Yahoo Tuning Groups Ultimate Backup tuning 123201/123200 chalmersia, 194481/194480 scintillisma

When it comes to small commas like the 13-limit chalmersisma
or chalmersia at 123201/123200 (0.01405 cents), there's no
question that John Chalmers has indeed surveyed this
universe of small superparticular or epimoric ratios very
thoroughly. And when, late last night, I came upon an even
smaller superparticular ratio, it was no surprise to find
that John had listed one also in "The Number of 23-Prime
Limit Superparticular Ratios Less than 10,000,000,"
Xenharmonikon_ 17 (Spring 1998), pp. 111-115 at 113. There
it was, #227, a 17-limit ratio: 194481/194480, which Scala
has informed me is 0.00890 cents.

By the way, here's a link to the complete text of John's
XH17 article (tuning-math #2239):

<http://robertinventor.com/tuning-math/s___3/msg_2225-2249.html>

Those who wish to cut to the chase on the 194481/194480 may
skip to Section 2.2 below, where a diagram shows how this
"supraminor scintillisma," as I call it, results from the
difference of either 441/440 and 442/441, or of the xenisma
at 2058/2057 and Ibn Sina's comma (which may have other
names also) at 2080/2079. Comparing the supraminor thirds at
63/52, 40/33, 17/14, and 147/121 will demonstrate all four
commas, and thus also the scintillisma at 194481/194480.

Before getting into the scintillisma, John's #227, however,
I'd like to suggest an alternative way of arriving at the
chalmersia, his #223, or 123201/123200, as the difference
between the 9801/9800 kalisma or Gauss's comma and the
10648/10647 harmonisma.

---------------------------------
1. The Chalmersia (123201/123200)
---------------------------------

One way to account for the Chalmersia which I've read is as
the difference of 351/350 and 352/351. Here's an alternative
route to 123201/123200 where there may be some interesting
connections with 351/350 that people can unravel, as well as
a connection with 352/351 that I can immediately see.

We can arrive at the Chalmersia as the difference between
Gauss's comma, also known as the kalisma, at 9801/9800
(0.17665 cents); and the harmonisma at 10648/10647 (0.16260
cents).

Gauss's comma or the kalisma at 9801/9800 may be defined as
the difference between two undecimal kleismas at 896/891 or
9.688 cents each, that is 19.376 cents; and a diaschisma in
its more recent sense of 2048/2025 or 19.553 cents. The
following demonstration may help:

| 896/891 | 896/891 |
81/64---------------- 14/11 ----------------- 12544/9801
407.820 9.688 417.508 9.688 427.196

2048/2025 |
|--------------------------------------------- 32/25
81/64 19.553 427.373
407.820

In the first example, we begin with a just 81/64 major third
from a chain of four 3/2 fifths. Adding one undecimal
kleisma at 896/891 results in a 14/11 major third, the
difference between 81/64 and 14/11 indeed one good
definition of this kleisma. If we then increase 14/11 by
another 896/891 kleisma, we arrive at 12544/9801 or 427.196
cents.

In the second, we likewise begin at 81/64 and add a
diaschisma at 2048/2025, an 81/80 syntonic comma (21.506
cents) less a 3-5 schisma at 32805/32768 (1.954 cents), and
arrive at 32/25 or 427.373 cents. This large major third is
wider than 12544/9801 by the minute amount of the Gauss
comma or kalisma at 9801/9800. See also:

</tuning/topicId_12989.html#12989>

The harmonisma at 10648/10647 routinely comes up in JI
tunings which seek just intervals of 14/11 and 13/11 while
keeping all usual fifths within five cents of pure. In such
a scheme, pure 3/2 fifths typically alternative with others
"virtually tempered" wide by either 352/351 or 4.925 cents
(the difference between 32/27 and 13/11, or 81/64 and 33/26),
or 364/363 or 4.763 cents (the difference between 13/11 and
33/28, or 33/26 and 14/11). Together, the 352/351 and
364/363 add up to the 896/891 undecimal kleisma we have
already encountered above, the difference of 9.688 cents
between 81/64 and 14/11:

(+352/351) (+364/363)
3/2 176/117 3/2 182/121
|-------------|-------------|-------------|------------|
C 701.955 G 706.880 D 701.955 A 706.718 E
1/1 3/2 44/39 22/13 14/11
0 701.955 208.835 910.790 417.508
|---------------------------|--------------------------|
C 44/39 D 273/242 E
208.835 208.673

The 10648/10647 at 0.163 cents is equal to the tiny
difference between 352/351 and 364/363. In the diagram, it
is the difference between the _almost_ equal tones C-D and
D-E at 44/39 (208.835 cents) and D-E (208.673 cents) making
up the major third C-E at a just 14/11 (417.508 cents).

So a "family tree" for the chalmersia at 123201/123200
might look like this. While the 351/350 (4.939 cents) or
ratwolfisma has been defined as the difference between 20/13
(745.786 cents) and 54/35 (750.725 cents), it is additionally
equal to the difference between supraminor thirds at 63/52
(332.208 cents) and 243/200 (337.148 cents), for example.

(54/35 - 20/13)
351/350
4.939
------------------|
| chalmersia
|--------------- 123201/123200
| 0.014
------------------|
(32/27 - 13/11)
352/351
4.925

(32/25 - 81/64)
19.553
2048/2025
---------------------| Gauss's comma
| or kalisma
|------------------|
| 9801/9800 | ---------------------| 0.177 |
(12544/9801 - 81/64) |
19.376 |
802816/793881 |
| chalmersia
|----------------
(32/27 - 13/11) | 123201/123200
4.925 | 0.014
352/351 | ---------------------| |
| harmonisma |
|------------------|
| 10648/10647
---------------------| 0.163
(13/11 - 33/28)
4.763
364/363

--------------------------------------
2. The 194481/194480: a 17-limit ratio
--------------------------------------

Like the 123201/123200 Chalmersia as 351/350 less 352/351,
the 194481/194480 may be defined as the difference of two
adjacent superparticular commas: 441/440 and 442/441. Both
of these commas seem relevant to the MET-24 tuning,
suggesting that 194481/194480 may also be relevant.

As with the Chalmersia with its alternative derivation from
the kalisma (9801/9800) less than harmonisma (10648/10647),
the 194481/194480 may be also be derived from the xenisma of
2058/2057 less Ibn Sina's comma, as it might called to
supplement rather than replace any existing names, at
2080/2079. This pair of commas may also both be relevant to
MET-24, again suggesting the relevance of 194481/194480 also.

--------------------------------
2.1 The 441/440 and 442/441 pair
--------------------------------

The 441/440 comma seems most relevant to MET-24, a parapyth
temperament with a canonical form of (2/1, 703.771, 57.422),
as the difference between 88/63 (578.582 cents) and a pure
7/5 (582.512 cents), or 3.930 cents, with the actual
tempered interval at 577.734 cents. Here I will use the
strictly regular 2048-EDO version to simplify things a bit.

Another definition of the 441/440 ties in the structure of
MET-24, where the tempered semitone is meant to represent
22/21 (80.537 cents), with a canonical size of 81.445 cents,
while a 4/3 fourth plus a 21/20 semitone (84.467 cents)
would yield a pure 7/5. Thus 441/440 is equal to the
difference of 22/21 and 21/20.

While MET-24 is designed to support primes 2-3-7-11-13,
sonorities suggesting prime 5 may sometimes arise. For
example, the following seventh sonority might well give a
total impression of 4:5:6:7, with 5:4 and 7:5 alike at about
441/440 wide or a bit more. Note that the major third is
close to 441/352, or 21/16 less 22/21, while 21/16 less
21/20 would yield a pure 5/4:

441/352
1/1 5/4 3/2 7/4
Eb ----------- F#* -------- Bb ----------- C*
0 390.829 703.711 968.555
|--------------------------|
88/63
7/5
577.734

Here the small major third could be taken as either 5/4 or a
near-just 441/352, and the diminished fifth as either 88/63
(e.g.26/21 plus 44/39, or 370.312 cents plus 207.422 cents)
or 7/5. This near-4:5:6:7 makes an interesting variant on a
more usual 44:56:66:77 or "rebounding seventh" chord with a
regular 14/11 major third (Eb-G-Bb-C#*), with the same
standard resolution, for example

C#* C D Eb C#* C D Eb
Bb C Bb Bb C Bb
F#* A Bb G A Bb
Eb F Eb or Eb F Eb

Here the outer minor seventh contracts to the fifth of
another unstable sonority with major third and fifth, the
highest voice then moving to the major sixth so that all
four voices may expand to a 2:3:4. The seventh is said to be
"rebounding" because the ultimate resolution most often
returns to the same note in the lowest voice on which this
seventh sonority is built.

A purely melodic context where 441/440 is relevant is the
tetrachord 10:9-8:7-21:20 or 1/1-8/7-80/63-4/3.

0 182.404 413.578 498.045
1/1 10/9 80/63 4/3
G G#* B C
0 183.398 414.844 496.289

While G-B represents 14/11 (417.508 cents) or 4/3 less
22/21, it even more closely approximates 80/63 (413.578
cents). These two major thirds differ by 441/440, showing
that this comma, if I am using the phrase correctly, is "in
the kernel" for MET-24.

This brings us now to the adjacent superparticular or
epimoric comma of 442/441 (3.921 cents) -- the difference,
notably between 63/52 (332.208 cents) and 17/14 (336.130
cents). In MET-24, this supraminor or small neutral third is
at 333.398 cents, and might represent either of these ratios
as well as al-Farabi's 40/33 (333.041 cents), to which it is
actually closest (more on this shortly).

Further, there may be a certain linkage between the 441/440
and 442/441 because a supraminor third such as Eb-F# (333.4
cents) plus the third generator of the tuning at 57.4 cents
yields a small major third such as Eb-F#* (390.829 cents).

Thus both members of this comma pair, 441/440 and 442/441,
are evidently relevant to MET-24. Now let us consider the
alternative pair -- and a situation where this pair is
involved as well!

------------------------------------
2.2 The 2058/2057 and 2080/2079 pair
------------------------------------

The 2058/2057 or xenisma (0.841 cents) may be defined, for
example, as the difference between a 17/14 supraminor third
and the supraminor third from 14/11 less 22/21, or 147/121
(336.971 cents).

In MET-24, where the tempered values for 14/11 and 22/21 are
414.844 cents and 81.445 cents, either 17/14 or 147/121 is
represented as 333.398 cents. Interestingly, 414.844 cents
can also reasonably represent either 33/26 (412.745 cents),
especially given the near-just 13/11 (289.210 cents) at a
tempered 288.867 cents; or, as we have seen 80/63 (413.578
cents).

Taking these three possible JI ratios for the regular major
third, and reducing each by 22/21, we have these related
ratios and commas for the supraminor third, plus 17/14:

Regular major third Eb-G (414.844 cents)

352:351 = 4.925
|------------------------------------------|
412.745 413.578 417.508
33/26 80/63 14/11
|--------------|---------------------------|
2080:2079 441:440
0.833 3.930

332.208 333.041 336.130 336.971
63/52 40/33 17/14 147/121
|--------------|-----------------|---------|
2080:2079 561:560 2058:2057
0.833 3.089 0.841
|--------------------------------|
63/52 442:441 17/14
3.921
|---------------------------|
40/33 441:440 147/121
3.930

Supraminor third Eb-F# (333.398 cents)

Comparing the sizes of the four just supraminor thirds at
63/52, 40/33, 17/14, and 147/121 accounts for the commas of
both relevant pairs for our 194481/194480. For the first
adjacent superparticular pair, we have 441/440 (40/33,
147/121) and 442/441 (63/52, 17/14). For the second pair, we
have both the xenisma at 2058/2057 (17/14, 147/121) and
2080/2079 (63/52, 40/33).

Note that to make the upper portion of the diagram fully
symmetrical with the lower, we could add a major third at
the ratio of 17/14 plus 22/21, or 187/147, 416.667 cents --
incidentally a superb approximation of 25/72 octave!

Let's focus on the second comma in the pair which, when
compared with the xenisma at 2058/2057, produces the
scintillisma: Ibn Sina's comma at 2080/2079.

Ibn Sina (980-1037) is the first theorist of whom I know to
discuss ratios of 13, and indeed to make them a central part
of his tuning systems for genera and modes. Quoting a genus
of al-Farabi at 1/1-9/8-99/80-4/3 or 9:8-11:10-320:297
(396:352:320:297), where "the interval of a tone is followed
by an interval of 11:10," he notes that "[t]he ratio of the
complementary interval will be 320:297, very close to
14:13." (Translation based on French version of Baron
Rodolphe d'Erlanger.)

The difference between 14:13 (128.298 cents) and the
slightly larger 320:297 (129.131 cents) is 2080/2079 (0.833
cents). Another way to define this comma is as the amount
that 11:10 plus 14:13, or 77/65 (293.302 cents) falls short
of 32:27 (294.135 cents). It is also the amount by which a
tetrachord with the three superparticular steps of 9:8,
11:10, and 14:13, adding up to 693/520 (497.212 cents) will
fall short of a just 4/3 fourth.

It is fascinating that one of the oldest 13-limit commas
alluded to in the literal, around a millennium ago, the
2080/2079, should have a connection with one of the newest
17-limit commas, perhaps first mentioned by John Chalmers,
and listed as his #227 in _Xenharmonikon_ 17, which I now
propose to call the supraminor scintillisma at 194481/194480 (with "supraminor" an optional adjective noting its
relevance to supraminor thirds).

Here, in conclusion, is a family tree based on the above
diagram of supraminor third sizes:

(147/121 - 40/33)
3.930
441/440
--------------------|
| scintillisma
|---------------------
| 194481/194480
--------------------| 0.009
(17/14 - 63/52)
3.921
442/441

(147/121 - 17/14)
0.841
2058/2057
----------------------|
xenisma | scintillisma
|--------------------
| 194481/194480
Ibn Sina's comma | 0.009
----------------------|
(40/33 - 63/52)
0.833
2080/2079

Peace and love,

Margo Schulter
mschulter@...
23 December 2012

123201/123200 chalmersia, 194481/194480 scintillisma

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