Re: 10-note scales from three tetrads posts

>[Paul Erlich:]
>I would _really_ love to understand what you're doing here, but I can't
>follow you at all.
>
>I took the three consecutive triads of F, C, and G, constructed with a
>major and minor third (@: 1/1*5/4*6/5) and separated by a fifth (the top
>note):
>
>4/3 5/3 2/1
>1/1 5/4 3/2
>3/2 15/8 9/8,
>
>and made them into three consecutive tetrads, constructed + 5/4 + 6/5 +
>5/4, and separated by the top note, a major seventh:
>
>16/15, 4/3, 8/5, 2/1
>1/1, 5/4, 3/2, 15/8
>15/8, 75/64, 45/32, 225/128

OK. I thought you were trying to do something like what Von Hoerner did.

>Then I 'ignored' [...subtracted it from the 15/8, 75/64, 45/32, 225/128...]
>the 225/224:
>16/15, 4/3, 8/5, 2/1
>1/1, 5/4, 3/2, 15/8
>[28/15], 7/6, 7/5, 7/4
>
>and I used this scale:
>
>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
>
>as a generating (I.) scale in the ('modal') example I gave (once again
>'ignoring' the 225/224's, as well as the 385/384's):
>
>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8,
>1/1, 12/11, 7/6, 5/4, 21/16, 7/5, 3/2, 18/11, 7/4, 15/8,
>1/1, 16/15, 8/7, 6/5, 9/7, 11/8, 3/2, 8/5, 12/7, 11/6,
>1/1, 16/15, 9/8, 6/5, 9/7, 7/5, 3/2, 8/5, 12/7, 15/8,
>1/1, 21/20, 9/8, 6/5, 21/16, 7/5, 3/2, 8/5, 7/4, 15/8,
>1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 32/21, 5/3, 25/14, 40/21,
>1/1, 16/15, 7/6, 5/4, 4/3, 10/7, 14/9, 5/3, 16/9, 15/8,
>1/1, 12/11, 7/6, 5/4, 4/3, 16/11, 14/9, 5/3, 7/4, 15/8,
>1/1, 16/15, 8/7, 11/9, 4/3, 10/7, 32/21, 8/5, 12/7, 11/6,
>1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 15/8,
>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8,
>
>
>What I primarily use this sort of a process for, is to recast various equal
>divisions of the octave (n-tET's) into the 'generating scale' (here, that
>'generating scale' would obviously be the three tetrad, 10-note scale,
>outlined above). I do this by 'turning' the ratios into equally divided
>scale steps (i.e., "n"-specific whole numbers):
>
>[(log16-log15)*(12/log2)]*n/12
>[(log4-log3)*(12/log2)]*n/12
>[(log8-log5)*(12/log2)]*n/12
>[(log1-log1)*(12/log2)]*n/12
>[(log5-log4)*(12/log2)]*n/12
>[(log3-log2)*(12/log2)]*n/12
>[(log15-log8)*(12/log2)]*n/12
>[(log7-log6)*(12/log2)]*n/12
>[(log7-log5)*(12/log2)]*n/12
>[(log7-log4)*(12/log2)]*n/12

OK. That's a valid move in some ETs satisfying certain consistency
conditions (more like Paul Hahn's than mine) _and_ in which 225/224 and
385/384 vanish. I think 31-tET is an example. Otherwise you might
unnecessarily destroy some consonances with your algorithm.

>Obviously the closer an n-tET is to the total number of scale degrees, the
>greater the chances are that you will encounter some pretty radical
>'fits...' but at the actual point of making music, I personally try to let
>my artistic intuitions and intentions (not to mention an endless repetition
>of empirical trial and tribulation!) be the final arbiter of (these scales,
>pretty radical 'fits...', etc., etc., etc.), musical 'worth.'

I think you're referring to ETs with a low number of notes? Well anyway, a
scale with "mistakes" in it can be musically more interesting that a
"correct" one.

>Earlier today (in a reply to Paul Erlich) I wrote...
>
>>Then I used this scale:
>>
>>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
>>
>>as a generating (I.) scale. In the ('modal') example I gave:
>>
>>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8
>>1/1, 12/11, 7/6, 5/4, 21/16, 7/5, 3/2, 18/11, 7/4, 15/8
>>1/1, 16/15, 8/7, 6/5, 9/7, 11/8, 3/2, 8/5, 12/7, 11/6
>>1/1, 16/15, 9/8, 6/5, 9/7, 7/5, 3/2, 8/5, 12/7, 15/8
>>1/1, 21/20, 9/8, 6/5, 21/16, 7/5, 3/2, 8/5, 7/4, 15/8
>>1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 32/21, 5/3, 25/14, 40/21
>>1/1, 16/15, 7/6, 5/4, 4/3, 10/7, 14/9, 5/3, 16/9, 15/8
>>1/1, 12/11, 7/6, 5/4, 4/3, 16/11, 14/9, 5/3, 7/4, 15/8
>>1/1, 16/15, 8/7, 11/9, 4/3, 10/7, 32/21, 8/5, 12/7, 11/6
>>1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 15/8
>>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8
>>
>>I once again 'ignored' the 225/224's, as well as the 385/384's
>
>
>The sixth degree of this example should have read:
>
>1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 32/21, 5/3, 16/9, 40/21, 2/1
>
>not:
>
>1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 32/21, 5/3, 25/14, 40/21, 2/1
>
>The 'modal' permutations of 3+4+3+3+2+3+3+4+3+4 (@: 0, 116, 271, 387, 503,
>581, 697, 813, 968, 1084, 1200 in 31-tET) would be the first 10-note scales
>[in an equal division of the octave] to give distinct
>approximations/representations of all of these ratios.

[Paul Erlich:]
>OK. That's a valid move in some ETs satisfying certain consistency
conditions (more like Paul Hahn's than mine) _and_ in which 225/224 and
385/384 vanish.

Couldn't your 22-tET "Alternate Pentachordal Major" scale be said to be
these three consecutive tetrads: 16/15, 4/3, 8/5, 1/1, 5/4, 3/2, 15/8,
75/64, 45/32, 225/128 (in which I'm assuming that the 225/224, and the
385/384 are ignored), where the 64/63 vanishes (64/63*225/224 &
64/63*385/384)?:

1/1, 16/15, 7/6, 5/4, 4/3, 7/5*, 3/2, 8/5, 7/4, 15/8, 2/1
1/1, 12/11, 7/6, 5/4, 4/3, 7/5, 3/2, 5/3, 7/4, 15/8, 2/1
1/1, 16/15, 8/7, 6/5, 9/7, 11/8, 3/2, 8/5, 12/7, 11/6, 2/1
1/1, 16/15, 8/7, 6/5, 9/7, 7/5, 3/2, 8/5, 12/7, 15/8, 2/1
1/1, 16/15, 8/7, 6/5, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 3/2, 5/3, 7/4, 15/8, 2/1
1/1, 16/15, 7/6, 5/4, 4/3, 10/7, 14/9, 5/3, 7/4, 15/8, 2/1
1/1, 12/11, 7/6, 5/4, 4/3, 16/11, 14/9, 5/3, 7/4, 15/8, 2/1
1/1, 16/15, 8/7, 6/5, 4/3, 10/7, 3/2, 8/5, 12/7, 11/6, 2/1
1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 15/8, 2/1
1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1

Dan

*If the 600c 11/22 half octave could be said to represent both the 7/5 and
the 10/7 by more than 'inversion,' in the context of 22-tET, could the
218&2/11thsc of the 4/22 also be said to approximate both the 8/7, and the
9/8 (as well as their 18/22 inversions, the 981&9/11thsc 7/4, and the
16/9)?

Dan Stearns' 10-note scale made by stacking 3 major 7th chords and ignoring
225/224 and 385/384 is a subset of Fokker's/Lumma's 12-note 7-limit
quasi-just scale.

Here are the three stacked major 7ths.

D#--------A#
/ \ /
/ \ /
/ \ /
/ \ /
E---------B---------F#
/ \ /
/ \ /
/ \ /
/ \ /
F---------C---------G
/ \ /
/ \ /
/ \ /
/ \ /
Db--------Ab

And here's what we get when we ignore (or distribute) the 225/224 and 385/384.

D#--------A#
,'/:\`. ,'/
F--/-:-\--C /
/|\/ : \/| /
/ |/\ : /\|/
E---------B---------F#
/|\`. /,'/ `.\:/,'
/ | \ Db-/------Ab
D#--------A# \ | /
,'/:\`. /,'/ `.\|/
F--/---\--C--/------G
/|\/ : \/| /
/ |/\ : /\|/
/ B---------F#
/,' `.\:/,'
Db--------Ab

When we notice where it repeats, we see it has 4 7-limit tetrads, 2 5-limit
triads (in a hexany) and a 6:8:11 (A#:D#:Ab).

Here's a corresponding repeating view of Lumma's 12. We see that the
addition of the D and A gives us 2 more 7-limit tetrads and 2 more 5-limit
triads.

D#--------A#
,'/:\`. ,'/
F--/-:-\--C /
/|\/ : \/| /
/ |/\ : /\|/
A---------E---------B---------F#
/|\ /|\`. /,'/ \`.\:/,'/
/ | \ / | \ Db-/---\--Ab /
/ D#--------A# \ | / \ | /
/,'/:\`.\ /,'/ `.\|/ \|/
F--/---\--C--/------G---------D
/|\/ : \/| /
/ |/\ : /\|/
/ B---------F#
/,' `.\:/,'
Db--------Ab

This view has the advantage of showing one 5-limit plane that contains the
whole scale.

Dan I don't see how your decatonic could ever work in 22-tET. 31-tET yes.

Regards,
-- Dave Keenan
http://dkeenan.com

[Dave Keenan:]
>Dan I don't see how your decatonic could ever work in 22-tET. 31-tET yes.

The main point [a point of interest rather than a point of contention] I
was interested in throwing out there pertained to Paul's quote:

"Many other theorists, including Schoenberg, have explained the diatonic
scale as arising from three consecutive triads in a chain of fifths. This
is another reversal of historical facts, as the chords were constructed
from the scale, and not the other way around. Von Hoerner constructed a
scale from three consecutive 7-limit tetrads in a chain of fifths, using
31-tone equal temperament. The structure of this scale is too bizarre for
it to function as a melodic entity."

Where if "arising" was put aside, I think you could say that Paul's 22-tET
"Alternate Pentachordal Major" scale:

sLsssssLss @
0, 2, 5, 7, 9, 11, 13, 15, 18, 20, 22

derived:

[(log16-log15)*(12/log2)]*22/12
[(log4-log3)*(12/log2)]*22/12
[(log8-log5)*(12/log2)]*22/12
[(log1-log1)*(12/log2)]*22/12
[(log5-log4)*(12/log2)]*22/12
[(log3-log2)*(12/log2)]*22/12
[(log15-log8)*(12/log2)]*22/12
[(log7-log6)*(12/log2)]*22/12
[(log7-log5)*(12/log2)]*22/12
[(log7-log4)*(12/log2)]*22/12,

could be said to 'be' (satisfy the conditions of) these three consecutive
tetrads: 16/15, 4/3, 8/5, 1/1, 5/4, 3/2, 15/8, 75/64, 45/32, 225/128 (in
which I'm already assuming that the 225/224, and the 385/384 are ignored),
where the 64/63 vanishes (64/63*225/224 & 64/63*385/384):

1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
1/1, 12/11, 7/6, 5/4, 4/3, 7/5, 3/2, 5/3, 7/4, 15/8, 2/1
1/1, 16/15, 8/7, 6/5, 9/7, 11/8, 3/2, 8/5, 12/7, 11/6, 2/1
1/1, 16/15, 8/7, 6/5, 9/7, 7/5, 3/2, 8/5, 12/7, 15/8, 2/1
1/1, 16/15, 8/7, 6/5, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
1/1, 16/15, 8/7, 5/4, 4/3, 7/5, 3/2, 5/3, 7/4, 15/8, 2/1
1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 14/9, 5/3, 7/4, 15/8, 2/1
1/1, 12/11, 7/6, 5/4, 4/3, 16/11, 14/9, 5/3, 7/4, 15/8, 2/1
1/1, 16/15, 8/7, 6/5, 4/3, 7/5, 3/2, 8/5, 12/7, 11/6, 2/1
1/1, 16/15, 8/7, 5/4, 4/3, 7/5, 3/2, 8/5, 12/7, 15/8, 2/1
1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1

Dan

Message text written by INTERNET:tuning@onelist.com
>
"Many other theorists, including Schoenberg, have explained the diatonic
scale as arising from three consecutive triads in a chain of fifths. This
is another reversal of historical facts, as the chords were constructed
from the scale, and not the other way around. <

I think that there is a sense in which Schoenberg is historically correct
but one which requires the rather more subtle understanding of musical
materials found throughout Schoenberg's writings.

Schoenberg was not offering a historical account of uninterpreted musical
materials but rather of tonal practice, which is characterized historically
more by the succession of alternative interpretations of known materials
than by the addition of new materials. While the diatonic collection for,
say, Josquin and Mozart may appear to be cohomologous if not identical in
content, the radically different immanent manifestations of the collections
suggest, if not demand, radically different constructions.

For Josquin a given diatonic collection is one of the possible collections
generated by hexachordal modulation, and triadic resources are still
essentially incidental to this melodic process. For Mozart, the diatonic
collection is heard simultaneously as projected into chains of fifths,
chains of fifths subtended by thirds, thirds subtended melodically by
seconds; the diatonic collection is frequently further subtended by the
chromatic. The torus-like lattice projection (for both the diatonic and
total chromatic collections*) implied by Mozart's tonal practice, which at
centre is a series of 3 triads, is not immanent for Josquin, so that one
may reasonable accept the premise that at some point between Josquin and
Mozart the diatonic scale _as we presently understand it_ was invented and
that the tri-triadic property was one of the properties central to its
construction.

Schoenberg's historical conjectures are not applied to any repertoire prior
to this scale; indeed, he scarcely knew any repertoire prior to Sebastian
Bach.

Daniel Wolf

*The possible relationship between closure of a collection over a torus and
at a given MOS is one worth further research. By "total chromatic
collection", I here intend to include chromatic collections of more than
twelve tones, such as those available in an extended meantone arrangement,
which would presumably not affect the toroid structure.

Dan Stearns wrote,

>Couldn't your 22-tET "Alternate Pentachordal Major" scale be said to be
>these three consecutive tetrads: 16/15, 4/3, 8/5, 1/1, 5/4, 3/2, 15/8,
>75/64, 45/32, 225/128

Sure, but my motivation was 7-limit tetrads, not 15-limit or extended
5-limit ones, which to me are already OK in 12-equal and superb in meantone.
Besides, 16:15 is not the usual interval by which roots of chords progress.

>*If the 600c 11/22 half octave could be said to represent both the 7/5 and
>the 10/7 by more than 'inversion,' in the context of 22-tET, could the
>218&2/11thsc of the 4/22 also be said to approximate both the 8/7, and the
>9/8 (as well as their 18/22 inversions, the 981&9/11thsc 7/4, and the
>16/9)?

Yes but my concern was 7-limit harmony and not 9-limit harmony. Since 8:7
and 7:4 have a stronger pull than 9:8 and 16:9, and since 8:7 and 7:4 are
closer to 22-tET, the 9-limit ratios do not pose a major threat to the
projection of a 7-limit standard of consonance in 22-tET. The 7:5 vs. 10:7
is a threat, but keeping the harmony to three or more voices allows other
intervals to clarify whether 7:5 or 10:7 is meant.

[Dan Stearns TD 217.5]
>Couldn't [Paul Erlich's] 22-tET "Alternate Pentachordal Major" scale be
said to be
>these three consecutive tetrads: 16/15, 4/3, 8/5, 1/1, 5/4, 3/2, 15/8,
>75/64, 45/32, 225/128 (in which I'm assuming that the 225/224, and the
>385/384 are ignored), where the 64/63 vanishes (64/63*225/224 &
>64/63*385/384)?:

I previously replied:
"Dan I don't see how your decatonic could ever work in 22-tET."

I didn't look hard enough. Sorry Dan.

When cast into 22-tET, your stack of 3 major 7th chords does indeed become
Paul Erlich's pentachordal (asymmetric) decatonic scale (no need to specify
any particular mode). It loses some 7-limit accuracy but picks up two more
complete 7-limit tetrads because the 64/63 vanishes (gets distributed), as
you said. These two extra tetrads are essential to Paul's determination of
its suitability as a generalised diatonic. i.e. it is not Paul's scale when
cast into 31-tET or anything other than double chains of fifths of a
similar size to those in 22-tET, e.g. 44, 54, 56, 66, 76 (34-tET if you can
stand the bad 4:7's and 32-tET if you can stand the bad 5:6's).

[Dan Stearns TD 215.21]
>The 'modal' permutations of 3+4+3+3+2+3+3+4+3+4 (@: 0, 116, 271, 387, 503,
>581, 697, 813, 968, 1084, 1200 in 31-tET) would be the first 10-note scales
>[in an equal division of the octave] to give distinct
>approximations/representations of all of these ratios.

That should have been
3+4+3+3+2+3+3+4+3+3 in 31-tET,
but in any case you've now shown that you can do it with
2+3+2+2+2+2+2+3+2+2 in 22-tET.
And yes, that particular mode (stating on C as I've shown it below) is what
Paul calls the Alternate Pentachordal Major.

Here's Paul's pentachordal decatonic showing the 3 stacked major 7th chords
and all 6 7-limit tetrads (with some repetition).

D#//-----A#//
,'/ \`. ,'/
A#//-------F--/- -\--C /
|\`. ,'/|\/ \/| /
| \ G / |/\ /\|/
E/--------B/-------F#/
/|\`.\|/,'/ `.\ /,'
/ | \ Db\-/-----Ab\
D#//------A#//\ | /
,'/ \`. /,'/|\`.\|/
F--/---\--C--/-|-\--G
/|\/ \/| / E/ \ |
/ |/\ /\|/,' `.\|
/ B/-------F#/-------Db\
/,' `.\ /,'
Db\-------Ab\

Note that
D#// = Eb
A#// = Bb
Db\ = C#/
Ab\ = G#/

So this could equally be notated as

Eb--------Bb
,'/ \`. ,'/
Bb--------F--/- -\--C /
|\`. ,'/|\/ \/| /
| \ G / |/\ /\|/
E/--------B/-------F#/
/|\`.\|/,'/ `.\ /,'
/ | \ C#/-/-----G#/
Eb--------Bb \ | /
,'/ \`. /,'/|\`.\|/
F--/---\--C--/-|-\--G
/|\/ \/| / E/ \ |
/ |/\ /\|/,' `.\|
/ B/-------F#/-------C#/
/,' `.\ /,'
C#/-------G#/

This makes the 7-limit tetrads more sensible but obscures the major 7th
chords.

Regards,

-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>I previously replied:
>"Dan I don't see how your decatonic could ever work in 22-tET."

>I didn't look hard enough. Sorry Dan.

>When cast into 22-tET, your stack of 3 major 7th chords does indeed become
>Paul Erlich's pentachordal (asymmetric) decatonic scale (no need to specify
>any particular mode). It loses some 7-limit accuracy but picks up two more
>complete 7-limit tetrads because the 64/63 vanishes (gets distributed), as
>you said. These two extra tetrads are essential to Paul's determination of
>its suitability as a generalised diatonic. i.e. it is not Paul's scale when
>cast into 31-tET or anything other than double chains of fifths of a
>similar size to those in 22-tET, e.g. 44, 54, 56, 66, 76 (34-tET if you can
>stand the bad 4:7's and 32-tET if you can stand the bad 5:6's).

You mean 64-tET, not 32-tET, right?

Anyway, my paper mentions obtaining major 7th chords (as well as minor 7th
chords) using the scale template 1, 4, 7, 10 as an alternate harmonization
of the decatonic scale (see footnote 32):

the pentachordal scale is 0 2 4 7 9 11 13 16 18 20 (22)

the template 1, 4, 7, 10 gives

0 7 13 20 (major 7th)
2 9 16 0 (dissonant)
4 11 18 2 (dissonant)
7 13 20 4 (minor 7th)
9 16 0 7 (major 7th)
11 18 2 9 (major 7th)
13 20 4 11 (major 7th)
16 0 7 13 (minor 7th)
18 2 9 16 (dissonant)
20 4 11 18 (dissonant)

the symmetrical scale is 0 2 4 7 9 11 13 15 18 20 (22)

the template 1, 4, 7, 10 gives

0 7 13 20 (major 7th)
2 9 15 0 (major 7th)
4 11 18 2 (dissonant)
7 13 20 4 (minor 7th ~10:12:15:18)
9 15 0 7 (dissonant)
11 18 2 9 (major 7th)
13 20 4 11 (major 7th)
15 0 7 13 (dissonant)
18 2 9 15 (minor 7th ~10:12:15:18)
20 4 11 18 (dissonant)

So either scale has four major seventh chords and two minor seventh chords
that conform well to 5-limit construction. In fact, just as the template 1,
4, 7, 9 produces, in most positions, tetrads with the maximum possible
number (6, or you might say 5 1/2 because of the ambiguous half-octave) of
consonances within the 7-limit (that's the main point of my paper,
http://www-math.cudenver.edu/~jstarret/22ALL.pdf -- see page 11 of to see
how these complete, consonant 7-limit tetrads occur in the scale), the
template 1, 4, 7, 10 produces, in most positions, tetrads with the maximum
possible number (5) of consonances within the 5-limit! But wait, there's
more:

the pentachordal scale is 0 2 4 7 9 11 13 16 18 20 (22)

the template 1, 3, 7, 9 gives

0 4 13 18 (dissonant)
2 7 16 20 (dissonant)
4 9 18 0 (dissonant)
7 11 20 2 (stacked-fifths tetrad)
9 13 0 4 (stacked-fifths tetrad)
11 16 2 7 (minor 7th ~ 12:14:18:21)
13 18 4 9 (minor 7th ~ 12:14:18:21)
16 20 7 11 (stacked-fifths tetrad)
18 0 9 13 (stacked-fifths tetrad)
20 2 11 16 (dissonant)

the symmetrical scale is 0 2 4 7 9 11 13 15 18 20 (22)

the template 1, 3, 7, 9 gives

0 4 13 18 (dissonant)
2 7 15 20 (minor 7th ~ 12:14:18:21)
4 9 18 0 (dissonant)
7 11 20 2 (stacked-fifths tetrad)
9 13 0 4 (stacked-fifths tetrad)
11 15 2 7 (dissonant)
13 18 4 9 (minor 7th ~ 12:14:18:21)
15 20 7 11 (dissonant)
18 0 9 13 (stacked-fifths tetrad)
20 2 11 15 (stacked-fifths tetrad)

The "stacked-fifths tetrad" is the tetrad with the maximum possible number
of 3-limit consonances (3) and the other three intervals can be considered
7-limit consonances in 22-equal if we don't mind the contradictory
ratio-implications of the different intervals (i.e, tuning the chord in JI,
something would have to break). The 12:14:18:21 minor seventh that arises
here, along with the 10:12:15:18 one that arises with the 1, 4, 7, 10
template, are the two saturated 9-limit tetrads (see
http://www.cix.co.uk/~gbreed/erlichs.htm).

Finally, consider:

the pentachordal scale is 0 2 4 7 9 11 13 16 18 20 (22)

the template 1, 4, 6, 9 gives

0 7 11 18 (French 6th)
2 9 13 20 (French 6th)
4 11 16 0 (dissonant)
7 13 18 2 (diminished 7th)
9 16 20 4 (dissonant)
11 18 0 7 (French 6th)
13 20 2 9 (French 6th)
16 0 4 11 (dissonant)
18 2 9 13 (diminished 7th)
20 4 11 16 (dissonant)

the symmetrical scale is 0 2 4 7 9 11 13 15 18 20 (22)

the template 1, 4, 6, 9 gives

0 7 11 18 (French 6th)
2 9 13 20 (French 6th)
4 11 15 0 (French 6th)
7 13 18 2 (diminished 7th)
9 15 20 4 (diminished 7th)
11 18 0 7 (French 6th)
13 20 2 9 (French 6th)
15 0 4 11 (French 6th)
18 2 7 13 (diminished 7th)
20 4 9 15 (diminished 7th)

In the French 6th and diminished 7th chords, all 6 intervals can be
considered 7-limit consonances in 22-equal (or half-consonances in the case
of the ambiguous half-octave), but these consonances would contradict one
another in JI.

I think all 22-equal tetrads that can be considered somewhat consonant are
either the original major and minor tetrads of my paper or one of the chords
mentioned above. Anyone disagree? (The chord 0 6 7 13 has 5 consonances
within the 5-limit but sounds a lot more dissonant than the produced by the
1, 4, 7, 10 template).