Hemififths based notation

Thinking about notating 99-et, I came up with this, which could also
be used to notate 58, 140 or 239.

The nominals come from a chain of neutral thirds, F-A-C-E-G-B-D. Here
of course FC, AE, CG, EB and GD are fifths. The sharp symbol
represents 28/27, and the chain continues D-F#-A# ... In 99, 28/27 is
5 steps. There is also a pair of symbols, } and (, say, for (either
of) 49/48 or 50/49, which are equated. Then the C major triad is
C-E}-G, and the minor triad is C-E(-G. In 99, this is 3 steps. A chain
of fifths up from C goes C-G-D-A#, so A# represents 27/16. A 28/27 up
from that is 7/4. Hence the major tetrad is C-E)-G-A#).

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> A chain
> of fifths up from C goes C-G-D-A#, so A# represents 27/16. A 28/27 up
> from that is 7/4. Hence the major tetrad is C-E)-G-A#).

Sorry, that should be C-E)-G-A## = C-E)-G-Ax, of course, since a 28/27
is a sharp.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> The nominals come from a chain of neutral thirds, F-A-C-E-G-B-D. Here
> of course FC, AE, CG, EB and GD are fifths. The sharp symbol
> represents 28/27, and the chain continues D-F#-A# ...

Since two of these 28/27 symbols make up an apotome, it probably makes
more sense to retain # as meaning the apotome, and use another symbol
for the half-apotome. Sagittal has an important symbol, an up arrow,
for an approximate half-apotome of 33/32, and another for the apotome
complement of 729/704. In the 11-limit hemififths (41&58) temperament,
these are the same, and we can notate them in ascii, sagittal style,
by v for the flat direction, and ^ for the sharp direction. Hence we
have a symbol pair which we can take to be various things in
different, related temperaments:

v, ^ equals:

27/28 and 28/27 in 7-limit hemififths

that, and 32/33 and 33/32 in 11-limit hemififths

392/405 and 405/392 in breed (2401/2400 planar) and ennealimmal
temperaments

In all these temperaments ^^ = #, vv = b. In hemififths, 15/14 is
equivalent to the apotome: (15/14)/(2187/2048) = 5120/5103. In breed
and ennealimmal of course it is not. If we used # as a symbol for
15/14 in breed and ennealimmal anyway, then in all of these systems
a 7/4 is notated as C-A#, precisely as in meantone. In breed and
ennealimmal, #vv would now become 10976/10935 ~ 5120/5103.

We now want a symbol pair for 49/50, 48/49 down and 50/49, 49/48 up.
We might use ? up and j down for that, I suppose, though Manuel's
choice of ), ( seems better for our purpose. In hemififths C-F# is an
interval of 10/7, and iterating this, a Bv## = B^# is a 50/49 in
hemififths, and ) is shorthand. In any of the above, C-E}-G-A# will be
the otonal tetrad, however.