Re: Eventone tunings

Hello, there, Herbert Kellner and Hermann Miller, and thank you for
very helpful comments about my "eventone" concept, to which I would
now propose to add a more general "evenstep" category.

First of all, Mr. Kellner, please let me agree that indeed my main motive
in proposing an "equitone" concept -- now changed to "eventone,"
because the other term is already in use for a system of notation --
was to include both meantone temperaments and Pythagorean intonation,
for example, in a common category of "tunings which form a regular
major third from two equal whole-tones."

As it happens, my special interest was in regular temperaments with
fifths _larger_ than pure, yielding regular major thirds of sizes such
as 14:11 (~417.51 cents) or 9:7 (~435.08 cents).

Since I use these tunings for music in a style largely derived from
the Gothic music of 13th-14th century Europe, I originally called them
"neo-Gothic meantones," but this raised the obvious historical
objection that "meantone" implies a tuning tempering the fifths in the
_narrow_ direction so as to achieve or approximate ratios for thirds
of 5:4 and 6:5.

Thus the more general term "eventone" covers regular tunings
including, for example, meantones with major thirds at or near 5:4 and
6:5; Pythagorean intonation with fifths at a pure 3:2; and neo-Gothic
temperaments with fifths wider than pure.

Hermann, this brings us to your question: might the term "eventone" be
used to describe any tuning where an interval such as a 320-cent minor
third in 15-tone equal temperament (15-tET) is derived from two equal
steps (here of 160 cents each)? Or, as you note, possibly from more than
two equal steps?

Here I would be inclined to propose the general term _evenstep_ for
this concept, and to define "eventone" proper as more specifically
referring to the derivation of the major third from two equal "tones"
in the more conventional diatonic sense. However, the best arbiter of
usage is experience and consensus, so I much regard this as an open
question.

If evenstep were used for the more general category, then "eventone"
might apply to tunings where a regular major third is formed from two
equal whole-tones with a size somewhere between 10:9 and 9:8 for
meantones; precisely at 9:8 for Pythagorean intonation; and somewhere
between 9:8 and 8:7 for neo-Gothic temperaments.

As I have remarked earlier, whether or not a tuning is regarded as an
eventone -- or "equitone" in my preceding usage -- depends not only on
its regular structure, but on the musical context.

For example, 22-tET is to me an eventone with two whole-tones of 4/22
octave (~218.18 cents) forming a regular major third of 8/22 octave
(~436.36 cents). Here the major third is very close to a pure 9:7, and
the whole-tone accordingly close to the mean of 9:8 and 8:7.

However, if one uses 22-tET for Paul Erlich's decatonic system based
on stable tetrads of 4:5:6:7, then the usual major third at 7/22
octave (near 5:4) is not formed from two equal whole-tones, and the
tuning is not an "eventone" in this stylistic context.

The same remark might apply to what I tend to view as the larger
category of evenstep tunings: here, again, a tuning might be open to
different interpretations of a given interval category based on style,
which might be derived from either equal or unequal steps depending on
the context.

Maybe one advantage of having an "evenstep" concept is that it might
avoid the specific diatonic implications of "tone" and suggest
application to a wide range of systems.

To sum up my tentative scheme, inviting much further dialogue:

MEANTONE: A tuning where major thirds approximating
a ratio of 5:4 are formed from two equal
whole-tones, not too far in size from the
mean of 10:9 and 9:8.

EVENTONE: A tuning where major thirds of some desired
size (e.g. 5:4, 81:64, 14:11, 9:7) are
derived from two equal whole-tones, with
some implication of a regular diatonic
structure (e.g. Easley Blackwood).

EVENSTEP: Any tuning where an interval of interest
is derived from two (or more?) equal steps,
e.g. a 15-tET minor third at 320 cents from
two equal steps of 160 cents each.

Most appreciatively,

Margo Schulter
mschulter@value.net