Septimal schisma as xenharmonic bridge?

Hello, there.

Recent Tuning List discussions have focused on making connections
between different currents of just intonation and other branches of
xenharmonics.

As someone with a special interest in Pythagorean tuning, while I
might not have any additional suggestions to contribute in the way of
organizational questions, it occurs to me to share a few reflections
on the relationship of this xenharmonic approach to others. Curiously,
the tuning if sufficiently extended also has an interesting musical
linkage in its very ratios with some of these other approaches, of
which more shortly.

An interesting feature of Pythagorean tuning is that it seems to serve
as a kind of crossroads for four trends in xenharmonics:

(1) Historical tunings. Like characteristic 15th-17th century meantone
and 18th-19th century unequal well-temperaments, the Pythagorean
tunings of the 13th-15th centuries are associated with a great
tradition of polyphonic composition. Understanding this tuning may play
a vital role in understanding Gothic harmony, as well as vice versa;
the tuning both influences and reflects the sonorous ideal of the
period.

(2) Cross-cultural tunings. Pythagorean systems based on pure fifths
or fourths, with various modifications, occur in many world musics
from China and India to Persia. Scholars such as Ervin Wilson have
begun to draw some connections which may lead to a richer
understanding of cross-cultural xenharmonics.

(3) Just intonation (JI). Pythagorean tuning is not only a system of
just intonation with a very long history, but also an amazingly
successful system not only ideally suited to much Gothic music but
inviting "Neo-Gothic extensions" approximating intervals in 5-based
and 7-based systems.

(4) N-tet and related systems. Like equal temperaments such as 17-tet,
Pythagorean tuning tends to have a non-tertian focus quite distinct
from "classical" European music of the post-Gothic era. Also, like
these systems, it has sometimes been dismissed as "purely
mathematical, not musical," or denounced as "intolerably dissonant."
Fortunately, the growing recognition promoted by xenharmonicists such
as Ivor Darreg that _all_ n-tet's have a potential for beautiful music
may lead to a more just appreciation of Pythagorean tuning also.

Now for the musical linkage of Pythagorean tuning, a 3-based JI
system, and both 5-based and 7-based systems.

This linkage involves two bridges, the first well known and the second
perhaps a bit more obscure -- but obvious once John Chalmers called it
to my attention.

These bridges are the regular schisma, and what I term the _septimal
schisma_, which permit extended Pythagorean tunings to emulate
intervals of 5-based and 7-based JI systems. While the use of these
intervals is, of course, at the discretion of the composer or
performer, one application is to supplement the basic 3-based
Pythagorean intervals with two new "flavors" of unstable intervals
which can serve as diversions or impel directed cadential action
toward stable octaves, fifths, and fourths.

The regular schisma of 32805:32768 or about 1.95 cents, as many
readers will be aware, is a bridge to a quasi-5-based world. It is the
difference between the Pythagorean comma (531441:524288, about 23.46
cents) and the syntonic comma (81:80, about 21.51 cents) which
separates a number of basic 3-based intervals from their 5-based
counterparts.

Possibly less well known, the septimal schisma of 33554432:33480783 or
about 3.80 cents is a bridge to a quasi-7-based world of
"superefficient" cadences and third-tone steps. It is the difference
between the Pythagorean comma and the septimal comma (64:63, about
27.26 cents) which separates basic Pythagorean intervals from their
7-based counterparts.

All we need to do is to extend Pythagorean tuning far enough, and both
schismas extend their welcoming doors into new xenharmonic
regions. More specifically, we generate new flavors of intervals
either a comma wider or a comma narrower than the usual forms. Here
are some key intervals as examples:

5-schisma Regular 3-based 7-schisma
(~1.95 cents) (~3.80 cents)
-----------------------------------------------------------------------
10 4ths or 5ths 2 5ths or 4ths 14 4ths or 5ths

M2 65536:59049 9:8 4782969:4194304
(180 cents; ~10:9) (204 cents) (227 cents; ~8:7)

m7 59049:32768 16:9 8388608:4782969
(1020 cents; ~9:5) (996 cents) (973 cents; ~7:4)
----------------------------------------------------------------------
9 5ths or 4ths 3 4ths or 5ths 15 4ths or 5ths

m3 19683:16384 32:27 16777216:14348907
(318 cents; ~6:5) (294 cents) (271 cents; ~7:6)

M6 32768:19683 27:16 14348907:8388608
(882 cents; ~5:3) (906 cents) (929 cents; ~12:7)
----------------------------------------------------------------------
8 4ths or 5ths 4 5ths or 4ths 16 5ths or 4ths

M3 8192:6561 81:64 43046721:33554432
(384 cents; ~5:4) (408 cents) (431 cents; ~9:7)

m6 6561:4096 128:81 67108864:43046721
(816 cents; ~8:5) (792 cents) (769 cents; ~14:9)
----------------------------------------------------------------------

Thus it might be said that Pythagorean tuning is not only the mother
of many systems but a cousin germane to many more. Also, as this chart
suggests, the usual Pythagorean intervals might be viewed as a kind of
"middle of the road" between 5-based and 7-based alternatives.

An open question: might an extended Pythagorean tuning with its
contrasts of basic 3-based, quasi-5-based, and quasi-7-based intervals
in some way have a kinship to the three genera (diatonic, chromatic,
enharmonic) or to Guido d'Arezzo's three hexachords (soft, natural,
and hard)?

In any case, the septimal schisma and the world of extended
Pythagorean tunings it opens may illustrate a common adage of the
xenharmonic movement: in exploring all the possibilities which music
has to offer, "12 is not enough."

Most respectfully,

Margo Schulter
mschulter@value.net