Re: Kalisma/xenisma (new names?) -- JI tuning and Scala file

Hello, there, and with special greetings to Manuel op de Coul, I am
writing to ask if two intervals with small and intriguing integer
ratios have yet received names, and if not, to propose possible names.
Both ratios themselves have already been documented and catalogued in
an article by John Chalmers in _Xenharmonikon_ 17, but I am not sure
if they have been named or related to tuning systems or musical
styles.[1]

In what follows, I first define the proposed kalisma (9801:9800) and
xenisma (2058:2057), and then present a 12-note multi-prime JI tuning
featuring some complex intervals related to these superparticular
ratios, with a Scala file appended after this main text and notes.

-----------------------------------------------------------
1. Defining the kalisma (9801:9800) and xenisma (2058:2057)
-----------------------------------------------------------

The smaller of these two intervals, my proposed _kalisma_, has a ratio
of 9801:9800 (~0.1766 cents), and is equal to the difference between a
large major third or ditone at 12544:9801 (14:11-plus-896:891) and a
diminished fourth at 32:25.

The other interval, my proposed _xenisma_, has a ratio of 2058:2057
(~0.8414 cents), and is equal to the difference between what might be
termed the supraminor third (following Scala's latest intnam.par) at
17:14 and the slightly larger 147:121; and also between what I would
term the "submajor third" (or "subditone") at 21:17 and the slightly
smaller 121:98.[2]

The term "kalisma" is from the Greek root _kal_, which I understand
means "beautiful," and might describe the energetic beauty of the
32:25 or 12544:9801 major thirds at around 427 cents, not too far from
Paul Erlich's point of "maximum entropy" or complexity at 423 cents,
or Dave Keenan's Golden Mediant at ~422.49 cents.

The "xenisma," from the Greek _xen_, "strange," may suggest the
strange and wonderful qualities in a neo-Gothic setting of supraminor
thirds or supraditones at 17:14 or 147:121, and likewise submajor
thirds or subditones at 21:17 or 121:98. These thirds, although
"alien" to the known medieval polyphony of Western Europe, are a
prominent feature of some favorite neo-Gothic tunings.

----------------------------------
2. A related multi-prime JI tuning
----------------------------------

To set these two intervals in some musical context, following this
text and notes I am also including a Scala file and keyboard diagram
for what is at least to me a new neo-Gothic just intonation (JI)
tuning using multiple primes, designed for use on a single standard
12-note keyboard. With two such keyboards available, tuning one of
them in this multi-prime JI set and the other in standard 12-note
Pythagorean or 3-limit JI (Eb-G#) might be a nice solution.

The 12-note JI tuning actually has an 11-note range of Bb-G#,
including two versions of D, one mapped to the usual "D" key of a
12-note keyboard and the other to the "Eb" key. These two keys provide
pure fifths for G and A respectively, and are separated by a comma of
896:891 (~9.6880 cents).

The basic idea of the tuning is to combine pure 3:2 fifths and 14:11
major thirds or ditones, the latter used as unstable but "relatively
blending" intervals in neo-Gothic styles, much like the usual
Pythagorean 81:64 of Gothic polyphony. These complex major thirds
often, for example, resolve by expanding to a stable fifth.

While the proposed kalisma and xenisma do not themselves directly
occur in this tuning, the tuning does include the complex large major
third or ditone 12544:9801 (defining the "kalisma" by its difference
from 32:25), and also the complex supraminor third 147:121 and
submajor third 121:98 (defining the "xenisma" by their differences
from 17:14 and 21:17 respectively).

In this neo-Gothic multi-prime JI scheme, the 12544:9801 (~427.1959
cents) occurs as a 14:11 major third or ditone enlarged by a comma of
896:891, or as an 81:64 Pythagorean major third or ditone enlarged by
two such commas -- the 896:891 defining the difference between the
81:64 (four pure 3:2 fifths) and the 14:11.

Here a crude lattice diagram, with an enthusiastic and appreciative
invitation for more refined and sophisticated versions to the various
exponents of this craft on the Tuning List, may be helpful.

Notes connected in horizontal rows form chains of pure 3:2 fifths,
while diagonally connected major thirds (e.g. F0-A+1, E+1-G#+2) form
pure 14:11 ratios, also expressible as the classic mediant between 5:4
and 9:7, or (5+9):(4+7).

F#+2 - C#+2 - G#+2
/ / /
/ / /
D+1 - A+1 - E+1 - B+1
/ / / /
/ / / /
Bb0 - F0 - C0 - G0 - D0

In the following discussion, as in the diagram, note numbers such as
"Bb0" or "F#2" refer to the number of 896:891 commas by which a note
is raised from its usual Pythagorean position, rather than to octave
numbers (my accustomed use).

The 12544:9801 occurs between the lower of the two versions of D and
the F#, or D0-F#2, a Pythagorean major third enlarged by two commas of
896:891.

Comparing the size of this 12544:9801 major third with that of the
very slightly larger 5-based diminished fourth at 32:25 or 12544:9800
(~427.3726 cents) yields the proposed kalisma at 9801:9800.

In this same 12-note tuning, the complex diminished fourth or submajor
third of 121:98 (~364.9841 cents) occurs as the difference between two
14:11 major thirds at ~417.5080 cents each and a 2:1 octave, as at
C#2-F0 or G#2-C0. This interval differs from the slightly larger 21:17
(~365.8255 cents) by the proposed xenisma of 2058:2057.

Similarly, a complex supraminor third at 147:121, equal to a usual 9:8
Pythagorean whole-tone plus a large chromatic semitone of 392:363,
occurs at F0-G#2 (F0-G0-G#2) and Bb-C#2 (Bb0-C0-C#2). This interval is
larger than the 17:14 by the same xenisma of 2058:2057.

More generally, the proposed kalisma and xenisma may illustrate how
stylistic approaches such as neo-Gothic may provide musical contexts
lending new meaning to complex integer ratios catalogued by devoted
xenharmonicists such as John Chalmers and Manuel op de Coul.

----
Note
----

1. John Chalmers, "The Number of 23-Prime-LImit Superparticular Ratios
Less than 10,000,000," _Xenharmonikon_ 17:111-115 (Spring 1998), where
in Table 1 the 9801:9800, here proposed as the "kalisma," is listed
under "Ratios of 11" as ratio number 186 (p. 112); 2058:2057, here
proposed as the "xenisma" is listed under "Ratios of 17" as ratio
number 148 (p. 113). As Chalmers explains, the ratio numbers of the
first column of the table are "the order of generation numbers of the
ratios as they are found by my program" written in the Microsoft
QuickBASIC language and run on a Macintosh SE/30 computer.

2. The latest version of Scala's intnam.par file lists 17/14 as
"supraminor third," but does not list the 21/17, for which I propose
the name "submajor third." The latter name seems both descriptive and
logical, since the 17/14 and 21/17 together form a pure 3/2 fifth, a
relationship fitting the symmetrical terms "supraminor" and
"submajor." In more neo-medieval terms, I have referred to these two
intervals or nearby approximations as the "suprasemitone" and
"subditone."

-------- Scala file begins on next line of text -------

! neogji12.scl
!
Neo-Gothic 12-note JI tuning (primes 2/3/7/11) F-F with Eb key as D+1
12
!
392/363
9/8
147/121
14/11
4/3
63/44
3/2
196/121
27/16
56/33
21/11
2/1

---------- Scale file concluded on previous line of text -------

Here is a keyboard diagram of the octave F-F in this tuning:

392:363 147:121 4:3 196:121 56:33
133.061 336.971 498.045 835.016 915.553
f# g# bb c#' d+1'
_133.1|70.8_133.1|80.5_80.5|123.4_ _133.1|70.8_9.6|203.9_
f g a b c' d' e' f'
1:1 9:8 14:11 63:44 3:2 27:16 21:11 2:1
0 203.910 417.508 621.418 701.955 905.865 1121.885 1200
9:8 112:99 9:8 22:21 9:8 112:99 22:21
203.910 213.598 203.910 80.537 203.920 213.598 80.537

This is, in effect, a tuning over an 11-note range (Bb-G#) with two
versions of D, one mapped to the usual "D" key and the other to the
"Eb" key. One plays the "D" key for a pure 3:2 fifth or 4:3 fourth
with G, and the "Eb" key for the same pure intervals with A.

For the structure of the tuning, please see also the lattice diagram
in the main text of this article.

Most respectfully,

Margo Schulter
mschulter@value.net