Schoenberg's 1910 overtone theory (was: optimizing octaves in MIRACLE scale)

Joe Pehrson recommended that I repost this to the "big list",
and he's right. It belonged here first anyway, because even
tho it features a lot of simple math, any further commentary
will probably be off-topic for that list but good for this one.
Enjoy.

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

/tuning-math/message/44

> --- 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"

--- End forwarded message ---