Re: Schoenberg, serliasim, 12-edo

If you're seriously interested in non-12-edo serialism,
that does exist too. Probably the prime proponent of it
is Ben Johnston, who composes in extended-JI (up to
31-limit AFAIK).

Johnston's _6th Quartet_ is in 11-limit JI, and is a
serial piece which arranges the 12-tone row as two
hexachords, each of which is a Partchian 1-3-5-7-9-11
hexad, one otonal and the other utonal. I've written
a lot about it here before, at various different times
over the years. If you have the patience to search the
archives, you'll find it.

-monz
http://tonalsoft.com
Tonescape microtonal music theory

This reminds me of a recurring idea of mine, one which has not yet made it past the giving-up-after-thinking-about-it-for-thirty-seconds stage - has anyone tried to make just versions of Messiaen modes, specifically the second and third? (Which would require sets of multiple sets of pitches a major diesis (for the 2nd mode) or minor diesis (for the third mode) apart, multiple tones for scale degrees to represent the various 11s, 13s, 17s, 19s, etc.)

Chris

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--- In tuning@yahoogroups.com, Christopher John Smith
<christopherjohn_smith@y...> wrote:

> This reminds me of a recurring idea of mine, one which has not
>yet made it past the giving-up-after-thinking-about-it-for-thirty-
>seconds stage - has anyone tried to make just versions of Messiaen
>modes, specifically the second and third?

Yup -- this is right up my alley, so thanks for asking! I'm pretty
sure Gene for example has posted some on the tuning-math list, but we
can procede without them . . .

>(Which would require sets
>of multiple sets of pitches a major diesis
>(for the 2nd mode) or
>minor diesis (for the third mode) apart,
>multiple tones for scale >degrees to represent the various 11s, 13s,
>17s, 19s, etc.)

It would? I guess that depends on your view and usage of just
intonation. Here's something explaining one possible view which I'll
basically use here:

http://lumma.org/tuning/erlich/erlich-tFoT.pdf

This is a preview discusses the diatonic scale and the decatonic (you
can think of the latter as a 7-limit version of Messaien's 10-note
MLT). The complete work will also cover the scales you're asking
about as well as some impossible in 12-equal, such as Porcupine.

You might want to read this too:

http://sonic-arts.org/td/erlich/intropblock1.htm

Are you amenable to these ways of thinking?

Not knowing quite what you mean by "the various 11s, 13s, 17s, 19s,
etc.", I'll give you some 5- and 7-limit realizations:

The diminished/octatonic scale (Messaien 2nd) in the 5-limit can be
considered a periodicity block with commatic unison vector 648:625
(like all scales in the 5-limit diminished family) and chromatic
unison vector 16:15, such as the Fokker block:

1/1
25/24
6/5
5/4
25/18
3/2
5/3
9/5
(2/1)

Since you are free to transpose any of the notes by the commatic
unison vector, we can allow for more consonant chords (4 major and 4
minor triads instead of 3 of each) by providing an alternate pitch
for each of two scale degrees by transposing by 648:625:

1/1
25/24 * 648/625 = 27/25
6/5
5/4
25/18 * 648/625 = 36/25
3/2
5/3
9/5
(2/1)

This allows the just major triads

1/1-5/4-3/2
6/5-3/2-9/5
36/25-9/5-27/25
5/3-25/24-5/4

and the just minor triads

1/1-6/5-3/2
6/5-36/25-9/5
25/18-5/3-25/24
5/3-1/1-5/4

The diminished/octatonic scale in the 7-limit can be considered a
periodicity block with commatic unison vectors 36:35 and 50:49 (like
all scales in the 7-limit diminished family) and chromatic unison
vector 16:15, such as the Fokker block:

1/1
15/14
6/5
5/4
7/5
3/2
12/7
7/4
(2/1)

Since you are free to transpose any of the notes by either commatic
unison vector (or both), we can allow for more consonant chords (4
major and 4 minor tetrads instead of 3 of each) by providing 1
alternate pitch for each of four and 2 alternate pitches for each of
two (maybe you can do it with less? i'll think about it) scale
degrees by transposing by 36:35 or 50:49, for example like this:

1/1
15/14 * 49/50 = 21/20
6/5 * 35/36 = 7/6
5/4 * 36/35 = 9/7
7/5 * 50/49 = 10/7
3/2
12/7 * 35/36 = 5/3; 12/7 * 49/50 = 42/25
7/4 * 36/35 = 9/5; 7/4 * 50/49 = 25/14
(2/1)

This allows the just major tetrads

1/1-5/4-3/2-7/4
6/5-3/2-9/5-21/20
10/7-25/14-15/14-5/4
12/7-15/14-9/7-3/2

and the just minor tetrads

1/1-6/5-3/2-12/7
7/6-7/5-7/4-1/1
7/5-42/25-21/20-6/5
5/3-1/1-5/4-10/7

The Tscherepnin scale (Messaien 3rd) in the 5-limit can be considered
a periodicity block with commatic unison vector 128:125 (like all
scales in the 5-limit augmented family) and chromatic unison vector
27:25, such as the Fokker block:

1/1
25/24
6/5
5/4
4/3
3/2
8/5
5/3
48/25

Since you are free to transpose any of the notes by the commatic
unison vector, we can allow for more consonant chords (6 major and 6
minor triads instead of 4 of each) by providing an alternate pitch
for each of three scale degrees by transposing by 128:125:

1/1
25/24
6/5
5/4
4/3 * 125/128 = 125/96
3/2
8/5 * 125/128 = 25/16
5/3
48/25 * 125/128 = 15/8

This allows the just major triads

1/1-5/4-3/2
25/24-125/96-25/16
5/4-25/16-15/8
4/3-5/3-1/1
8/5-1/1-6/5
5/3-25/24-5/4

and the just minor triads

1/1-6/5-3/2
25/24-5/4-25/16
5/4-3/2-15/8
4/3-8/5-1/1
8/5-48/25-6/5
5/3-1/1-5/4

The Tcherepnin scale in the 7-limit can be considered a periodicity
block with commatic unison vectors 35:36 and 128:125 (like all scales
in the 7-limit augmented family) and chromatic unison vector 21:20,
such as the Fokker block:

1/1
35/32
6/5
5/4
7/5
3/2
25/16
7/4
15/8

Since you are free to transpose any of the notes by either commatic
unison vector (or both), we can allow for more consonant chords (3
major and 3 minor tetrads instead of 2 major and 0 minor) by
providing an alternate pitch for each of five scale degrees by
transposing by 35:36 and/or 128:125, for example like this:

1/1
35/32 * 128/125 = 28/25
6/5 * 35/36 = 7/6
5/4
7/5
3/2 * 35/36 = 35/24
25/16 * 128/125 = 8/5
7/4
15/8 * 128/125 * 35/36 = 28/15

This allows the just major tetrads

1/1-5/4-3/2-7/4
5/4-25/16-15/8-35/32
8/5-1/1-6/5-7/5

and the just minor tetrads

7/6-7/5-7/4-1/1
35/24-7/4-35/32-5/4
28/15-28/25-7/5-8/5

What do you think?

Questions/comments/corrections?