Is it a chain, a circle -- or a loop? (for Joe Pehrson)

Hello, there, Joseph Pehrson and everyone, and it looks like we need a
new word to complement "chain" and "circle" -- how about "loop," as in
"a 53-note Pythagorean loop."

What a loop says to me is that you _can_, musically, treat a tuning
set like 53-note Pythagorean as an approximate circle: but you also,
if you want, can keep spiralling on with more fifths.

In 53-EDO, by definition, the last fifth brings you precisely to an
octave of the first note; it's not only circulating, it's closed.

In 53-note Pythagorean, you tune pure 52 fifths, and decide to define
your "53rd fifth," as it were, by tuning the octave of the first
note. That's a loop, which you've decided to treat as a circle -- a
nice bicycle chain, also, which you can ride about town musically
while others debate it's not quite circular geometry.

In this 53-note loop, the last fifth is about 3.62 cents narrower than
the others (all pure), and a neat consequence is that you get two
slightly different "flavors" intervals separated by this amount in
their sizes, especially notable for certain intervals with chains
including large numbers of fifths.

For example, in a complete 53-note loop, you get small semitones at
both 66.76 cents (the usual 90.22-cent semitone less a 23.46-cent
Pythagorean comma) and 70.38 cents (precisely three of those commas).

In 53-EDO, you'd get one semitone of around this "thirdtone" size,
sort of splitting the difference between the two Pythagorean sizes.

With an EDO circle, the closure is defined mathematically. With a
loop, it's a matter of choice: does this look like a good place to
"call it a day," tune a pure octave (if this is a 2:1 octave tuning),
and treat the resulting set as a musical "circle"?

A 53-note Pythagorean loop, treated as a complete system, seems to me
just as much a "tuning circle" or "bicycle chain" as an unequal
well-temperament (for example, Paul Erlich's 22-note unequal system).
The circle does in effect close, but not quite "equally": one fifth is
a different size than those others -- or, in a typical
well-temperament, we have two or more sets or groups of unequally
sized fifths.

One way to explain this is that in a 53-Pythagorean loop, one fifth is
3.62 cents narrower than the rest; in 53-EDO, this 3.62 cents gets
equally distributed to every fifth. Either way, it adds up: we have a
bicycle chain, or circle. However, with 53-EDO, the circle closes
itself; with a Pythagorean loop, we close it by tuning that pure
octave and "virtually tempering" that 53rd odd fifth.

The term "loop" appeals to me because maybe it suggests this element
of choice. Similarly, a 31-meantone loop (1/4-comma), or a 46-loop
(the equivalent of a meantone for just 14:11 thirds), is musically
very close to 31-EDO or 46-EDO: the mathematics of the loop permit us,
but do not require us (as with an EDO) to "call it a circle."

Now for the question of those schismic thirds, early 15th-century
Pythagorean tunings, and the like.

First of all, as people have noted, you can get near-5-limit thirds in
either Pythagorean or 53-EDO, not surprising since these two tunings
are so _close_ to being identical. The basic spellings and mappings
are the same.

One point about known early 15th-century tunings: the near-5-limit or
"schisma" thirds are generally those involving written sharps, for
example A-C#, E-G#, and D-F# tuned as A-Db, E-Ab, and D-Gb. The
diatonic notes are tuned in a usual Pythagorean fashion, so that we
get C-E (a usual Pythagorean third) rather than C-Fb.

In the late 15th-century however, a _theoretical_ scheme like that of
Bartoleme Ramos for a 5-limit monochord could be interpreted as a
slight modification of some more thorough "skhismic" tuning (to use
the Helmholtz-Ellis spelling) where regular thirds involving the usual
_musica recta_ notes or basic gamut (Bb-B) are also of this type. By
this point, Ramos himself seems to describe a practical keyboard in
some kind of meantone, avoiding the "Wolf" interval among the musica
recta notes that such a skhismic temperament would involve in a simple
12-note scheme. In a Gb-B tuning of the early 15th-century variety, we
have the Wolf at B-Gb, or written B-F# -- sometimes an issue, but a
lot less often than something like A-Fb (written A-E) would be in the
C-Fb scenario on a 12-note keyboard.

Anyway, Pythagorean has lots of neat thirds and other intervals to
offer. We can get intervals about 3.80 cents from pure ratios of 7
(9:7, 7:6, 7:4), and various kinds of neutral or semineutral thirds,
for example.

The same remarks mostly apply to 53-EDO, where the 5-limit thirds are
a bit more accurate, and the 7-flavor (2-3-7-9) thirds a bit less so.
Here are some pairings:

18/53 13/53 Pythagorean type (~81:64, ~32:27)
19/53 12/53 7-flavor type (~9:7, ~7:6)
17/53 14/53 5-limit/schismic type (~5:4, ~6:5)
16/53 15/53 "Semineutral" type (~69:56, ~28:23)
20/53 11/53 "Ultra-Gothic" type (~13:10, ~15:13)

That's five very useful flavors of thirds, and in a 53-note
Pythagorean loop, we'd get variations of 3.62 cents in these size
categories when we looked at the whole cycle, a touch distinguishing
the two tunings.

Anyway, I hope that the "loop" can maybe provide a third alternative
in the "circle vs. chain" discussion: a loop is a chain that you
choose to make into a circle by permitting an odd fifth closing the
octave to be a bit unequal. With a loop, it's by choice; with an EDO
circle, it's by definition.

Most appreciatively, and happy composing,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

/tuning/topicId_33170.html#33170

> Hello, there, Joseph Pehrson and everyone, and it looks like we
need a new word to complement "chain" and "circle" -- how
about "loop," as in "a 53-note Pythagorean loop."
>

Thank you so much, Margo, for your incredibly informative post about
Pythagorean tuning and 53-EDO.

Certainly the concept of "loop" seems appropriate for your tuning
and, frankly, seems like a more interesting tuning than trying
to "homogenize" everything into 53 equal parts. It's nice to know,
too, that *most* of the tuning is truly 3-limit Pythagorean and, it
seems, more "authentic" for Medieval performance practice, which is
most probably why you like it as well...

The term "loop" sounds very much like "Lupus" which is, of course,
the *wolf* in Latin and astronomical constellations. In this case,
the "wolf" wouldn't have much bite, since the deviation of the very
last fifth is under 4 cents... so some people might not even *notice*
the "wolf at the door..."

Thanks again!

JP