--- 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:

> > 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:

> --- 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:

> 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:

> 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:

>

> 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:

> 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