[CAUT] Shulze tuning research

[Fred Sturm]

On Oct 24, 2008, at 1:28 PM, Fred Sturm wrote:

> 	A tuning based on the 12th (nominally, setting aside inharmonicity)  
> will produce an octave wider by 1.2 cents than a tuning based on the  
> octave. (Inharmonicity will have an effect, but this is a good  
> starting point - it gives a good idea of the relative size we are  
> looking at).

	Actually, the model I described above compares 2:1 and 3:1 "tuning  
systems." In fact, none of us (to the best of my knowledge) tunes a  
2:1, either theoretical or accounting for inharmonicity. A better  
comparison would be to a tuning many if not most of us are familiar  
with: the original FAC tuning of the SAT. I think that most of us  
would agree that that tuning is fairly conservative in overall  
stretch. It can be described pretty accurately as based on a somewhat  
expanded 4:1 double octave (expanded by somewhere in the range of 0.5  
- 1.0 cents for the most part) in the top 2/3 of the piano, a 6:1 in  
the bass.
	A 3:1 tuning accounting for inharmonicity will be very close to the  
FAC tuning in overall stretch. There will be differences in detail,  
largely depending on how the tuning is executed (if done with ETD,  
along what partial the mathematical curve is placed will be a major  
factor).
	Bottom line, the beat rates of M3, M10, M17 will be "generally about  
the same" in both the FAC and the 3:1 tuning models. Differences, if  
any, would be quite small.
Regards,
Fred Sturm
University of New Mexico
fssturm at unm.edu