Septimal schisma as xenharmonic bridge?

from the Mills College Tuning Digest, 1998

edited, annotated, and appended by Joseph L. Monzo

# 1559

Topic No. 8

Date: Tue, 20 Oct 1998 22:39:32 -0700 (PDT)
From: Margo Schulter
To:Tuning Digest
Subject: 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

Here is my Lattice Diagram illustrating the "schisma bridges" to approximate 5- and 7-Limit ratios as shown by Schulter in the table above. The 3-Limit ratios with higher exponents - the "bridges" -- are placed on the Lattice in the positions of the 5- and 7-Limit ratios which they approximate, with the 5- and 7-Limit equivalents indicated.