temperament interval widths

[Billbrpt@AOL.COM]

In a message dated 9/20/00 11:22:28 PM Central Daylight Time, 
drwoodwind@hotmail.com (Ron Koval) writes:

<< But... 
 there are sometimes that a chord that 'should' have more energy will sound 
 smoother than it should.  Hmmmmm... So I'm trying to check ratios and 
 differences between width and beat rates of the M3, m3 and P5 relationships 
 in the triad, to see if I can link that to cents offsets.
 
 So far my guess is the equal-beating phenom. that Bill Bremmer has used in 
 his tunings. >>

Sorry it took me so long to get to this.  This would be my guess too.  In the 
1/7 Comma Meantone, there are 11 Major triads in 2nd inversion (example, C 
Major spelled G3-C4-E4) in which the 3rd on top beats exactly the same as the 
6th.  The remaining key is the so-called "Wolf" key but it is not really a 
"Wolf" because it is not dissonant enough to qualify as one.  It has a 
super-energized, "electrified" sound.  The 3rd is fairly wide (I forget how 
wide exactly but around the size which is considered borderline harsh 20-22 
cents, I think),  and the 5th is wide by about 8 cents (the exact width  of 
both intervals depends upon the inharmonicity of the piano.)

The keys at the top of the Cycle of 5ths sound much more pure than they 
really are in many musical contexts and the keys at the bottom which should 
sound harsher are mitigated by the same effect.  That is the "magic" of this 
temperament.  It is no hocus-pocus or baloney.  The Equal Beating (EB) 
intervals have a canceling out effect much like the idea behind noise 
reduction systems.

When I designed the EBVT, I had this effect in mind and used it for all it is 
worth.  I see a lot of people trying another Victorian style temperament upon 
the consistent recomendation by a particular individual simply because you 
can get it by dialing in some numbers on an Accu-Tuner.  The designer of that 
temperament however, did not try for or focus on the properties that EB has 
at all.  It is simply an arrangement that avoids harshness by starting with a 
particular width of 3rd at the top of the cycle of 5ths.  There is also no 
way of producing the slightly irregular octaves that can make the whole 
tuning "shine" (as Franz Mohr puts it) by using a smooth curve calculation.

Using these dial-a-temperament methods may be fine up to a point but as an 
artist, you learn nothing by doing it that way.  There will always be choices 
and opportunities to do something better.  Learning to tune from the book of 
EB temperaments that Owen Jorgensen wrote was the most beneficial thing I 
ever did for myself as a technician.  Dialing in a bunch of numbers and 
standing back and saying, "Wow, does that ever sound weird" has never, and 
will never do anything constructive for me.

Bill Bremmer RPT
Madison, Wisconsin