Schoenberg: 11-edo and 7,11-prime-space

it's been said relatively often
that Schoenberg would have done
better with his "atonal" style
if he had chosen to use 11-edo
rather than 12-edo.

for example, from p 15 of
Paul Erlich's _Tuning, Tonality,
and Twenty-Two-Tone Temperament_:

>> Finally, it is worth noting
>> that 11-tone equal temperament,
>> contained within 22-equal, is a
>> ridiculously dissonant tuning,[35]
>> containing hardly any tonal
>> (root-defining) sounds, and none
>> whatsoever within the 5-limit.
>> The serial composer who is
>> willing to subtract one from
>> the number of notes in the
>> row could be freed from the
>> constant effort to avoid those
>> intervals in 12-equal which define
>> a key center.

and Paul's footnote 35 says:

>> [35] Of all equal tunings, Blackwood says, "The most
>> effective one for random dissonance is eleven notes."
>> Keislar, Douglas. 1991. "Six American Composers On
>> Nonstandard Tunings." _Perspectives of New Music_
>> Vol. 29 No.1 p. 177.

in fact, 11-edo really would have served Schoenberg's goals
excellently, but for a quite different reason:

in his 1911 _Harmonielehre_, Schoenberg derives the
12-edo chromatic scale from the first 11 partials of
C, F, and G, which he characterizes as the I, IV, and V
of a diatonic major scale, and which, according to his
explanation, can be analyzed as the euler-genus 3^-1...1.

Schoenberg opens his 1923 paper _Twelve-tone Composition_ with:

>> In twelve-tone composition consonances (major and minor triads)
>> and also the simpler dissonances (diminished triads and seventh
>> chords) -- in fact almost everything that used to make up the
>> ebb and flow of harmony -- are, as far as possible, avoided.
>> ... At the root of all this is the unconscious urge to try
>> out the new resources independently, to wrest from them
>> possibilities of constructing forms, to produce with them
>> alone all the effects of a clear style, of a compact, lucid
>> and comprehensive presentation of the musical idea. ... the
>> chance of enjoying whatever can *only* be attained by the
>> new resources when the old ones are excluded!

putting these two together, it seems to me that Schoenberg
was saying that his new "atonal" style was an attempt to
produce music illustrating primarily ratios-of-11 in various
combinations with 7-limit ratios.

an analysis of a bingo-card lattice of 11-edo as a
representation of 7,11-prime-space shows that it represents
ratios of this prime-space *very* well!

this can also be seen on the error graphs on my webpage:
http://tonalsoft.com/enc/edo-prime-error.htm

-monz
http://tonalsoft.com
microtonal music software

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> an analysis of a bingo-card lattice of 11-edo as a
> representation of 7,11-prime-space shows that it represents
> ratios of this prime-space *very* well!
>
> this can also be seen on the error graphs on my webpage:
> http://tonalsoft.com/enc/edo-prime-error.htm

in his 1927 lecture (published in Enlgish in 1934 as
_Problems of Harmony_), Schoenberg derives the 12-edo
chromatic scale from the first 13 partials of C, F, and G.

an 11,13-prime-space bingo-card lattice, and my
EDO prime-error page, show that 13-edo is a good choice
for representing 11,13-prime-space.

-monz
http://tonalsoft.com
microtonal music software

>> an analysis of a bingo-card lattice of 11-edo as a
>> representation of 7,11-prime-space shows that it represents
>> ratios of this prime-space *very* well!
>>
>> this can also be seen on the error graphs on my webpage:
>> http://tonalsoft.com/enc/edo-prime-error.htm
>
>
>in his 1927 lecture (published in Enlgish in 1934 as
>_Problems of Harmony_), Schoenberg derives the 12-edo
>chromatic scale from the first 13 partials of C, F, and G.
>
>an 11,13-prime-space bingo-card lattice, and my
>EDO prime-error page, show that 13-edo is a good choice
>for representing 11,13-prime-space.

I don't know what you mean by 11,13-prime-space. Do you
mean factors of 11 and 13 only? In deriving 12 from the
first 13 partials of C, F, and G, does it not follow that
he wants good approximations to more than just 11 and 13?
And by using C, F, and G, does he not place special
importance on 3:2?

-Carl

hi Carl,

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> > [me, monz:]
> > in his 1927 lecture (published in Enlgish in 1934 as
> > _Problems of Harmony_), Schoenberg derives the 12-edo
> > chromatic scale from the first 13 partials of C, F, and G.
> >
> > an 11,13-prime-space bingo-card lattice, and my
> > EDO prime-error page, show that 13-edo is a good choice
> > for representing 11,13-prime-space.
>
> I don't know what you mean by 11,13-prime-space. Do you
> mean factors of 11 and 13 only?

yes, exactly.

> In deriving 12 from the first 13 partials of C, F, and G,
> does it not follow that he wants good approximations to
> more than just 11 and 13?

yes, but ...

> And by using C, F, and G, does he not place special
> importance on 3:2?

yes, but ...

in the message i posted immediately previous to the one
you quoted, i gave a bit of intro as to why i'm making
these speculations:

>> [quote from Schoenberg:]
>> ... At the root of all this is the unconscious urge to try
>> out the new resources independently, to wrest from them
>> possibilities of constructing forms, to produce with them
>> alone all the effects of a clear style, of a compact, lucid
>> and comprehensive presentation of the musical idea. ... the
>> chance of enjoying whatever can *only* be attained by the
>> new resources when the old ones are excluded!

so he's saying here -- in my interpretation, anyway --
that the music he was writing at this time was intended
to exploit the ratios-of-11, and possibly also ratios-of-13,
*without* using anything that was clearly audible as a
representation of 3, 5, or possibly even 7.

i originally intended to have both of these posts in one,
but decided to separate them since they discuss two different
tunings (11-edo and 13-edo) which each represent a different
prime-space. so the introductory comments quoting Schoenberg
only appeared in the first post.

-monz
http://tonalsoft.com
microtonal music software

>>> an analysis of a bingo-card lattice of 11-edo as a
>>> representation of 7,11-prime-space shows that it represents
>>> ratios of this prime-space *very* well!
>>>
>>> this can also be seen on the error graphs on my webpage:
>>> http://tonalsoft.com/enc/edo-prime-error.htm
>>
>>
>>in his 1927 lecture (published in Enlgish in 1934 as
>>_Problems of Harmony_), Schoenberg derives the 12-edo
>>chromatic scale from the first 13 partials of C, F, and G.
>>
>>an 11,13-prime-space bingo-card lattice, and my
>>EDO prime-error page, show that 13-edo is a good choice
>>for representing 11,13-prime-space.
>
>I don't know what you mean by 11,13-prime-space. Do you
>mean factors of 11 and 13 only? In deriving 12 from the
>first 13 partials of C, F, and G, does it not follow that
>he wants good approximations to more than just 11 and 13?
>And by using C, F, and G, does he not place special
>importance on 3:2?

P.S. Got your off-list mail, will be getting back to you
soon.

-Carl

>in the message i posted immediately previous to the one
>you quoted, i gave a bit of intro as to why i'm making
>these speculations:
>
>>> [quote from Schoenberg:]
>>> ... At the root of all this is the unconscious urge to try
>>> out the new resources independently, to wrest from them
>>> possibilities of constructing forms, to produce with them
>>> alone all the effects of a clear style, of a compact, lucid
>>> and comprehensive presentation of the musical idea. ... the
>>> chance of enjoying whatever can *only* be attained by the
>>> new resources when the old ones are excluded!
>
>so he's saying here -- in my interpretation, anyway --
>that the music he was writing at this time was intended
>to exploit the ratios-of-11, and possibly also ratios-of-13,
>*without* using anything that was clearly audible as a
>representation of 3, 5, or possibly even 7.

I guess I feel like I don't have enough context to understand
this quote. What is he referring to as "the root of all this"?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> > in the message i posted immediately previous to the one
> > you quoted, i gave a bit of intro as to why i'm making
> > these speculations:
> >
> >>> [quote from Schoenberg:]
> >>> ... At the root of all this is the unconscious urge to try
> >>> out the new resources independently, to wrest from them
> >>> possibilities of constructing forms, to produce with them
> >>> alone all the effects of a clear style, of a compact, lucid
> >>> and comprehensive presentation of the musical idea. ... the
> >>> chance of enjoying whatever can *only* be attained by the
> >>> new resources when the old ones are excluded!
> >
> > so he's saying here -- in my interpretation, anyway --
> > that the music he was writing at this time was intended
> > to exploit the ratios-of-11, and possibly also ratios-of-13,
> > *without* using anything that was clearly audible as a
> > representation of 3, 5, or possibly even 7.
>
> I guess I feel like I don't have enough context to understand
> this quote. What is he referring to as "the root of all this"?

i had more of the quote in the first post, but even there,
had snipped out some of it. here's the whole opening paragraph
of Schoenberg's paper:

>> In twelve-tone composition consonances (major and minor triads)
>> and also the simpler dissonances (diminished triads and seventh
>> chords) -- in fact almost everything that used to make up the
>> ebb and flow of harmony -- are, as far as possible, avoided.
>> But this is not because of any natural law of the new art.
>> It is, presumably, just one manifestation of a reaction,
>> one that does not have its own special causes but derives
>> from another manifestation -- which it tries to contradict,
>> and whose laws are therefore the same, basically, as its own.
>> At the root of all this is the unconscious urge to try
>> out the new resources independently, to wrest from them
>> possibilities of constructing forms, to produce with them
>> alone all the effects of a clear style, of a compact, lucid
>> and comprehensive presentation of the musical idea. To use
>> here the old resources in the old sense saves trouble --
>> the trouble of cultivating the new -- but also means passing
>> up the the chance of enjoying whatever can *only* be attained
>> by the new resources when the old ones are excluded!

-monz
http://tonalsoft.com
microtonal music software

>i had more of the quote in the first post, but even there,
>had snipped out some of it. here's the whole opening paragraph
>of Schoenberg's paper:
>
>>> In twelve-tone composition consonances (major and minor triads)
>>> and also the simpler dissonances (diminished triads and seventh
>>> chords) -- in fact almost everything that used to make up the
>>> ebb and flow of harmony -- are, as far as possible, avoided.
>>> But this is not because of any natural law of the new art.
>>> It is, presumably, just one manifestation of a reaction,
>>> one that does not have its own special causes but derives
>>> from another manifestation -- which it tries to contradict,
>>> and whose laws are therefore the same, basically, as its own.
>>> At the root of all this is the unconscious urge to try
>>> out the new resources independently, to wrest from them
>>> possibilities of constructing forms, to produce with them
>>> alone all the effects of a clear style, of a compact, lucid
>>> and comprehensive presentation of the musical idea. To use
>>> here the old resources in the old sense saves trouble --
>>> the trouble of cultivating the new -- but also means passing
>>> up the the chance of enjoying whatever can *only* be attained
>>> by the new resources when the old ones are excluded!

Thanks, monz. I'd say this gives reasonable justification for
your statement. But personally, I don't think Schonberg was
nearly as good of a theorist as he was a composer. In fact,
based on the limited listening I've done, his theorizing was
remarkably detrimental to his composing.

-Carl