Re: TD 836: Monz lattice diagrams for neo-Gothic JI tuning

Hello, there, and thanks to Monz for a couple of comments about my
higher-prime neo-Gothic JI tuning for which he's now designed two
beautiful lattices -- the new ASCII-based version which I can view in
real time with the Lynx browser, as well as the most impressive
graphical version.

> One is the bridge spanning one exponent each of prime-factors 7 and
> 11, separating 14/11 (= ~418 cents) and 81/64 (== [3^4] = ~408
> cents). This has the ratio 896/891, which is [3^-4 * 7^1 * 11^-1]
> in prime-factor notation, and is ~9.688 cents.

An interesting question is what to call this ratio of 896:891, the
difference between the usual Pythagorean or 3-limit major third at
81:64 (four 3:2 fifths up), and a 14:11. It's also the difference
between the usual Pythagorean minor third at 32:27 and 33:28, the
interval which together with 14:11 makes up a pure 3:2 fifth.

One possible name is the "undecimal syntonic comma," although I'm not
sure how descriptive (or confusing) this might be. Anyway, the 896:891
seems a "kinder and gentler" divergence than the syntonic comma at
81:80 (~21.506 cents) or the Pythagorean comma at 531441:524288
(~23.460 cents).

Melodically, it may involve less "stress" (maybe a gentler term than
"pain"); vertically, at a tad less than 10 cents, it makes a fifth
quite "beatful," but not an outright Wolf -- people maybe tending to
draw the line here at around 10-12 cents.

Of course, in a neo-Gothic style where this tuning would be
appropriate, we're likely looking for even more active thirds (and
sixths) than Pythagorean, and also can "fall back" on standard
Pythagorean without any problem.

Thus the situation is more forgiving than in a 5-limit JI system, for
example, where an 81:64 where a 5:4 was intended would be a
"mistuning" or "near-Wolf. An 81:64 in place of a 14:11 is a
"variation of color," and the 12-note tuning takes advantage of this
by using regular Pythagerean thirds and sixths (e.g. Bb-D, Bb-G) at
the flat end of the chain to make things a bit simpler.

This comparison of the 896:891 to the syntonic and Pythagorean commas
brings us to Monz's next point, or "bridge."

> The other is the bridge spanning two exponents each of prime-factors
> 7 and 11, separating 12544:9801 (== [3^-4 * 7^2 * 11^-2] = ~427
> cents) and 3^16 (= ~431 cents, and a ratio with numbers too large
> for me to bother with). This bridge is [3^20* 7^-2 * 11^2] in
> prime-factor notation, and is ~4.084 cents.

This is a comparison which didn't occur to me, but is a very
interesting angle once pointed out. Here's how I might explain it.

The 12544:9801 is equal to a Pythagorean major third enlarged by _two_
of those 896:891 commas we've been discussing, or 802816:793881,
~19.376 cents, giving us a size of ~427.195 cents.

The Pythagorean interval of 16 fifths up (e.g. Gb-A#), much favored as
a wide major third in Xeno-Gothic music with a 24-note Pythagorean
tuning, is equal to a Pythagorean major third enlarged by one
Pythagorean comma, or ~23.460 cents, giving us a size of ~431.280
cents.

Thus what we're comparing here is the size of two 896:891 commas at
around 9.7 cents each versus one Pythagorean comma at around 23.5
cents; the Pythagorean comma is slightly larger, a difference more
precisely of ~4.084 cents.

This 4-cent schisma, or whatever, has the somewhat imposing integer
ratio of 421900912521:420906795008, equal to the Pythagorean comma of
531441:524288 divided by (896:891)^2 or 802816:793881.

Thanking Monz for calling my attention to this comparison, I might add
that I still need to write more about the musical uses of this tuning.

For now, I might just offer a general caution: in typical JI schemes,
one of the goals is to "maximize consonance" or "minimize dissonance"
of intervals across various limits.

Here, however, in a neo-Gothic setting with a 3-limit of stability,
what we're doing by trying out different sizes of unstable major
thirds (e.g. 81:64, 14:11, 12544:9801) is mainly going for some
contrast and variety. Note that fifths or fourths impure by around 9.7
cents can have a pleasant cadential effect, adding another flavor of
tension building our anticipation of the resolution.

Most appreciatively,

Margo Schulter
mschulter@value.net