Yahoo Tuning Groups Ultimate Backup tuning Just temperament -- another meaning (2)

Hello, everyone, and this article will delve a bit more into the
"virtual tempering" patterns that make a just temperament possible.

Let us consider again the justly tempered diatonic of C-C in my
previous post:

! msdiat7.scl
!
Diatonic scale with symmetrical tetrachords based on 14:11 and 13:11 thirds
7
!
44/39
14/11
4/3
3/2
22/13
21/11
2/1

Looking at the chain of fifths for the pure 14:11 major third C-E
gives an overview of much of the structure:

+896:891
(+9.688)
|-------------------- 14:11 (417.508) -----------------|
|------------- 56:33 (915.553) ----------|
|-------------- 22:13 (910.790) ---------|
|------ 44:39 (208.835) ----|---- 273:243 (208.673) ---|
C ----------- G ----------- D ---------- A ----------- E
3:2 176:117 3:2 182:121
701.955 706.880 701.955 706.718
pure +352:351 pure +364:363
(+4.925) (+4.763)

As shown in this diagram, a 14:11 third is made up of two pure fifths
plus two "virtually tempered" ones: here G-D, wide by 352:351; and
A-E, wide by 364:363. These two small ratios add up to 896:891, the
amount by which a 14:11 third exceeds an 81:64 Pythagorean third
formed from four pure 3:2 fifths (~407.820 cents).

In effect, we have divided the 896:891 comma or kleisma into two
almost equal parts of 352:351 and 364:363. Thus the whole-tone steps
of C-D and D-E are almost identical in size at 44:39 and 273:243,
differing only by the harmonisma of 10648:10647 or about 0.163 cents.
These whole-tones are formed from one pure and one wide fifth.

This "quasi-bisection" of the 896:891 results also in some variations
of sizes for major sixths or minor thirds, formed from chains of three
fifths up or down.

In our chain above, the major sixth C-A is formed from two pure fifths
plus G-D at 352:351 wide, producing a size of 22:13 -- or a minor
third A-C of 13:11 (~289.210 cents).

The major sixth G-E, however, is formed from one pure fifth and the
two virtually tempered fifths G-D at 352:351 wide and A-E at 364:363,
making larger than Pythagorean (27:16, ~905.865 cents) by a full
896:891, at 56:33, or a minor third E-G at 33:28 (~284.447 cents), the
fifth's complement of 14:11.

Additional varieties of thirds and sixths arise at other points in the
diatonic chain of fifths, for example:

+352:351
(+4.925)
|-------------------- 33:26 (412.745) -----------------|
|------------- 22:13 (910.628) ----------|
|------------- 22:13 (910.628) ----------|
|------- 9:8 (203.910) -----|---- 273:242 (208.763) ---|
F ----------- C ----------- G ---------- D ----------- A
3:2 3:2 273:242 3:2
701.955 701.955 706.880 701.955
pure pure +352:351 pure
(+4.763)

Here the chain for the major third F-A consists of three pure fifths,
plus one fifth (G-D) at 352:351 wide, producing a size of 33:26, or
about 412.745 cents, the fifth's complement of 13:11.

Here we have the 9:8 major third F-G plus G-A at 273:242, together
making up 33:26. The major sixths F-D and C-A both consist of two pure
fifths, plus the wide G-D, and thus have sizes of 22:13, or 352:351
wider than the Pythagorean 27:16 (with C-A also appearing in the
previous example).

Another size of major sixth or minor third appears in the chain
forming the major third G-B:

+896:891
(+9.688)
|-------------------- 14:11 (417.508) -----------------|
|------------ 819:484 (915.553) ---------|
|-------------- 56:33 (915.553) ---------|
|------ 44:39 (208.835) ----|---- 273:243 (208.673) ---|
G ----------- D ----------- A ---------- E ----------- B
176:117 3:2 182:121 3:2
706.880 701.955 706.718 701.955
+352:351 pure +364:363 pure
(+4.925) (+4.763)

Here, as in the first chain of C-E, there are two pure and two
virtually tempered fifths forming a 14:11 major third. Likewise, there
are two whole-tones making a near-equal division of the 896:891 comma
into 352:351 (G-A) and 364:363 (A-B), with sizes larger than 9:8 by
these amounts.

As with the chain C-E, the chain G-B has the 56:33 major sixth G-E
made up of one pure and two wide fifths, again together accounting for
the full comma or kleisma of 896:891.

However, the major sixth D-B differs minutely from the 22:13 sixths in
our previous examples: it is made up of two pure fifths plus the fifth
A-E at 364:363 wide, or 819:484 (~910.628), while 22:13 (e.g C-A in
the first example) is formed from two pure fifths plus one fifth wide
by 352:351. Similarly, we have minutely differing minor thirds at
13:11 (e.g. A-C) and 968:819 (~289.372 cents).

This is another illustration of the distinction of the harmonisma at
10648:10647.

Having considered these ratios as they shape the patterns of virtual
temperaments, let us consider the compromises involved from a more
general philosophical viewpoint.

An important factor in "just temperament" is the relatively small size
of the 896:891 comma or undecimal kleisma involved in obtaining pure
14:11 major thirds, as opposed to the larger commas for 9:7 (64:63) or
5:4 (81:80).

Happily, the 896:891 divides conveniently into almost equal parts of
352:351 (the tredecimal kleisma involved in obtaining a pure 13:11)
and 364:363 (the 11-13 schisma by which 13:11 differs from 33:28).

Thus just temperament mixes pure fifths with others about as impure as
in meantone or George Secor's 17-note well-temperament in its nearer
portion of Ab-B, and produces a delightful mixture of thirds at 14:11,
13:11, 33:26, 33:28 -- and also 968:819 (almost identical to 13:11).

Just temperament contrasts with two more familiar arrangements for
obtaining fifths at or near 3:2, and 14:11 thirds:

Usual just intonation:

14:11 (417.508)
|------------------------------------------------------|
|----- 112:99 (213.598) ----|----- 9:8 (203.910) ------|
G ----------- D ----------- A ---------- E ----------- B
3:2 448:297 3:2 3:2
701.955 711.643 701.955 701.955
pure +896:891 pure pure
+9.688

Usual eventone temperament:

14:11 (417.508)
|------------------------------------------------------|
|----- 209.254 (+ 4.844) ---|---- 209.254 (+4.844) ----|
G ----------- D ----------- A ---------- E ----------- B
704.377 704.377 704.377 704.377
+2.422 +2.422 +2.422 +2.422

In conventional just intonation, as many fifths as possible are made
pure, so that a 14:11 third is formed by three such fifths plus a
fifth wide by the full 896:891, or about 9.688 cents. Such a fifth,
although not quite an outright "Wolf" by traditional European
compositional standards, is nevertheless quite rough, especially in
contrast to the pure fifths of the tuning, and in likely styles where
fifths and fourths are the main stable concords.

In a conventional regular or eventone tuning, all fifths are tempered
by 1/4 of the 896:891 comma or kleisma, or about 2.422 cents, a
compromise considerably milder than in meantone, and slightly greater
than in 12n-tone equal temperament (fifths at 700 cents, ~1.955 cents
narrow).

The delight of just temperament is its mixture of pure and moderately
impure fifths, producing a subtle variety of thirds and sixths, and
also the intellectual charms of the 10648:10647.

Most appreciatively,

Margo Schulter
mschulter@value.net

Just temperament -- another meaning (2)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗M. Schulter <MSCHULTER@VALUE.NET>