Re: For Graham, on schismas and tunings

Hello, there, Graham, and thank you for your very extensive remarks
and examples on the "septimal schisma" question. It is fascinating to
see how the same kind of intonational scheme -- a Pythagorean or
quasi-Pythagorean tuning in 17 or more notes per octave -- can lead in
various musical directions.

> I haven't worked out the minimum number of notes from the spiral of
> fifths required for this to work, but I suspect it will be more than
> 17. Margo, how many do you have? Is it 17 tuned to 24 keys, or do
> you get all 24?

Xeno-Gothic definitely uses all 24, mapped to two keyboards tuned a
Pythagorean comma apart -- a "generalized" keyboard where the two
manuals have identical intervals.

In fact, with only 17 (a tuning described in Europe in the early 15th
century), for example, we get only major third a comma wide, between
the two extreme notes of the tuning, Gb-A# in the medieval example.

On a Xeno-Gothic keyboard, by the way, the notation often rather
nicely reflects the fingering. For example, in

D@4 C@4
B3 C@4
G@3 F@3
E3 F@3

we play the fifth E3-B3 on one keyboard and the fifth G@3-D@4 on the
other in the first sonority, and these two fifths "merge" into F@3-C@4
on the keyboard where we played the fifth G@3-D@4.

Curiously, my original convention was to have the "usual" keyboard as
the physically lower one, and the keyboard a Pythagorean comma lower
as the upper one -- but this is totally arbitrary. Psychologically, I
suspect that this convention reflects the idea that the lower keyboard
has a "classic" 14th-century range of Eb-G#, so the upper one will
have the less common Ab (making a pure fifth with Eb on the lower
keyboard) on the key corresponding to G# on the lower keyboard.

> As the distinction between the 3- and 7-limit progressions is lost,
> it seems that this breaks the spirit of Xeno-Gothic: that you can
> choose the commas. Although the 5-7 distinction is still there.

Yes, I agree that "Xeno-Gothic" does imply a choice of commas, so the
more general term "neo-Gothic" might be appropriate for a medievalist
approach to 17-tet.

Might I ask more about this "5-7" distinction in 17-tet? The loss of
the comma distinctions is, of course, clear -- maybe analogous to the
loss in meantone of the 5-limit distinction between 9:8 and 10:9, or
16:9 and 9:5. However, unlike either Xeno-Gothic or 22-tet, 17-tet
doesn't have any close 5-odd-limit approximations, although on a more
complex level the 17-tet step is almost identical to a 25:24.

Please let me thank you again very warmly for your experiments, and
for sharing a very subtle keyboard layout which I need to study more
closely. I am especially moved that, using your own expertise and
feeling for fine distinctions in schisma tunings, you took my
progression and optimized it for what you found a moving musical
experience.

Dialogues like this may make a point that people can look at the same
tunings and mathematical relationships from many musical points of
view, and that sharing these points of view can lend perspective to
all.

Again, thank you for your thoughtful words, your patient experiments,
and your helpful Web site.

Most appreciatively,

Margo