Just temperament -- another meaning

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proper text formatting of tuning diagrams and tables -- M.S.]

Hello, there, everyone, and the recent discussions of "Just
Temperament" provide me with an occasion to share a seven-note
diatonic tuning which might actually fit that description.

By this I mean a just or rational intonation system based on integer
ratios only in which certain fifths and fourths are "tempered" much as
in a usual temperament based on irrational ratios.

Here is a favorite example of mine, a seven-note diatonic scale
actually a subset of a 17-note system, here show as an octave C-C:

! 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

This scale, combining pure 3:2 fifths or 4:3 fourths with pure 14:11
major thirds (C-E, G-B) and 13:11 minor thirds (D-F, A-C), also
includes other intervals differing slightly in size from these.

Thus the mildly unstable sonority C-E-G, for example, combines a 14:11
third below (~417.51 cents) with its fifth's complement at 33:28 above
(~284.45 cents).

Similarly, the mildly unstable D-F-A has 13:11 below (~289.21 cents)
and its fifth's complement of 33:26 (~412.75 cents) above.

The most striking feature of this "just temperament," however, is the
fact that it compares four just fifths (C-G, D-A, E-B, F-C) with two
"virtually tempered" ones: G-D at 176:117 (~706.880 cents), wide by
352:351 or about 4.93 cents; and A-E at 182:121 (~706.718 cents), wide
by 364:363 or about 4.76 cents.

The compromise of these two fifths is comparable to that of the nearer
fifths (Ab-B) in George Secor's 17-note well-temperament at ~707.22
cents (~5.27 cents wide), or to that in the other direction in
1/4-comma meantone (~696.58 cents, ~5.38 cents narrow).

Comparing this variety of just temperament to a more familiar just
intonation scale will confirm the distinctive feature of four pure
fifths, plus two "moderately impure ones" as judged by historical
standards of the Western European compositional tradition. In
contrast, a usual JI system has five pure fifths plus a more
dramatically impure one:

Just temperament (3/2, 13/11, 14/11)

1/1 44/39 14/11 4/3 3/2 22/13 21/11 2/1
0 208.83 417.51 498.04 701.96 910.79 1119.46 1200
44:39 273:242 22:21 9:8 44:39 273:242 22:21
208.83 208.67 80.54 203.91 208.83 208.67 80.54

Archytas diatonic (3/2, 7/6, 9/7)

1/1 9/8 9/7 4/3 3/2 27/16 27/14 2/1
0 203.91 435.08 498.04 701.96 905.87 1137.04 1200
9:8 8:7 28:27 9:8 9:8 8:7 28:27
203.91 231.17 62.96 203.91 203.91 231.17 62.96

Ptolemy syntonic diatonic (3/2, 5/4, 6/5)

1/1 9/8 5/4 4/3 3/2 27/16 15/8 2/1
0 203.91 386.31 498.04 701.96 905.87 1088.27 1200
9:8 10:9 16:15 9:8 9:8 10:9 16:15
203.91 182.40 111.73 203.91 203.91 182.40 111.73

Thus in the Archytas diatonic based on prime factors of 3 and 7, we
find that the fifth A-E at 32:21 or ~729.22 cents is wide by a full
64:63 (~27.26 cents), the comma of Archytas as it has aptly been named
by George Secor.

In the syntonic diatonic of Ptolemy, based on the primes 3 and 5, we
find that the fifth A-E at 40:27 (~680.45 cents) is narrow by a full
81:80 (~21.51 cents), the syntonic comma or comma of Didymus.

Looking more closely at the sizes of melodic steps in these just
scales, we see that either the Archytas or Ptolemy diatonic involves
two sizes of whole-tone steps plus a single diatonic semitone step
(Archytas 9:8 or 8:7, 28:27; Ptolemy 9:8 or 10:9, 16:15).

In the just temperament, however, we have _three_ sizes of whole-tones
at 9:8 (~203.91 cents), 273:242 (~208.67 cents), and 44:39 (~208.83
cents), along with the diatonic semitone at 22:21 (~80.54 cents).

For now, I will note that the just temperament is therefore based on
four commas, which might also be termed kleismas or schismas by those
preferring to reserve the term "comma" for larger intervals like the
familiar 81:80 of Didymus and 64:63 of Archytas.

These four small intervals are 896:891 (~9.69 cents); 352:351;
364:363; and 10648:10647 (~0.162595 cents). Here are proposed names
and descriptions for these superparticular intervals:

---------------------------------------------------------------------
Ratio Cents Possible names Description
---------------------------------------------------------------------
896:891 ~9.688 undecimal kleisma 14:11 vs. 81:64
or 3-11 kleisma (417.51 vs. 407.82)
.....................................................................
352:351 ~4.925 tredecimal kleisma 32:27 vs. 13:11
or 3-13 kleisma (294.13 vs. 289.21)
.....................................................................
364:363 ~4.763 11-13 schisma 13:11 vs. 33:28
(289.21 vs. 284.45)
.....................................................................
10648:10647 ~0.163 harmonisma 352:351 vs. 364:363
(4.93 vs. 4.76)
---------------------------------------------------------------------

Here the 896:891 kleisma defines the difference between a pure 14:11
third and a smaller Pythagorean major third at 81:64 formed by four
pure fifths.

The 352:351 kleisma defines the difference between a Pythagorean minor
third at 32:27 from a chain of three pure fifths, and a pure 13:11.

The 364:363 schisma defines the difference of these two ratios, and is
also equal to the difference between 13:11 and 33:28, or 14:11 and
33:26; or the amount by which a pure 14:11 plus a pure 13:11 exceeds a
pure 3:2. The last definition is illustrated in our just temperament
by the sonority A-C-E, with 13:11 below and 14:11 above, producing an
outer fifth at 182:121, wide by 364:363 (i.e. 121:143:182).

The 10648:10647 harmonisma, which I name in honor of Kathleen
Schlesinger and her _harmoniai_ or just arithmetic divisions sometimes
featuring the ratios 14:11 and 13:11, is equal to the difference
between 352:351 and 364:363.

In our scale, the harmonisma occurs for example as the slight
difference in the whole-tone steps of 44:39 (larger than 9:8 by
352:351) and 273:242 (larger than 9:8 by 364:363).

In a following post, I will look more closely at how the pattern of
these four ratios makes possible the artful compromises of a just
temperament. To conclude for the moment, I give Scala data for the
scale and its complete set of intervals, illustrating the "virtually
tempered" fifths or fourths and also some differences of 10648:10647
or about 0.16 cents:

Diatonic scale with symmetrical tetrachords based on 14:11 and 13:11 thirds
0: 1/1 0.000000 unison, perfect prime
1: 44/39 208.8353
2: 14/11 417.5081 undecimal diminished fourth
3: 4/3 498.0452 perfect fourth
4: 3/2 701.9553 perfect fifth
5: 22/13 910.7907
6: 21/11 1119.463
7: 2/1 1200.000 octave
|
Interval class, Number of incidences, Size:
1: 2 22/21 80.537 cents
1: 1 9/8 203.910 cents major whole tone
1: 2 273/242 208.673 cents
1: 2 44/39 208.835 cents
2: 1 33/28 284.447 cents
2: 2 13/11 289.210 cents
2: 1 968/819 289.372 cents
2: 1 33/26 412.745 cents
2: 2 14/11 417.508 cents undecimal diminished fourth
3: 1 117/88 493.120 cents
3: 1 121/91 493.282 cents
3: 4 4/3 498.045 cents perfect fourth
3: 1 63/44 621.418 cents
4: 1 88/63 578.582 cents
4: 4 3/2 701.955 cents perfect fifth
4: 1 182/121 706.718 cents
4: 1 176/117 706.880 cents
5: 2 11/7 782.492 cents undecimal augmented fifth
5: 1 52/33 787.255 cents
5: 1 819/484 910.628 cents
5: 2 22/13 910.790 cents
5: 1 56/33 915.553 cents
6: 2 39/22 991.165 cents
6: 2 484/273 991.327 cents
6: 1 16/9 996.090 cents Pythagorean minor seventh
6: 2 21/11 1119.463 cents

Most appreciatively,

Margo Schulter
mschulter@value.net

yes, margo, _that_ *is* an example of just temperament, or at least
rational temperament. thank you for jogging our foggy memories.

"M. Schulter" <MSCHULTER@VALUE.NET> writes:

> By this I mean a just or rational intonation system based on integer
> ratios only in which certain fifths and fourths are "tempered" much
> as
> in a usual temperament based on irrational ratios.

Another reasonable definition would be a JI scale which has
approximations so nearly JI that they can be considered just. One
astonishing example of this I have discovered twice now, in two different
ways, so I think it is definitively worth taking note of as a real
phenom. It is the 19-note Fokker block based on the two commas 81/80 and
78732/78125. Considered merely as a 5-limit scale, it is about as
ordinary as it gets, and sports eight major and eight minor triads.
However, if we allow differences of 4375/4374 to count as just, we have
in addition five subminor and five supermajor triads, three subminor and
three supermajor tetrads, and one major and one minor tetrad. Since
4375/4374 is so tiny (a mere 2/5 of a cent) we may regard these
additional chords and intervals as effectively just. Here's the scale:

! sc19_justtemper
Fokker block from commas <81/80, 78732/78125>
!
19
!
250/243
27/25
10/9
125/108
6/5
5/4
162/125
4/3
25/18
36/25
3/2
125/81
8/5
5/3
216/125
9/5
50/27
243/125
2/1

The major tetrad, for an example, is 10/9-25/18-5/3-243/125; hardly
distinguishable from the JI
10/9-25/18-5/3-35/18. If we want, we can temper by the planar temperament
4375/4374, for instance by using the 1547 et, but this really isn't
necessary; it works excellently as it is.

In a message dated 10/9/02 5:50:09 PM Central Daylight Time,
genewardsmith@juno.com writes:

> Another reasonable definition would be a JI scale which has
> approximations so nearly JI that they can be considered just.

And my scale within +/- 1-2 cents the theoretical values accomplishes this.
But I call it JT since the notes aren't in phase. That's one reason for
detuning them.

Pauline