So, then we are talking about an exact 3:1 after all eh ? :)
Sorry Ric, but you are completely wrong here. The difference of my
temperament compared with Gary´s over a telfth is 3:2.998688889
(without inharmonicity), so that equals to 0,75 cent. That is not
at all an academic difference for me.
Cheers,
Bernhard Stopper
Hust so folks have it clear here. The difference referred to above if
taken over the D3-A4 twelfth amounts to about 0.575 bps over the entire
twelfth. This is a simple theoretical difference between what results by
increasing D3 (without any inharmonicity figured in) by a factor of the
19th root of 3 vs the 19th root of 6^(19/31). In the former you get
(using 440hz as D3's 3rd partial starting frequency) 1320 hz for of A4
fundemental, and in the later 1319.431~ hz for A4 fundemental. Neither
of these numbers are in real life useful as in real pianos you MUST deal
with inharmonicity. At the heart of both articles is the idea of using
12ths as a priority for tuning instead of octaves. Theoretical
justifications for why one should use the twelfth may vary, but the
central point remain the same.
Once past the theoretical one comes to the practical... which demands
using each individuals inharmonicity instead of numbers like those
above. One matches coincident partials in real pianos to do this. In my
case, I simply use Tunelab Pocket PC to do this and its easy enough to
do. Aural tuners relying on 12ths tests to help in setting octaves have
ended up in the same basic arena for years, perhaps without knowing it.
Thats as about as far as I'm going to bother going into theoretical this
and thats. Its about using the 12th and not the octave as a tuning
priority.... and why that ends up working, and what that exactly does to
the overall stretch of a piano in comparison to octave priorities. And
for that matter how that again compares to Virgils method.
At least thats what its about for me.
Cheers
RicB
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