Yahoo Tuning Groups Ultimate Backup tuning Tetrachordal explorations on the 22 tet guitar

This is relevant to both groups I hope - blast away if I've overstepped
the mark!

I've been working through John Chalmer's excellent and comprehensive
"Divisions of the Tetrachord" which I would thoroughly recommend to
beginners and active composers alike. I've always had respect for the
great theorists and musicians of past and there is great beauty in the
simplicity of some of these historical tetrachordal structures.

The tetrachord, for those who haven't come across the term, is the span
between the tonic, say, C or 1/1 and the perfect (or tempered) fourth or
4/3. Then there is a step, 9/8, to the fifth , or G, 3/2. Then comes the
second tetrachord which is often the same as the first. This completes
the octave. This is a very simple explanation but suffices to show the
basic structure.

As my particular interest at the moment is in writing for acoustic
instruments and voices I looked at what I could do with the resources of
a 22 tone equal temperament (22-tet) guitar in relation to approximating
Just tetrachords. This has proved to be beneficial first as a
compositional and improvisational exercise per se and secondly has
whetted my appetite for building simple wood, metal or string
instruments that could be tuned to the Just versions of these scales.

I'd like to look at two structures. The step size of 22-tet is 54.55
cents. So the easiest way to find tetrachords that can be approximated
by 22 tet is to browse through the lists given in "Divisions" and to
pick out those with cents between steps as close as possible to 55 cents
or multiples thereof.

The first that popped out was Didymos' Enharmonic, aka Archytas'
Enharmonic, ratios 32/31, 31/30 and 5/4. The enharmonic is generally
characterised by two small steps, quartertones, and a major third (or
5/4) to the tetrachord. You don't get enharmonic structures in 12 tet
which has been our loss in the West for many years. Cents between steps
are as follows:

Just 55 57 386
22 tet (rounded up) 55 55 382

Pretty close, given that the fourth in 22 tet is close to the limits of
acceptability. This gives us the very exotic scale steps in 22 tet of 0
1 2 9 13 14 15 22. And it sounds beautiful against the open drone on E.
Given that the standard tuning has a bias towards adjacent strings in
fifths/fourths finding your way around the fingering of this scale is
straightforward. I'll put up some patterns soon. If I might digress, the
tuning of any fretted instrument in fifths/fourths lends well to
figuring out tetrachord fingering. Take DAGDAD (high to low) on the
guitar. Lots of Ds and As so the first four notes on the D, the 1/1,
will be at exactly the same fretting positions as the four notes from A,
the 3/2, to the octave D. And you have a 4/3 open G available. I'm sure
this has all been said before but I'm very excited about it all the
same, so there :-)

The second one to catch my eyes and ears was a tetrachord attributed to
the great Arabic theorist Al Farabi. This is referred to as a chromatic
tetrachord, which in 12 tetspeak has two semitones and a minor third of
some sort. The ratios are 16/15, 15/14 and 7/6, all superparticular
(numerator one higher than denominator) like the previous. For a cents
between step sizes comparison:-

Just 112 119 267
22 tet 109 109 273

Again fairly good with the maximum deviation at 10 cents, though
psychoacoustically I think these deviations are less important than the
perception of the tetrachords as a whole with a distinctive mood. I'd be
interested to hear of any research in this area. The steps in 22 tet are
0 2 4 9 13 15 17 22.

Several compositional methods suggest themselves. One is a form of modal
modulation, moving between a scale that has, say, the root on E to one
with the root on A . Additionally there are four modes of each
tetrachord to explore. I am attracted to the idea of marking sections of
a piece with a change from scale one to scale two. The changeover can be
very intriguing. Morphing into the Al Farabi scale can feel initially
like a move to a major tonality despite the minor third. And of course
there is the option of combining the two different tetrachords into a
hybrid scale. At the moment I'm improvising before settling on some
compositional ideas and I hope to be able to share the fruits of these
noodlings with you all later.

Still to come. Avicenna's, Ptolemy's Equable and Xenakis' Soft
Chromatic.

Happy Hunting.

🔗Graham Breed <graham@microtonal.co.uk>

Alison wrote:

> The first that popped out was Didymos' Enharmonic, aka Archytas'
> Enharmonic, ratios 32/31, 31/30 and 5/4. The enharmonic is
generally
> characterised by two small steps, quartertones, and a major third
(or
> 5/4) to the tetrachord. You don't get enharmonic structures in 12
tet
> which has been our loss in the West for many years. Cents between
steps
> are as follows:
>
> Just 55 57 386
> 22 tet (rounded up) 55 55 382
>
> Pretty close, given that the fourth in 22 tet is close to the
limits of
> acceptability. This gives us the very exotic scale steps in 22 tet
of 0
> 1 2 9 13 14 15 22. And it sounds beautiful against the open drone
on E.

From your definition of "enharmonic" couldn't you have gone straight
to this scale?

> The second one to catch my eyes and ears was a tetrachord
attributed to
> the great Arabic theorist Al Farabi. This is referred to as a
chromatic
> tetrachord, which in 12 tetspeak has two semitones and a minor
third of
> some sort. The ratios are 16/15, 15/14 and 7/6, all superparticular
> (numerator one higher than denominator) like the previous. For a
cents
> between step sizes comparison:-
>
> Just 112 119 267
> 22 tet 109 109 273

I thought I'd point out a Miracle connection. In decimal notation,
this would be 0 1 2 4^ 6 7 8 0. As you always need to raise the 4 in
a tetrachordal scale, this is the simplest decimal chromatic. You
lose that in 22 because it's not part of the Miracle family, but it
may be useful to think of if you're going to move to JI.

> Again fairly good with the maximum deviation at 10 cents, though
> psychoacoustically I think these deviations are less important than
the
> perception of the tetrachords as a whole with a distinctive mood.
I'd be
> interested to hear of any research in this area. The steps in 22
tet are
> 0 2 4 9 13 15 17 22.

That looks like a subset of one of Paul Erlich's decatonics.

Graham

🔗paul@stretch-music.com

🔗Alison Monteith <alison.monteith3@which.net>

Graham Breed wrote:

> Alison wrote:
>
> > The first that popped out was Didymos' Enharmonic, aka Archytas'
> > Enharmonic, ratios 32/31, 31/30 and 5/4. The enharmonic is
> generally
> > characterised by two small steps, quartertones, and a major third
> (or
> > 5/4) to the tetrachord. You don't get enharmonic structures in 12
> tet
> > which has been our loss in the West for many years. Cents between
> steps
> > are as follows:
> >
> > Just 55 57 386
> > 22 tet (rounded up) 55 55 382
> >
> > Pretty close, given that the fourth in 22 tet is close to the
> limits of
> > acceptability. This gives us the very exotic scale steps in 22 tet
> of 0
> > 1 2 9 13 14 15 22. And it sounds beautiful against the open drone
> on E.
>
> >From your definition of "enharmonic" couldn't you have gone straight
> to this scale?

I suppose I could.

> > The second one to catch my eyes and ears was a tetrachord
> attributed to
> > the great Arabic theorist Al Farabi. This is referred to as a
> chromatic
> > tetrachord, which in 12 tetspeak has two semitones and a minor
> third of
> > some sort. The ratios are 16/15, 15/14 and 7/6, all superparticular
> > (numerator one higher than denominator) like the previous. For a
> cents
> > between step sizes comparison:-
> >
> > Just 112 119 267
> > 22 tet 109 109 273
>
> I thought I'd point out a Miracle connection. In decimal notation,
> this would be 0 1 2 4^ 6 7 8 0. As you always need to raise the 4 in
> a tetrachordal scale, this is the simplest decimal chromatic. You
> lose that in 22 because it's not part of the Miracle family, but it
> may be useful to think of if you're going to move to JI.

Thanks. I still haven't studied decimal notation yet. Soon, though.

> > Again fairly good with the maximum deviation at 10 cents, though
> > psychoacoustically I think these deviations are less important than
> the
> > perception of the tetrachords as a whole with a distinctive mood.
> I'd be
> > interested to hear of any research in this area. The steps in 22
> tet are
> > 0 2 4 9 13 15 17 22.
>
> That looks like a subset of one of Paul Erlich's decatonics.

> Graham

Yes, another compositional possibility, ie thinning out from the decatonic to the tetrachordal or
vice versa. Thanks for the comments.

Best Wishes.

🔗paul@stretch-music.com

🔗mschulter <MSCHULTER@VALUE.NET>

Hello, there, Alison Monteith, and I thought that I might mention
another kind of tetrachord for which 22-tET has a very interesting
approximation: what I'd call the "Archytas/al-Farabi diatonic" based
on ratios of 7.

Here's a kind of sketch I tried to write today: I'm not sure how
understandable or readable it is, and would warmly invite comments
both from you and from others, for example Paul Erlich. Needless to
say, I'm viewing 22-tET here from my own angle, only one way of
approaching a most versatile scale.

Also, someone like John Chalmers (or Joe Monzo) is a much better
source on Greek theory than I am, so my remarks should be taken with
this caution.

- - -

This kind of tetrachord, described (among others) by the Greek
theorist Archytas, and later made the basis of a scale by the medieval
theorist al-Farabi, features whole-tones at 9:8 and 8:7, and semitones
at 28:27.

Here's the tetrachord with pure 7-based ratios, if we place the 9:8
whole-tone first and the 28:27 semitone last, with the intervals above
the lowest note (here G in a G-C tetrachord) and the steps between
notes shown as ratios and in rounded cents:

G A B C
1:1 9:8 9:7 4:3
0 204 435 498
9:8 8:7 28:27
204 231 63

It's easy to find this type of tetrachord in 22-tET -- although with
an important modification as regards the two whole-tones -- if we
arrange the scale in what Paul Erlich terms a "Pythagorean pattern,"
my standard pattern for neo-Gothic music, by the way.

To get this pattern, we simply take four 22-tET steps as a whole-tone,
and one step as a semitone -- giving us step sizes of around 218 cents
and 55 cents (or more precisely about 218.18 cents and 54.55 cents).

If we start from C, for example, we get a "Pythagorean" diatonic scale
like this:

C D E F G A B C
0 4 8 9 13 17 21 22
4 4 1 4 4 4 1

In this 22-tET tuning, our G-C tetrachord looks like this, with
intervals given in 22-tET steps and rounded cents:

G A B C
0 4 8 9
0 218 436 491
4 4 1
218 218 55

Here the interval between G and B, 8 steps, is about 436.46 cents,
very close to the size of the 9:7 ratio between these steps in the
original Archytas/al-Farabi tetrachord. The fourth is somewhat
narrower, tempered from a pure 4:3 (about 498.04 cents) to about 7.14
cents narrower, or 490.90 cents.

An important difference is that in the original tetrachord, our 9:7
interval G-B was formed from two unequal whole-tones of 9:8 and 8:7.
In 22-tET, however, our major third of almost the same size is formed
from two _equal_ whole-tones at 4 steps or 218 cents, very close to
the average or "mean" between 9:8 (204 cents) and 8:7 (231 cents).
Actually the 22-tET whole-tone is very slightly larger than this mean,
so that the 9:7 major third is about 1.28 cents wider than a pure 9:7
at around 435.08 cents.

Also, the 22-tET diatonic semitone, here B-C, is a bit smaller than in
the Archytas/al-Farabi tetrachord: it has a size of about 54.55 cents
in comparison to the 28:27 semitone at around 62.96 cents.

In this respect, 22-tET might be said further to accentuate one of the
characteristics of this tetrachord in a pleasing way: the contrast
between the wide whole-tones and narrow semitones.

In fact, our usual semitone in the 22-tET version of this tetrachord
is literally a "quartertone" -- exactly 1/4 the size of the
whole-tone. Yet to my ears, and evidently to others, the effect is a
very nice diatonic scale, and one with interesting proportions.

Another point this tetrachord illustrates: the 22-tET interval of one
step, or 55 cents, can serve either as a diatonic semitone, or as some
"enharmonic" kind of interval in a scale leaning toward larger
semitones (e.g. a semitone of 2 steps, or around 109.09 cents, one of
the basic sizes in Paul's decatonic scales).

Most appreciatively,

Margo Schulter
mschulter@value.net

Tetrachordal explorations on the 22 tet guitar

🔗Alison Monteith <alison.monteith3@which.net>

🔗Graham Breed <graham@microtonal.co.uk>

🔗paul@stretch-music.com

🔗paul@stretch-music.com

🔗Alison Monteith <alison.monteith3@which.net>

🔗paul@stretch-music.com

🔗mschulter <MSCHULTER@VALUE.NET>

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

🔗paul@stretch-music.com