Inversional Matrii

The following piece of theory goes back to one of my first
compositions, so i'll give you the full story here.

In 1995, i was reading a book about my then-favourite composer (and
still one of my favourites) Alban Berg, including some theoretical
notes for the Concerto for Piano, Violin and Winds. As a theme for
this work, he _extracted the musical note-names_ from the names of
himself, Anton Webern and Arnold Schönberg. Interpreting H as B, B as
B-flat and S as E-flat,ingeniously.

What i found most interesting, however, was that Arnold Schönberg's
theme-which came out as A D Eb C B Bb E G, contained no repeated
notes. Which was an amazing coincidence for the grand father of twelve
tone theory.

So this name-theme could itself be turned into a true twelve tone
series with the addition of four other notes. Which would be C#, F, F#
and Ab. I examined this for a while, concluding that there wasn't
really any permutation of those four notes that made more sense than
any other to add to the original eight notes, which made it rather
meaningless to try. Until i considered Schönberg's own favored method
of the inversional matrix.

In a inversional matrix, the basic 12-tone row is printed out
horisontally in twelve lines, transposed to each note of it's
_inversion_ . This gives an easy overview over both inverted and
original forms for the composer, allowing the viewer to find whatever
chords and interval series he's looking for- as long as they're
present in any form of the series.

So i decided to apply this process to the original eight note row of
Schönberg's name, giving this matrix:

A D Eb C B Bb E G
E A Bb G F# F B D
Eb Ab A F# F E Bb C#
F# B C A Ab G C# E
G C C# Bb A Ab D F
Ab C# D B Bb A Eb F#
D G Ab F E Eb A C
B E F D C# C F# A

This eventually resulted in a finished work for cello and piano, a
clever homage to Schönberg in multiple ways. Curiously enough, i ended
up using a lot of the "diagonal" chords, which wouldn't be considered
part of any proper serial treatment.

So what is the relevance to microtonality here? Well, the inversional
matrix can be applied to ANY set of intervals, not only tone rows,
they don't even have to be in a equal temperament. In fact... if 8
notes from the overtone series is used, the result is a proper
Tonality Diamond!

1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8
16/9 1/1 10/9 11/9 4/3 13/9 14/9 5/3
8/5 9/5 1/1 11/10 6/5 13/10 7/5 3/2
16/11 18/11 20/11 1/1 12/11 13/11 14/11 15/11
4/3 3/2 5/3 11/6 1/1 13/12 7/6 5/4
16/13 18/13 20/13 22/13 24/13 1/1 14/13 15/13
8/7 9/7 10/7 11/7 12/7 13/7 1/1 15/14
16/15 6/5 3/2 22/15 5/4 26/15 15/14 1/1

But when the overtones (which are still just an example of what
intervals could be used) are put in a different order, it gets much
more interesting:

1/1 11/8 5/4 7/4 9/8 13/8 3/2 15/8
16/11 1/1 20/11 14/11 18/11 13/11 24/11 15/11
8/5 11/10 1/1 7/5 9/5 13/10 6/5 3/2
8/7 11/7 10/7 1/1 9/7 13/7 12/7 15/7
16/9 11/9 10/9 14/9 1/1 13/9 4/3 5/3
16/13 22/13 20/13 14/13 18/13 1/1 24/13 15/13
4/3 11/6 5/3 7/6 3/2 13/12 1/1 5/4
16/15 22/15 4/3 28/15 6/5 26/15 8/5 1/1

I hope i'm making clear the compositional possibilities available here.

/Mats Öljare
Eskilstuna, Sweden
http://oljare.iuma.com

--- In tuning@yahoogroups.com, Mats Öljare <oljare@h...> wrote:
>
> The following piece of theory goes back to one of my first
> compositions, so i'll give you the full story here.
>
> In 1995, i was reading a book about my then-favourite
composer (and
> still one of my favourites) Alban Berg, including some
theoretical
> notes for the Concerto for Piano, Violin and Winds. As a theme
for
> this work, he _extracted the musical note-names_ from the
names of
> himself, Anton Webern and Arnold Schönberg. Interpreting H
as B, B as
> B-flat and S as E-flat,ingeniously.
>
> What i found most interesting, however, was that Arnold
Schönberg's
> theme-which came out as A D Eb C B Bb E G, contained no
repeated
> notes. Which was an amazing coincidence for the grand father
of twelve
> tone theory.
>
> So this name-theme could itself be turned into a true twelve
tone
> series with the addition of four other notes. Which would be
C#, F, F#
> and Ab. I examined this for a while, concluding that there
wasn't
> really any permutation of those four notes that made more
sense than
> any other to add to the original eight notes, which made it
rather
> meaningless to try. Until i considered Schönberg's own
favored method
> of the inversional matrix.
>
> In a inversional matrix, the basic 12-tone row is printed out
> horisontally in twelve lines, transposed to each note of it's
> _inversion_ . This gives an easy overview over both inverted
and
> original forms for the composer, allowing the viewer to find
whatever
> chords and interval series he's looking for- as long as they're
> present in any form of the series.
>
> So i decided to apply this process to the original eight note row
of
> Schönberg's name, giving this matrix:
>
> A D Eb C B Bb E G
> E A Bb G F# F B D
> Eb Ab A F# F E Bb C#
> F# B C A Ab G C# E
> G C C# Bb A Ab D F
> Ab C# D B Bb A Eb F#
> D G Ab F E Eb A C
> B E F D C# C F# A
>
> This eventually resulted in a finished work for cello and piano,
a
> clever homage to Schönberg in multiple ways. Curiously
enough, i ended
> up using a lot of the "diagonal" chords, which wouldn't be
considered
> part of any proper serial treatment.
>
> So what is the relevance to microtonality here? Well, the
inversional
> matrix can be applied to ANY set of intervals, not only tone
rows,
> they don't even have to be in a equal temperament. In fact... if 8
> notes from the overtone series is used, the result is a proper
> Tonality Diamond!
>
> 1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8
> 16/9 1/1 10/9 11/9 4/3 13/9 14/9 5/3
> 8/5 9/5 1/1 11/10 6/5 13/10 7/5 3/2
> 16/11 18/11 20/11 1/1 12/11 13/11 14/11 15/11
> 4/3 3/2 5/3 11/6 1/1 13/12 7/6 5/4
> 16/13 18/13 20/13 22/13 24/13 1/1 14/13 15/13
> 8/7 9/7 10/7 11/7 12/7 13/7 1/1 15/14
> 16/15 6/5 3/2 22/15 5/4 26/15 15/14 1/1
>
> But when the overtones (which are still just an example of what
> intervals could be used) are put in a different order, it gets
much
> more interesting:
>
> 1/1 11/8 5/4 7/4 9/8 13/8 3/2 15/8
> 16/11 1/1 20/11 14/11 18/11 13/11 24/11 15/11
> 8/5 11/10 1/1 7/5 9/5 13/10 6/5 3/2
> 8/7 11/7 10/7 1/1 9/7 13/7 12/7 15/7
> 16/9 11/9 10/9 14/9 1/1 13/9 4/3 5/3
> 16/13 22/13 20/13 14/13 18/13 1/1 24/13 15/13
> 4/3 11/6 5/3 7/6 3/2 13/12 1/1 5/4
> 16/15 22/15 4/3 28/15 6/5 26/15 8/5 1/1
>
> I hope i'm making clear the compositional possibilities
available here.
>
> /Mats Öljare
> Eskilstuna, Sweden
> http://oljare.iuma.com

yup, this sounds like the kind of stuff daniel wolf in particular
might be interested in -- would you join/post it to the specmus
list?

It occurs to me now-is it possible that Partch actually got the idea
of the tonality diamond from reading about Schönberg's matrix? /Ö

--- In tuning@yahoogroups.com, Mats Öljare <oljare@h...> wrote:
>
> It occurs to me now-is it possible that Partch actually got the
idea
> of the tonality diamond from reading about Schönberg's
matrix? /Ö

daniel wolf will have an opinion on this subject . . .

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Sunday, March 30, 2003 9:55 PM
Subject: [tuning] Re: Inversional Matrii

--- In tuning@yahoogroups.com, Mats �ljare <oljare@h...> wrote:
>
> It occurs to me now-is it possible that Partch actually got the
idea
> of the tonality diamond from reading about Sch�nberg's
matrix? /�

daniel wolf will have an opinion on this subject . . .

my guess: it's *possible*, since Partch showed by
what he wrote in _Genesis of a Music_ that he had
at least a little familiarity with Schoenberg's
theoretical writings.

but i'm absolutely convinced that Partch's tonality
diamond concept was derived from Max Meyer. Partch
had obviously read Meyer's book _The Musician's Arithmetic_
from cover to cover -- he cites it frequently in his
own book -- and in Meyer's book you will actually
find the diagonal diagram which is but one small
step from the Tonality Diamond, the only difference
being that Meyer used integers which relate all
pitches to the unity, and Partch used ratios (fractions)
instead.

PS -- Mats, if you're seriously interested in Schoenberg's
theory and music, especially c. 1905-1912 and especially
the connections it might have with microtonality, i've done
a ton of research on that. the plan was to write a whole
book about it, but i'm still in the very eary preliminary
stages of that ... mainly because my Tuning Dictionary
and software project are eating up all my free time these
days. the meatiest webpage i have available about it
so far is this one:

http://sonic-arts.org/monzo/schoenberg/harm/1911-1922.htm

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, March 30, 2003 9:55 PM
> Subject: [tuning] Re: Inversional Matrii
>
>
> --- In tuning@yahoogroups.com, Mats Öljare <oljare@h...>
wrote:
> >
> > It occurs to me now-is it possible that Partch actually got the
> idea
> > of the tonality diamond from reading about Schönberg's
> matrix? /Ö
>
> daniel wolf will have an opinion on this subject . . .
>
>
> my guess: it's *possible*, since Partch showed by
> what he wrote in _Genesis of a Music_ that he had
> at least a little familiarity with Schoenberg's
> theoretical writings.
>
> but i'm absolutely convinced that Partch's tonality
> diamond concept was derived from Max Meyer. Partch
> had obviously read Meyer's book _The Musician's Arithmetic_
> from cover to cover -- he cites it frequently in his
> own book -- and in Meyer's book you will actually
> find the diagonal diagram which is but one small
> step from the Tonality Diamond, the only difference
> being that Meyer used integers which relate all
> pitches to the unity, and Partch used ratios (fractions)
> instead.

a larger difference is that meyer never thought to reinterpret the
*interval matrix* of the basic consonant chord as a *pitch matrix*
with far more pitches . . . the diamond as a _scale_ was partch's
idea . . .