More on equal beating tunings

Before exhausting this thread, I want to clarify one statement I
made in my last post about equal beating temperaments. The
equal and equal & opposite labels (which I now see as ill-chosen)
refer to the equations, not the intervals. I've not checked to see
whether the beating intervals themselves are each flatter or sharper
than their just values in each case.

While Paul Erlich's point about dividing the ditonic comma is
well-taken, historically linear and "irregular" temperaments were
constructed by dividing it as well as the syntonic comma. See
J. M. Barbour's "Tuning and Temperament."

The usual term for a regular cyclic tuning whose fifths are sharper
than 12-tet (700 cents) is "positive." Examples are Helmholtz's
1/8-th skhisma tuning with just major thirds and Sabat-Garibaldi's
1/9-th shkhisma tuning with just minor thirds. These are the positive
analogs of the 1/4 and 1/3 comma "negative" meantone tunings.

--John



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