Yahoo Tuning Groups Ultimate Backup tuning 22

The 22-et is well-known, and deserve to be, so I will expand on it a
bit from the point of view of white-keys, black-keys. Just as we can
think of the 12-et (actually, the 12-et meantone) as 7+5, we can
start by chopping up 22 notes into white keys and black keys; if each
of these is reasonable as an et the result will be also.

If we allow that the 12-et is, after all, a reasonable et, the
obvious place to start is with 12+10. We have 12/10 = 6/5 = 1+1/5, so
we may approximate it by 1/1; using these as numerators and 5 and 6
as denominators, we get 1/6 < 2/11 < 1/5, where the 2/11 is the
mediant, obtained by adding (1+1)/(6+5) = 2/11. We can find how many
generator steps it takes to reach a given interval q (modulo half-
octaves) by the formula

gen(q) = 6 g10(q) - 5 g12(q),

where g12 measures how many steps are taken in the 12-et, and g10 how
many in the 10-et, where these should be chosen so as to add up to
the steps in the 22-et.

We have the following table:

12 10 22 0
19 16 35 1
28 23 51 -2
34 28 62 -2
41 35 76 5

The first column is the 12-et, counting steps for 2,3,5,7, and 11;
the second is the 10-et and the third is their sum, 22, for the same
intervals. The last column is gen(q), 6 times the 10 column minus 5
times the 12 column. So this tells us that to get a 3, we go up one
generator--which might seem wrong, until we recall we are working
modulo half-octaves, so that a half-octave plus a semitone (our
generator) gives us a fifth--which we may also view as the generator,
so that this is a circle-of-fifths system. To get a 7 we go down 2,
so that 7/4 is a tone below the octave, indicative of the fact that
this is a septimal comma system, where 9/8 and 8/7 are represented by
the same interval. Since 5/7 is represented by the half-octave (we
have 50/49 in our system of commas also), the 5 is also 2 tones down,
as the table tells us. The final column can also be obtained by
modulo-11 arithmetic, so that 1 = 35/2 mod 11, -2 = 51/2 mod 11, and
so forth.

The last row, for 11, has 11 represented by 2^(41/12); we need this
value if we are to use 2^(35/10) for the 10-et; otherwise, we must
use 2^(34/10) since in any event the two must sum to 76. One way
gives us 5, as above, and the other way gives us -6; but 76/2 = 5 =
-6 mod 11, so both are acceptable. Either way we get a span of
7 between the 11 and the 5 or 7, so a complexity of 2*7 = 14 for the
11-limit. Similarly, we have a complexity of 8 for the 9-limit (since
a 3 is 1, a 9 is 2) and 6 for the 7-limit. Because the 10 and 12 ets
are relatively reasonable in the 7-limit, gen(q) cannot be too large;
11 is not as well approximated and we find gen(11) to be larger.
Another way to look at the generator calculation is in fact in terms
of the relative error; consider the following table:

-23 180 157 2400
164 -263 -99 -4800
374 -88 286 -4800
-616 487 -129 12000

The first column is the error of the 12 et in relative cents, so that
the 3 is 23 rc flat, the 5 is 164 rc sharp, and so forth. The second
column is the error of the 10-et, and the third column is the error
of the 22-et, which in relative cents is simply the sum of 12 and 10;
to get the absolute error in cents you divide by 22. The fourth
column is 12 times the 10 error minus 10 times the 12 error; because
of round-off error the results we get by using the tabulated values
are slightly different; but they are close enough that dividing
through by 1200 clearly gives us 2, -4, -4, 10; and from these
|-4-10| = 14 as the 11-complexity, and so forth.

From our 10+12=22 starting point, we know that 10 and 12 note scales
will be sensible choices, which means 5 or 6 notes out of 11. We get
either (22223)x2 or (222221)x2 for our scales, and a good supply of
7-harmony. Of course we are not required to be as symmetrical as
possible, and Paul is quite fond of 6 fifths vs 4 fifths. Paul tells
us the decatonic 22-et is the first scale he constructed, and he
takes a sort of fatherly interest, I think. I'm fond of 7 fifths vs 5
fifths myself, and that was the outcome of my first efforts--
suffering under the delusion that the diatonic scale arose by
tempering the just diatonic via equal temperament, I attempted to
replicate the process--and succeeded. You can get these scales, among
other ways, by constructing scales analogous to the diatonic (for 10,
12, 15, 22, 31 etc notes) with superparticular ratios and all the
rest of it, and then introducing approximations--a similar but less
sophisticated process than tempering a block.

After 12+10 the next most obvious choice is 15+7; the 15-division is
not as precise as 12 in its tuning of triads but both the 15 and the
7 ets do reasonably well up to 11, so this should be an interesting
choice for exploring the 11-limit possibilities of the 22-et. From
15/7 = 2+1/7 we get numerators of 2 and 1, respectively, and locate
our generator by 2/15 < 3/22 < 1/7. Using either 15h7 - 7h15 or
mod 22 arithmetic, we find a 5-limit complexity of 6, 7-limit of 11,
9-limit of 12, 11-limit of 12 and even 13-limit of 12, though the
22-et is not especially good up to the 13-limit. From considering 15
and 7 together, we see that this system is associated to the commas
64/63 (septimal comma), 385/384 (Keenan's kleisma) as well as the
as-yet-unnamed intervals of 875/864, 55/54, 100/99, 121/120 and
176/175. The 7-note scale has pattern 3333334, 15 notes gives us the
pattern 212121212121211 and a good supply of 11-limit chords, and we
can use 8 notes and 33333331 if we like as well.

From 19/3 = 6+1/3 we get 6/19 < 7/22 < 1/3; since this generator is
about 1/pi, the numerologically inclined might consider precisely pi
generators to the octave; perhaps Lucy will try it if he ever gets
bored with Lucy-tuning. The generator is a flattened major third,
whether in the 22-et version or the slightly less flat 2^(1/pi)
version, and as we would expect from its genesis from 19 and 3, this
is a good system if we want to focus on the triad-generating powers
of the 22-et. It has a 5-limit complexity of 5, representing 3 by
five successive major thirds. Hence the fundamental comma for this
system is the small diesis of (1/3)(5/4)^5 = 3125/3072. While it is a
bit of a stretch to claim 19 notes out of 22 constitute a scale, we
could try instead a 7-note alternative to the diatonic scale, with
pattern 6161611, a 10-note scale with pattern 5115115111, or a
13-note scale with pattern 4111411141111. We can construct a 7-note
PB for the 7-note scale by using either the augmented second of 75/64
or the diminished third of 144/125 as our chroma, and the small
diesis as a comma. We get from 75/64 the rotation of the Pythagorean
scale by 120 degrees; that is, by sending 3-->16/5, and from 144/125
we get a few minor thirds as well as major thirds:

1--25/24--5/4--125/96--192/125--8/5--48/25--(2)

With the small diesis approximation, this gives us some triads, and
both scales are the same mode in 22-et: 1616161, what you might call
the moment-of-symmetry mode, symmetrical around the middle 6. However
the first mode given, 6161611, has both a major and a minor triad on
the first scale degree, and a major and minor triad on the third
scale degree, a major third above it, which might serve as this
scale's version of tonic and dominant.

Other possibilities are the 18+4 system, with generator
4/9 < 5/11 < 1/2, with a 7-limit complexity of 8 and scales of
patterns (41411)x2 and (3113111)x2, the 14+8 system, with generator
1/4 < 3/11 < 2/7 and a 5-complexity of 10, a 7-complexity of 10, a
9-complexity of 12 and an 11-complexity of 12, and scales of pattern
(3332)x2 and (1212122)x2, and the 17+5 system, which as we might
expect does well for 3-complexity, being in fact the Margo scales, of
22-et fifths. We have the Margo diatonic, 4414441, and the hyper-
Pythagorean temperament, 313113131311.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., genewardsmith@j... wrote:

[lots of stuff skipped]

Gene, you said 22 is well-known, but . . . very little has been
written about it compared with 19, 24, 31, and 53. 22 is not nearly
as well-known as it deserves to be -- the main obstacle having been
the 'comma problem' -- but this being the 21st century, it's time
make music outside the common-practice language, out where the 'comma
problem' can be either a non-issue or a desirable feature. BTW,
Blackwood's 22-tET piece is my least favorite of his microtonal
etudes.

> Paul tells
> us the decatonic 22-et is the first scale he constructed, and he
> takes a sort of fatherly interest, I think. I'm fond of 7 fifths vs
5
> fifths myself,

If I understand what that means, it's identical to the 12-tone
keyboard tuning I suggested in my paper -- put the 7 "fifths" on the
white keys, and the opposing 5 "fifths" on the black keys. If you
published this first, I certainly should give you credit for it. The
piece "Decatonic Swing" is performed with exactly this arrangement on
the keyboard, and (aside from mistakes), the same 12 notes on the
guitar.

> After 12+10 the next most obvious choice is 15+7; the 15-division
is
> not as precise as 12 in its tuning of triads but both the 15 and
the
> 7 ets do reasonably well up to 11, so this should be an interesting
> choice for exploring the 11-limit possibilities of the 22-et.

Cool -- I'll have to keep this in mind in my next musical endeavors:

> The 7-note scale has pattern 3333334,

That's the tuning of the white keys (starting with A; G is the tonic)
in "Glassic" and the 11-limit resources are exploited in said piece.
The tuning of the black keys is ingenious (if I may say so) because
it adds additional 11-limit resources as well as providing a 9-
note "Pythagorean" chain, while keeping the keyboard in order of
pitch. Can you figure out the tuning?

> 15 notes gives us the
> pattern 212121212121211

I'll have to work with this next. Thanks, Gene!

> It has a 5-limit complexity of 5, representing 3 by
> five successive major thirds.

I noticed this a long time ago when tuning my 22-tET guitar in major
thirds. Cool!

> mode in 22-et: 1616161, what you might call
> the moment-of-symmetry mode, symmetrical around the middle 6.
However
> the first mode given, 6161611, has both a major and a minor triad
on
> the first scale degree, and a major and minor triad on the third
> scale degree, a major third above it, which might serve as this
> scale's version of tonic and dominant.

I'll have to give this a try.

> We have the Margo diatonic, 4414441,

The 6th mode of this is what Margo calls "Paul's Aeolian tour de
force".

> and the hyper-
> Pythagorean temperament, 313113131311.

This is fun because you get some "just" major triads that look like
minor triads on the keyboard, and vice versa.

🔗genewardsmith@juno.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> If I understand what that means, it's identical to the 12-tone
> keyboard tuning I suggested in my paper -- put the 7 "fifths" on
the
> white keys, and the opposing 5 "fifths" on the black keys. If you
> published this first, I certainly should give you credit for it.

I wouldn't worry about it--unlike some of my other stuff, I didn't
even try to publish it, nor did I talk about it to an audience of
mathematicians, which I've also done with some other things. I think
the method might be worth writing up, however--it is essentially what
I was doing in the "interesting notation" posts on the math group,
where I point out how scale steps with square numerators break apart
into triangle numerators, and vice-verse:

(4/3)^2 9/8 = 2, 2+1 = 3, 4/3 = 6/5 10/9 square --> triangle triangle

(6/5)^2 9/8 (10/9)^2 = 2 2+1+=5 (pentatonic) 6/5 = 9/8 16/15
triangle --> square square, leading to:

(9/8)^3 (10/9)^2 (16/15)^2 = 2 3+2+2=7 (diatonic) 9/8 = 15/14 21/20

(10/9)^2 (15/14)^3 (16/15)^2 (21/20)^3 = 3 2+3+2+3=10
10/9 = 16/15 25/24 (an interesting alternative being
10/9 = 15/14 28/27, not part of the general pattern, which leads into
some worthwhile sidesteets)

(15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 = 2 3+4+3+2=12
15/14 = 25/24 36/35

(16/15)^4 (21/20)^3 (25/24)^5 (36/35)^3 = 2 4+3+5+3=15
16/15 = 28/27 36/35

(21/20)^3 (25/24)^5 (28/27)^4 (36/35)^7 = 2 3+5+4+7=19
21/20 = 36/35 49/48

(25/24)^5 (28/27)^4 (36/35)^10 (49/48)^3 = 2 5+4+10+3=22
25/24 = 45/44 55/54 (or 49/48 50/49, not part of this pattern)

(28/27)^4 (36/35)^10 (45/44)^5 (49/48)^3 (55/54)^5 = 2
4+10+5+3+5=27 28/27 = 49/48 64/63 (or 55/54 56/55)

(36/35)^10 (45/44)^5 (49/48)^7 (55/54)^5 (64/63)^4 = 2
10+5+7+5+4=31

At this point it seems prudent to stop; I've ignored
10/9 = 15/14 28/27, 28/27 = 55/54 56/55 and 25/24 = 49/48 50/49, and
am now faced with the fact that 36 is both triangular and square,
leading to 36/35 = 66/65 78/77 = 64/63 81/80. By constructing scales
each of which is contained in the scale of the previous stage, one
can get things analogous to JI diatonic, and by introducing
equivalences between scale steps, which makes their ratio into a
comma, one can get temperings.

> > The 7-note scale has pattern 3333334,

> That's the tuning of the white keys (starting with A; G is the
tonic)
> in "Glassic" and the 11-limit resources are exploited in said
piece.
> The tuning of the black keys is ingenious (if I may say so) because
> it adds additional 11-limit resources as well as providing a 9-
> note "Pythagorean" chain, while keeping the keyboard in order of
> pitch. Can you figure out the tuning?

You seem to be asking me to figure out the tuning of something
already in evidence--what should I figure out?

22

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.com>

🔗Gerald Eskelin <stg3music@earthlink.net>