Re: Sesquisexta post (Web-friendly?) -- thanks, Joseph Pehrson

Hello, everyone, and with an invaluable assist from Joseph Pehrson,
here is what I hope may be a more "Web-friendly" version of my
original post on pitch drift in the Sesquisexta tuning, avoiding ASCII
characters whose definite side-effects I now know about thanks to his
timely caution.

Generally seeing these posts in the e-mail Digest mode, I wouldn't
have guessed there was any problem -- although now and then, when
visiting Yahoo, I now realized that I did see some curious line breaks
or "links" from certain symbols. However, Joseph, it took you to focus
my attention on these "anomalies," and to realize that they strongly
suggest the use of other symbols.

Here's a repost which I hope may solve these problems for Web users,
including a correction in one of the cadential diagrams also noted in
another recent post.

Again, Joseph, thank you very being the messenger carrying the news
that nowadays e-mail address symbols may be treated as e-mail links in
an HTML setting, so that other inflection symbols are better.

- - -

Hello, there, everyone, and it seems that exploring a new scale can
sometimes involve the need for a few corrections on interval sizes or
spellings -- in the case of my (new?) Sesquisexta tuning with two
Pythagorean keyboards a pure 7:6 apart, at least the latter.

Despite the quite imperfect state of my keyboard technique, this is
one instance where readers might better have trusted my hands than my
theory.

Beyond some _very_ obvious corrections, this article provides an
opportunity to report a feature of my first encounter with this tuning
which I hadn't noticed, but confirmed in the process of reviewing my
note spellings: interesting "shifts" in pitch level between the start
and end of a phrase.

---------------------------
1. Correcting the spellings
---------------------------

Reproducing for convenience the quick keyboard diagram from my earlier
post on "comparing notes" for the week, and adding somewhat more
"conventional" spellings for the keys on the upper manual, I will
offer corrected spellings for the cadences I was apparently too
excited and/or tired to get right the first time:

381 561 879 1083 1263
C#}/Ev Eb}/Gbv F#}/Av G#}/Bv Bb}/Dbv
C}/Ebv D}/Fv E}/Gv F}/Abv G}/Bbv A}/Cv B}/Dv C}/Ebv
267 470 675 765 969 1173 1377 1467
-------------------------------------------------------------------
114 294 612 816 996
C# Eb F# G# Bb
C D E F G A B C
0 204 408 498 702 906 1110 1200

The first style of notation, with the ampersand sign "}" showing a note
raised by a 7:6, reflects how I tend to conceptualize the keyboard when
moving around it. However, a conventional neo-Gothic notation with the
"at" sign "v" showing a note lowered by a comma (here septimal, 64:63
or ~27.26 cents) is also very useful, e.g. D}=Fv.

Recalling a comment on notations by Joseph Pehrson, I might add that
the use of the "v" sign could be interpreted to mean that this tuning
is worth e-mailing home about.

Anyway, it's time for me to get the spellings right for those 7-flavor
cadences on D, and here I'll give these spellings using both the
Sesquisexta-style notation with } and the conventional neo-Gothic
notation with v.

In spelling a standard cadence on D, for example as the final or note
of repose in the Dorian mode, I obviously got the two styles of
spelling confused in a curious way -- among other things. Here a major
sixth sonority at 7:9:12 (0-435-933 cents) is expanding to a 2:3:4
trine (0-702-1200 cents) in an intensive manner (ascending melodic
semitones):

incorrect spelling[1] correction
---------------------- ------------------------
E}3-G3-C4 to D}3-A3-D4 E}3-B3-E4 to D}3-A}3-D}4
(Gv3-G3-C4 to Fv3-A3-D4) Gv3-B3-E4 to Fv3-Cv4-Fv4

Similarly, in giving a variation of this cadence where the penultimate
sonority also includes a fifth above the lowest voice for a "full
neo-Gothic quad" of 14:18:21:24 (0-267-702-933 cents), I got the two
spellings confused:

incorrect spelling[2] correction
-------------------------- ----------------------------
E}3-G3-B}-C4 to D}3-A3-D4 E}3-B3-B}3-E4 to D}3-A}3-D}4
(Gv3-G3-C4-Cv4 to Fv3-A3-D4) Gv3-B3-Dv3-E4 to Fv3-Cv4-Fv4

For people who might be curious about how just how comical my initial
misspellings are, at least for the intended progressions, I've added
some footnotes as indicated above.

Fortunately, I got the spelling right for at least the last example of
a 7-flavor minor seventh sonority at 12:14:18:21 (0-267-702-969 cents)
contracting to a fifth: E3-E}3-B3-B}3 to D}3-A}3., or in conventional
notation, E3-Gv3-B3-Dv4 to F#v3-C#v4.

To clarify the steps in these cadences, I'll provide more detailed
diagrams below, but first let's note what may be a rather amazing
facet of these progressions which only occurred to me this morning,
and which I verified at the keyboard. Maybe Johnny Reinhard and John
deLaubenfels may find this especially amusing.

-------------------------------------
2. A rather large "microtonal" shift?
-------------------------------------

Suppose I want to do a passage with parallel 7:9:12 major sixth
sonorities in D Dorian. Here I find it natural to start with D3-A3-D4,
for example, on the lower keyboard, and then move to my first 7:9:12
by relocating my lower hand only to D}3 (or Fv3) on the upper
keyboard. This gives me D}3-A3-D4 (or Fv3-A3-D4), with the major third
and sixth a septimal comma wider than Pythagorean, or in other words
at a pure 9:7 and 12:7, just as intended.

Now, let's suppose I play a string of parallel 7:9:12 sonorities and
then make a cadence on the mode's final. In the styles of the 14th and
early 15th centuries, this is generally an _intensive_ cadence, with
the upper voices ascending by semitones and the lower voice descending
by a whole-tone. As my corrected spelling shows, this cadence goes to
a trine the _upper_ manual: E}3-B3-E3 to D}3-A}3-D}4 (or Gv3-B3-E3 to
Fv3-Cv4-Fv4).

When I played it on Monday, it all seemed at once so new and so
natural that I didn't realize a nuance that occurred to me this
morning in reviewing my spellings: I am starting on D3-A3-D4, and
ending on the _visually_ corresponding D}3-A}3-D}4 on the upper
keyboard -- actually a 7:6 higher! Spelling the latter sonority
conventionally as Fv3-Cv4-Fv4 makes this "shift" more explicit.

One might say that it took me awhile to "get the drift."

Possibly one factor explaining the "natural" quality of this shift or
drift for me is the _literal_ parallelism of the 7:9:12 or other
7-flavor sonorities. This could lead to lots of interesting
ramifications in practice and theory.[3]

----------------------------------------
3. Cadential progressions in more detail
----------------------------------------

One way to explore a few features of interval spellings and steps in
the Sesquisexta notation is to consider all four varieties of most
proximate 7-flavor cadences on D -- or, more precisely, on the notes
mapped to the "D" key of either manual as the intensive or remissive
context may indicate. Notes on the upper manual are given both a
Sesquisexta keyboard-style spelling and a more conventional neo-Gothic
spelling:

Expansive 7-flavor quad (14:18:21:24)

Intensive Remissive

E4 --- +63 --- D}4/Fv4 C4 --- +204 --- D4
(231) (0) (231) (0)
B}4/Dv4 --- -204 --- A}3/Cv4 G}4/Bv4 --- -63 --- A3
(498,267) (702,0) (498,267) (702,0)
B3 --- +63 --- A}3/Cv4 G3 --- +204 --- A3
(933,702,435) (1200,702,702) (933,702,435) (1200,702,702)
E}3/Gv3 --- -204 --- D}3/Fv3 C}3/Ev3 --- -63 --- D3

Contractive 7-flavor quad (12:14:18:21)

Intensive Remissive

B}3/Dv4 --- -204 --- A}3/Cv4 G}3/Bbv3 --- -63 --- A3
(267) (0) (267) (0)
B3 --- +63 --- A}3/Cv4 G3 --- +204 --- A3
(498,435) (702,702) (498,435) (702,702)
E}3/Gv3 --- -204 --- D}3/Fv3 C}3/Ebv3 --- -63 --- D3
(969,702,267) (702,702,0) (969,702,267) (702,702,0)
E3 --- +63 --- D}3/Fv3 C3 --- +204 --- D3

In these cadences, ascending or descending whole steps are played in
the usual manner on the same keyboard (e.g. C3-D3, G3-A3, E}3-D}3,
B}3-A}3). This yields a usual Pythagorean step of 9:8 (~203.91
cents).

Ascending semitones, however, involve moving in visual terms from a
note on the lower keyboard to one appearing a whole-tone _down_ on the
upper keyboard: e.g. B3-A}3, E3-D}4 (see intensive cadences). What is
happening here is that the motion in the "horizontal" dimension brings
us down by a 204-cent whole-tone, but the "vertical" motion to the
upper manual takes us up by a 267-cent minor third -- so that the net
effect of (267 - 204) cents is to ascend by a 7-flavor semitone of
28:27 or ~62.96 cents.

We can also spell these semitone motions as B3-Cv4 and E3-Fv4, showing
that they are indeed a septimal comma smaller than Pythagorean, or
28:27.

For descending 7-flavor semitones, we move from the lower to the upper
manual while visually _descending_ by a whole-tone, a net motion of
(204 - 267) cents or 63 cents down. The remissive cadences above
provide examples: C}3-D3, G}3-A3. The spellings Ev3-D3 and Bv3-A3
again confirm that these semitones are a septimal comma narrower than
Pythagorean.

The frequent tendency to associate "visually equivalent" notes or
vertical centers on the two manuals which are actually a 7:6 apart may
have various artistic potentials, especially if one follows Ivor
Darreg's advice not to let the niceties of spelling get in the way of
making good music.

-----
Notes
-----

1. As my more conventional notation may suggest, E}3-G3-C4 or
Gv3-G3-C4 would actually represent a sonority of 0-27-525 cents, with
the lower pair of voices E}3-G3 or Gv3-G3 at a 27-cent septimal comma,
and the outer voices at a fourth of Gv3-C4, a septimal comma wide,
256:189 or ~525.31 cents. This isn't to exclude such a sonority, only
to recognize that it is distinct from the intended 7:9:12. Obviously I
was juxtaposing new and more familiar spellings in a curious fashion.
The goal of this standard resolution from the intended E}3-B3-E4
should be D}3-A}3-D}4; my given notation D}3-A3-D4 would produce
another 7:9:12 sonority. One could, of course, end a phrase or section
on D}3-A3-D4; in conventional neo-Gothic style this would be a kind of
"half-cadence," possibly inviting an eventual resolution to
C}3-G}3-C}4 (intensive) or E3-B3-E3 (remissive).

2. For this version, E}3-G3-B}3-C4 or Gv3-G3-Dv4-C4 -- in order of
ascending pitches, E}3-G3-C4-B}4 or Gv3-G3-C4-Dv4 -- would actually
represent a sonority of 0-27-525-702 cents. We have a narrow fifth at
189:128 or ~674.69 cents at G3-B}3 or G3-Dv4, and the small whole-tone
C4-B}4 or C4-Dv4 at 567:512 or ~176.65 cents, in addition to the three
intervals of the E}3-G3-C4 sonority discussed in the previous note,
and the usual fifth E}3-B}3 or Gv3-Dv4.

3. Should one wish to take a more conventional approach to pitch
levels, what I might term 64:63 "exchange vectors" from one keyboard
to the other are available for any note on the upper manual with a
regularly spelled minor third above, as the alternate spellings
indicate. For example, E}3/Gv3 is a septimal comma below G3, the note
on the lower keyboard _visually_ a minor third higher. Thus if one
starts on D3-A3-D4 and wishes to conclude a phrase in parallel 7:9:12
sonorities with a standard intensive cadence to the nearest
corresponding pitch on the upper manual. one could use the progression
C#}3-G3-C3 to B}2-F#}3-B}3, i.e. Ev3-G3-C4 to Dv3-Av3-Dv4, only a
septimal comma below the starting pitch of D. Conversely, any note on
the lower keyboard with a regularly spelled minor third _below_ it has
an exchange vector with a note on the upper keyboard a septimal comma
lower, in this example D3 to B}2/Dv3. The "odd" keys without exchange
vectors are C#, F#, and G# on the lower keyboard; and Eb/Gbv, Bb/Dbv,
and F/Abv on the upper keyboard. While one _can_ use exchange vectors
to seek corresponding pitch levels, whether one _should_ do in a given
context is a matter of artistic judgment, with people's preferences in
different musical contexts a fascinating open question.

Most respectfully,

Margo Schulter
mschulter@value.net