Just "pelog-style pentatonic"

Hello, everyone, and in response to a recent and most ingenious pelog
tuning based on two 9-tET scales at a fifth apart, I'd like to share a
kind of tuning that occurred to me after reading that proposal, and
which I then found was almost identical to one of Lou Harrison's
tunings included in the Scala archive.

My "just pelog-style Phyrgian pentatonic" is like this:

E F G B C E
1:1 28:27 7:6 3:2 14:9 2:1
0.0 62.96 266.87 701.96 764.92 1200
28:27 9:8 9:7 28:27 9:7
62.96 203.91 435.08 62.96 435.08

While I really like the just version, note that there's an excellent
approximation of this available in 36-tET, for example:

E F G B C E
0.0 66-2/3 266-2/3 700.00 766-2/3 1200
66-2/3 200.00 433-1/3 66-2/3 433-1/3

Harrison's version, as mentioned, is almost identical to my just
tuning:

E F A B C E
1:1 28:27 4:3 3:2 14:9 2:1
0.0 62.96 266.87 701.96 764.92 1200
28:27 9:7 9:8 28:27 9:7
62.96 203.91 203.91 62.96 435.08

A feature of a "pelog-style" tuning is the contrast between the small
"semitone" and large "major third" steps, and tuning these steps at
around 27:28 and 7:9 further accentuates the effect.

In 22-tET, by the way, versions of these tunings are available with
ratios between steps of 1:4:8 for approximate ratios of 27:28, 8:9,
and 7:9; the near-6:7 ratio for my version with E-G , or Harrison's
with A-C, is 5 tuning steps in this temperament.

There are all kind of neat possibilities for this beautiful type of
scale. For example, in my "Phrygian" E-F-G-B-C-E, there's a
Gothic-style European cadence G-B-E to F-C-F on the second step, and
also cadences like this, with C4 showing middle C:

C4 E4
G3 B3
F3 E3

In Harrison's tuning, with its own delightful patterns, the major third
F-A invites expansion to E-B, for example

C4 E4
A3 B3
F3 E3

and also gets me into some fast melodic patterns, for example over a
drone, that remind me of gamelan.

Anyway, while my "just pelog-style pentatonic" turns out to be mostly
a "rediscovery" of Harrison's, this tuning in either version is a
treat I really enjoy, and it's a convenient subset of a range of
tunings with pure or approximate ratios of 2-3-7-9.

Most appreciatively, with peace, love, and happy holidays,

Margo

Margo Schulter wrote:
My "just pelog-style Phyrgian pentatonic" is like this:

E F G B C E
1:1 28:27 7:6 3:2 14:9 2:1
and
Harrison's version, as mentioned, is almost identical to my just
tuning:

E F A B C E
1:1 28:27 4:3 3:2 14:9 2:1

I just want to mention to Gene, it's there a nice case of pentatonic gammier
whose odd generator is
<7 9 21 81>
It is well approximated by 17, 36, 41, 58, 77, 94, 113... The tempered generator
are, for instance, with 17-tET and 36-tET:
[0 3 4 10]

[0 6 8 21]
and I give here the corresponding diamond for 17-tET:
.0 .3 .4 10
14 .0 .1 .7
13 16 .0 .6
.7 10 11 .0
You could read easily the treillis of modes, seeing that as a torus.

Pierre

http://www.aei.ca/~plamothe/pix/schul.gif

Margo,

You qualifie your scale as "pelog-style pentatonic". Maybe it's the case with Harrison.
I would like ask to you if it would not be appropriate to qualifie it as "japanese-style".
Maybe the distance of 1/4 tone relatively to the japanese standard is too high, but
I give here some arguments you could examine.

I mentionned for Gene in a precedent message that your two pentatonic scales are
modes of the maximal gammier generated by the odds <7 9 21 81>.

I forgot to mention that the minimal gammier in which these modes are sui generis is
the gammier <3 7 9 27>.

It's that simpler generator which gives the diamond 17-tET shown there and copied
here:
.0 .3 .4 10
14 .0 .1 .7
13 16 .0 .6
.7 10 11 .0
Now my point.

Here are three "japanese-style" scales in regard of your two scales, with the
minimal generator and some treillis exhibiting all possible modes.
1 16/15 4/3 3/2 8/5 2 <1 3 9 15>
http://www.aei.ca/~plamothe/gam/tg8-japonais.gif
1 19/18 4/3 3/2 19/12 2 <1 3 9 19>
http://www.aei.ca/~plamothe/gam/tg43-japonais.gif
1 18/17 4/3 3/2 27/17 2 <3 9 17 27>
1 28/27 4/3 3/2 14/9 2 <3 7 9 27>
1 28/27 7/6 3/2 14/9 2 <3 7 9 27>
http://www.aei.ca/~plamothe/pix/schul2.gif

What merit consideration is the global transformation of all these structures one
to another by a simple operator. For instance the operator transforming the third
5/4 in third 9/7:
T : 5 --> 9/7
applying to <1 3 9 15> gives <3 7 9 27>.
T<1 3 9 15> = <1 3 9 27/7> = <27/27 27/9 27/3 27/7>
What is equivalent, as generator by duality, to <3 7 9 27>

What means that comparing the scale of a gammier to a scale of another gammier
we have equality of intervals or distance of 36/35.

Another example.
T : 17 --> 7

T<3 9 17 27> = <3 9 7 27> = <3 7 9 27>
Sure a similar transformation T: 5 --> 7 acting on chinese <1 3 5 9> gives the
slendro <1 3 7 9> and the style change. But if you refer to my text
http://www.aei.ca/~plamothe/gammes-gsp.htm
I had found 4 pentatonic styles being approximately:
Chinese : 2 2 3 2 3 - (in 12-tET)
Slendro : 1 1 1 1 1 - (in 5-tET)
Pelog : 2 5 3 2 5 - (in 17-tET)
Japanese : 1 4 2 1 4 - (in 12-tET)
It seems that the approximative Harrison 1 6 3 1 6 - (in 17-tET) is nearest from
Japanese than from Pelog. Or, would have to correct my styles?

Best regards,

Pierre Lamothe

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> I had found 4 pentatonic styles being approximately:
> Chinese : 2 2 3 2 3 - (in 12-tET)

How about Thai: 1 1 2 1 2 - (in 7-tET, where 25:24 vanishes).

> Slendro : 1 1 1 1 1 - (in 5-tET)
> Pelog : 2 5 3 2 5 - (in 17-tET)

Pelog and some similar East African scales are usually given as

1 3 1 1 3 (in 9-tET, where 135:128 vanishes),
or 2 5 2 2 5 (in 16-tET, where 135:128 vanishes),
or 3 7 3 3 7 (in 23-tET, where 135:128 vanishes).

> Japanese : 1 4 2 1 4 - (in 12-tET)

I'd call this a planar tuning of a PB, while all of the above (save
17-tET pelog) are linear tunings of PBs.

> It seems that the approximative Harrison 1 6 3 1 6 - (in 17-tET) is
nearest from
> Japanese than from Pelog. Or, would have to correct my styles?

Clearly there's a lot of room for interpretation! But, I haven't seen
much evidence for intervals less than 100 cents in real Gamelan
tunings.

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> I had found 4 pentatonic styles being approximately:

Note that, in the rotations (= modes) given, these all have two
identical tetrachords (actually "trichords"), either a 3:2 or a 4:3
apart:

> Chinese : 2 2 3 2 3 - (in 12-tET)

2 + 2 3 + 2 3

Thai: 1 + 1 2 + 1 2 (in 7-tET)

> Slendro : 1 1 1 1 1 - (in 5-tET)

1 + 1 1 + 1 1 or
1 1 + 1 + 1 1

> Pelog : 2 5 3 2 5 - (in 17-tET)

2 5 + 3 + 2 5 (though this doesn't look much like a Gamelan tuning to
me -- I'd say 2 5 + 2 + 2 5 in 16-tET)

> Japanese : 1 4 2 1 4 - (in 12-tET)

1 4 + 2 + 1 4

Lou Harrison's: ~ 1 6 + 3 + 1 6 in 17-tET

Tetrachordality is an important property, not to be taken lightly.