P12's discussion

[Richard Brekne]

Bernhard Stopper wrote:

>There is probably some misunderstanding of the P12 in your post.
>What you say about beat rates of 3rds 5ths and 4ths is correct, they serve
>us aural tuners as anchor points for spacing the temperament octave (and
>more).
>A P12 is a octave + fifth and is pure, when the 3rd harmonic of then lower
>key has the same frequency of the base frequency of the upper key.
>You can check this aurally by checking a M6th below the lower key and a
>Double octave + M3rd with the same beat rate.
>Spacing this P12 interval evenly, will result in stretched octaves of about
>1,2 cent + inharmonicity stretch.
>
>regards,
>
>Bernhard Stopper
>
>  
>
Bernard... I think you need to explain your last statement in terms of 
which coincidents you are talking about.  (Lowest coincident pair I 
imagine).  You must remember that Americans are used to thinking in 
octave types, and many will take your last sentence to mean that the 
octave is quite stretched indeed.

For edification, my own direct measurements show the resulting octaves 
to be just barely on the narrow side of a pure 6:3 octave type in the 
selected temperament 12th range.  That is to say, that the 6th partial 
of the lower note has very very nearly the identical frequency of the 
3rd partial of the upper note in case there should be any confusion.

As one moves upwards chromatically the octaves move towards 4:2 types, 
then in the end wind up just wide of just 2:1 's

Thinking in octave types always struck me as really more a bps type of 
thinking then a relative to cents.  But thats just my opinion.

Cheers
RicB