Hello cyber-land techs.... New on the tuning front is the focus on the ratio between the major 3rd and minor 3rd of a chord. Jason Kanter wrote about this last month, and has included some experiments on his website. (www.rollingball.com) I began tuning alternate temperaments years ago out of curiosity. After attending a class taught by Ed Foote, I had the tools to begin graphing the temperaments to try to predict how they would sound. Along the way, I've come in contact with a few people who have helped me to gather enough information to ask questions to continue the learning process. I began to wonder how the major third and the minor third that co-exist in a major triad interact. I used a spreadsheet to start looking at the m3-M3 beats, as well as the m3/M3 beats. As I was focusing on the subtraction, Paul Bailey was looking into the division, or ratio between these intervals. Along came Robert Wendell, who has the math skills to do a lot of calculations in his head when creating temperaments. A few of us were searching for a really nice mild temperament, while others were developing tunings with much more contrast. What was discovered is that there are a few ratios of m3/M3 that create a pleasing beat sychronicity. Let's look, for example at a C chord. (C-E-G) the interval of C-E is called a major third, while the E-G is called a minor third. In equal temperament, one C-E beats around 10bps, while the related E-G beats around 17bps. Combining those two beats gives......lots of beats! (27/second) Imagine a drummer tapping 10 times with the left hand and 17 times with the right hand. Only the first beats in each hand would fall together. This, I think is the source of the ever present "shimmer", or "waver" used to describe equal temperament, by alternate temperament technicians. Dividing those beat rates gives us a ratio of around 1.7 What happens if we could keep the major third beat @10 times/sec, while change the m3 beat to 20 times/sec? Each beat of the major third would be matched with a beat of the minor third, with a minor third beat coming in the middle. (adding up to 20/second) Dividing these beat rates gives a ratio of 2. Ooooooohh, we added beats to one interval to come up with less beats combined. Coool! Another way to get 20beats/second is the combination of M3=10 and m3=15,(triplets vs. duplets) with a ratio of 1.5. This synchronizing of beats was noticed randomly appearing in historical temperaments, but not used to create new temperaments until recently. Robert Wendell just discovered (his words) a temperament that has all the M3/m3 ratios set to either 1.5 or 2. He describes it as just falling into place, once he got started, kindof working backwards from F#. (I don't pretend to understand how he got there!) For those of you willing to try this mild well temperament, here are the offsets from equal temperament. Wendell Natural Well temperament: (M3rds range from 7.4 – 17.6 cents from pure thirds) C 4.24 C# -1.59 D 2.41 D# 2.32 E -2.08 F 2.29 F# 1.83 G 3.74 G# 0.37 A 0 A# 4.28 B -0.12 There is a little retrograde progression (D-A) to make the ratios work out. Also, since the modern ear seems to be very sensitive to the thirds between 17 and 18 cents wide of pure, he has altered the above temperament to fix the retrograde progression and lessen the widest thirds a little. This effects the ratios. Though they do not equal 1.5 or 2, they remain very close. Wendell Very Mild Well (M3rds from 7.8 to 17.2 cents from pure) C 4.24 C# -0.83 D 3.04 D# 3.08 E -1.65 F 2.69 F# 2.27 G 4.29 G# 1.13 A 0.0 A# 5.04 B 0.31 These are worth a listen.... Hope you get a chance to try them! Ron Koval Chicagoland _________________________________________________________________ MSN 8 helps eliminate e-mail viruses. Get 2 months FREE*. http://join.msn.com/?page=features/virus
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