Reply to Valentine re Positive systems

Bob:

>Maybe this is the same thing, but I though positive simply meant that the
>"3/2" of the temperment was "above" of the "5/4" after four trips, rather
>than dead-one as in a meantone system.

If I understand your statement, the answer is "no." Meantone fifths are
negative as are those 5ths larger than meantone but less than those of
12-tet, such those of the 1/6 and 1/5 comma systems even though the
corresponding major thirds are larger than 5/4. The just 5th 3/2, the 17,
22, 29, 34, 41 and 53-tet fifths, as well as those of the 1/8 and 1/9
skhisma systems, are all positive.

Four fifths less than or or equal to 700 cents (minus 2 octaves) generate a
major third of 400 cents or less. The meantone 5th generates a perfect 5/4
and fifths whose sizes fall between the meantone and 12-tet generate thirds
between 5/4 and 400 cents. Fifths larger than 700 cents generate still
wider major thirds and the just fifth of 3/2 produces the Pythagorean
ditone 81/64.

However, eight 4ths less than or equal to 500 cents (or downward fifths
larger than 700 cents) minus 3 octaves also generate major thirds which are
correctly notated as diminished 4ths aka "skhismic major thirds." In
12-tet, the two generated thirds are both equal to 400 cents, so the 12-tet
5th is a convenient standard and Bosanquet so used it.

--John

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End of TUNING Digest 1466
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