunings are more complex > arrangements of the pitches, and the music that results is also more >complex. The study of historical temperaments is an academic pursuit, and as such must adhere to the rules of academia. Thus the musicians, pianists, and musicologists can study the effects of temperament on music. The tuner of course will have to render these temperaments in a consistant manner. The historian will try to determine what actually was the tuning Bach had in mind for WTC, or Chopin for his Perludes and Etudes. What they look for is direct evidence such as words from the Masters themselves, "I wrote this piece for this temperament" Or I recommend performing this piece on an instrument according to _____" As it stands I have not seen any such evidence at least in the English language. One approach to historiography is the asking of questions and formulating the answers according to the evidence. One will naturally ask "What does the 'Well' mean in Well Tempered Clavier?" Of all the answers presented, the student will ask, "What did Bach himself say about Well Tempered?" History is full of unanswered questions many appear waiting to be answered if and when the evidence is uncovered. One can argue the possibilites that Bach meant the K or the W, or any of the variations there of. The music student faced with parctical issues will want to evaluate WTC on at least two. And with some authority can say, here is what this piece sounds like in W and here is what it sounds like in K. I perfer W to K for these reasons. Of course he and hopefully the lister mighyt wonders what the "other" temperament was that WTC was NOT written for. That would imply other Bach music was written in the "other" temp. And another endeavor in the study of historical temperaments will occur. In our scientific age the student will ask "what is meant by key color?" and in the process of scrutany and study might also ask, "What did the composers say" and "can this be demonstrated". And that leads to a BIG consideration of temperament, how practical is it to play and hear Bach for example anyone of a few besides ET? Someone has to come and tune it. With the "revolution" of digital keyboards this might provide the quick and simple fix. With a push of a button ET can go to Meantone, or a WT. The player can hear in an instant the differences. With the record and playback features the listener can hear this also. The benefit for tuners is that player who decides to perform or record on a specially tuned piano. The tuners who can do this get the call. Hmm I have an idea that some pop style female singers would fall in love with Meantone. I bet a few Jazz players would really dig flat fifths in modern pure fifths temp. Now if 12 5ths need to be narrow by 2 cents.... would that give pure fifths in an octave stretched 3.77 cents? Why not make it 4? The fourths should be ok if it were 12th root of 2.x ? Ric Means Well
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