Re: Web-friendly repost (reply to Monz on Sesquisexta lattices)

[Here is another repost of an article originally posted before Joseph
Pehrson's very helpful message alerting me to problems with certain
notational symbols on the Web.]

Hello, there, Monz and everyone; this reply continues a response to
your patent lattice diagrams of the Sesquisexta tuning with two
Pythagorean manuals a 7:6 apart.

> If one considers the very good 5-limit approximations (with an error
> of a skhisma of ~2 cents) given by this scale, a nice 5-limit
> dimension emerges on the lattice:

[neat lattice diagram]

> Note, however, that there is a notational inconsistency in regard
> to these schismatic near-equivalents.

Here I wonder if the notational inconsistency (in 72-tET notation)
might reflect the actual difference in size between two types of
"schisma" or "schisma-like" thirds in this tuning, respectively a
Pythagorean or a septimal comma smaller than the regular Pythagorean
81:64 major third (~407.82 cents).

Each Pythagorean keyboard has the usual schisma thirds -- diminished
fourths or augmented seconds, for example C#-F or F-G# -- at 8192:6561
(~384.36 cents) and 19683:16384 (~317.60 cents). As you note, these
thirds are only about 2 cents from 5-based ratios of 5:4 and 6:5,
differing by a 3-5 schisma of 32805:32768 (~1.95 cents).

Additionally, between the two manuals we find some major thirds
a _septimal_ comma smaller than usual Pythagorean, e.g. C-Ev, at
5103:4096 (~380.56 cents). These thirds are smaller than 5:4 by what
might be termed the 5-7 schisma of 5120:5103 (~5.75 cents), also the
difference between a Pythagorean diminished fifth at 729:512 and a
7:5.

From a usual neo-Gothic perspective these "5-flavor" thirds are
somewhat specialized intervals in comparison to the usual ratios in
this tuning of 3-prime and 7-prime, but one very characteristic use
would be a cadence like the following. I've spelled notes on the upper
manual in both the conventional and "Sesquisexta-style" manners, to
show what the actual keyboard motions are like:

G4 A4 Bb4 G#}4/Bv4 C#4 B}4/Dv4
C#}4/Ev4 E4 F4 F}4/Abv4 G#4 F#}4/Av4
C4 Bb3 or C#}4/Ev4 B}3/Dv4

In the first example, the schisma major third -- between C4 on the
lower keyboard and the note _visually_ an apotome or chromatic
semitone higher on the upper keyboard (C#}4 or Ev4)! -- moves to a
usual Pythagorean or "3-flavor" third above the same lowest note,
while the fifth of the first sonority moves to a major sixth, leading
to a regular Pythagorean cadence expanding to a stable trine (2:3:4).
Note the 27-cent septimal comma shift in the middle voice (Ev4-E4)
moving from the "5-7 schisma third" to the usual Pythagorean third.

In the second example, we start with a schisma third sonority on the
upper manual, and move to a 7-flavor version of a cadential sixth
sonority at a pure 7:9:12, with a standard 7-flavor cadence
following. Here we have a very interesting diesis-like shift of
Abv-G#, equal in size to a septimal comma plus a Pythagorean comma
(531441:52488, ~23.46 cents), or ~50.72 cents.

Checking in Manuel Op de Coul's Scala for MS-DOS, I find that this
ratio of 59049:57344 has a name: "Harrison's comma." This progression
reminds me of a very similar cadence in 22-tET (or 22-EDO, if you
like) where we move from a "schisma-like" third to a near-9:7 above
the same lowest note while a third voice moves from fifth to major
sixth, leading again to a usual cadential expansion to a trine.

Either Harrison's comma or a 22-tET step is close to 36:35 (~48.77
cents), which Scala terms the "septimal diesis," the difference
between 5:4 and 9:7. In 22-tET, this step also happens to be the usual
neo-Gothic diatonic semitone; in Sesquisexta, the 7-flavor cadential
semitone at 28:27 (~62.96 cents) is slightly larger.

I'd describe the "5-flavor" thirds and also the neutral thirds found
in this tuning as "supplementary" intervals, adding variety to the
main themes of 3-prime and 7-prime.

Most appreciatively,

Margo Schulter
mschulter@value.net