trick with tuning 12-out-of-N

The piano keyboard is symmetric on the note D. If a 12-out-of-N tuning
is symmetric, then putting the mirror point on D will allow one to hear
exact inversions using the identical fingering with the other hand. (Of
course 12tet is a subset of symmetric 12-out-of-N tunings).

Bob Valentine

>Bob Valentine

>The piano keyboard is symmetric on the note D. If a 12-out-of-N tuning
>is symmetric, then putting the mirror point on D will allow one to hear
>exact inversions using the identical fingering with the other hand. (Of
>course 12tet is a subset of symmetric 12-out-of-N tunings).

Unfortunately, most symmetric 12-tone scales do not have a mirror point. For
example, the two meantone scales

C C# D Eb E F F# G Ab A Bb B

C C# D Eb E F F# G G# A Bb B

are centered around the fifths G-D and D-A, respectively.

On Tue, 31 Aug 1999, Paul H. Erlich wrote:
> Unfortunately, most symmetric 12-tone scales do not have a mirror point.

If one did, it would have to have another one exactly half an octave
away, otherwise it would have an odd number of notes. That rules out
all 12-tone JI scales, and all 12-tone scales embedded in odd ETs.

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
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