RE: active meantones (was: Re: Re: Robert Smith Equal Harmony)

Hi Aaron, and list members,

Aaron Krister Johnson wrote:
> I call all meantones where the fifth is flatter than 1/3 comma (or
19-equal)
> 'active meantones'. They are an acquired taste, but I rather like them.
> 1/2-comma is close to 26-equal, and is a bit more active and approaching
> pelog. 2/5-comma is close to 45-equal, and is a compromise between
19-equal
> and 26-equal. You can try ever more active and flatter fifths by testing
5th
> sizes from 26-equal and smaller by adding 4 to the numerator and 7 to the
> denominator of each n-tet in a series:
>
> 15 steps of 26-tet, 19 of 33-tet, 23 of 40-tet, 27 of 47-tet, 31 of
> 54-tet......
>
> of course, the resulting thirds (after going 4 fifths up) become more and
more
> neutral sounding and harsh with harmonic timbres. 26 (1/2-comma) for me is
a
> limit to how far *I* will go in these negative 'active meantones' and
still
> use traditional Western harmonic progressions. Going beyond gets more
> 'pelog-ish', but if that's you thing, you should enjoy it very much. It's
a
> different kind of beauty--very spicy and restless.

Aaron,
Thanks for this! I appreciate the chance to learn from
your experience.

Excuse a little maths? I just want to get a handle on your
series of tunings.

* Start maths *
Filling in the first three terms of your series of fifths
backwards gives, of course, 11 of 19; 7 of 12; and 3 of 5.
So the series has nth term = (3+4n) of (5+7n), and for large
enough n, this approaches 4 of 7. All these fifths thus lie
between 720 cents (3 of 5) - the sharpest of them all - and
~686 cents (4 of 7) - the flattest of them all. The nearest
to a just fifth of 702 cents is given by 7 of 12.

Four fifths minus two octaves gives us the thirds of this
series of tunings; the third thus ranges from 480 cents
down to ~343 cents - that is, from a flattish fourth down to
a sharpish minor third.

At some point, I guess, that third will be SO neutral that it
becomes hard to distinguish major third from minor third -
Let's see - the fifth is (3+4n) steps of (5+7n), the (major)
third is 4 x (3+4n) - 2 x (5+7n) = (2+2n) steps, the minor
third is (3+4n) - (2+2n) = (1+2n) steps. So the difference
between major and minor third is just 1 step of (5+7n); and
so they become closer together the larger we make n - the
larger EDO we choose. Of course, they won't become
indistinguishable until they're within a few cents of each
other - for example, we don't get a 5-cent difference until
243-EDO, when the fifth is 139 steps; and we don't get a
2-cent difference until 600-EDO, when the fifth is 343
steps.
* End maths *

But long before that point, you say you won't "still use
traditional Western harmonic progressions". That's very
likely a better test of what I was looking for than my
vague expression "fairly just". I guess that for "traditional
Western harmonic progressions" to be useful, we need to
have a very clear distinction between major and minor -
they need to be so far apart there's no likelihood of
hearers confusing them.

Just revisiting something you wrote:
> of course, the resulting thirds (after going 4 fifths up)
> become more and more neutral sounding and harsh with
> harmonic timbres ...

Presumably, much of this harshness would disappear if
one were to synthesise inharmonic timbres with overtones
from the tuning?

> On Friday 12 August 2005 11:57 am, Paul Erlich wrote:
> > --- In MakeMicroMusic@yahoogroups.com, "Yahya Abdal-Aziz"
> >
> > > Here's my question, on which I seek your collective wisdom based
> > > on actual experience of alternative tunings (rather than theory):
> > > How far can one take this narrowing of the chromatic semitones,
> > > while still retaining a fairly just flavour to the intonation?
> >
> > Hi Yahya,
> >
> > Perhaps Aaron Johnson is uniquely qualified to answer this question,
> > considering the extensive work he's just done in 2/5-comma meantone,
> > which could be one possible answer to your query.
...etc

Regards,
Yahya