HOLD THE PRESSES!

I wrote:

> From: monz <joemonz@yahoo.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, January 15, 2002 11:10 PM

> Subject: Re: badly tuned remote overtones

>

>

> c = 12*4 = 48 c = 8*6 = 48

> b = 5*9 = 45

> b = 11*4 = 44

> (bb= 7*6 = 42)

> a = 10*4 = 40

> g = 9*4 = 36 g = 6*6 = 36 g = 4*9 = 36

>

> ...

>

> These are all the unison-vectors implied by Schoenberg's diagram:

>

> E 5*6=30 : Eb 4*7=28 = 15:14

> B 11*4=44 : Bb 7*6=42 = 22:21

> B 5*9=45 : B 11*4=44 = 45:44

> B 5*9=45 : Bb 7*6=42 = 15:14

> F 16*4=64 : F 7*9=63 = 64:63

> F 11*6=66 : F 16*4=64 = 33:32

> F 11*6=66 : F 7*9=63 = 22:21

> A 9*9=81 :(A 20*4=80) = 81:80

> C 11*9=99 :(C 24*4=96) = 33:32

I've been writing again and again about the inconsistency

of Scheonberg's notation of the 11th harmonic in the diagram

on p 24 in the Carter translation. Well, I was just stunned

to find out that in the original 1911 German edition, the

11th harmonic of F is written as "b", *which in German means

b-flat*! (what we call "b" is written as "h" in German).

It has been changed *without comment* in the Carter translation

(or perhaps it's simply an easily-overlooked error), and

so my writings on Schoenberg's theories must all be revised.

The corrupt diagram appears on p 24 of Carter's 1978 English

translation, published as _Theory of Harmony_. The original

diagram is on p 23 of the 1911 edition of _Harmonielehre_.

So in the above diagram, 22:21 is both a unison-vector

(F 11*6=66 : F 7*9=63) and a chromatic-vector used as

a unison-vector (B 11*4=44 : Bb 7*6=42), and 45:44 is

a unison vector (B 5*9=45 : B 11*4=44) -- this is incorrect.

The note I've been giving as B 11*4=44 must be called

Bb 11*4=44. This has the effect of changing 45:44 into

a chromatic-vector, and making 22:21 consistently a

unison-vector.

Since Schoenberg spends a good deal of space in

_Harmonielehre_ on ways to use the 12-EDO scale to

construct harmonies that lie outside traditional tonal

theory, we must assume that he intends to distinguish

pairs of notes that are separated by approximately a

semitone, so that the 15:14 chromatic-vector cannot be

considered a unison-vector. Therefore, the total list

of unison-vectors implied by Schoenberg's 1911 diagram is:

Bb 11*4=44 : Bb 7*6=42 = 22:21

F 16*4=64 : F 7*9=63 = 64:63

F 11*6=66 : F 16*4=64 = 33:32

F 11*6=66 : F 7*9=63 = 22:21

A 9*9=81 :(A 20*4=80) = 81:80

C 11*9=99 :(C 24*4=96) = 33:32

But because 22:21, 33:32, and 64:63 form a dependent triplet

(any one of them can be found by multiplying the other two),

this does not suffice to create a periodicity-block, which

needs another independent unison-vector.

So I will give Paul Erlich the benefit of the doubt

and assume that Schoenberg was following tradition as

closely as possible in his note naming, by thinking in

terms of meantone, and therefore the most likely candidate

for the other independent UV is the 5-limit diesis

128:125 = [ 7 0 -3 ] .

So ...

matrix

2 3 5 7 11 unison-vectors ~cents

[ 1 0 0 0 0 ] = 2:1 0

[ 7 0 -3 0 0 ] = 128:125 41.05885841

[ -5 1 0 0 1 ] = 33:32 53.27294323

[ 6 -2 0 -1 0 ] = 64:63 27.2640918

[ -4 4 -1 0 0 ] = 81:80 21.5062896

fractional inverse

[ 12 0 0 0 0 ]

[ 19 -1 0 0 3 ]

[ 28 -4 0 0 0 ]

[ 34 2 0 -12 -6 ]

[ 41 1 12 0 -3 ]

determinant = | 12 |

The only cardinality given by this matrix is 12,

with the following mapping:

h12(2) = 12

h12(3) = 19

h12(5) = 28

h12(7) = 34

h12(11) = 41

So with references of "F", "C", and "G",

our 12-EDO mapping is:

2 3 5 7 11

G D B F C

C G E Bb F

F C A Eb Bb

and 11 is given consistently as the 12-EDO "4th" above

the fundamental, and never as the "augmented 4th", as

Carter's edition implies.

-monz

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

> Therefore, the total list

> of unison-vectors implied by Schoenberg's 1911 diagram is:

>

> Bb 11*4=44 : Bb 7*6=42 = 22:21

> F 16*4=64 : F 7*9=63 = 64:63

> F 11*6=66 : F 16*4=64 = 33:32

> F 11*6=66 : F 7*9=63 = 22:21

> A 9*9=81 :(A 20*4=80) = 81:80

> C 11*9=99 :(C 24*4=96) = 33:32

OK . . .

>

>

> But because 22:21, 33:32, and 64:63 form a dependent triplet

> (any one of them can be found by multiplying the other two),

> this does not suffice to create a periodicity-block, which

> needs another independent unison-vector.

> So I will give Paul Erlich the benefit of the doubt

> and assume that Schoenberg was following tradition as

> closely as possible in his note naming, by thinking in

> terms of meantone, and therefore the most likely candidate

> for the other independent UV is the 5-limit diesis

> 128:125 = [ 7 0 -3 ] .

This is not like anything I would say. Giving me the benefit of the

doubt, huh? This reminds me of where you recently told Klaus that you

disagree with me about sustained notes changing their

intonation . . . if your interpretations of Schoenberg and other

theorists are as good as your interpretation of me . . . well,

whatever, I love you Monz, you get a big ice cream with a cherry on

top.

You better tell the tuning list if you believe that Partch's

criticism of Schoenberg was based on a mistranslation!!!

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Wednesday, January 16, 2002 4:09 PM

> Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly tuned

remote overtones)

>

>

> > So I will give Paul Erlich the benefit of the doubt

> > and assume that Schoenberg was following tradition as

> > closely as possible in his note naming, by thinking in

> > terms of meantone, and therefore the most likely candidate

> > for the other independent UV is the 5-limit diesis

> > 128:125 = [ 7 0 -3 ] .

>

> This is not like anything I would say. Giving me the benefit of the

> doubt, huh? This reminds me of where you recently told Klaus that you

> disagree with me about sustained notes changing their

> intonation . . . if your interpretations of Schoenberg and other

> theorists are as good as your interpretation of me . . . well,

> whatever, I love you Monz, you get a big ice cream with a cherry on

> top.

OK, so I misunderstand things sometimes too ... like most people.

Sorry. :(

All I'm saying is that *if* Schoenberg had any kind of meantone

conception in mind -- which I think is quite likely, given its

ubiquity in European music, right down to our current notation,

(*this* is why I'm giving a nod to you, Paul!!) -- then the

unison-vector he'll bump into, in 1/4-comma meantone, is 128:125.

> You better tell the tuning list if you believe that Partch's

> criticism of Schoenberg was based on a mistranslation!!!

Wow, I hadn't even *thought* of *that*! I'll have to get some

sleep and check those details tomorrow. But thanks for the

suggestion.

-monz

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

>

> > From: paulerlich <paul@s...>

> > To: <tuning-math@y...>

> > Sent: Wednesday, January 16, 2002 4:09 PM

> > Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly

tuned

> remote overtones)

> >

> >

> > > So I will give Paul Erlich the benefit of the doubt

> > > and assume that Schoenberg was following tradition as

> > > closely as possible in his note naming, by thinking in

> > > terms of meantone, and therefore the most likely candidate

> > > for the other independent UV is the 5-limit diesis

> > > 128:125 = [ 7 0 -3 ] .

> >

> > This is not like anything I would say. Giving me the benefit of

the

> > doubt, huh? This reminds me of where you recently told Klaus that

you

> > disagree with me about sustained notes changing their

> > intonation . . . if your interpretations of Schoenberg and other

> > theorists are as good as your interpretation of me . . . well,

> > whatever, I love you Monz, you get a big ice cream with a cherry

on

> > top.

>

>

> OK, so I misunderstand things sometimes too ... like most people.

> Sorry. :(

>

> All I'm saying is that *if* Schoenberg had any kind of meantone

> conception in mind -- which I think is quite likely, given its

> ubiquity in European music, right down to our current notation,

> (*this* is why I'm giving a nod to you, Paul!!) -- then the

> unison-vector he'll bump into, in 1/4-comma meantone, is 128:125.

No way, dude. This is your "rational implications" business rearing

its ugly head again -- in no way do I endorse that view.

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Thursday, January 17, 2002 1:06 PM

> Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly tuned

remote overtones)

>

>

> > All I'm saying is that *if* Schoenberg had any kind of meantone

> > conception in mind -- which I think is quite likely, given its

> > ubiquity in European music, right down to our current notation,

> > (*this* is why I'm giving a nod to you, Paul!!) -- then the

> > unison-vector he'll bump into, in 1/4-comma meantone, is 128:125.

>

> No way, dude. This is your "rational implications" business rearing

> its ugly head again -- in no way do I endorse that view.

I think you misunderstand me, Paul. I just mean that there's

probably a good chance that at least some of the time, Schoenberg

thought of the "Circle of 5ths" in a meantone rather than a

Pythagorean sense.

This reference to you is only meant to credit you for opening

my eyes to the strong meantone basis behind a good portion of

the "common-practice" European musical tradition.

-monz

_________________________________________________________

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

>

> I think you misunderstand me, Paul. I just mean that there's

> probably a good chance that at least some of the time, Schoenberg

> thought of the "Circle of 5ths" in a meantone rather than a

> Pythagorean sense.

I doubt it. For Schoenberg, the circle of 5ths closes after 12

fifths -- which is closer to being true in Pythagorean than in most

meantones.

> This reference to you is only meant to credit you for opening

> my eyes to the strong meantone basis behind a good portion of

> the "common-practice" European musical tradition.

OK -- but you're confusing two completely unrelated facts -- that

128:125 is just in 1/4-comma meantone, and that 128:125 is one of the

simplest unison vectors for defining a 12-tone periodicity block.

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, January 18, 2002 1:04 PM

> Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly tuned

remote overtones)

>

>

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

> >

> > I think you misunderstand me, Paul. I just mean that there's

> > probably a good chance that at least some of the time, Schoenberg

> > thought of the "Circle of 5ths" in a meantone rather than a

> > Pythagorean sense.

>

> I doubt it. For Schoenberg, the circle of 5ths closes after 12

> fifths -- which is closer to being true in Pythagorean than in most

> meantones.

I understand that, Paul ... but if one is trying to ascertain

the potential rational basis behind Schoenberg's work, how does

one decide which unison-vectors are valid and which are not?

Schoenberg was very clear about what he felt were the "overtone"

implications of the diatonic scale (and later, the chromatic

as well), but as I showed in my posts, the only "obvious"

5-limit unison-vector is the syntonic comma, and it seemed

to me that there always needed to be *two* 5-limit unison-vectors

in order to have a matrix of the proper size (so that it's square).

(I realize that by transposition it need not be a 5-limit UV,

but I'm not real clear on what else *could* be used, except for

the 56:55 example Gene used.)

> > This reference to you is only meant to credit you for opening

> > my eyes to the strong meantone basis behind a good portion of

> > the "common-practice" European musical tradition.

>

> OK -- but you're confusing two completely unrelated facts -- that

> 128:125 is just in 1/4-comma meantone, and that 128:125 is one of the

> simplest unison vectors for defining a 12-tone periodicity block.

OK, I'm willing to take note of your point, but ... *why* are

these two facts "completely unrelated"? Isn't it possible that

there *is* some relation between them that no-one has noticed

before?

-monz

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

>

> > From: paulerlich <paul@s...>

> > To: <tuning-math@y...>

> > Sent: Friday, January 18, 2002 1:04 PM

> > Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly

tuned

> remote overtones)

> >

> >

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

> > >

> > > I think you misunderstand me, Paul. I just mean that there's

> > > probably a good chance that at least some of the time,

Schoenberg

> > > thought of the "Circle of 5ths" in a meantone rather than a

> > > Pythagorean sense.

> >

> > I doubt it. For Schoenberg, the circle of 5ths closes after 12

> > fifths -- which is closer to being true in Pythagorean than in

most

> > meantones.

>

>

> I understand that, Paul ... but if one is trying to ascertain

> the potential rational basis behind Schoenberg's work, how does

> one decide which unison-vectors are valid and which are not?

Only ones he specifically pointed to are valid. If it's not enough to

give you a PB, it'll still give you a linear temperament, which

should be interesting enough!

> Schoenberg was very clear about what he felt were the "overtone"

> implications of the diatonic scale (and later, the chromatic

> as well), but as I showed in my posts, the only "obvious"

> 5-limit unison-vector is the syntonic comma, and it seemed

> to me that there always needed to be *two* 5-limit unison-vectors

> in order to have a matrix of the proper size (so that it's square).

In 5-limit, yes.

>

> (I realize that by transposition it need not be a 5-limit UV,

By transposition? Not sure what you're getting at, but none of the

unison vectors need to necessarily be 5-limit when constructing an 11-

limit PB.

> > > This reference to you is only meant to credit you for opening

> > > my eyes to the strong meantone basis behind a good portion of

> > > the "common-practice" European musical tradition.

> >

> > OK -- but you're confusing two completely unrelated facts -- that

> > 128:125 is just in 1/4-comma meantone, and that 128:125 is one of

the

> > simplest unison vectors for defining a 12-tone periodicity block.

>

>

> OK, I'm willing to take note of your point, but ... *why* are

> these two facts "completely unrelated"? Isn't it possible that

> there *is* some relation between them that no-one has noticed

> before?

128:125 is just in 1/4-comma meantone. If 128:125 is tempered out,

the meantone is transformed into 12-tET. Then again, you could just

as well temper 2048:2025 out of meantone and get 12-tET, or temper

32768:32805 out of meantone and get 12-tET. These intervals are not

just in 1/4-comma meantone. 128:125 is a UV of choice for this

purpose _only_ because it's the simplest, _not_ because it's just in

1/4-comma meantone.