<HTML>In a message dated 8/18/02 12:47:27 PM Central Daylight Time, drwoodwind@hotmail.com (Ron Koval) writes: <BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">Thank you Jason and Bill for coming up with some new numbers and publishing them in this forum.  Yes, it looks different from what's been published in the past, but that may reflect the ongoing evolution of this temperament.   Did a quick graph and the E maj third (?!) is the strongest at 19.5 cents from pure.  Most everything else floats around 14-15 cents, with the C,F and G thirds closer to pure.  I'll be tuning this one this week some time to see how it plays.... Ron Koval Chicagoland </BLOCKQUOTE> The basic idea of the EBVT hasn't really changed much and I really never could understand why those who figure these things out were having so much trouble.  Maybe it would help to know that I originally designed my "Victorianized Modified Meantone Temperament" from an 18th Century Modified Meantone Temperament published by Owen Jorgensen. The original had the C4-E4 3rd as pure.  Of course, that meant extreme harshness elsewhere.  To mitigate that, I began opening up that 3rd and the consequences of that would change depending on how wide I opened it. Nowadays, I have settled on the 3rds beating at 6 beats per second and have confirmed that the F3-A3 beats exactly the same, 6 beats per second as does the G3-B3 3rd and the G3-E4 6th.  Knowing this, I can start the temperament from A3 (after setting a 6:3 octave with A4) and tune the F3 as the first interval, the way most people tune ET.  Just making that interval slightly sweeter, 1 beat per second slower is easy to do.  It can be confirmed by setting a metronome at 120 and counting 3 beats for each tick. But there is always room for variations.  I can set that first interval at 5 beats per second and as slow as 4 beats per second (or really anywhere in between and maybe this has been the problem, the aural perception of exactly what that speed should be).  At 4 beats per second, there would be 2  beats per tick of a metronome set at 120.  The 5 beats per second would fall in between. Regardless of the initial starting interval speed, all other steps remain the same.  Those first few intervals are all tuned to exactly the same chosen beat rate.  When it comes to the pure intervals, those are tuned according to the steps laid out.  However, the slower the initial interval speed, the more tempered the 4ths & 5ths among the white keys will become.  It is a trade off but an acceptable one.  It just depends on how much "color" or contrast is desired. I have even been able to carry the idea one step further and create a Quasi Equal Temperament by setting the initial F3-A3 3rd at 7 beats per second, the same as ET.  The F3-C4 5th is still pure and the C4-E4 3rd will also beat at 7 beats per second, considerably milder than its ET counterpart but not so slow as to sound "dead" to those who are used to the sound of ET (or the usual attempts).  This gives a fairly smooth sounding temperament with a few pure 5ths but no obviously jarring contrasts and still manages to make a few keys, the ones the ear really wants to hear as being milder, have that satisfying quality. There's no reason why all 4 ideas could not have a set of "correction" figures but I am certainly not the one who knows how to do that.  But thanks much to all who have put their efforts into this. Bye the way, having the E3-G#3 third be the largest *size* interval is OK.  Consider it's placement within the scale.  It doesn't make it the *fastest* interval and that's what matters.  Werkmeister did not know the concept of interval size, only speed or rate of beating.  Besides, this never was intended to be a Well Tempered Tuning anyway.  It is an excessively modified, Modified Meantone Temperament (MMT).  Generally, MMT's do *not* meet Werkmeister's rules regarding what are known as "imbalances" and they also have slightly *wide* 5ths, both of  which he considered improper and counterproductive. However, in the highly modified state that the original MMT evolved to, those wide 5ths became pure, which does meet Werkmeister's rules.  If there do remain any slight aurally detectable (not interval size) imbalances, they are slight enough to be ignored, the way parallel 5ths are sometimes allowed to stand in modern composition.  So, says Owen Jorgensen (and me too). Jason's graph shows an irregular but fairly smooth curve which avoids both extremes which can cause a temperament to be rejected by the pianist.  There are plenty of temperaments which look good on paper, take the Thomas Young #1, for example.  It creates a perfectly uniform looking curve.  But does that necessarily mean it is the best or expresses all music appropriately?  Hardly.  Other mild Well Tempered Tunings have smooth looking curves too but none of the pure 5ths we all like to hear and only a few  instances of Equal Beating by happenstance, not by design. Long ago, I realized when trying various temperaments in Owen Jorgensen's publications that often, the most odd ball, irregular ones seemed to be the most appealing.  I've always kept that in mind and carry it far beyond the temperament sequence octave when tuning. For more information on Werkmeister's Rules for Well Tempered Tuning, see that page in my website. Bill Bremmer RPT Madison, Wisconsin <A HREF="http://www.billbremmer.com/">Click here: -=w w w . b i l l b r e m m e r . c o m =-</A> </HTML>