Fwd: optimizing octaves in MIRACLE scale.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

Monz wrote,

> I'm interested in seeing what results
> when we include the 2:1 in the optimization, and I can't
> do that calculation.

OK -- we now have to do a two-parameter optimization, where the two
parameters are the size of the generator (let's call it G), and the
size of the approximate 2:1 (let's call it O). So this requires
multivariate calculus rather than the freshman univariate kind. Also,
we'll now be minimizing the sum-of-squares of the errors of all the
intervals in an integer limit, rather than an odd limit. This integer
limit can be 7, 8, 9, 10, 11, or 12. Pick one and I'll try to work it
out (it _will_ be hairy). If you will allow me to use MATLAB, I can
perform the optimization numerically (that is, the computer will make
repeated guesses and converge on the correct solution), which will at
least reduce _somewhat_ the amount of work I have to do.

--- End forwarded message ---

OK, cool. MATLAB is fine with me. (If I didn't "allow" you
to use it, does that mean that you'd go thru the drudgery of
the hand calculations just to show me how it's done? If so,
Paul, you're a really beautiful person. You can relax and
use MATLAB.)

Did we ever take a serious look at 11-odd-limit approximations
in the MIRACLE family? I don't recall much about 11, other
than Graham's discussions of neutral "3rds".

So if most of our work so far is 7- or 9-based, I guess
that's OK. I'd like to include 11 if you have no preference.

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> OK, cool. MATLAB is fine with me. (If I didn't "allow" you
> to use it, does that mean that you'd go thru the drudgery of
> the hand calculations just to show me how it's done?

I might try and then give up.
>
> Did we ever take a serious look at 11-odd-limit approximations
> in the MIRACLE family?

Oh yes . . . Dave Keenan has been thinking 11-limit all along. He posted some 7-limit and
11-limit optimization results, and I posted a 9-limit one, fully worked out step-by-step
(remember?). We've talked about the hexads in Canasta, and these are 11-limit hexads, of
course . . . etc. etc..

> I'd like to include 11 if you have no preference.

So shall we call our integer limit 12?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

/tuning-math/message/18

> > Did we ever take a serious look at 11-odd-limit approximations
> > in the MIRACLE family?
>
> Oh yes . . . Dave Keenan has been thinking 11-limit all along.
> He posted some 7-limit and 11-limit optimization results, and
> I posted a 9-limit one, fully worked out step-by-step
> (remember?). We've talked about the hexads in Canasta, and
> these are 11-limit hexads, of course . . . etc. etc..

Of course... duh! I knew all this. Guess it's just
information overload.

> > I'd like to include 11 if you have no preference.
>
> So shall we call our integer limit 12?

Sure! Guess what?... that ties this in nicely with
Schoenberg's alleged integer-limit of 12 in his
_Harmonielehre_ (the explanation disparaged by Partch).

Interesting...

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning-math/message/18
>
> > > Did we ever take a serious look at 11-odd-limit approximations
> > > in the MIRACLE family?
> >
> > Oh yes . . . Dave Keenan has been thinking 11-limit all along.
> > He posted some 7-limit and 11-limit optimization results, and
> > I posted a 9-limit one, fully worked out step-by-step
> > (remember?). We've talked about the hexads in Canasta, and
> > these are 11-limit hexads, of course . . . etc. etc..
>
> Of course... duh! I knew all this. Guess it's just
> information overload.
>
>
> > > I'd like to include 11 if you have no preference.
> >
> > So shall we call our integer limit 12?
>
>
> Sure! Guess what?... that ties this in nicely with
> Schoenberg's alleged integer-limit of 12 in his
> _Harmonielehre_ (the explanation disparaged by Partch).

Umm . . . I thought that explanation used a _prime-limit_ of 13, not an _integer-limit_ of 12. In
particular, Partch showed that Schoenberg's two derivations of the note C# -- as the 11th
harmonic of G and as the 13th harmonic of F -- hence as 33/32 and 13/12 -- differed by virtually
an entire semitone (i.e., Schoenberg assumed a "unison vector" of 143:128).

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

/tuning-math/message/20

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > [Paul:]
> > > So shall we call our integer limit 12?
> >
> > [monz:]
> > Sure! Guess what?... that ties this in nicely with
> > Schoenberg's alleged integer-limit of 12 in his
> > _Harmonielehre_ (the explanation disparaged by Partch).
>
>
> Umm . . . I thought that explanation used a _prime-limit_
> of 13, not an _integer-limit_ of 12. In particular, Partch
> showed that Schoenberg's two derivations of the note C# --
> as the 11th harmonic of G and as the 13th harmonic of F --
> hence as 33/32 and 13/12 -- differed by virtually an entire
> semitone (i.e., Schoenberg assumed a "unison vector" of 143:128).

Damn, Paul! Duh again!

I had signed off for the night, and just realized this error
and came back to the computer to correct it, and you've
already explained it sufficiently!

Here's the full scoop:

The incorrect part of my statement was the mention of
Partch's analysis.

The Schoenberg work Partch cites is a lecture given in
1934 called in the English translation in _Style and Idea_
"Problems of Harmony".

I was correct in saying that an alleged 12-integer-limit
would connect our optimization with Schoenberg's in his
_Harmonielehre_ of 1911. That's precisely how he explains
the origin of the diatonic scale, plus the first couple
of chromatic alterations which suggest the 12-EDO paradigm
he hints at in a couple of sections of the 1911 edition.

(In the more commonly found 1922 edition he expands quite
a bit at these points and presents fully 12-EDO outlines.)

He obviously decided on a prime-limit of 13 some time later.

I'm interested now in whether Schoenberg thought of his
1934 analysis as a prime-limit or an odd-limit, because
my hazy immediate recollection suggests the latter.

I'll take a closer look at the Schoenberg article to see
if it's possible to determine this, and also make sure that
my dates are accurate.

But for sure, the 12-integer-limit is in _Harmonielehre_.
FTR, Schoenberg actually wrote it during the summer of
1910. It was published in 1911.

Hmmm... 1910 was the same summer Mahler composed his 10th
Symphony, probably reflecting a good deal of the influence
I believe Schoenberg was having on Mahler, who supported
Schoenberg (financially and otherwise) for years past the
point when he could no longer understand Schoenberg's work.

Mahler wrote to Schoenberg in 1909 that "I have the score of
your [Schoenberg's 2nd] Quartet with me here [in New York]
and study it from time to time, but it's difficult for me."

I believe that Mahler's work shows the influence of Schoenberg
as early as the _7th Symphony_, 1905.

So research into this kind of tuning paradigm may have
some bearing on my attempts to experimentally retune
Mahler's work.

Interesting.

-monz
http://www.monz.org
"All roads lead to n^0"

I wrote,

> > hence as 33/32 and 13/12 -- differed by virtually an entire
> > semitone (i.e., Schoenberg assumed a "unison vector" of 143:128).

Oops! That should be 104:99, not 143:128!

> But for sure, the 12-integer-limit is in _Harmonielehre_.

Really? So ratios such as 16:9 would have fallen outside it?

Oh, Monz . . . you're not expecting the result to be a stretched or
squashed 72-tET, are you? 'Cause if you are, then it's a one-
parameter optimization -- much easier.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I wrote,
>
> > [monz]
> > But for sure, the 12-integer-limit is in _Harmonielehre_.
>
> Really? So ratios such as 16:9 would have fallen outside it?

Paul, I started a response to this but it is getting
long and interesting. I'll post it tonight.

-monz

I wrote:

> Oh, Monz . . . you're not expecting the result to be a stretched or
> squashed 72-tET, are you? 'Cause if you are, then it's a one-
> parameter optimization -- much easier.

And if it is, the answer is 71.959552-tET, or 72-tET with the octave
stretched to 1200.6745¢.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

/tuning-math/message/31

> I wrote:
>
> > Oh, Monz . . . you're not expecting the result to be a
> > stretched or squashed 72-tET, are you? 'Cause if you are,
> > then it's a one-parameter optimization -- much easier.
>
> And if it is, the answer is 71.959552-tET, or 72-tET with
> the octave stretched to 1200.6745¢.

With a step-size of 16.67603472 cents.

Thanks, Paul. Uh... I don't think "expecting" is the way
I'd say it, but yes, I *was* *guessing* that it would be
a stretched 72-EDO.

But I'm unclear on why my expectation would have any effect
on the type of optimization. ...?

Also, on the asking of what are probably elementary questions
like this to the rest of you on this list: is it OK for me to
ask questions like this here? Or will it be perceived as a
nuisance to those of you who are ready to discuss nitty-gritty
tuning math? I know that Paul is generous with his help, and so
I can keep this stuff relegated to private email if others prefer.

My hope is that the Tuning-math List can be a place for people
of all mathematical levels to be able discuss aspects of this
subject, but I would perfectly understand if most subscribers
want to keep discussion on a high level.

-monz
http://www.monz.org
"All roads lead to n^0"

monz wrote:

> My hope is that the Tuning-math List can be a place for people
> of all mathematical levels to be able discuss aspects of this
> subject, but I would perfectly understand if most subscribers
> want to keep discussion on a high level.

I agree (with the first bit)

Graham

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

/tuning-math/message/24

> I wrote,
>
> > > hence as 33/32 and 13/12 -- differed by virtually an entire
> > > semitone (i.e., Schoenberg assumed a "unison vector" of
143:128).
>
> Oops! That should be 104:99, not 143:128!
>
> > But for sure, the 12-integer-limit is in _Harmonielehre_.
>
> Really? So ratios such as 16:9 would have fallen outside it?

(early response:)

Oops... Schoenberg doesn't actually claim that the 12th
harmonic is any kind of limit... it's simply where his
musical illustration and its accompanying explanation end.
I suppose he implies that it continues beyond into
inaudibility.

The musical illustration uses the 1st thru 12th harmonics
on F, C, and G. So using a 12-integer-limit here would
relate it to Schoenberg's illustration, but not necessarily
to his actual theory.

In the later article, "Problems of Harmony", which BTW
was written in 1927 then revised in 1934 for presentation
in America, Schoenberg definitely explains harmony as
being based on a 13-integer-limit as harmonics 1 thru 13
on F, C, and G.

I would label this system as (1...13)/(3^(-1...1)).

Is there a better notation for that?

Still later, in _Structural Functions of Harmony_ [1949],
his "Chart of the Regions" (2 versions, in major and minor)
uses terms such as "mediant" which imply more extended
5-limit derivations for some notes than the ratios implied
by the overtone model.

====

That was my first response to this.

I was going to concede to Paul that I had been in error,
and to some extent I *was*, but guess what?... The scale
of approximated ratios implied by Schoenberg's diagram
provides only one 16:9!, between d-27 and c-48.

There are 3 other varieties of "minor 7th":

11:6 (really a "neutral 7th") between d-54 and c-99,

9:5 between b-45 and a-80, and

7:4 between g-36 and f-63.

I was getting concerned that this thread was veering
off-topic, but this gives me the opportunity to remedy
that situation. :)

(My quotes of Schoenberg are from the English translation
of _Harmonielehre_ by Roy Carter, and the page numbers
refer to that edition.)

Schoenberg [p 23] posits the existences of two "forces", one
pulling downward and one pulling upward around the tonic,
which he illustrates as: F <- C -> G and likens to resistance
against gravity. In mathematical terms, he is referring to
the harmonic relationships of 3^-1 and 3^1, respectively.

> [Schoenberg, p 24:]
>
> ...thus it is explained how the scale that finally emerged
> is put together from the most important components of a
> fundamental tone and its nearest relatives. These nearest
> relatives are just what gives the fundamental tone stability;
> for it represents the point of balance between their opposing
> tendencies. This scale appears as the residue of the properties
> of the three factors, as a vertical projection, as addition:

Schoenberg then presents a diagram of the overtones and the
resulting scale, which I have adaptated, adding the partial-numbers
which relate all the overtones together as a single set:

b-45
g-36
e-30
d-27
c-24
a-20
g-18 g-18
f-16
c-12 c-12
f-8

f c g a d e b
8 12 18 20 27 30 45

> [Schoenberg:]
>
> Adding up the overtones (omitting repetitions) we get the seven
> tones of our scale. Here they are not yet arranged consecutively.
> But even the scalar order can be obtained if we assume that the
> further overtones are also in effect. And that assumption is
> in fact not optional; we must assume the presence of the other
> overtones. The ear could also have defined the relative pitch
> of the tones discovered by comparing them with taut strings,
> which of course become longer or shorter as the tone is lowered
> or raised. But the more distant overtones were also a
> dependable guide. Adding these we get the following:

Schoenberg then extends the diagram to include the
following overtones:

fundamental partials

F 2...12, 16
C 2...11
G 2...12

(Note, therefore, that he is not systematic in his employment
of the various partials.)

Again, I adapt the diagram by adding partial-numbers:

d-108
c-99
b-90
a-81
g-72
f-66
f-64
(f-63)
e-60
d-54 d-54
c-48 c-48
b-45
b-44
(bb-42)
a-40
g-36 g-36 g-36
f-32
e-30
(eb-28)
d-27
c-24 c-24
a-20
g-18 g-18
f-16
c-12 c-12
f-8

(eb) (bb)
c d e f g a b c d e f g a b c d
[44] [64]
(28) (42) [66]
24 27 30 32 36 40 45 48 54 60 63 72 81 90 99 108

(Note also that Schoenberg was unsystematic in his naming
of the nearly-1/4-tone 11th partials, calling 11th/F by the
higher of its nearest 12-EDO relatives, "b", while calling
11th/C and 11th/G by the lower, "f" and "c" respectively.
This, ironically, is the reverse of the actual proximity
of these overtones to 12-EDO: ~10.49362941, ~5.513179424,
and ~0.532729432 Semitones, respectively).

The partial-numbers are also given for the resulting scale
at the bottom of the diagram, showing that 7th/F (= eb-28)
is weaker than 5th/C (= e-30), and 7th/C (= bb-42) is weaker
than 5th/G (= b-45).

Also note that 11th/F (= b-44), 16th/F (= f-64) and 11th/C
(= f-66) are all weaker still, thus I have included them in
square brackets. These overtones are not even mentioned by
Schoenberg.

Schoenberg does take note of the ambiguity present in this
collection of ratios, in his later article _Problems of Harmony_.
I won't go into that here because this is focusing on his
1911 theory.

Here is an interval matrix of Schoenberg's scale
(broken in half to fit the screen), with implied
proportions given along the left and the bottom,
and Semitone values of the intervals in the body.

Because Schoenberg's implied proportions form an
"octave"-specific pitch-set in his presentation
(not necessarily in his theory), this matrix has
no "bottom" half.

Interval Matrix of Schoenberg's implied JI scale:

108 26.04 24.00 23.37 22.18 21.06 19.02 17.20 16.35 15.55 15.16 14.04
99 24.53 22.49 21.86 20.67 19.55 17.51 15.69 14.84 14.04 13.65 12.53
90 22.88 20.84 20.21 19.02 17.90 15.86 14.04 13.19 12.39 12.00 10.88
81 21.06 19.02 18.39 17.20 16.08 14.04 12.22 11.37 10.57 10.18 9.06
72 19.02 16.98 16.35 15.16 14.04 12.00 10.18 9.33 8.53 8.14 7.02
66 17.51 15.47 14.84 13.65 12.53 10.49 8.67 7.82 7.02 6.63 5.51
64 16.98 14.94 14.31 13.12 12.00 9.96 8.14 7.29 6.49 6.10 4.98
63 16.71 14.67 14.04 12.84 11.73 9.69 7.86 7.02 6.21 5.83 4.71
60 15.86 13.82 13.19 12.00 10.88 8.84 7.02 6.17 5.37 4.98 3.86
54 14.04 12.00 11.37 10.18 9.06 7.02 5.20 4.35 3.55 3.16 2.04
48 12.00 9.96 9.33 8.14 7.02 4.98 3.16 2.31 1.51 1.12 0.00
45 10.88 8.84 8.21 7.02 5.90 3.86 2.04 1.19 0.39 0.00
44 10.49 8.45 7.82 6.63 5.51 3.47 1.65 0.81 0.00
42 9.69 7.65 7.02 5.83 4.71 2.67 0.84 0.00
40 8.84 6.80 6.17 4.98 3.86 1.82 0.00
36 7.02 4.98 4.35 3.16 2.04 0.00
32 4.98 2.94 2.31 1.12 0.00
30 3.86 1.82 1.19 0.00
28 2.67 0.63 0.00
27 2.04 0.00
24 0.00
24 27 28 30 32 36 40 42 44 45 48

---

108 12.00 10.18 9.33 9.06 8.53 7.02 4.98 3.16 1.51 0.00
99 10.49 8.67 7.82 7.55 7.02 5.51 3.47 1.65 0.00
90 8.84 7.02 6.17 5.90 5.37 3.86 1.82 0.00
81 7.02 5.20 4.35 4.08 3.55 2.04 0.00
72 4.98 3.16 2.31 2.04 1.51 0.00
66 3.47 1.65 0.81 0.53 0.00
64 2.94 1.12 0.27 0.00
63 2.67 0.84 0.00
60 1.82 0.00
54 0.00
54 60 63 64 66 72 81 90 99 108

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> There are 3 other varieties of "minor 7th":
>
> 11:6 (really a "neutral 7th") between d-54 and c-99,
>
> 9:5 between b-45 and a-80, and
>
> 7:4 between g-36 and f-63.

7:4 (really a subminor 7th) between g-36 and f-63
:-)

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning-math/message/31
>
> > I wrote:
> >
> > > Oh, Monz . . . you're not expecting the result to be a
> > > stretched or squashed 72-tET, are you? 'Cause if you are,
> > > then it's a one-parameter optimization -- much easier.
> >
> > And if it is, the answer is 71.959552-tET, or 72-tET with
> > the octave stretched to 1200.6745¢.
>
>
> With a step-size of 16.67603472 cents.
>
> Thanks, Paul. Uh... I don't think "expecting" is the way
> I'd say it, but yes, I *was* *guessing* that it would be
> a stretched 72-EDO.
>
> But I'm unclear on why my expectation would have any effect
> on the type of optimization. ...?

Well, because it's easier to solve the problem of how best to stretch
or squashed 72-tET for the 12-integer-limit (a univariate
optimization), than to solve the problem of what the best size of
generator _and_ the best size of octave are for MIRACLE in the 12-
integer limit (a multivariate optimization). If you _want_ a
stretched or squashed 72-tET, then there you have it, I'm done. If
not, I'm hitting the Matlab Optimization Toolbox manual.
>
> Also, on the asking of what are probably elementary questions
> like this to the rest of you on this list: is it OK for me to
> ask questions like this here?

You better believe it!

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

/tuning-math/message/21

>
> Mahler wrote to Schoenberg in 1909 that "I have the score of
> your [Schoenberg's 2nd] Quartet with me here [in New York]
> and study it from time to time, but it's difficult for me."
>

Can you imagine this? And, it's one of his "easier" works, in the
overview...

________ ______ _______
Joseph Pehrson

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

/tuning-math/message/44

> Schoenberg then presents a diagram of the overtones and the
> resulting scale, which I have adaptated, adding the partial-numbers
> which relate all the overtones together as a single set:
>
> b-45
> g-36
> e-30
> d-27
> c-24
> a-20
> g-18 g-18
> f-16
> c-12 c-12
> f-8
>
>
> f c g a d e b
> 8 12 18 20 27 30 45
>

<cut> etc.

This is an extremely interesting post, Monz, and I would recommend
that it be reposted to the "big" list as well...

_________ _______ ______
Joseph Pehrson