About the Piagui scale

Mike Battaglia wrote:

Para: tuning@yahoogroups.com
De: "Mike Battaglia" <battaglia01@...>
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Fecha: 06 Aug 2008, 07:15:47 PM
Asunto: Re: [tuning] Re: Why are scales with a minimal number of different intervals more

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Interesting. I do like 7 ET myself, although I don't know if I'd
gravitate towards that for music that doesn't modulate.

One thing I notice is if a singer jumps from C-E, she'll probably sing
that pretty close to a 5/4 major third. But, if she goes from C-D and
then from D-E, she might end up closer to 81/64 than 5/4. I think it's
just because singers usually have in mind a list of different
intervals - a major third, a whole step, etc... And the one they'll
pick out for the whole step is usually around 9/8, and the one they
pick out for the major third is usually 5/4. If they go from C-D and
then from D-E, they move up two whole steps. I don't often hear 9/8
tempered flat to accomodate 5/4 or 5/4 tempered sharp to accomodate
9/8, except the latter in certain cases, such as when two whole steps
are sung in a row (which isn't really a tempering at all).
--------------------------------The above message is related to the study I did recently on an imperfect scale whose major third is 1,25707872211 = K(2) P(2), where (2) is exponent of the Piagui semitone factors K and P. The Piagui semitone factors K and P produced the variants I, II and III. Note that this major third is close to 1,25 = 5/4.K=(9/8) with exponent (1/2) equals to 1,06066017178 and P =(8/9) [(2) with exponent (1/4)] gives 1,05707299111. This imperfect scale shows interesting features.The experimental major third of 1,25707872211 is the relative frequency of note E with respect to Do = 1 and works in Piagui III scale. I reach to the conclusion that it is not possible to get a scale better than any of the three Piagui scales, whatever the method employed. The + and minus 2 cents of Piagui II deviations from the equal tempered tones make a small discrepance between the Piagui II and the equal tempered scale. Since the imperfections of the equal tempered are small though detectable, the tone frequency differences have also to be small like the mentioned 2 cents. I say this because one or two members of the tuning group declared that the only two cents of the Piagui tone deviations with respect to the equal tempered is rather low. They didn´t notice that since the imperfection of the equal tempered scale is also rather low, a better scale necessarily has to be close to the e- tempered one.Lima, August 08, 2008MARIO PIZARRO< piagui@... >